Kinetic theory of gases
The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles (atoms or molecules), all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no  other interactions between the particles. The kinetic theory of gases explains the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity. The model also accounts for related phenomena, such as Brownian motion. Historically, the kinetic theory of gases was the first explicit exercise of the ideas of statistical mechanics. History. In about 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This Epicurean atomistic point of view was  rarely considered in the subsequent centuries, when Aristotlean ideas were dominant. In 1738 Daniel Bernoulli published "Hydrodynamica", which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the pressure of the gas, and that their average kinetic energy determines the temperature of the gas. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could  be perfectly elastic. Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816) and John James Waterston (1843), which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational motion of the particles. In 1857 Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also rotational and vibrational molecular motions. In this same work  he introduced the concept of mean free path of a particle. In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested  and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases." In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. The logarithmic connection between entropy and probability was also first stated by Boltzmann. At the beginning of the 20th century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905) and Marian Smoluchowski's (1906) papers on Brownian motion, which succeeded in  making certain accurate quantitative predictions based on the kinetic theory. Assumptions. The application of kinetic theory to ideal gases makes the following assumptions: Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible. As a simplifying assumption, the particles are usually assumed to have the same mass as one another; however, the theory can be generalized to a mass distribution, with each mass type contributing to the gas properties independently of one another in agreement with Dalton's Law of partial pressures. Many of the model's predictions are the same whether  or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below). More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation. Equilibrium properties. Pressure and kinetic energy. In the kinetic theory of gases, the pressure is assumed to be equal to the force (per unit area) exerted by  the atoms hitting and rebounding from the gas container's surface. Consider a gas of a large number "N" of molecules, each of mass "m", enclosed in a cube of volume "V" = "L". When a gas molecule collides with the wall of the container perpendicular to the "x" axis and bounces off in the opposite direction with the same speed (an elastic collision), the change in momentum is given by: formula_1 where "p" is the momentum, "i" and "f" indicate initial and final momentum (before and after collision), "x" indicates that only the "x" direction is being considered, and  formula_2 is the speed of the particle in the "x" direction (which is the same before and after the collision). The particle impacts one specific side wall once during the time interval formula_3 formula_4 where "L" is the distance between opposite walls. The force of this particle's collision with the wall is formula_5 The total force on the wall due to collisions by molecules impacting the walls with a range of possible values of formula_2 is formula_7 where the bar denotes an average over the possible velocities of the "N" particles. Since the motion of the particles is random  and there is no bias applied in any direction, the average squared speed in each direction is identical: formula_8 By the Pythagorean theorem, in three dimensions the average squared speed formula_9 is given by formula_10 Therefore formula_11 and formula_12 and so the force can be written as formula_13 This force is exerted uniformly on an area "L". Therefore, the pressure of the gas is formula_14 where "V" = "L" is the volume of the box. In terms of the translational kinetic energy "K" of the gas, since formula_15 we have formula_16 This is an important, non-trivial result of the  kinetic theory because it relates pressure, a macroscopic property, to the translational kinetic energy of the molecules, which is a microscopic property. Temperature and kinetic energy. Rewriting the above result for the pressure as formula_17, we may combine it with the ideal gas law where formula_18 is the Boltzmann constant and formula_19 the absolute temperature defined by the ideal gas law, to obtain formula_20 which leads to a simplified expression of the average kinetic energy per molecule, formula_21 The kinetic energy of the system is formula_22 times that of a molecule, namely formula_23. Then the temperature formula_19 takes the form  which becomes Equation () is one important result of the kinetic theory: "The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature". From equations () and (), we have Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy. Equations () and () are called the "classical results", which could also be derived from statistical mechanics; for more details, see: Since there are formula_25 degrees of freedom in a monatomic-gas system with formula_26 particles, the kinetic energy per degree of freedom per molecule is In the kinetic energy per degree of  freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant or R/2 per mole. This result is related to the equipartition theorem. Thus the kinetic energy per Kelvin of one mole of (monatomic ideal gas) is 3 [R/2] = 3R/2. Thus the kinetic energy per Kelvin can be calculated easily: At standard temperature (273.15 K), the kinetic energy can also be obtained: Although monatomic gases have 3 (translational) degrees of freedom per atom, diatomic gases should have 6 degrees of freedom per molecule (3 translations, two rotations, and one vibration). However, the lighter diatomic gases (such as diatomic  oxygen) may act as if they have only 5 due to the strongly quantum-mechanical nature of their vibrations and the large gaps between successive vibrational energy levels. Quantum statistical mechanics is needed to accurately compute these contributions. Collisions with container wall. For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated based on naive kinetic theory, and the results can be used for analyzing effusive flow rates, which is useful in applications such as the gaseous diffusion method for isotope separation. Assume that  in the container, the number density (number per unit volume) is formula_27 and that the particles obey Maxwell's velocity distribution: formula_28 Then for a small area formula_29 on the container wall, a particle with speed formula_30 at angle formula_31 from the normal of the area formula_29, will collide with the area within time interval formula_33, if it is within the distance formula_34 from the area formula_29. Therefore, all the particles with speed formula_30 at angle formula_31 from the normal that can reach area formula_29 within time interval formula_33 are contained in the tilted pipe with a height of formula_40  and a volume of formula_41. The total number of particles that reach area formula_29 within time interval formula_33 also depends on the velocity distribution; All in all, it calculates to be:formula_44 Integrating this over all appropriate velocities within the constraint formula_45 yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time: formula_46 This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed formula_47 of the Maxwell's velocity distribution, one has to integrate overformula_48. The momentum transfer to the container  wall from particles hitting the area formula_29 with speed formula_30 at angle formula_31 from the normal, in time interval formula_33 is:formula_53Integrating this over all appropriate velocities within the constraint formula_45 yields the pressure (consistent with Ideal gas law):formula_55If this small area formula_56 is punched to become a small hole, the effusive flow rate will be: formula_57 Combined with the ideal gas law, this yields formula_58 The above expression is consistent with Graham's law. To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with formula_59 that hit the  area formula_29 within the time interval formula_33 are contained in the tilted pipe with a height of formula_40 and a volume of formula_41; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of formula_64: formula_65 with the constraint formula_66. The constant formula_67 can be determined by the normalization condition formula_68 to be formula_69, and overall:formula_70 Speed of molecules. From the kinetic energy formula it can be shown that formula_71 formula_72 formula_73 where "v" is in m/s, "T" is in kelvins, and "m" is the mass of one molecule of gas. The most probable  (or mode) speed formula_74 is 81.6% of the root-mean-square speed formula_75, and the mean (arithmetic mean, or average) speed formula_47 is 92.1% of the rms speed (isotropic distribution of speeds). See: Transport properties. The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means using Kinetic Theory to consider what are known as "transport properties", such as viscosity, thermal conductivity and mass diffusivity. Viscosity and kinetic momentum. In books on elementary kinetic theory one can find results for dilute gas modeling that are used  in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force "F". The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component formula_77 which increase uniformly with distance formula_78 above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium  distribution of molecular motions. Let formula_79 be the collision cross section of one molecule colliding with another. As in the previous section, the number density formula_80 is defined as the number of molecules per (extensive) volume, or formula_81. The collision cross section per volume or collision cross section density is formula_82, and it is related to the mean free path formula_83 by formula_84 Notice that the unit of the collision cross section per volume formula_82 is reciprocal of length. The mean free path is the average distance traveled by a molecule, or a number of molecules per volume, before they  make their first collision. Let formula_86 be the forward velocity of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area formula_29 on one side of the gas layer, with speed formula_30 at angle formula_31 from the normal, in time interval formula_33 is formula_91 These molecules made their last collision at a distance formula_92 above and below the gas layer, and each will contribute a forward momentum of formula_93 where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient formula_94 can be considered  to be constant over a distance of mean free path. Integrating over all appropriate velocities within the constraint formula_95 yields the forward momentum transfer per unit time per unit area (also known as shear stress): formula_96 The net rate of momentum per unit area that is transported across the imaginary surface is thus formula_97 Combining the above kinetic equation with Newton's law of viscosity formula_98 gives the equation for shear viscosity, which is usually denoted formula_99 when it is a dilute gas: formula_100 Combining this equation with the equation for mean free path gives formula_101 Maxwell-Boltzmann distribution gives  the average (equilibrium) molecular speed as formula_102 where formula_103 is the most probable speed. We note that formula_104 and insert the velocity in the viscosity equation above. This gives the well known equation for shear viscosity for dilute gases: formula_105 and formula_106 is the molar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the kinetic translational energy dominates over rotational and vibrational molecule energies. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core  particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by formula_107 The radius formula_108 is called collision cross section radius or kinetic radius, and the diameter formula_109 is called collision cross section diameter or kinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collision cross section and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e.  a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the Lennard-Jones potential or Morse potential which have a negative part that attracts the other molecule from distances longer than the hard core radius. The radius for zero Lennard-Jones potential is then appropriate to use as an estimate for the kinetic radius. Thermal conductivity and heat flux. Following a similar logic as above, one can derive the kinetic model for thermal conductivity of a dilute gas: Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so  massive compared to the gas layer that they can be treated as thermal reservoirs. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy formula_110 which increases uniformly with distance formula_78 above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions. Let formula_112 be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area formula_29 on one side of the gas layer, with speed  formula_30 at angle formula_31 from the normal, in time interval formula_33 is formula_117 These molecules made their last collision at a distance formula_92 above and below the gas layer, and each will contribute a molecular kinetic energy of formula_119 where formula_120 is the specific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient formula_121 can be considered to be constant over a distance of mean free path. Integrating over all appropriate velocities within the constraint formula_95 yields the energy transfer per unit time per unit area (also known as heat  flux): formula_123 Note that the energy transfer from above is in the formula_124 direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus formula_125 Combining the above kinetic equation with Fourier's law formula_126 gives the equation for thermal conductivity, which is usually denoted formula_127 when it is a dilute gas: formula_128 Diffusion Coefficient and diffusion flux. Following a similar logic as above, one can derive the kinetic model for mass diffusivity of a dilute gas: Consider a steady diffusion between two regions of the same gas with perfectly flat  and parallel boundaries separated by a layer of the same gas. Both regions have uniform number densities, but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density formula_129 in the layer increases uniformly with distance formula_78 above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions. Let formula_131 be the number density of the gas at an imaginary horizontal surface inside the layer. The number of  molecules arriving at an area formula_29 on one side of the gas layer, with speed formula_30 at angle formula_31 from the normal, in time interval formula_33 is formula_136 These molecules made their last collision at a distance formula_92 above and below the gas layer, where the local number density is formula_138 Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient formula_139 can be considered to be constant over a distance of mean free path. Integrating over all appropriate velocities within the constraint formula_95 yields the molecular transfer per unit time per  unit area (also known as diffusion flux): formula_141 Note that the molecular transfer from above is in the formula_124 direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus formula_143 Combining the above kinetic equation with Fick's first law of diffusion formula_144 gives the equation for mass diffusivity, which is usually denoted formula_145 when it is a dilute gas: formula_146