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-[{"problem_text": "The change in molar internal energy when $\\mathrm{CaCO}_3(\\mathrm{~s})$ as calcite converts to another form, aragonite, is $+0.21 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Calculate the difference between the molar enthalpy and internal energy changes when the pressure is 1.0 bar given that the densities of the polymorphs are $2.71 \\mathrm{~g} \\mathrm{~cm}^{-3}$ and $2.93 \\mathrm{~g} \\mathrm{~cm}^{-3}$, respectively.", "answer_latex": " -0.28", "answer_number": "-0.28", "unit": " $\\mathrm{~Pa} \\mathrm{~m}^3 \\mathrm{~mol}^{-1}$", "source": "atkins", "problemid": " 2.2", "comment": " ", "solution": "The change in enthalpy when the transition occurs is\r\n$$\r\n\\begin{aligned}\r\n\\Delta H_{\\mathrm{m}} & =H_{\\mathrm{m}}(\\text { aragonite })-H_{\\mathrm{m}}(\\text { calcite }) \\\\\r\n& =\\left\\{U_{\\mathrm{m}}(\\mathrm{a})+p V_{\\mathrm{m}}(\\mathrm{a})\\right\\}-\\left\\{U_{\\mathrm{m}}(\\mathrm{c})+p V_{\\mathrm{m}}(\\mathrm{c})\\right\\} \\\\\r\n& =\\Delta U_{\\mathrm{m}}+p\\left\\{V_{\\mathrm{m}}(\\mathrm{a})-V_{\\mathrm{m}}(\\mathrm{c})\\right\\}\r\n\\end{aligned}\r\n$$\r\nwhere a denotes aragonite and c calcite. It follows by substituting $V_{\\mathrm{m}}=M / \\rho$ that\r\n$$\r\n\\Delta H_{\\mathrm{m}}-\\Delta U_{\\mathrm{m}}=p M\\left(\\frac{1}{\\rho(\\mathrm{a})}-\\frac{1}{\\rho(\\mathrm{c})}\\right)\r\n$$\r\nSubstitution of the data, using $M=100 \\mathrm{~g} \\mathrm{~mol}^{-1}$, gives\r\n$$\r\n\\begin{aligned}\r\n\\Delta H_{\\mathrm{m}}-\\Delta U_{\\mathrm{m}} & =\\left(1.0 \\times 10^5 \\mathrm{~Pa}\\right) \\times\\left(100 \\mathrm{~g} \\mathrm{~mol}^{-1}\\right) \\times\\left(\\frac{1}{2.93 \\mathrm{~g} \\mathrm{~cm}^{-3}}-\\frac{1}{2.71 \\mathrm{~g} \\mathrm{~cm}^{-3}}\\right) \\\\\r\n& =-2.8 \\times 10^5 \\mathrm{~Pa} \\mathrm{~cm}{ }^3 \\mathrm{~mol}^{-1}=-0.28 \\mathrm{~Pa} \\mathrm{~m}^3 \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Estimate the molar volume of $\\mathrm{CO}_2$ at $500 \\mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.", "answer_latex": " 0.366", "answer_number": "0.366", "unit": " $\\mathrm{dm}^3\\mathrm{~mol}^{-1}$", "source": "atkins", "problemid": " 1.4", "comment": " ", "solution": "According to Table 1.6, $a=3.610 \\mathrm{dm}^6 \\mathrm{~atm} \\mathrm{~mol}^{-2}$ and $b=4.29 \\times 10^{-2} \\mathrm{dm}^3$ $\\mathrm{mol}^{-1}$. Under the stated conditions, $R T / p=0.410 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. The coefficients in the equation for $V_{\\mathrm{m}}$ are therefore\r\n$$\r\n\\begin{aligned}\r\n& b+R T / p=0.453 \\mathrm{dm}^3 \\mathrm{~mol}^{-1} \\\\\r\n& a / p=3.61 \\times 10^{-2}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^2 \\\\\r\n& a b / p=1.55 \\times 10^{-3}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^3\r\n\\end{aligned}\r\n$$\r\nTherefore, on writing $x=V_{\\mathrm{m}} /\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)$, the equation to solve is\r\n$$\r\nx^3-0.453 x^2+\\left(3.61 \\times 10^{-2}\\right) x-\\left(1.55 \\times 10^{-3}\\right)=0\r\n$$\r\nThe acceptable root is $x=0.366$ (Fig. 1.18), which implies that $V_{\\mathrm{m}}=0.366 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$."}, {"problem_text": "Suppose the concentration of a solute decays exponentially along the length of a container. Calculate the thermodynamic force on the solute at $25^{\\circ} \\mathrm{C}$ given that the concentration falls to half its value in $10 \\mathrm{~cm}$.", "answer_latex": " 17", "answer_number": "17", "unit": " $\\mathrm{kN} \\mathrm{mol}^{-1}$", "source": "atkins", "problemid": " 20.3", "comment": " ", "solution": "The concentration varies with position as\r\n$$\r\nc=c_0 \\mathrm{e}^{-x / \\lambda}\r\n$$\r\nwhere $\\lambda$ is the decay constant. Therefore,\r\n$$\r\n\\frac{\\mathrm{d} c}{\\mathrm{~d} x}=-\\frac{c}{\\lambda}\r\n$$\r\nEquation 20.45 then implies that\r\n$$\r\n\\mathcal{F}=\\frac{R T}{\\lambda}\r\n$$\r\nWe know that the concentration falls to $\\frac{1}{2} c_0$ at $x=10 \\mathrm{~cm}$, so we can find $\\lambda$ from $\\frac{1}{2}=\\mathrm{e}^{-(10 \\mathrm{~cm}) / \\lambda}$. That is $\\lambda=(10 \\mathrm{~cm} / \\ln 2)$. It follows that\r\n$$\r\n\\mathcal{F}=\\left(8.3145 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\times(298 \\mathrm{~K}) \\times \\ln 2 /\\left(1.0 \\times 10^{-1} \\mathrm{~m}\\right)=17 \\mathrm{kN} \\mathrm{mol}^{-1}\r\n$$\r\nwhere we have used $1 \\mathrm{~J}=1 \\mathrm{~N} \\mathrm{~m}$.\r\n"}, {"problem_text": "A container is divided into two equal compartments (Fig. 5.8). One contains $3.0 \\mathrm{~mol} \\mathrm{H}_2(\\mathrm{~g})$ at $25^{\\circ} \\mathrm{C}$; the other contains $1.0 \\mathrm{~mol} \\mathrm{~N}_2(\\mathrm{~g})$ at $25^{\\circ} \\mathrm{C}$. Calculate the Gibbs energy of mixing when the partition is removed. Assume perfect behaviour.\r\n", "answer_latex": " -6.9", "answer_number": "-6.9", "unit": " $\\mathrm{~kJ}$", "source": "atkins", "problemid": " 5.2", "comment": " ", "solution": "Given that the pressure of nitrogen is $p$, the pressure of hydrogen is $3 p$; therefore, the initial Gibbs energy is\r\n$$\r\nG_{\\mathrm{i}}=(3.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{H}_2\\right)+R T \\ln 3 p\\right\\}+(1.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{N}_2\\right)+R T \\ln p\\right\\}\r\n$$\r\nWhen the partition is removed and each gas occupies twice the original volume, the partial pressure of nitrogen falls to $\\frac{1}{2} p$ and that of hydrogen falls to $\\frac{3}{2} p$. Therefore, the Gibbs energy changes to\r\n$$\r\nG_{\\mathrm{f}}=(3.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{H}_2\\right)+R T \\ln \\frac{3}{2} p\\right\\}+(1.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{N}_2\\right)+R T \\ln \\frac{1}{2} p\\right\\}\r\n$$\r\nThe Gibbs energy of mixing is the difference of these two quantities:\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\text {mix }} G & =(3.0 \\mathrm{~mol}) R T \\ln \\left(\\frac{\\frac{3}{2} p}{3 p}\\right)+(1.0 \\mathrm{~mol}) R T \\ln \\left(\\frac{\\frac{1}{2} p}{p}\\right) \\\\\r\n& =-(3.0 \\mathrm{~mol}) R T \\ln 2-(1.0 \\mathrm{~mol}) R T \\ln 2 \\\\\r\n& =-(4.0 \\mathrm{~mol}) R T \\ln 2=-6.9 \\mathrm{~kJ}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "What is the mean speed, $\\bar{c}$, of $\\mathrm{N}_2$ molecules in air at $25^{\\circ} \\mathrm{C}$ ?", "answer_latex": " 475", "answer_number": "475", "unit": " $\\mathrm{~m} \\mathrm{~s}^{-1}$", "source": "atkins", "problemid": " 20.1", "comment": " ", "solution": "The integral required is\r\n$$\r\n\\begin{aligned}\r\n\\bar{c} & =4 \\pi\\left(\\frac{M}{2 \\pi R T}\\right)^{3 / 2} \\int_0^{\\infty} v^3 \\mathrm{e}^{-M v^2 / 2 R T} \\mathrm{~d} v \\\\\r\n& =4 \\pi\\left(\\frac{M}{2 \\pi R T}\\right)^{3 / 2} \\times \\frac{1}{2}\\left(\\frac{2 R T}{M}\\right)^2=\\left(\\frac{8 R T}{\\pi M}\\right)^{1 / 2}\r\n\\end{aligned}\r\n$$\r\nwhere we have used the standard result from tables of integrals (or software) that\r\n$$\r\n\\int_0^{\\infty} x^3 \\mathrm{e}^{-a x^2} \\mathrm{~d} x=\\frac{1}{2 a^2}\r\n$$\r\nSubstitution of the data then gives\r\n$$\r\n\\bar{c}=\\left(\\frac{8 \\times\\left(8.3141 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\times(298 \\mathrm{~K})}{\\pi \\times\\left(28.02 \\times 10^{-3} \\mathrm{~kg} \\mathrm{~mol}^{-1}\\right)}\\right)^{1 / 2}=475 \\mathrm{~m} \\mathrm{~s}^{-1}\r\n$$\r\nwhere we have used $1 \\mathrm{~J}=1 \\mathrm{~kg} \\mathrm{~m}^2 \\mathrm{~s}^{-2}$.\r\n"}, {"problem_text": "Caesium (m.p. $29^{\\circ} \\mathrm{C}$, b.p. $686^{\\circ} \\mathrm{C}$ ) was introduced into a container and heated to $500^{\\circ} \\mathrm{C}$. When a hole of diameter $0.50 \\mathrm{~mm}$ was opened in the container for $100 \\mathrm{~s}$, a mass loss of $385 \\mathrm{mg}$ was measured. Calculate the vapour pressure of liquid caesium at $500 \\mathrm{~K}$.", "answer_latex": " 8.7", "answer_number": "8.7", "unit": " $\\mathrm{kPa}$", "source": "atkins", "problemid": " 20.2", "comment": " ", "solution": " The mass loss $\\Delta m$ in an interval $\\Delta t$ is related to the collision flux by\r\n$$\r\n\\Delta m=Z_{\\mathrm{W}} A_0 m \\Delta t\r\n$$\r\nwhere $A_0$ is the area of the hole and $m$ is the mass of one atom. It follows that\r\n$$\r\nZ_{\\mathrm{W}}=\\frac{\\Delta m}{A_0 m \\Delta t}\r\n$$\r\nBecause $Z_{\\mathrm{W}}$ is related to the pressure by eqn 20.14 , we can write\r\n$$\r\np=\\left(\\frac{2 \\pi R T}{M}\\right)^{1 / 2} \\frac{\\Delta m}{A_0 \\Delta t}\r\n$$\r\nBecause $M=132.9 \\mathrm{~g} \\mathrm{~mol}^{-1}$, substitution of the data gives $p=8.7 \\mathrm{kPa}$ (using $1 \\mathrm{~Pa}=$ $\\left.1 \\mathrm{~N} \\mathrm{~m}^{-2}=1 \\mathrm{~J} \\mathrm{~m}^{-1}\\right)$."}, {"problem_text": "To get an idea of the distance dependence of the tunnelling current in STM, suppose that the wavefunction of the electron in the gap between sample and needle is given by $\\psi=B \\mathrm{e}^{-\\kappa x}$, where $\\kappa=\\left\\{2 m_{\\mathrm{e}}(V-E) / \\hbar^2\\right\\}^{1 / 2}$; take $V-E=2.0 \\mathrm{eV}$. By what factor would the current drop if the needle is moved from $L_1=0.50 \\mathrm{~nm}$ to $L_2=0.60 \\mathrm{~nm}$ from the surface?", "answer_latex": " 0.23", "answer_number": "0.23", "unit": " ", "source": "atkins", "problemid": " 8.2", "comment": " ", "solution": " When $L=L_1=0.50 \\mathrm{~nm}$ and $V-E=2.0 \\mathrm{eV}=3.20 \\times 10^{-19} \\mathrm{~J}$ the value of $\\kappa L$ is\r\n$$\r\n\\begin{aligned}\r\n\\kappa L_1 & =\\left\\{\\frac{2 m_{\\mathrm{e}}(V-E)}{\\hbar^2}\\right\\}^{1 / 2} L_1 \\\\\r\n& =\\left\\{\\frac{2 \\times\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right) \\times\\left(3.20 \\times 10^{-19} \\mathrm{~J}\\right)}{\\left(1.054 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)^2}\\right\\}^{1 / 2} \\times\\left(5.0 \\times 10^{-10} \\mathrm{~m}\\right) \\\\\r\n& =\\left(7.25 \\times 10^9 \\mathrm{~m}^{-1}\\right) \\times\\left(5.0 \\times 10^{-10} \\mathrm{~m}\\right)=3.6\r\n\\end{aligned}\r\n$$\r\nBecause $\\kappa L_1>1$, we use eqn $8.19 \\mathrm{~b}$ to calculate the transmission probabilities at the two distances. It follows that\r\n$$\r\n\\text { current at } \\begin{aligned}\r\n\\frac{L_2}{\\text { current at } L_1} & =\\frac{T\\left(L_2\\right)}{T\\left(L_1\\right)}=\\frac{16 \\varepsilon(1-\\varepsilon) \\mathrm{e}^{-2 \\kappa L_2}}{16 \\varepsilon(1-\\varepsilon) \\mathrm{e}^{-2 \\kappa L_1}}=\\mathrm{e}^{-2 \\kappa\\left(L_2-L_1\\right)} \\\\\r\n& =\\mathrm{e}^{-2 \\times\\left(7.25 \\times 10^{-9} \\mathrm{~m}^{-1}\\right) \\times\\left(1.0 \\times 10^{-10} \\mathrm{~m}\\right)}=0.23\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Calculate the typical wavelength of neutrons that have reached thermal equilibrium with their surroundings at $373 \\mathrm{~K}$.\r\n", "answer_latex": " 226", "answer_number": "226", "unit": " $\\mathrm{pm}$", "source": "atkins", "problemid": " 19.4", "comment": " ", "solution": "From the equipartition principle, we know that the mean translational kinetic energy of a neutron at a temperature $T$ travelling in the $x$-direction is $E_{\\mathrm{k}}=\\frac{1}{2} k T$. The kinetic energy is also equal to $p^2 / 2 m$, where $p$ is the momentum of the neutron and $m$ is its mass. Hence, $p=(m k T)^{1 / 2}$. It follows from the de Broglie relation $\\lambda=h / p$ that the neutron's wavelength is\r\n$$\r\n\\lambda=\\frac{h}{(m k T)^{1 / 2}}\r\n$$\r\nTherefore, at $373 \\mathrm{~K}$,\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left\\{\\left(1.675 \\times 10^{-27} \\mathrm{~kg}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\times(373 \\mathrm{~K})\\right\\}^{1 / 2}} \\\\\r\n& =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left(1.675 \\times 1.381 \\times 373 \\times 10^{-50}\\right)^{1 / 2}\\left(\\mathrm{~kg}^2 \\mathrm{~m}^2 \\mathrm{~s}^{-2}\\right)^{1 / 2}} \\\\\r\n& =2.26 \\times 10^{-10} \\mathrm{~m}=226 \\mathrm{pm}\r\n\\end{aligned}"}, {"problem_text": "Calculate the separation of the $\\{123\\}$ planes of an orthorhombic unit cell with $a=0.82 \\mathrm{~nm}, b=0.94 \\mathrm{~nm}$, and $c=0.75 \\mathrm{~nm}$.", "answer_latex": " 0.21", "answer_number": "0.21", "unit": " $\\mathrm{~nm}$", "source": "atkins", "problemid": " 19.1", "comment": " ", "solution": " For the first part, simply substitute the information into eqn 19.3. For the second part, instead of repeating the calculation, note that, if all three Miller indices are multiplied by $n$, then their separation is reduced by that factor (Fig. 19.12):\r\n$$\r\n\\frac{1}{d_{n h, n k, n l}^2}=\\frac{(n h)^2}{a^2}+\\frac{(n k)^2}{b^2}+\\frac{(n l)^2}{c^2}=n^2\\left(\\frac{h^2}{a^2}+\\frac{k^2}{b^2}+\\frac{l^2}{c^2}\\right)=\\frac{n^2}{d_{h k l}^2}\r\n$$\r\nwhich implies that\r\n$$\r\nd_{n h, n k, n l}=\\frac{d_{h k l}}{n}\r\n$$\r\nAnswer Substituting the indices into eqn 19.3 gives\r\n$$\r\n\\frac{1}{d_{123}^2}=\\frac{1^2}{(0.82 \\mathrm{~nm})^2}+\\frac{2^2}{(0.94 \\mathrm{~nm})^2}+\\frac{3^2}{(0.75 \\mathrm{~nm})^2}=0.22 \\mathrm{~nm}^{-2}\r\n$$\r\nHence, $d_{123}=0.21 \\mathrm{~nm}$.\r\n"}, {"problem_text": "Calculate the moment of inertia of an $\\mathrm{H}_2 \\mathrm{O}$ molecule around the axis defined by the bisector of the $\\mathrm{HOH}$ angle (3). The $\\mathrm{HOH}$ bond angle is $104.5^{\\circ}$ and the bond length is $95.7 \\mathrm{pm}$.", "answer_latex": " 1.91", "answer_number": "1.91", "unit": " $10^{-47} \\mathrm{~kg} \\mathrm{~m}^2$", "source": "atkins", "problemid": "12.1", "comment": " ", "solution": "From eqn 12.2,\r\n$$\r\nI=\\sum_i m_i x_i^2=m_{\\mathrm{H}} x_{\\mathrm{H}}^2+0+m_{\\mathrm{H}} x_{\\mathrm{H}}^2=2 m_{\\mathrm{H}} x_{\\mathrm{H}}^2\r\n$$\r\nIf the bond angle of the molecule is denoted $2 \\phi$ and the bond length is $R$, trigonometry gives $x_{\\mathrm{H}}=R \\sin \\phi$. It follows that\r\n$$\r\nI=2 m_{\\mathrm{H}} R^2 \\sin ^2 \\phi\r\n$$\r\nSubstitution of the data gives\r\n$$\r\n\\begin{aligned}\r\nI & =2 \\times\\left(1.67 \\times 10^{-27} \\mathrm{~kg}\\right) \\times\\left(9.57 \\times 10^{-11} \\mathrm{~m}\\right)^2 \\times \\sin ^2\\left(\\frac{1}{2} \\times 104.5^{\\circ}\\right) \\\\\r\n& =1.91 \\times 10^{-47} \\mathrm{~kg} \\mathrm{~m}^2\r\n\\end{aligned}\r\n$$\r\nNote that the mass of the $\\mathrm{O}$ atom makes no contribution to the moment of inertia for this mode of rotation as the atom is immobile while the $\\mathrm{H}$ atoms circulate around it.\r\n"}, {"problem_text": "The data below show the temperature variation of the equilibrium constant of the reaction $\\mathrm{Ag}_2 \\mathrm{CO}_3(\\mathrm{~s}) \\rightleftharpoons \\mathrm{Ag}_2 \\mathrm{O}(\\mathrm{s})+\\mathrm{CO}_2(\\mathrm{~g})$. Calculate the standard reaction enthalpy of the decomposition.\r\n$\\begin{array}{lllll}T / \\mathrm{K} & 350 & 400 & 450 & 500 \\\\ K & 3.98 \\times 10^{-4} & 1.41 \\times 10^{-2} & 1.86 \\times 10^{-1} & 1.48\\end{array}$", "answer_latex": " +80", "answer_number": "+80", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "atkins", "problemid": " 6.3", "comment": " ", "solution": "We draw up the following table:\r\n$\\begin{array}{lllll}T / \\mathrm{K} & 350 & 400 & 450 & 500 \\\\ \\left(10^3 \\mathrm{~K}\\right) / T & 2.86 & 2.50 & 2.22 & 2.00 \\\\ -\\ln K & 7.83 & 4.26 & 1.68 & -0.39\\end{array}$\r\nThese points are plotted in Fig. 6.9. The slope of the graph is $+9.6 \\times 10^3$, so\r\n$$\r\n\\Delta_{\\mathrm{r}} H^{\\ominus}=\\left(+9.6 \\times 10^3 \\mathrm{~K}\\right) \\times R=+80 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n$$"}, {"problem_text": "The osmotic pressures of solutions of poly(vinyl chloride), PVC, in cyclohexanone at $298 \\mathrm{~K}$ are given below. The pressures are expressed in terms of the heights of solution (of mass density $\\rho=0.980 \\mathrm{~g} \\mathrm{~cm}^{-3}$ ) in balance with the osmotic pressure. Determine the molar mass of the polymer.\r\n$\\begin{array}{llllll}c /\\left(\\mathrm{g} \\mathrm{dm}^{-3}\\right) & 1.00 & 2.00 & 4.00 & 7.00 & 9.00 \\\\ h / \\mathrm{cm} & 0.28 & 0.71 & 2.01 & 5.10 & 8.00\\end{array}$\r\n", "answer_latex": " 1.2", "answer_number": "1.2", "unit": " $10^2 \\mathrm{~kg} \\mathrm{~mol}^{-1}$", "source": "atkins", "problemid": " 5.4", "comment": " ", "solution": "The data give the following values for the quantities to plot:\r\n$$\r\n\\begin{array}{llllll}\r\nc /\\left(\\mathrm{g} \\mathrm{dm}^{-3}\\right) & 1.00 & 2.00 & 4.00 & 7.00 & 9.00 \\\\\r\n(h / c) /\\left(\\mathrm{cm} \\mathrm{g}^{-1} \\mathrm{dm}^3\\right) & 0.28 & 0.36 & 0.503 & 0.729 & 0.889\r\n\\end{array}\r\n$$\r\nThe points are plotted in Fig. 5.28. The intercept is at 0.21 . Therefore,\r\n$$\r\n\\begin{aligned}\r\nM & =\\frac{R T}{\\rho g} \\times \\frac{1}{0.21 \\mathrm{~cm} \\mathrm{~g}^{-1} \\mathrm{dm}^3} \\\\\r\n& =\\frac{\\left(8.3145 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\times(298 \\mathrm{~K})}{\\left(980 \\mathrm{~kg} \\mathrm{~m}^{-1}\\right) \\times\\left(9.81 \\mathrm{~m} \\mathrm{~s}^{-2}\\right)} \\times \\frac{1}{2.1 \\times 10^{-3} \\mathrm{~m}^4 \\mathrm{~kg}^{-1}} \\\\\r\n& =1.2 \\times 10^2 \\mathrm{~kg} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Estimate the wavelength of electrons that have been accelerated from rest through a potential difference of $40 \\mathrm{kV}$.", "answer_latex": " 6.1", "answer_number": "6.1", "unit": " $10^{-12} \\mathrm{~m}$", "source": "atkins", "problemid": " 7.2", "comment": " ", "solution": " The expression $p^2 / 2 m_{\\mathrm{e}}=e \\Delta \\phi$ solves to $p=\\left(2 m_{\\mathrm{e}} e \\Delta \\phi\\right)^{1 / 2}$; then, from the de Broglie relation $\\lambda=h / p$,\r\n$$\r\n\\lambda=\\frac{h}{\\left(2 m_{\\mathrm{e}} e \\Delta \\phi\\right)^{1 / 2}}\r\n$$\r\nSubstitution of the data and the fundamental constants (from inside the front cover) gives\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left\\{2 \\times\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right) \\times\\left(1.602 \\times 10^{-19} \\mathrm{C}\\right) \\times\\left(4.0 \\times 10^4 \\mathrm{~V}\\right)\\right\\}^{1 / 2}} \\\\\r\n& =6.1 \\times 10^{-12} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "The standard enthalpy of formation of $\\mathrm{H}_2 \\mathrm{O}(\\mathrm{g})$ at $298 \\mathrm{~K}$ is $-241.82 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Estimate its value at $100^{\\circ} \\mathrm{C}$ given the following values of the molar heat capacities at constant pressure: $\\mathrm{H}_2 \\mathrm{O}(\\mathrm{g}): 33.58 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} ; \\mathrm{H}_2(\\mathrm{~g}): 28.82 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} ; \\mathrm{O}_2(\\mathrm{~g})$ : $29.36 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$. Assume that the heat capacities are independent of temperature.", "answer_latex": "-242.6 ", "answer_number": "-242.6", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "atkins", "problemid": " 2.6", "comment": " ", "solution": "The reaction is $\\mathrm{H}_2(\\mathrm{~g})+\\frac{1}{2} \\mathrm{O}_2(\\mathrm{~g}) \\rightarrow \\mathrm{H}_2 \\mathrm{O}(\\mathrm{g})$, so\r\n$$\r\n\\Delta_{\\mathrm{r}} C_p^{\\ominus}=C_{p, \\mathrm{~m}}^{\\ominus}\\left(\\mathrm{H}_2 \\mathrm{O}, \\mathrm{g}\\right)-\\left\\{C_{p, \\mathrm{~m}}^{\\ominus}\\left(\\mathrm{H}_2, \\mathrm{~g}\\right)+\\frac{1}{2} C_{p, \\mathrm{~m}}^{\\ominus}\\left(\\mathrm{O}_2, \\mathrm{~g}\\right)\\right\\}=-9.92 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\r\n$$\r\nIt then follows that\r\n$$\r\n\\Delta_{\\mathrm{r}} H^{\\ominus}(373 \\mathrm{~K})=-241.82 \\mathrm{~kJ} \\mathrm{~mol}^{-1}+(75 \\mathrm{~K}) \\times\\left(-9.92 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right)=-242.6 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n$$\r\n"}, {"problem_text": "The standard potential of the cell $\\mathrm{Pt}(\\mathrm{s})\\left|\\mathrm{H}_2(\\mathrm{~g})\\right| \\mathrm{HBr}(\\mathrm{aq})|\\operatorname{AgBr}(\\mathrm{s})| \\mathrm{Ag}(\\mathrm{s})$ was measured over a range of temperatures, and the data were found to fit the following polynomial:\r\n$$\r\nE_{\\text {cell }}^{\\bullet} / \\mathrm{V}=0.07131-4.99 \\times 10^{-4}(T / \\mathrm{K}-298)-3.45 \\times 10^{-6}(\\mathrm{~T} / \\mathrm{K}-298)^2\r\n$$\r\nThe cell reaction is $\\operatorname{AgBr}(\\mathrm{s})+\\frac{1}{2} \\mathrm{H}_2(\\mathrm{~g}) \\rightarrow \\mathrm{Ag}(\\mathrm{s})+\\mathrm{HBr}(\\mathrm{aq})$. Evaluate the standard reaction Gibbs energy, enthalpy, and entropy at $298 \\mathrm{~K}$.\r\n", "answer_latex": " -21.2", "answer_number": "-21.2", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "atkins", "problemid": " 6.5", "comment": " ", "solution": "At $T=298 \\mathrm{~K}, E_{\\text {cell }}^{\\ominus}=+0.07131 \\mathrm{~V}$, so\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\mathrm{r}} G^{\\ominus} & =-v F E_{\\text {cell }}^{\\ominus}=-(1) \\times\\left(9.6485 \\times 10^4 \\mathrm{Cmol}^{-1}\\right) \\times(+0.07131 \\mathrm{~V}) \\\\\r\n& =-6.880 \\times 10^3 \\mathrm{~V} \\mathrm{Cmol}^{-1}=-6.880 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe temperature coefficient of the cell potential is\r\n$$\r\n\\frac{\\mathrm{d} E_{\\text {cell }}^{\\ominus}}{\\mathrm{d} T}=-4.99 \\times 10^{-4} \\mathrm{~V} \\mathrm{~K}^{-1}-2\\left(3.45 \\times 10^{-6}\\right)(T / \\mathrm{K}-298) \\mathrm{V} \\mathrm{K}^{-1}\r\n$$\r\nAt $T=298 \\mathrm{~K}$ this expression evaluates to\r\n$$\r\n\\frac{\\mathrm{d} E_{\\text {cell }}^{\\ominus}}{\\mathrm{d} T}=-4.99 \\times 10^{-4} \\mathrm{~V} \\mathrm{~K}^{-1}\r\n$$\r\nSo, from eqn 6.36 , the reaction entropy is\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\mathrm{r}} S^{\\ominus} & =1 \\times\\left(9.6485 \\times 10^4 \\mathrm{Cmol}^{-1}\\right) \\times\\left(-4.99 \\times 10^{-4} \\mathrm{~V} \\mathrm{~K}^{-1}\\right) \\\\\r\n& =-48.1 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe negative value stems in part from the elimination of gas in the cell reaction. It then follows that\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\mathrm{r}} H^{\\ominus} & =\\Delta_{\\mathrm{r}} G^{\\ominus}+T \\Delta_{\\mathrm{r}} S^{\\ominus}=-6.880 \\mathrm{~kJ} \\mathrm{~mol}^{-1}+(298 \\mathrm{~K}) \\times\\left(-0.0482 \\mathrm{~kJ} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\\\\r\n& =-21.2 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "In an industrial process, nitrogen is heated to $500 \\mathrm{~K}$ in a vessel of constant volume. If it enters the vessel at $100 \\mathrm{~atm}$ and $300 \\mathrm{~K}$, what pressure would it exert at the working temperature if it behaved as a perfect gas?", "answer_latex": " 167", "answer_number": "167", "unit": " $\\mathrm{atm}$", "source": "atkins", "problemid": " 1.2", "comment": " ", "solution": "Cancellation of the volumes (because $V_1=V_2$ ) and amounts (because $\\left.n_1=n_2\\right)$ on each side of the combined gas law results in\r\n$$\r\n\\frac{p_1}{T_1}=\\frac{p_2}{T_2}\r\n$$\r\nwhich can be rearranged into\r\n$$\r\np_2=\\frac{T_2}{T_1} \\times p_1\r\n$$\r\nSubstitution of the data then gives\r\n$$\r\np_2=\\frac{500 \\mathrm{~K}}{300 \\mathrm{~K}} \\times(100 \\mathrm{~atm})=167 \\mathrm{~atm}\r\n$$\r\n"}]
+[
+ {
+  "problem_text": "The change in molar internal energy when $\\mathrm{CaCO}_3(\\mathrm{~s})$ as calcite converts to another form, aragonite, is $+0.21 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Calculate the difference between the molar enthalpy and internal energy changes when the pressure is 1.0 bar given that the densities of the polymorphs are $2.71 \\mathrm{~g} \\mathrm{~cm}^{-3}$ and $2.93 \\mathrm{~g} \\mathrm{~cm}^{-3}$, respectively.",
+  "answer_latex": " -0.28",
+  "answer_number": "-0.28",
+  "unit": " $\\mathrm{~Pa} \\mathrm{~m}^3 \\mathrm{~mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 2.2",
+  "comment": " ",
+  "solution": "The change in enthalpy when the transition occurs is\r\n$$\r\n\\begin{aligned}\r\n\\Delta H_{\\mathrm{m}} & =H_{\\mathrm{m}}(\\text { aragonite })-H_{\\mathrm{m}}(\\text { calcite }) \\\\\r\n& =\\left\\{U_{\\mathrm{m}}(\\mathrm{a})+p V_{\\mathrm{m}}(\\mathrm{a})\\right\\}-\\left\\{U_{\\mathrm{m}}(\\mathrm{c})+p V_{\\mathrm{m}}(\\mathrm{c})\\right\\} \\\\\r\n& =\\Delta U_{\\mathrm{m}}+p\\left\\{V_{\\mathrm{m}}(\\mathrm{a})-V_{\\mathrm{m}}(\\mathrm{c})\\right\\}\r\n\\end{aligned}\r\n$$\r\nwhere a denotes aragonite and c calcite. It follows by substituting $V_{\\mathrm{m}}=M / \\rho$ that\r\n$$\r\n\\Delta H_{\\mathrm{m}}-\\Delta U_{\\mathrm{m}}=p M\\left(\\frac{1}{\\rho(\\mathrm{a})}-\\frac{1}{\\rho(\\mathrm{c})}\\right)\r\n$$\r\nSubstitution of the data, using $M=100 \\mathrm{~g} \\mathrm{~mol}^{-1}$, gives\r\n$$\r\n\\begin{aligned}\r\n\\Delta H_{\\mathrm{m}}-\\Delta U_{\\mathrm{m}} & =\\left(1.0 \\times 10^5 \\mathrm{~Pa}\\right) \\times\\left(100 \\mathrm{~g} \\mathrm{~mol}^{-1}\\right) \\times\\left(\\frac{1}{2.93 \\mathrm{~g} \\mathrm{~cm}^{-3}}-\\frac{1}{2.71 \\mathrm{~g} \\mathrm{~cm}^{-3}}\\right) \\\\\r\n& =-2.8 \\times 10^5 \\mathrm{~Pa} \\mathrm{~cm}{ }^3 \\mathrm{~mol}^{-1}=-0.28 \\mathrm{~Pa} \\mathrm{~m}^3 \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Estimate the molar volume of $\\mathrm{CO}_2$ at $500 \\mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.",
+  "answer_latex": " 0.366",
+  "answer_number": "0.366",
+  "unit": " $\\mathrm{dm}^3\\mathrm{~mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 1.4",
+  "comment": " ",
+  "solution": "\nAccording to Table $$\n\\begin{array}{lll}\n\\hline & \\boldsymbol{a} /\\left(\\mathbf{a t m} \\mathbf{d m}^6 \\mathrm{~mol}^{-2}\\right) & \\boldsymbol{b} /\\left(\\mathbf{1 0}^{-2} \\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right) \\\\\n\\hline \\mathrm{Ar} & 1.337 & 3.20 \\\\\n\\mathrm{CO}_2 & 3.610 & 4.29 \\\\\n\\mathrm{He} & 0.0341 & 2.38 \\\\\n\\mathrm{Xe} & 4.137 & 5.16 \\\\\n\\hline\n\\end{array}\n$$, $a=3.610 \\mathrm{dm}^6 \\mathrm{~atm} \\mathrm{~mol}^{-2}$ and $b=4.29 \\times 10^{-2} \\mathrm{dm}^3$ $\\mathrm{mol}^{-1}$. Under the stated conditions, $R T / p=0.410 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. The coefficients in the equation for $V_{\\mathrm{m}}$ are therefore\r\n$$\r\n\\begin{aligned}\r\n& b+R T / p=0.453 \\mathrm{dm}^3 \\mathrm{~mol}^{-1} \\\\\r\n& a / p=3.61 \\times 10^{-2}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^2 \\\\\r\n& a b / p=1.55 \\times 10^{-3}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^3\r\n\\end{aligned}\r\n$$\r\nTherefore, on writing $x=V_{\\mathrm{m}} /\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)$, the equation to solve is\r\n$$\r\nx^3-0.453 x^2+\\left(3.61 \\times 10^{-2}\\right) x-\\left(1.55 \\times 10^{-3}\\right)=0\r\n$$\r\nThe acceptable root is $x=0.366$, which implies that $V_{\\mathrm{m}}=0.366 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$."
+ },
+ {
+  "problem_text": "Suppose the concentration of a solute decays exponentially along the length of a container. Calculate the thermodynamic force on the solute at $25^{\\circ} \\mathrm{C}$ given that the concentration falls to half its value in $10 \\mathrm{~cm}$.",
+  "answer_latex": " 17",
+  "answer_number": "17",
+  "unit": " $\\mathrm{kN} \\mathrm{mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 20.3",
+  "comment": " ",
+  "solution": "\nThe concentration varies with position as\r\n$$\r\nc=c_0 \\mathrm{e}^{-x / \\lambda}\r\n$$\r\nwhere $\\lambda$ is the decay constant. Therefore,\r\n$$\r\n\\frac{\\mathrm{d} c}{\\mathrm{~d} x}=-\\frac{c}{\\lambda}\r\n$$\r\nFrom Equation $$\n\n\\mathcal{F}=-\\frac{R T}{c}\\left(\\frac{\\partial c}{\\partial x}\\right)_{p, T}\n\n$$ then implies that\r\n$$\r\n\\mathcal{F}=\\frac{R T}{\\lambda}\r\n$$\r\nWe know that the concentration falls to $\\frac{1}{2} c_0$ at $x=10 \\mathrm{~cm}$, so we can find $\\lambda$ from $\\frac{1}{2}=\\mathrm{e}^{-(10 \\mathrm{~cm}) / \\lambda}$. That is $\\lambda=(10 \\mathrm{~cm} / \\ln 2)$. It follows that\r\n$$\r\n\\mathcal{F}=\\left(8.3145 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\times(298 \\mathrm{~K}) \\times \\ln 2 /\\left(1.0 \\times 10^{-1} \\mathrm{~m}\\right)=17 \\mathrm{kN} \\mathrm{mol}^{-1}\r\n$$\r\nwhere we have used $1 \\mathrm{~J}=1 \\mathrm{~N} \\mathrm{~m}$."
+ },
+ {
+  "problem_text": "A container is divided into two equal compartments. One contains $3.0 \\mathrm{~mol} \\mathrm{H}_2(\\mathrm{~g})$ at $25^{\\circ} \\mathrm{C}$; the other contains $1.0 \\mathrm{~mol} \\mathrm{~N}_2(\\mathrm{~g})$ at $25^{\\circ} \\mathrm{C}$. Calculate the Gibbs energy of mixing when the partition is removed. Assume perfect behaviour.",
+  "answer_latex": " -6.9",
+  "answer_number": "-6.9",
+  "unit": " $\\mathrm{~kJ}$",
+  "source": "atkins",
+  "problemid": " 5.2",
+  "comment": " ",
+  "solution": "Given that the pressure of nitrogen is $p$, the pressure of hydrogen is $3 p$; therefore, the initial Gibbs energy is\r\n$$\r\nG_{\\mathrm{i}}=(3.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{H}_2\\right)+R T \\ln 3 p\\right\\}+(1.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{N}_2\\right)+R T \\ln p\\right\\}\r\n$$\r\nWhen the partition is removed and each gas occupies twice the original volume, the partial pressure of nitrogen falls to $\\frac{1}{2} p$ and that of hydrogen falls to $\\frac{3}{2} p$. Therefore, the Gibbs energy changes to\r\n$$\r\nG_{\\mathrm{f}}=(3.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{H}_2\\right)+R T \\ln \\frac{3}{2} p\\right\\}+(1.0 \\mathrm{~mol})\\left\\{\\mu^{\\ominus}\\left(\\mathrm{N}_2\\right)+R T \\ln \\frac{1}{2} p\\right\\}\r\n$$\r\nThe Gibbs energy of mixing is the difference of these two quantities:\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\text {mix }} G & =(3.0 \\mathrm{~mol}) R T \\ln \\left(\\frac{\\frac{3}{2} p}{3 p}\\right)+(1.0 \\mathrm{~mol}) R T \\ln \\left(\\frac{\\frac{1}{2} p}{p}\\right) \\\\\r\n& =-(3.0 \\mathrm{~mol}) R T \\ln 2-(1.0 \\mathrm{~mol}) R T \\ln 2 \\\\\r\n& =-(4.0 \\mathrm{~mol}) R T \\ln 2=-6.9 \\mathrm{~kJ}\r\n\\end{aligned}\r\n$$\r\n"
+ },
+ {
+  "problem_text": "What is the mean speed, $\\bar{c}$, of $\\mathrm{N}_2$ molecules in air at $25^{\\circ} \\mathrm{C}$ ?",
+  "answer_latex": " 475",
+  "answer_number": "475",
+  "unit": " $\\mathrm{~m} \\mathrm{~s}^{-1}$",
+  "source": "atkins",
+  "problemid": " 20.1",
+  "comment": " ",
+  "solution": "The integral required is\r\n$$\r\n\\begin{aligned}\r\n\\bar{c} & =4 \\pi\\left(\\frac{M}{2 \\pi R T}\\right)^{3 / 2} \\int_0^{\\infty} v^3 \\mathrm{e}^{-M v^2 / 2 R T} \\mathrm{~d} v \\\\\r\n& =4 \\pi\\left(\\frac{M}{2 \\pi R T}\\right)^{3 / 2} \\times \\frac{1}{2}\\left(\\frac{2 R T}{M}\\right)^2=\\left(\\frac{8 R T}{\\pi M}\\right)^{1 / 2}\r\n\\end{aligned}\r\n$$\r\nwhere we have used the standard result from tables of integrals (or software) that\r\n$$\r\n\\int_0^{\\infty} x^3 \\mathrm{e}^{-a x^2} \\mathrm{~d} x=\\frac{1}{2 a^2}\r\n$$\r\nSubstitution of the data then gives\r\n$$\r\n\\bar{c}=\\left(\\frac{8 \\times\\left(8.3141 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\times(298 \\mathrm{~K})}{\\pi \\times\\left(28.02 \\times 10^{-3} \\mathrm{~kg} \\mathrm{~mol}^{-1}\\right)}\\right)^{1 / 2}=475 \\mathrm{~m} \\mathrm{~s}^{-1}\r\n$$\r\nwhere we have used $1 \\mathrm{~J}=1 \\mathrm{~kg} \\mathrm{~m}^2 \\mathrm{~s}^{-2}$.\r\n"
+ },
+ {
+  "problem_text": "Caesium (m.p. $29^{\\circ} \\mathrm{C}$, b.p. $686^{\\circ} \\mathrm{C}$ ) was introduced into a container and heated to $500^{\\circ} \\mathrm{C}$. When a hole of diameter $0.50 \\mathrm{~mm}$ was opened in the container for $100 \\mathrm{~s}$, a mass loss of $385 \\mathrm{mg}$ was measured. Calculate the vapour pressure of liquid caesium at $500 \\mathrm{~K}$.",
+  "answer_latex": " 8.7",
+  "answer_number": "8.7",
+  "unit": " $\\mathrm{kPa}$",
+  "source": "atkins",
+  "problemid": " 20.2",
+  "comment": " ",
+  "solution": "\nThe mass loss $\\Delta m$ in an interval $\\Delta t$ is related to the collision flux by\r\n$$\r\n\\Delta m=Z_{\\mathrm{W}} A_0 m \\Delta t\r\n$$\r\nwhere $A_0$ is the area of the hole and $m$ is the mass of one atom. It follows that\r\n$$\r\nZ_{\\mathrm{W}}=\\frac{\\Delta m}{A_0 m \\Delta t}\r\n$$\r\nBecause $Z_{\\mathrm{W}}$ is related to the pressure by eqn $$\nZ_{\\mathrm{W}}=\\frac{p}{(2 \\pi m k T)^{1 / 2}}\n$$, we can write\r\n$$\r\np=\\left(\\frac{2 \\pi R T}{M}\\right)^{1 / 2} \\frac{\\Delta m}{A_0 \\Delta t}\r\n$$\r\nBecause $M=132.9 \\mathrm{~g} \\mathrm{~mol}^{-1}$, substitution of the data gives $p=8.7 \\mathrm{kPa}$ (using $1 \\mathrm{~Pa}=$ $\\left.1 \\mathrm{~N} \\mathrm{~m}^{-2}=1 \\mathrm{~J} \\mathrm{~m}^{-1}\\right)$.\n"
+ },
+ {
+  "problem_text": "To get an idea of the distance dependence of the tunnelling current in STM, suppose that the wavefunction of the electron in the gap between sample and needle is given by $\\psi=B \\mathrm{e}^{-\\kappa x}$, where $\\kappa=\\left\\{2 m_{\\mathrm{e}}(V-E) / \\hbar^2\\right\\}^{1 / 2}$; take $V-E=2.0 \\mathrm{eV}$. By what factor would the current drop if the needle is moved from $L_1=0.50 \\mathrm{~nm}$ to $L_2=0.60 \\mathrm{~nm}$ from the surface?",
+  "answer_latex": " 0.23",
+  "answer_number": "0.23",
+  "unit": " ",
+  "source": "atkins",
+  "problemid": " 8.2",
+  "comment": " ",
+  "solution": "\nWhen $L=L_1=0.50 \\mathrm{~nm}$ and $V-E=2.0 \\mathrm{eV}=3.20 \\times 10^{-19} \\mathrm{~J}$ the value of $\\kappa L$ is\r\n$$\r\n\\begin{aligned}\r\n\\kappa L_1 & =\\left\\{\\frac{2 m_{\\mathrm{e}}(V-E)}{\\hbar^2}\\right\\}^{1 / 2} L_1 \\\\\r\n& =\\left\\{\\frac{2 \\times\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right) \\times\\left(3.20 \\times 10^{-19} \\mathrm{~J}\\right)}{\\left(1.054 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)^2}\\right\\}^{1 / 2} \\times\\left(5.0 \\times 10^{-10} \\mathrm{~m}\\right) \\\\\r\n& =\\left(7.25 \\times 10^9 \\mathrm{~m}^{-1}\\right) \\times\\left(5.0 \\times 10^{-10} \\mathrm{~m}\\right)=3.6\r\n\\end{aligned}\r\n$$\r\nBecause $\\kappa L_1>1$, we use eqn $T\\approx 16 \\varepsilon(1-\\varepsilon) \\mathrm{e}^{-2 \\kappa L}$ to calculate the transmission probabilities at the two distances. It follows that\r\n$$\r\n\\text { current at } \\begin{aligned}\r\n\\frac{L_2}{\\text { current at } L_1} & =\\frac{T\\left(L_2\\right)}{T\\left(L_1\\right)}=\\frac{16 \\varepsilon(1-\\varepsilon) \\mathrm{e}^{-2 \\kappa L_2}}{16 \\varepsilon(1-\\varepsilon) \\mathrm{e}^{-2 \\kappa L_1}}=\\mathrm{e}^{-2 \\kappa\\left(L_2-L_1\\right)} \\\\\r\n& =\\mathrm{e}^{-2 \\times\\left(7.25 \\times 10^{-9} \\mathrm{~m}^{-1}\\right) \\times\\left(1.0 \\times 10^{-10} \\mathrm{~m}\\right)}=0.23\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Calculate the typical wavelength of neutrons that have reached thermal equilibrium with their surroundings at $373 \\mathrm{~K}$.\r\n",
+  "answer_latex": " 226",
+  "answer_number": "226",
+  "unit": " $\\mathrm{pm}$",
+  "source": "atkins",
+  "problemid": " 19.4",
+  "comment": " ",
+  "solution": "From the equipartition principle, we know that the mean translational kinetic energy of a neutron at a temperature $T$ travelling in the $x$-direction is $E_{\\mathrm{k}}=\\frac{1}{2} k T$. The kinetic energy is also equal to $p^2 / 2 m$, where $p$ is the momentum of the neutron and $m$ is its mass. Hence, $p=(m k T)^{1 / 2}$. It follows from the de Broglie relation $\\lambda=h / p$ that the neutron's wavelength is\r\n$$\r\n\\lambda=\\frac{h}{(m k T)^{1 / 2}}\r\n$$\r\nTherefore, at $373 \\mathrm{~K}$,\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left\\{\\left(1.675 \\times 10^{-27} \\mathrm{~kg}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\times(373 \\mathrm{~K})\\right\\}^{1 / 2}} \\\\\r\n& =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left(1.675 \\times 1.381 \\times 373 \\times 10^{-50}\\right)^{1 / 2}\\left(\\mathrm{~kg}^2 \\mathrm{~m}^2 \\mathrm{~s}^{-2}\\right)^{1 / 2}} \\\\\r\n& =2.26 \\times 10^{-10} \\mathrm{~m}=226 \\mathrm{pm}\r\n\\end{aligned}"
+ },
+ {
+  "problem_text": "Calculate the separation of the $\\{123\\}$ planes of an orthorhombic unit cell with $a=0.82 \\mathrm{~nm}, b=0.94 \\mathrm{~nm}$, and $c=0.75 \\mathrm{~nm}$.",
+  "answer_latex": " 0.21",
+  "answer_number": "0.21",
+  "unit": " $\\mathrm{~nm}$",
+  "source": "atkins",
+  "problemid": " 19.1",
+  "comment": " ",
+  "solution": "\nFor the first part, simply substitute the information into eqn $$\n\\frac{1}{d_{h k l}^2}=\\frac{h^2}{a^2}+\\frac{k^2}{b^2}+\\frac{l^2}{c^2}\n$$. For the second part, instead of repeating the calculation, note that, if all three Miller indices are multiplied by $n$, then their separation is reduced by that factor:\r\n$$\r\n\\frac{1}{d_{n h, n k, n l}^2}=\\frac{(n h)^2}{a^2}+\\frac{(n k)^2}{b^2}+\\frac{(n l)^2}{c^2}=n^2\\left(\\frac{h^2}{a^2}+\\frac{k^2}{b^2}+\\frac{l^2}{c^2}\\right)=\\frac{n^2}{d_{h k l}^2}\r\n$$\r\nwhich implies that\r\n$$\r\nd_{n h, n k, n l}=\\frac{d_{h k l}}{n}\r\n$$\r\nAnswer Substituting the indices into eqn $$\n\\frac{1}{d_{h k l}^2}=\\frac{h^2}{a^2}+\\frac{k^2}{b^2}+\\frac{l^2}{c^2}\n$$ gives\r\n$$\r\n\\frac{1}{d_{123}^2}=\\frac{1^2}{(0.82 \\mathrm{~nm})^2}+\\frac{2^2}{(0.94 \\mathrm{~nm})^2}+\\frac{3^2}{(0.75 \\mathrm{~nm})^2}=0.22 \\mathrm{~nm}^{-2}\r\n$$\r\nHence, $d_{123}=0.21 \\mathrm{~nm}$.\n"
+ },
+ {
+  "problem_text": "Calculate the moment of inertia of an $\\mathrm{H}_2 \\mathrm{O}$ molecule around the axis defined by the bisector of the $\\mathrm{HOH}$ angle (3). The $\\mathrm{HOH}$ bond angle is $104.5^{\\circ}$ and the bond length is $95.7 \\mathrm{pm}$.",
+  "answer_latex": " 1.91",
+  "answer_number": "1.91",
+  "unit": " $10^{-47} \\mathrm{~kg} \\mathrm{~m}^2$",
+  "source": "atkins",
+  "problemid": "12.1",
+  "comment": " ",
+  "solution": "\n$$\r\nI=\\sum_i m_i x_i^2=m_{\\mathrm{H}} x_{\\mathrm{H}}^2+0+m_{\\mathrm{H}} x_{\\mathrm{H}}^2=2 m_{\\mathrm{H}} x_{\\mathrm{H}}^2\r\n$$\r\nIf the bond angle of the molecule is denoted $2 \\phi$ and the bond length is $R$, trigonometry gives $x_{\\mathrm{H}}=R \\sin \\phi$. It follows that\r\n$$\r\nI=2 m_{\\mathrm{H}} R^2 \\sin ^2 \\phi\r\n$$\r\nSubstitution of the data gives\r\n$$\r\n\\begin{aligned}\r\nI & =2 \\times\\left(1.67 \\times 10^{-27} \\mathrm{~kg}\\right) \\times\\left(9.57 \\times 10^{-11} \\mathrm{~m}\\right)^2 \\times \\sin ^2\\left(\\frac{1}{2} \\times 104.5^{\\circ}\\right) \\\\\r\n& =1.91 \\times 10^{-47} \\mathrm{~kg} \\mathrm{~m}^2\r\n\\end{aligned}\r\n$$\r\nNote that the mass of the $\\mathrm{O}$ atom makes no contribution to the moment of inertia for this mode of rotation as the atom is immobile while the $\\mathrm{H}$ atoms circulate around it.\n"
+ },
+ {
+  "problem_text": "The data below show the temperature variation of the equilibrium constant of the reaction $\\mathrm{Ag}_2 \\mathrm{CO}_3(\\mathrm{~s}) \\rightleftharpoons \\mathrm{Ag}_2 \\mathrm{O}(\\mathrm{s})+\\mathrm{CO}_2(\\mathrm{~g})$. Calculate the standard reaction enthalpy of the decomposition.\r\n$\\begin{array}{lllll}T / \\mathrm{K} & 350 & 400 & 450 & 500 \\\\ K & 3.98 \\times 10^{-4} & 1.41 \\times 10^{-2} & 1.86 \\times 10^{-1} & 1.48\\end{array}$",
+  "answer_latex": " +80",
+  "answer_number": "+80",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 6.3",
+  "comment": " ",
+  "solution": "\nWe draw up the following table:\r\n$\\begin{array}{lllll}T / \\mathrm{K} & 350 & 400 & 450 & 500 \\\\ \\left(10^3 \\mathrm{~K}\\right) / T & 2.86 & 2.50 & 2.22 & 2.00 \\\\ -\\ln K & 7.83 & 4.26 & 1.68 & -0.39\\end{array}$\r\nBy plotting these points, the slope of the graph is $+9.6 \\times 10^3$, so\r\n$$\r\n\\Delta_{\\mathrm{r}} H^{\\ominus}=\\left(+9.6 \\times 10^3 \\mathrm{~K}\\right) \\times R=+80 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n$$\n"
+ },
+ {
+  "problem_text": "The osmotic pressures of solutions of poly(vinyl chloride), PVC, in cyclohexanone at $298 \\mathrm{~K}$ are given below. The pressures are expressed in terms of the heights of solution (of mass density $\\rho=0.980 \\mathrm{~g} \\mathrm{~cm}^{-3}$ ) in balance with the osmotic pressure. Determine the molar mass of the polymer.\r\n$\\begin{array}{llllll}c /\\left(\\mathrm{g} \\mathrm{dm}^{-3}\\right) & 1.00 & 2.00 & 4.00 & 7.00 & 9.00 \\\\ h / \\mathrm{cm} & 0.28 & 0.71 & 2.01 & 5.10 & 8.00\\end{array}$\r\n",
+  "answer_latex": " 1.2",
+  "answer_number": "1.2",
+  "unit": " $10^2 \\mathrm{~kg} \\mathrm{~mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 5.4",
+  "comment": " ",
+  "solution": "\nThe data give the following values for the quantities to plot:\r\n$$\r\n\\begin{array}{llllll}\r\nc /\\left(\\mathrm{g} \\mathrm{dm}^{-3}\\right) & 1.00 & 2.00 & 4.00 & 7.00 & 9.00 \\\\\r\n(h / c) /\\left(\\mathrm{cm} \\mathrm{g}^{-1} \\mathrm{dm}^3\\right) & 0.28 & 0.36 & 0.503 & 0.729 & 0.889\r\n\\end{array}\r\n$$\r\nBy plotting these points,the intercept is at 0.21 . Therefore,\r\n$$\r\n\\begin{aligned}\r\nM & =\\frac{R T}{\\rho g} \\times \\frac{1}{0.21 \\mathrm{~cm} \\mathrm{~g}^{-1} \\mathrm{dm}^3} \\\\\r\n& =\\frac{\\left(8.3145 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\times(298 \\mathrm{~K})}{\\left(980 \\mathrm{~kg} \\mathrm{~m}^{-1}\\right) \\times\\left(9.81 \\mathrm{~m} \\mathrm{~s}^{-2}\\right)} \\times \\frac{1}{2.1 \\times 10^{-3} \\mathrm{~m}^4 \\mathrm{~kg}^{-1}} \\\\\r\n& =1.2 \\times 10^2 \\mathrm{~kg} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Estimate the wavelength of electrons that have been accelerated from rest through a potential difference of $40 \\mathrm{kV}$.",
+  "answer_latex": " 6.1",
+  "answer_number": "6.1",
+  "unit": " $10^{-12} \\mathrm{~m}$",
+  "source": "atkins",
+  "problemid": " 7.2",
+  "comment": " ",
+  "solution": " The expression $p^2 / 2 m_{\\mathrm{e}}=e \\Delta \\phi$ solves to $p=\\left(2 m_{\\mathrm{e}} e \\Delta \\phi\\right)^{1 / 2}$; then, from the de Broglie relation $\\lambda=h / p$,\r\n$$\r\n\\lambda=\\frac{h}{\\left(2 m_{\\mathrm{e}} e \\Delta \\phi\\right)^{1 / 2}}\r\n$$\r\nSubstitution of the data and the fundamental constants (from inside the front cover) gives\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left\\{2 \\times\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right) \\times\\left(1.602 \\times 10^{-19} \\mathrm{C}\\right) \\times\\left(4.0 \\times 10^4 \\mathrm{~V}\\right)\\right\\}^{1 / 2}} \\\\\r\n& =6.1 \\times 10^{-12} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\n"
+ },
+ {
+  "problem_text": "The standard enthalpy of formation of $\\mathrm{H}_2 \\mathrm{O}(\\mathrm{g})$ at $298 \\mathrm{~K}$ is $-241.82 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Estimate its value at $100^{\\circ} \\mathrm{C}$ given the following values of the molar heat capacities at constant pressure: $\\mathrm{H}_2 \\mathrm{O}(\\mathrm{g}): 33.58 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} ; \\mathrm{H}_2(\\mathrm{~g}): 28.82 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} ; \\mathrm{O}_2(\\mathrm{~g})$ : $29.36 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$. Assume that the heat capacities are independent of temperature.",
+  "answer_latex": "-242.6 ",
+  "answer_number": "-242.6",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 2.6",
+  "comment": " ",
+  "solution": "The reaction is $\\mathrm{H}_2(\\mathrm{~g})+\\frac{1}{2} \\mathrm{O}_2(\\mathrm{~g}) \\rightarrow \\mathrm{H}_2 \\mathrm{O}(\\mathrm{g})$, so\r\n$$\r\n\\Delta_{\\mathrm{r}} C_p^{\\ominus}=C_{p, \\mathrm{~m}}^{\\ominus}\\left(\\mathrm{H}_2 \\mathrm{O}, \\mathrm{g}\\right)-\\left\\{C_{p, \\mathrm{~m}}^{\\ominus}\\left(\\mathrm{H}_2, \\mathrm{~g}\\right)+\\frac{1}{2} C_{p, \\mathrm{~m}}^{\\ominus}\\left(\\mathrm{O}_2, \\mathrm{~g}\\right)\\right\\}=-9.92 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\r\n$$\r\nIt then follows that\r\n$$\r\n\\Delta_{\\mathrm{r}} H^{\\ominus}(373 \\mathrm{~K})=-241.82 \\mathrm{~kJ} \\mathrm{~mol}^{-1}+(75 \\mathrm{~K}) \\times\\left(-9.92 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right)=-242.6 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n$$\r\n"
+ },
+ {
+  "problem_text": "The standard potential of the cell $\\mathrm{Pt}(\\mathrm{s})\\left|\\mathrm{H}_2(\\mathrm{~g})\\right| \\mathrm{HBr}(\\mathrm{aq})|\\operatorname{AgBr}(\\mathrm{s})| \\mathrm{Ag}(\\mathrm{s})$ was measured over a range of temperatures, and the data were found to fit the following polynomial:\r\n$$\r\nE_{\\text {cell }}^{\\bullet} / \\mathrm{V}=0.07131-4.99 \\times 10^{-4}(T / \\mathrm{K}-298)-3.45 \\times 10^{-6}(\\mathrm{~T} / \\mathrm{K}-298)^2\r\n$$\r\nThe cell reaction is $\\operatorname{AgBr}(\\mathrm{s})+\\frac{1}{2} \\mathrm{H}_2(\\mathrm{~g}) \\rightarrow \\mathrm{Ag}(\\mathrm{s})+\\mathrm{HBr}(\\mathrm{aq})$. Evaluate the standard reaction enthalpy at $298 \\mathrm{~K}$.",
+  "answer_latex": " -21.2",
+  "answer_number": "-21.2",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "atkins",
+  "problemid": " 6.5",
+  "comment": " ",
+  "solution": "\nAt $T=298 \\mathrm{~K}, E_{\\text {cell }}^{\\ominus}=+0.07131 \\mathrm{~V}$, so\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\mathrm{r}} G^{\\ominus} & =-v F E_{\\text {cell }}^{\\ominus}=-(1) \\times\\left(9.6485 \\times 10^4 \\mathrm{Cmol}^{-1}\\right) \\times(+0.07131 \\mathrm{~V}) \\\\\r\n& =-6.880 \\times 10^3 \\mathrm{~V} \\mathrm{Cmol}^{-1}=-6.880 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe temperature coefficient of the cell potential is\r\n$$\r\n\\frac{\\mathrm{d} E_{\\text {cell }}^{\\ominus}}{\\mathrm{d} T}=-4.99 \\times 10^{-4} \\mathrm{~V} \\mathrm{~K}^{-1}-2\\left(3.45 \\times 10^{-6}\\right)(T / \\mathrm{K}-298) \\mathrm{V} \\mathrm{K}^{-1}\r\n$$\r\nAt $T=298 \\mathrm{~K}$ this expression evaluates to\r\n$$\r\n\\frac{\\mathrm{d} E_{\\text {cell }}^{\\ominus}}{\\mathrm{d} T}=-4.99 \\times 10^{-4} \\mathrm{~V} \\mathrm{~K}^{-1}\r\n$$\r\nSo, from eqn $$\n\\frac{\\mathrm{d} E_{\\text {cell }}^{\\Theta}}{\\mathrm{d} T}=\\frac{\\Delta_{\\mathrm{r}} S^{\\Theta}}{v F}\n$$ , the reaction entropy is\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\mathrm{r}} S^{\\ominus} & =1 \\times\\left(9.6485 \\times 10^4 \\mathrm{Cmol}^{-1}\\right) \\times\\left(-4.99 \\times 10^{-4} \\mathrm{~V} \\mathrm{~K}^{-1}\\right) \\\\\r\n& =-48.1 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe negative value stems in part from the elimination of gas in the cell reaction. It then follows that\r\n$$\r\n\\begin{aligned}\r\n\\Delta_{\\mathrm{r}} H^{\\ominus} & =\\Delta_{\\mathrm{r}} G^{\\ominus}+T \\Delta_{\\mathrm{r}} S^{\\ominus}=-6.880 \\mathrm{~kJ} \\mathrm{~mol}^{-1}+(298 \\mathrm{~K}) \\times\\left(-0.0482 \\mathrm{~kJ} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right) \\\\\r\n& =-21.2 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "In an industrial process, nitrogen is heated to $500 \\mathrm{~K}$ in a vessel of constant volume. If it enters the vessel at $100 \\mathrm{~atm}$ and $300 \\mathrm{~K}$, what pressure would it exert at the working temperature if it behaved as a perfect gas?",
+  "answer_latex": " 167",
+  "answer_number": "167",
+  "unit": " $\\mathrm{atm}$",
+  "source": "atkins",
+  "problemid": " 1.2",
+  "comment": " ",
+  "solution": "Cancellation of the volumes (because $V_1=V_2$ ) and amounts (because $\\left.n_1=n_2\\right)$ on each side of the combined gas law results in\r\n$$\r\n\\frac{p_1}{T_1}=\\frac{p_2}{T_2}\r\n$$\r\nwhich can be rearranged into\r\n$$\r\np_2=\\frac{T_2}{T_1} \\times p_1\r\n$$\r\nSubstitution of the data then gives\r\n$$\r\np_2=\\frac{500 \\mathrm{~K}}{300 \\mathrm{~K}} \\times(100 \\mathrm{~atm})=167 \\mathrm{~atm}\r\n$$\r\n"
+ }
+]

calculus.json

@@ -1 +1,380 @@
-[{"problem_text": "A fluid has density $870 \\mathrm{~kg} / \\mathrm{m}^3$ and flows with velocity $\\mathbf{v}=z \\mathbf{i}+y^2 \\mathbf{j}+x^2 \\mathbf{k}$, where $x, y$, and $z$ are measured in meters and the components of $\\mathbf{v}$ in meters per second. Find the rate of flow outward through the cylinder $x^2+y^2=4$, $0 \\leqslant z \\leqslant 1$.\r\n", "answer_latex": " 0", "answer_number": "0", "unit": " $\\mathrm{kg}/\\mathrm{s}$", "source": "calculus", "problemid": " 16.7.43", "comment": " "}, {"problem_text": "Suppose that $2 \\mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \\mathrm{~cm}$ to a length of $42 \\mathrm{~cm}$. How far beyond its natural length will a force of $30 \\mathrm{~N}$ keep the spring stretched?", "answer_latex": " 10.8", "answer_number": "10.8", "unit": " $\\mathrm{cm}$", "source": "calculus", "problemid": " 6.4.9(b)", "comment": " "}, {"problem_text": "Find the work done by a force $\\mathbf{F}=8 \\mathbf{i}-6 \\mathbf{j}+9 \\mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons.", "answer_latex": " 144", "answer_number": "144", "unit": " $\\mathrm{J}$", "source": "calculus", "problemid": " 12.3.49", "comment": " "}, {"problem_text": "A ball is thrown at an angle of $45^{\\circ}$ to the ground. If the ball lands $90 \\mathrm{~m}$ away, what was the initial speed of the ball?\r\n", "answer_latex": " 30", "answer_number": "30", "unit": " $\\mathrm{m}/\\mathrm{s}$", "source": "calculus", "problemid": " 13.4.25", "comment": " "}, {"problem_text": "Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of $12 \\mathrm{~m}$ at a constant speed with a rope that weighs $0.8 \\mathrm{~kg} / \\mathrm{m}$. Initially the bucket contains $36 \\mathrm{~kg}$ of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?\r\n", "answer_latex": " 3857", "answer_number": "3857", "unit": " $\\mathrm{J}$", "source": "calculus", "problemid": " 6.4.17", "comment": " "}, {"problem_text": "Find the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base.", "answer_latex": " 24", "answer_number": "24", "unit": " ", "source": "calculus", "problemid": " 6.2.55", "comment": " "}, {"problem_text": "A swimming pool is circular with a $40-\\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \\mathrm{ft}$ at the south end to $7 \\mathrm{ft}$ at the north end. Find the volume of water in the pool.", "answer_latex": " $1800\\pi$", "answer_number": "5654.86677646", "unit": " $\\mathrm{ft}^3$", "source": "calculus", "problemid": "15.4.35", "comment": " "}, {"problem_text": "The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \\mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?", "answer_latex": " 35.64", "answer_number": "35.64", "unit": " $\\mathrm{AU}$", "source": "calculus", "problemid": " 10.6.27", "comment": " "}, {"problem_text": " If a ball is thrown into the air with a velocity of $40 \\mathrm{ft} / \\mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.", "answer_latex": " -32", "answer_number": "-32", "unit": "$\\mathrm{ft} / \\mathrm{s}$", "source": "calculus", "problemid": " 2.1.5(a)", "comment": " "}, {"problem_text": "A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \\mathrm{~cm}$ apart. The liver is $15 \\mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63$, 39 , and 0 . Use the Midpoint Rule to estimate the volume of the liver.\r\n", "answer_latex": " 1110", "answer_number": "1110", "unit": " $\\mathrm{cm}^3$", "source": "calculus", "problemid": " 6.2.43", "comment": " "}, {"problem_text": "A manufacturer of corrugated metal roofing wants to produce panels that are $28 \\mathrm{in}$. wide and $2 \\mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\\sin (\\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)\r\n", "answer_latex": " 29.36", "answer_number": "29.36", "unit": " $\\mathrm{in}$", "source": "calculus", "problemid": " 8.1.39", "comment": " "}, {"problem_text": "The dye dilution method is used to measure cardiac output with $6 \\mathrm{mg}$ of dye. The dye concentrations, in $\\mathrm{mg} / \\mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \\leqslant t \\leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.", "answer_latex": " 6.6", "answer_number": "6.6", "unit": " $\\mathrm{L}/\\mathrm{min}$", "source": "calculus", "problemid": " 8.4.17", "comment": " "}, {"problem_text": "A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \\mathrm{ft}$ high with a radius of $200 \\mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \\pi \\mathrm{~ft}^3 / \\mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \\mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?", "answer_latex": " 9.8", "answer_number": "9.8", "unit": " $\\mathrm{h}$", "source": "calculus", "problemid": " 9.RP.11(a)", "comment": " Review Plus Problem"}, {"problem_text": "A boatman wants to cross a canal that is $3 \\mathrm{~km}$ wide and wants to land at a point $2 \\mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \\mathrm{~km} / \\mathrm{h}$ and the speed of his boat is $13 \\mathrm{~km} / \\mathrm{h}$. How long will the trip take?\r\n", "answer_latex": " 20.2", "answer_number": "20.2", "unit": " $\\mathrm{min}$", "source": "calculus", "problemid": " 12.2.39", "comment": " "}, {"problem_text": "Find the area bounded by the curves $y=\\cos x$ and $y=\\cos ^2 x$ between $x=0$ and $x=\\pi$.", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "calculus", "problemid": " 7.R.73", "comment": " "}, {"problem_text": "A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\\circ}$ above the horizontal moves the sled $80 \\mathrm{ft}$. Find the work done by the force.", "answer_latex": " $2400\\cos({40}^{\\circ})$", "answer_number": "1838.50666349", "unit": " $\\mathrm{ft-lb}$", "source": "calculus", "problemid": " 12.3.51", "comment": " "}, {"problem_text": " If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then\r\n$$\r\n\\frac{1}{R}=\\frac{1}{R_1}+\\frac{1}{R_2}+\\frac{1}{R_3}\r\n$$\r\nIf the resistances are measured in ohms as $R_1=25 \\Omega$, $R_2=40 \\Omega$, and $R_3=50 \\Omega$, with a possible error of $0.5 \\%$ in each case, estimate the maximum error in the calculated value of $R$.", "answer_latex": " $\\frac{1}{17}$", "answer_number": "0.05882352941", "unit": " $\\Omega$", "source": "calculus", "problemid": " 14.4.39", "comment": " "}, {"problem_text": "The length and width of a rectangle are measured as $30 \\mathrm{~cm}$ and $24 \\mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \\mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.\r\n", "answer_latex": " 5.4", "answer_number": "5.4", "unit": " $\\mathrm{cm^2}$", "source": "calculus", "problemid": " 14.4.33", "comment": " "}, {"problem_text": "The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \\times 10^7 \\mathrm{~km}$. Find its maximum distance from the sun.", "answer_latex": " 7", "answer_number": "7", "unit": " $\\mathrm{10^7} \\mathrm{~km}$", "source": "calculus", "problemid": " 10.6.29", "comment": " "}, {"problem_text": "Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \\mathrm{~cm}$ and height $12 \\mathrm{~cm}$ if the tin is $0.04 \\mathrm{~cm}$ thick.", "answer_latex": " 16", "answer_number": "16", "unit": " $\\mathrm{cm^3}$", "source": "calculus", "problemid": " 14.4.35", "comment": " "}, {"problem_text": "Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.  A cable that weighs $2 \\mathrm{~lb} / \\mathrm{ft}$ is used to lift $800 \\mathrm{~lb}$ of coal up a mine shaft $500 \\mathrm{~ft}$ deep. Find the work done.\r\n", "answer_latex": " 650000", "answer_number": "650000", "unit": " $\\mathrm{ft-lb}$", "source": "calculus", "problemid": " 6.4.15", "comment": " "}, {"problem_text": " A patient takes $150 \\mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? ", "answer_latex": " 157.875", "answer_number": "157.875", "unit": " $\\mathrm{mg}$", "source": "calculus", "problemid": "11.2.69(a) ", "comment": " "}, {"problem_text": "A $360-\\mathrm{lb}$ gorilla climbs a tree to a height of $20 \\mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.", "answer_latex": " 7200", "answer_number": "7200", "unit": " $\\mathrm{ft-lb}$", "source": "calculus", "problemid": " 6.4.1(b)", "comment": " "}, {"problem_text": "Find the area of triangle $A B C$, correct to five decimal places, if\r\n$$\r\n|A B|=10 \\mathrm{~cm} \\quad|B C|=3 \\mathrm{~cm} \\quad \\angle A B C=107^{\\circ}\r\n$$", "answer_latex": " 14.34457", "answer_number": "14.34457", "unit": " $\\mathrm{cm^2}$", "source": "calculus", "problemid": " D.89", "comment": " "}, {"problem_text": "Use Stokes' Theorem to evaluate $\\int_C \\mathbf{F} \\cdot d \\mathbf{r}$, where $\\mathbf{F}(x, y, z)=x y \\mathbf{i}+y z \\mathbf{j}+z x \\mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.\r\n", "answer_latex": " $-\\frac{1}{2}$", "answer_number": "-0.5", "unit": " ", "source": "calculus", "problemid": " 16.R.33", "comment": " "}, {"problem_text": "A hawk flying at $15 \\mathrm{~m} / \\mathrm{s}$ at an altitude of $180 \\mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation\r\n$$\r\ny=180-\\frac{x^2}{45}\r\n$$\r\nuntil it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.", "answer_latex": " 209.1", "answer_number": "209.1", "unit": " $\\mathrm{m}$", "source": "calculus", "problemid": " 8.1.37", "comment": " "}, {"problem_text": "The intensity of light with wavelength $\\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\\theta$ is given by $I(\\theta)=N^2 \\sin ^2 k / k^2$, where $k=(\\pi N d \\sin \\theta) / \\lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\\lambda=632.8 \\times 10^{-9} \\mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \\mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\\int_{-10^{-6}}^{10^{-6}} I(\\theta) d \\theta$ emerging from the grating.", "answer_latex": " 59.4", "answer_number": "59.4", "unit": " ", "source": "calculus", "problemid": " 7.7.43", "comment": " "}, {"problem_text": "A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \\%$, use differentials to estimate the maximum percentage error in the calculated surface area.", "answer_latex": " 2.3", "answer_number": "2.3", "unit": " $\\%$", "source": "calculus", "problemid": " 14.4.41", "comment": " "}, {"problem_text": "The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \\leqslant x \\leqslant 4$", "answer_latex": "$1248\\pi$", "answer_number": "3920.70763168", "unit": " ", "source": "calculus", "problemid": "16.7.47 ", "comment": " "}, {"problem_text": "If a ball is thrown into the air with a velocity of $40 \\mathrm{ft} / \\mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.", "answer_latex": " -24", "answer_number": "-24", "unit": " $\\mathrm{ft} / \\mathrm{s}$", "source": "calculus", "problemid": " 2.7.13", "comment": " "}, {"problem_text": "A woman walks due west on the deck of a ship at $3 \\mathrm{mi} / \\mathrm{h}$. The ship is moving north at a speed of $22 \\mathrm{mi} / \\mathrm{h}$. Find the speed of the woman relative to the surface of the water.", "answer_latex": " $\\sqrt{493}$", "answer_number": "22.2036033112", "unit": " $\\mathrm{mi}/\\mathrm{h}$", "source": "calculus", "problemid": " 12.2.35", "comment": " "}, {"problem_text": "A $360-\\mathrm{lb}$ gorilla climbs a tree to a height of $20 \\mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. ", "answer_latex": " 7200", "answer_number": "7200", "unit": "$\\mathrm{ft-lb}$", "source": "calculus", "problemid": " 6.4.1(a)", "comment": " "}, {"problem_text": " A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \\mathrm{i}+80 \\mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \\mathrm{ft} / \\mathrm{s}^2$, so the acceleration vector is $\\mathbf{a}=-4 \\mathbf{j}-32 \\mathbf{k}$. What speed does the ball land?\r\n", "answer_latex": " $10\\sqrt{93}$", "answer_number": "96.4365076099", "unit": " $\\mathrm{ft}/\\mathrm{s}$", "source": "calculus", "problemid": " 13.4.31", "comment": " "}, {"problem_text": "The demand function for a commodity is given by\r\n$$\r\np=2000-0.1 x-0.01 x^2\r\n$$\r\nFind the consumer surplus when the sales level is 100 .", "answer_latex": " 7166.67", "answer_number": "7166.67", "unit": " $\\$$", "source": "calculus", "problemid": " 8.R.17", "comment": " "}, {"problem_text": "The linear density in a rod $8 \\mathrm{~m}$ long is $12 / \\sqrt{x+1} \\mathrm{~kg} / \\mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.", "answer_latex": " 6", "answer_number": "6", "unit": " $\\mathrm{~kg} / \\mathrm{m}$", "source": "calculus", "problemid": " 6.5.19", "comment": " "}, {"problem_text": "A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \\mathrm{~ft}$ to $x=10 \\mathrm{~ft}$.", "answer_latex": " 4.5", "answer_number": "4.5", "unit": " $\\mathrm{ft-lb}$", "source": "calculus", "problemid": " 6.4.3", "comment": " "}, {"problem_text": "One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \\%$ of the population to become infected?", "answer_latex": " 15", "answer_number": "15", "unit": " $\\mathrm{days}$", "source": "calculus", "problemid": " 9.R.19", "comment": " "}, {"problem_text": "Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \\mathrm{~cm}$, $4 \\mathrm{~cm}$, and $5 \\mathrm{~cm}$", "answer_latex": " 10", "answer_number": "10", "unit": " $\\mathrm{cm}^3$", "source": "calculus", "problemid": " 6.2.53", "comment": " "}, {"problem_text": "The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.", "answer_latex": " 36", "answer_number": "36", "unit": " ", "source": "calculus", "problemid": " 6.R.23", "comment": " review problem"}, {"problem_text": "A projectile is fired with an initial speed of $200 \\mathrm{~m} / \\mathrm{s}$ and angle of elevation $60^{\\circ}$. Find the speed at impact.", "answer_latex": " 200", "answer_number": "200", "unit": " $\\mathrm{m}/\\mathrm{s}$", "source": "calculus", "problemid": " 13.4.23(c)", "comment": " "}, {"problem_text": "A force of $30 \\mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \\mathrm{~cm}$ to a length of $15 \\mathrm{~cm}$. How much work is done in stretching the spring from $12 \\mathrm{~cm}$ to $20 \\mathrm{~cm}$ ?\r\n", "answer_latex": " 3.2", "answer_number": "3.2", "unit": " $\\mathrm{J}$", "source": "calculus", "problemid": " 6.R.27", "comment": " review problem"}, {"problem_text": "Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\\eta=0.027, R=0.008 \\mathrm{~cm}$, $I=2 \\mathrm{~cm}$, and $P=4000$ dynes $/ \\mathrm{cm}^2$.", "answer_latex": " 1.19", "answer_number": "1.19", "unit": " $\\mathrm{10^{-4}} \\mathrm{~cm}^3/\\mathrm{s}$", "source": "calculus", "problemid": " 8.4.15", "comment": " "}]
+[
+ {
+  "problem_text": "A fluid has density $870 \\mathrm{~kg} / \\mathrm{m}^3$ and flows with velocity $\\mathbf{v}=z \\mathbf{i}+y^2 \\mathbf{j}+x^2 \\mathbf{k}$, where $x, y$, and $z$ are measured in meters and the components of $\\mathbf{v}$ in meters per second. Find the rate of flow outward through the cylinder $x^2+y^2=4$, $0 \\leqslant z \\leqslant 1$.\r\n",
+  "answer_latex": " 0",
+  "answer_number": "0",
+  "unit": " $\\mathrm{kg}/\\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 16.7.43",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that $2 \\mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \\mathrm{~cm}$ to a length of $42 \\mathrm{~cm}$. How far beyond its natural length will a force of $30 \\mathrm{~N}$ keep the spring stretched?",
+  "answer_latex": " 10.8",
+  "answer_number": "10.8",
+  "unit": " $\\mathrm{cm}$",
+  "source": "calculus",
+  "problemid": " 6.4.9(b)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the work done by a force $\\mathbf{F}=8 \\mathbf{i}-6 \\mathbf{j}+9 \\mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons.",
+  "answer_latex": " 144",
+  "answer_number": "144",
+  "unit": " $\\mathrm{J}$",
+  "source": "calculus",
+  "problemid": " 12.3.49",
+  "comment": " "
+ },
+ {
+  "problem_text": "A ball is thrown at an angle of $45^{\\circ}$ to the ground. If the ball lands $90 \\mathrm{~m}$ away, what was the initial speed of the ball?\r\n",
+  "answer_latex": " 30",
+  "answer_number": "30",
+  "unit": " $\\mathrm{m}/\\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 13.4.25",
+  "comment": " "
+ },
+ {
+  "problem_text": "Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of $12 \\mathrm{~m}$ at a constant speed with a rope that weighs $0.8 \\mathrm{~kg} / \\mathrm{m}$. Initially the bucket contains $36 \\mathrm{~kg}$ of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?\r\n",
+  "answer_latex": " 3857",
+  "answer_number": "3857",
+  "unit": " $\\mathrm{J}$",
+  "source": "calculus",
+  "problemid": " 6.4.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base.",
+  "answer_latex": " 24",
+  "answer_number": "24",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 6.2.55",
+  "comment": " "
+ },
+ {
+  "problem_text": "A swimming pool is circular with a $40-\\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \\mathrm{ft}$ at the south end to $7 \\mathrm{ft}$ at the north end. Find the volume of water in the pool.",
+  "answer_latex": " $1800\\pi$",
+  "answer_number": "5654.86677646",
+  "unit": " $\\mathrm{ft}^3$",
+  "source": "calculus",
+  "problemid": "15.4.35",
+  "comment": " "
+ },
+ {
+  "problem_text": "The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \\mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] By finding a polar equation for the orbit of Halley's comet, what is the maximum distance from the comet to the sun?",
+  "answer_latex": " 35.64",
+  "answer_number": "35.64",
+  "unit": " $\\mathrm{AU}$",
+  "source": "calculus",
+  "problemid": " 10.6.27",
+  "comment": " "
+ },
+ {
+  "problem_text": " If a ball is thrown into the air with a velocity of $40 \\mathrm{ft} / \\mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.",
+  "answer_latex": " -32",
+  "answer_number": "-32",
+  "unit": "$\\mathrm{ft} / \\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 2.1.5(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \\mathrm{~cm}$ apart. The liver is $15 \\mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63, 39, 0$. Use the Midpoint Rule to estimate the volume of the liver.\r\n",
+  "answer_latex": " 1110",
+  "answer_number": "1110",
+  "unit": " $\\mathrm{cm}^3$",
+  "source": "calculus",
+  "problemid": " 6.2.43",
+  "comment": " "
+ },
+ {
+  "problem_text": "A manufacturer of corrugated metal roofing wants to produce panels that are $28 \\mathrm{in}$. wide and $2 \\mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\\sin (\\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)\r\n",
+  "answer_latex": " 29.36",
+  "answer_number": "29.36",
+  "unit": " $\\mathrm{in}$",
+  "source": "calculus",
+  "problemid": " 8.1.39",
+  "comment": " "
+ },
+ {
+  "problem_text": "The dye dilution method is used to measure cardiac output with $6 \\mathrm{mg}$ of dye. The dye concentrations, in $\\mathrm{mg} / \\mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \\leqslant t \\leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.",
+  "answer_latex": " 6.6",
+  "answer_number": "6.6",
+  "unit": " $\\mathrm{L}/\\mathrm{min}$",
+  "source": "calculus",
+  "problemid": " 8.4.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \\mathrm{ft}$ high with a radius of $200 \\mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \\pi \\mathrm{~ft}^3 / \\mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \\mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?",
+  "answer_latex": " 9.8",
+  "answer_number": "9.8",
+  "unit": " $\\mathrm{h}$",
+  "source": "calculus",
+  "problemid": " 9.RP.11(a)",
+  "comment": " Review Plus Problem"
+ },
+ {
+  "problem_text": "A boatman wants to cross a canal that is $3 \\mathrm{~km}$ wide and wants to land at a point $2 \\mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \\mathrm{~km} / \\mathrm{h}$ and the speed of his boat is $13 \\mathrm{~km} / \\mathrm{h}$. How long will the trip take?\r\n",
+  "answer_latex": " 20.2",
+  "answer_number": "20.2",
+  "unit": " $\\mathrm{min}$",
+  "source": "calculus",
+  "problemid": " 12.2.39",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the area bounded by the curves $y=\\cos x$ and $y=\\cos ^2 x$ between $x=0$ and $x=\\pi$.",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 7.R.73",
+  "comment": " "
+ },
+ {
+  "problem_text": "A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\\circ}$ above the horizontal moves the sled $80 \\mathrm{ft}$. Find the work done by the force.",
+  "answer_latex": " $2400\\cos({40}^{\\circ})$",
+  "answer_number": "1838.50666349",
+  "unit": " $\\mathrm{ft-lb}$",
+  "source": "calculus",
+  "problemid": " 12.3.51",
+  "comment": " "
+ },
+ {
+  "problem_text": " If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then\r\n$$\r\n\\frac{1}{R}=\\frac{1}{R_1}+\\frac{1}{R_2}+\\frac{1}{R_3}\r\n$$\r\nIf the resistances are measured in ohms as $R_1=25 \\Omega$, $R_2=40 \\Omega$, and $R_3=50 \\Omega$, with a possible error of $0.5 \\%$ in each case, estimate the maximum error in the calculated value of $R$.",
+  "answer_latex": " $\\frac{1}{17}$",
+  "answer_number": "0.05882352941",
+  "unit": " $\\Omega$",
+  "source": "calculus",
+  "problemid": " 14.4.39",
+  "comment": " "
+ },
+ {
+  "problem_text": "The length and width of a rectangle are measured as $30 \\mathrm{~cm}$ and $24 \\mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \\mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.\r\n",
+  "answer_latex": " 5.4",
+  "answer_number": "5.4",
+  "unit": " $\\mathrm{cm^2}$",
+  "source": "calculus",
+  "problemid": " 14.4.33",
+  "comment": " "
+ },
+ {
+  "problem_text": "The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \\times 10^7 \\mathrm{~km}$. Find its maximum distance from the sun.",
+  "answer_latex": " 7",
+  "answer_number": "7",
+  "unit": " $\\mathrm{10^7} \\mathrm{~km}$",
+  "source": "calculus",
+  "problemid": " 10.6.29",
+  "comment": " "
+ },
+ {
+  "problem_text": "Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \\mathrm{~cm}$ and height $12 \\mathrm{~cm}$ if the tin is $0.04 \\mathrm{~cm}$ thick.",
+  "answer_latex": " 16",
+  "answer_number": "16",
+  "unit": " $\\mathrm{cm^3}$",
+  "source": "calculus",
+  "problemid": " 14.4.35",
+  "comment": " "
+ },
+ {
+  "problem_text": "Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.  A cable that weighs $2 \\mathrm{~lb} / \\mathrm{ft}$ is used to lift $800 \\mathrm{~lb}$ of coal up a mine shaft $500 \\mathrm{~ft}$ deep. Find the work done.\r\n",
+  "answer_latex": " 650000",
+  "answer_number": "650000",
+  "unit": " $\\mathrm{ft-lb}$",
+  "source": "calculus",
+  "problemid": " 6.4.15",
+  "comment": " "
+ },
+ {
+  "problem_text": " A patient takes $150 \\mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? ",
+  "answer_latex": " 157.875",
+  "answer_number": "157.875",
+  "unit": " $\\mathrm{mg}$",
+  "source": "calculus",
+  "problemid": "11.2.69(a) ",
+  "comment": " "
+ },
+ {
+  "problem_text": "A $360-\\mathrm{lb}$ gorilla climbs a tree to a height of $20 \\mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.",
+  "answer_latex": " 7200",
+  "answer_number": "7200",
+  "unit": " $\\mathrm{ft-lb}$",
+  "source": "calculus",
+  "problemid": " 6.4.1(b)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the area of triangle $A B C$, correct to five decimal places, if\r\n$$\r\n|A B|=10 \\mathrm{~cm} \\quad|B C|=3 \\mathrm{~cm} \\quad \\angle A B C=107^{\\circ}\r\n$$",
+  "answer_latex": " 14.34457",
+  "answer_number": "14.34457",
+  "unit": " $\\mathrm{cm^2}$",
+  "source": "calculus",
+  "problemid": " D.89",
+  "comment": " "
+ },
+ {
+  "problem_text": "Use Stokes' Theorem to evaluate $\\int_C \\mathbf{F} \\cdot d \\mathbf{r}$, where $\\mathbf{F}(x, y, z)=x y \\mathbf{i}+y z \\mathbf{j}+z x \\mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.\r\n",
+  "answer_latex": " $-\\frac{1}{2}$",
+  "answer_number": "-0.5",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 16.R.33",
+  "comment": " "
+ },
+ {
+  "problem_text": "A hawk flying at $15 \\mathrm{~m} / \\mathrm{s}$ at an altitude of $180 \\mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation\r\n$$\r\ny=180-\\frac{x^2}{45}\r\n$$\r\nuntil it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.",
+  "answer_latex": " 209.1",
+  "answer_number": "209.1",
+  "unit": " $\\mathrm{m}$",
+  "source": "calculus",
+  "problemid": " 8.1.37",
+  "comment": " "
+ },
+ {
+  "problem_text": "The intensity of light with wavelength $\\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\\theta$ is given by $I(\\theta)=N^2 \\sin ^2 k / k^2$, where $k=(\\pi N d \\sin \\theta) / \\lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\\lambda=632.8 \\times 10^{-9} \\mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \\mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\\int_{-10^{-6}}^{10^{-6}} I(\\theta) d \\theta$ emerging from the grating.",
+  "answer_latex": " 59.4",
+  "answer_number": "59.4",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 7.7.43",
+  "comment": " "
+ },
+ {
+  "problem_text": "A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \\%$, use differentials to estimate the maximum percentage error in the calculated surface area.",
+  "answer_latex": " 2.3",
+  "answer_number": "2.3",
+  "unit": " $\\%$",
+  "source": "calculus",
+  "problemid": " 14.4.41",
+  "comment": " "
+ },
+ {
+  "problem_text": "The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \\leqslant x \\leqslant 4$",
+  "answer_latex": "$1248\\pi$",
+  "answer_number": "3920.70763168",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": "16.7.47 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "If a ball is thrown into the air with a velocity of $40 \\mathrm{ft} / \\mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.",
+  "answer_latex": " -24",
+  "answer_number": "-24",
+  "unit": " $\\mathrm{ft} / \\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 2.7.13",
+  "comment": " "
+ },
+ {
+  "problem_text": "A woman walks due west on the deck of a ship at $3 \\mathrm{mi} / \\mathrm{h}$. The ship is moving north at a speed of $22 \\mathrm{mi} / \\mathrm{h}$. Find the speed of the woman relative to the surface of the water.",
+  "answer_latex": " $\\sqrt{493}$",
+  "answer_number": "22.2036033112",
+  "unit": " $\\mathrm{mi}/\\mathrm{h}$",
+  "source": "calculus",
+  "problemid": " 12.2.35",
+  "comment": " "
+ },
+ {
+  "problem_text": "A $360-\\mathrm{lb}$ gorilla climbs a tree to a height of $20 \\mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. ",
+  "answer_latex": " 7200",
+  "answer_number": "7200",
+  "unit": "$\\mathrm{ft-lb}$",
+  "source": "calculus",
+  "problemid": " 6.4.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": " A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \\mathrm{i}+80 \\mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \\mathrm{ft} / \\mathrm{s}^2$, so the acceleration vector is $\\mathbf{a}=-4 \\mathbf{j}-32 \\mathbf{k}$. What speed does the ball land?\r\n",
+  "answer_latex": " $10\\sqrt{93}$",
+  "answer_number": "96.4365076099",
+  "unit": " $\\mathrm{ft}/\\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 13.4.31",
+  "comment": " "
+ },
+ {
+  "problem_text": "The demand function for a commodity is given by\r\n$$\r\np=2000-0.1 x-0.01 x^2\r\n$$\r\nFind the consumer surplus when the sales level is 100 .",
+  "answer_latex": " 7166.67",
+  "answer_number": "7166.67",
+  "unit": " $\\$$",
+  "source": "calculus",
+  "problemid": " 8.R.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "The linear density in a rod $8 \\mathrm{~m}$ long is $12 / \\sqrt{x+1} \\mathrm{~kg} / \\mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.",
+  "answer_latex": " 6",
+  "answer_number": "6",
+  "unit": " $\\mathrm{~kg} / \\mathrm{m}$",
+  "source": "calculus",
+  "problemid": " 6.5.19",
+  "comment": " "
+ },
+ {
+  "problem_text": "A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \\mathrm{~ft}$ to $x=10 \\mathrm{~ft}$.",
+  "answer_latex": " 4.5",
+  "answer_number": "4.5",
+  "unit": " $\\mathrm{ft-lb}$",
+  "source": "calculus",
+  "problemid": " 6.4.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \\%$ of the population to become infected?",
+  "answer_latex": " 15",
+  "answer_number": "15",
+  "unit": " $\\mathrm{days}$",
+  "source": "calculus",
+  "problemid": " 9.R.19",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \\mathrm{~cm}$, $4 \\mathrm{~cm}$, and $5 \\mathrm{~cm}$",
+  "answer_latex": " 10",
+  "answer_number": "10",
+  "unit": " $\\mathrm{cm}^3$",
+  "source": "calculus",
+  "problemid": " 6.2.53",
+  "comment": " "
+ },
+ {
+  "problem_text": "The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.",
+  "answer_latex": " 36",
+  "answer_number": "36",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 6.R.23",
+  "comment": " review problem"
+ },
+ {
+  "problem_text": "A projectile is fired with an initial speed of $200 \\mathrm{~m} / \\mathrm{s}$ and angle of elevation $60^{\\circ}$. Find the speed at impact.",
+  "answer_latex": " 200",
+  "answer_number": "200",
+  "unit": " $\\mathrm{m}/\\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 13.4.23(c)",
+  "comment": " "
+ },
+ {
+  "problem_text": "A force of $30 \\mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \\mathrm{~cm}$ to a length of $15 \\mathrm{~cm}$. How much work is done in stretching the spring from $12 \\mathrm{~cm}$ to $20 \\mathrm{~cm}$ ?\r\n",
+  "answer_latex": " 3.2",
+  "answer_number": "3.2",
+  "unit": " $\\mathrm{J}$",
+  "source": "calculus",
+  "problemid": " 6.R.27",
+  "comment": " review problem"
+ },
+ {
+  "problem_text": "Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\\eta=0.027, R=0.008 \\mathrm{~cm}$, $I=2 \\mathrm{~cm}$, and $P=4000$ dynes $/ \\mathrm{cm}^2$.",
+  "answer_latex": " 1.19",
+  "answer_number": "1.19",
+  "unit": " $\\mathrm{10^{-4}} \\mathrm{~cm}^3/\\mathrm{s}$",
+  "source": "calculus",
+  "problemid": " 8.4.15",
+  "comment": " "
+ }
+]

calculus_sol.json

@@ -1 +1,102 @@
-[{"problem_text": "How large should we take $n$ in order to guarantee that the Trapezoidal and Midpoint Rule approximations for $\\int_1^2(1 / x) d x$ are accurate to within 0.0001 ?", "answer_latex": " 41", "answer_number": "41", "unit": " ", "source": "calculus", "problemid": " 7.7.2", "comment": " with solution", "solution": "We saw in the preceding calculation that $\\left|f^{\\prime \\prime}(x)\\right| \\leqslant 2$ for $1 \\leqslant x \\leqslant 2$, so we can take $K=2$, $a=1$, and $b=2$ in 3 . Accuracy to within 0.0001 means that the size of the error should be less than 0.0001 . Therefore we choose $n$ so that\r\n$$\r\n\\frac{2(1)^3}{12 n^2}<0.0001\r\n$$\r\nSolving the inequality for $n$, we get\r\nor\r\n$$\r\n\\begin{aligned}\r\n& n^2>\\frac{2}{12(0.0001)} \\\\\r\n& n>\\frac{1}{\\sqrt{0.0006}} \\approx 40.8\r\n\\end{aligned}\r\n$$\r\nThus $n=41$ will ensure the desired accuracy."}, {"problem_text": "Find the length of the cardioid $r=1+\\sin \\theta$.", "answer_latex": " 8", "answer_number": "8", "unit": " ", "source": "calculus", "problemid": " 10.4.4", "comment": " with solution", "solution": "The cardioid is shown in Figure 8. (We sketched it in Example 7 in Section 10.3.) Its full length is given by the parameter interval $0 \\leqslant \\theta \\leqslant 2 \\pi$, so Formula 5 gives\r\n$$\r\n\\begin{aligned}\r\nL & =\\int_0^{2 \\pi} \\sqrt{r^2+\\left(\\frac{d r}{d \\theta}\\right)^2} d \\theta=\\int_0^{2 \\pi} \\sqrt{(1+\\sin \\theta)^2+\\cos ^2 \\theta} d \\theta \\\\\r\n& =\\int_0^{2 \\pi} \\sqrt{2+2 \\sin \\theta} d \\theta\r\n\\end{aligned}\r\n$$\r\nWe could evaluate this integral by multiplying and dividing the integrand by $\\sqrt{2-2 \\sin \\theta}$, or we could use a computer algebra system. In any event, we find that the length of the cardioid is $L=8$.\r\n"}, {"problem_text": "Estimate the volume of the solid that lies above the square $R=[0,2] \\times[0,2]$ and below the elliptic paraboloid $z=16-x^2-2 y^2$. Divide $R$ into four equal squares and choose the sample point to be the upper right corner of each square $R_{i j}$. ", "answer_latex": " 34", "answer_number": "34", "unit": " ", "source": "calculus", "problemid": "15.1.1 ", "comment": " with solution", "solution": "The squares are shown in Figure 6. The paraboloid is the graph of $f(x, y)=16-x^2-2 y^2$ and the area of each square is $\\Delta A=1$. Approximating the volume by the Riemann sum with $m=n=2$, we have\r\n$$\r\n\\begin{aligned}\r\nV & \\approx \\sum_{i=1}^2 \\sum_{j=1}^2 f\\left(x_i, y_j\\right) \\Delta A \\\\\r\n& =f(1,1) \\Delta A+f(1,2) \\Delta A+f(2,1) \\Delta A+f(2,2) \\Delta A \\\\\r\n& =13(1)+7(1)+10(1)+4(1)=34\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Find the average value of the function $f(x)=1+x^2$ on the interval $[-1,2]$.", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "calculus", "problemid": " 6.5.1", "comment": " with equation", "solution": "With $a=-1$ and $b=2$ we have\r\n$$\r\n\\begin{aligned}\r\nf_{\\text {ave }} & =\\frac{1}{b-a} \\int_a^b f(x) d x=\\frac{1}{2-(-1)} \\int_{-1}^2\\left(1+x^2\\right) d x \\\\\r\n& =\\frac{1}{3}\\left[x+\\frac{x^3}{3}\\right]_{-1}^2=2\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Find the area of the region enclosed by the parabolas $y=x^2$ and $y=2 x-x^2$", "answer_latex": " $\\frac{1}{3}$", "answer_number": "0.333333333333333", "unit": " ", "source": "calculus", "problemid": " 6.1.2", "comment": "with solution ", "solution": "We first find the points of intersection of the parabolas by solving their equations simultaneously. This gives $x^2=2 x-x^2$, or $2 x^2-2 x=0$. Thus $2 x(x-1)=0$, so $x=0$ or 1 . The points of intersection are $(0,0)$ and $(1,1)$.\r\nWe see from Figure 6 that the top and bottom boundaries are\r\n$$\r\ny_T=2 x-x^2 \\quad \\text { and } \\quad y_B=x^2\r\n$$\r\nThe area of a typical rectangle is\r\n$$\r\n\\left(y_T-y_B\\right) \\Delta x=\\left(2 x-x^2-x^2\\right) \\Delta x\r\n$$\r\nand the region lies between $x=0$ and $x=1$. So the total area is\r\n$$\r\n\\begin{aligned}\r\nA & =\\int_0^1\\left(2 x-2 x^2\\right) d x=2 \\int_0^1\\left(x-x^2\\right) d x \\\\\r\n& =2\\left[\\frac{x^2}{2}-\\frac{x^3}{3}\\right]_0^1=2\\left(\\frac{1}{2}-\\frac{1}{3}\\right)=\\frac{1}{3}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "The region $\\mathscr{R}$ enclosed by the curves $y=x$ and $y=x^2$ is rotated about the $x$-axis. Find the volume of the resulting solid.", "answer_latex": " $\\frac{2\\pi}{15}$", "answer_number": "0.41887902047", "unit": " ", "source": "calculus", "problemid": " 6.2.4", "comment": " with solution", "solution": "The curves $y=x$ and $y=x^2$ intersect at the points $(0,0)$ and $(1,1)$. The region between them, the solid of rotation, and a cross-section perpendicular to the $x$-axis are shown in Figure. A cross-section in the plane $P_x$ has the shape of a washer (an annular ring) with inner radius $x^2$ and outer radius $x$, so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle:\r\n$$\r\nA(x)=\\pi x^2-\\pi\\left(x^2\\right)^2=\\pi\\left(x^2-x^4\\right)\r\n$$\r\nTherefore we have\r\n$$\r\n\\begin{aligned}\r\nV & =\\int_0^1 A(x) d x=\\int_0^1 \\pi\\left(x^2-x^4\\right) d x \\\\\r\n& =\\pi\\left[\\frac{x^3}{3}-\\frac{x^5}{5}\\right]_0^1=\\frac{2 \\pi}{15}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Use Simpson's Rule with $n=10$ to approximate $\\int_1^2(1 / x) d x$.", "answer_latex": " 0.693150", "answer_number": "0.693150", "unit": " ", "source": "calculus", "problemid": " 7.7.4", "comment": " with solution", "solution": "Putting $f(x)=1 / x, n=10$, and $\\Delta x=0.1$ in Simpson's Rule, we obtain\r\n$$\r\n\\begin{aligned}\r\n\\int_1^2 \\frac{1}{x} d x & \\approx S_{10} \\\\\r\n& =\\frac{\\Delta x}{3}[f(1)+4 f(1.1)+2 f(1.2)+4 f(1.3)+\\cdots+2 f(1.8)+4 f(1.9)+f(2)] \\\\\r\n& =\\frac{0.1}{3}\\left(\\frac{1}{1}+\\frac{4}{1.1}+\\frac{2}{1.2}+\\frac{4}{1.3}+\\frac{2}{1.4}+\\frac{4}{1.5}+\\frac{2}{1.6}+\\frac{4}{1.7}+\\frac{2}{1.8}+\\frac{4}{1.9}+\\frac{1}{2}\\right) \\\\\r\n& \\approx 0.693150\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Use the Midpoint Rule with $m=n=2$ to estimate the value of the integral $\\iint_R\\left(x-3 y^2\\right) d A$, where $R=\\{(x, y) \\mid 0 \\leqslant x \\leqslant 2,1 \\leqslant y \\leqslant 2\\}$.", "answer_latex": " -11.875", "answer_number": "-11.875", "unit": " ", "source": "calculus", "problemid": " 15.1.3", "comment": " ", "solution": "In using the Midpoint Rule with $m=n=2$, we evaluate $f(x, y)=x-3 y^2$ at the centers of the four subrectangles shown in Figure 10. So $\\bar{X}_1=\\frac{1}{2}, \\bar{X}_2=\\frac{3}{2}, \\bar{y}_1=\\frac{5}{4}$, and $\\bar{y}_2=\\frac{7}{4}$. The area of each subrectangle is $\\Delta A=\\frac{1}{2}$. Thus\r\n$$\r\n\\begin{aligned}\r\n\\iint_R\\left(x-3 y^2\\right) d A & \\approx \\sum_{i=1}^2 \\sum_{j=1}^2 f\\left(\\bar{x}_i, \\bar{y}_j\\right) \\Delta A \\\\\r\n& =f\\left(\\bar{x}_1, \\bar{y}_1\\right) \\Delta A+f\\left(\\bar{x}_1, \\bar{y}_2\\right) \\Delta A+f\\left(\\bar{x}_2, \\bar{y}_1\\right) \\Delta A+f\\left(\\bar{x}_2, \\bar{y}_2\\right) \\Delta A \\\\\r\n& =f\\left(\\frac{1}{2}, \\frac{5}{4}\\right) \\Delta A+f\\left(\\frac{1}{2}, \\frac{7}{4}\\right) \\Delta A+f\\left(\\frac{3}{2}, \\frac{5}{4}\\right) \\Delta A+f\\left(\\frac{3}{2}, \\frac{7}{4}\\right) \\Delta A \\\\\r\n& =\\left(-\\frac{67}{16}\\right) \\frac{1}{2}+\\left(-\\frac{139}{16}\\right) \\frac{1}{2}+\\left(-\\frac{51}{16}\\right) \\frac{1}{2}+\\left(-\\frac{123}{16}\\right) \\frac{1}{2} \\\\\r\n& =-\\frac{95}{8}=-11.875\r\n\\end{aligned}\r\n$$\r\nThus we have\r\n$$\r\n\\iint_R\\left(x-3 y^2\\right) d A \\approx-11.875\r\n$$\r\n"}, {"problem_text": "The base radius and height of a right circular cone are measured as $10 \\mathrm{~cm}$ and $25 \\mathrm{~cm}$, respectively, with a possible error in measurement of as much as $0.1 \\mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated volume of the cone.", "answer_latex": " $20\\pi$", "answer_number": "62.8318530718", "unit": " $\\mathrm{~cm}^3$", "source": "calculus", "problemid": " 14.4.5", "comment": "with solution ", "solution": "The volume $V$ of a cone with base radius $r$ and height $h$ is $V=\\pi r^2 h / 3$. So the differential of $V$ is\r\n$$\r\nd V=\\frac{\\partial V}{\\partial r} d r+\\frac{\\partial V}{\\partial h} d h=\\frac{2 \\pi r h}{3} d r+\\frac{\\pi r^2}{3} d h\r\n$$\r\nSince each error is at most $0.1 \\mathrm{~cm}$, we have $|\\Delta r| \\leqslant 0.1,|\\Delta h| \\leqslant 0.1$. To estimate the largest error in the volume we take the largest error in the measurement of $r$ and of $h$. Therefore we take $d r=0.1$ and $d h=0.1$ along with $r=10, h=25$. This gives\r\n$$\r\nd V=\\frac{500 \\pi}{3}(0.1)+\\frac{100 \\pi}{3}(0.1)=20 \\pi\r\n$$\r\nThus the maximum error in the calculated volume is about $20 \\pi \\mathrm{cm}^3 \\approx 63 \\mathrm{~cm}^3$.\r\n"}, {"problem_text": "A force of $40 \\mathrm{~N}$ is required to hold a spring that has been stretched from its natural length of $10 \\mathrm{~cm}$ to a length of $15 \\mathrm{~cm}$. How much work is done in stretching the spring from $15 \\mathrm{~cm}$ to $18 \\mathrm{~cm}$ ?", "answer_latex": " 1.56", "answer_number": "1.56", "unit": " $\\mathrm{~J}$", "source": "calculus", "problemid": " 6.4.3", "comment": " with equation", "solution": "According to Hooke's Law, the force required to hold the spring stretched $x$ meters beyond its natural length is $f(x)=k x$. When the spring is stretched from $10 \\mathrm{~cm}$ to $15 \\mathrm{~cm}$, the amount stretched is $5 \\mathrm{~cm}=0.05 \\mathrm{~m}$. This means that $f(0.05)=40$, so\r\n$$\r\n0.05 k=40 \\quad k=\\frac{40}{0.05}=800\r\n$$\r\nThus $f(x)=800 x$ and the work done in stretching the spring from $15 \\mathrm{~cm}$ to $18 \\mathrm{~cm}$ is\r\n$$\r\n\\begin{aligned}\r\nW & \\left.=\\int_{0.05}^{0.08} 800 x d x=800 \\frac{x^2}{2}\\right]_{0.05}^{0.08} \\\\\r\n& =400\\left[(0.08)^2-(0.05)^2\\right]=1.56 \\mathrm{~J}\r\n\\end{aligned}\r\n$$"}]
+[
+ {
+  "problem_text": "How large should we take $n$ in order to guarantee that the Trapezoidal and Midpoint Rule approximations for $\\int_1^2(1 / x) d x$ are accurate to within 0.0001 ?",
+  "answer_latex": " 41",
+  "answer_number": "41",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 7.7.2",
+  "comment": " with solution",
+  "solution": "We saw in the preceding calculation that $\\left|f^{\\prime \\prime}(x)\\right| \\leqslant 2$ for $1 \\leqslant x \\leqslant 2$, so we can take $K=2$, $a=1$, and $b=2$ in 3 . Accuracy to within 0.0001 means that the size of the error should be less than 0.0001 . Therefore we choose $n$ so that\r\n$$\r\n\\frac{2(1)^3}{12 n^2}<0.0001\r\n$$\r\nSolving the inequality for $n$, we get\r\nor\r\n$$\r\n\\begin{aligned}\r\n& n^2>\\frac{2}{12(0.0001)} \\\\\r\n& n>\\frac{1}{\\sqrt{0.0006}} \\approx 40.8\r\n\\end{aligned}\r\n$$\r\nThus $n=41$ will ensure the desired accuracy."
+ },
+ {
+  "problem_text": "Find the length of the cardioid $r=1+\\sin \\theta$.",
+  "answer_latex": " 8",
+  "answer_number": "8",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 10.4.4",
+  "comment": " with solution",
+  "solution": "\nThe cardioid's full length is given by the parameter interval $0 \\leqslant \\theta \\leqslant 2 \\pi$, $$\r\n\\begin{aligned}\r\nL & =\\int_0^{2 \\pi} \\sqrt{r^2+\\left(\\frac{d r}{d \\theta}\\right)^2} d \\theta=\\int_0^{2 \\pi} \\sqrt{(1+\\sin \\theta)^2+\\cos ^2 \\theta} d \\theta \\\\\r\n& =\\int_0^{2 \\pi} \\sqrt{2+2 \\sin \\theta} d \\theta\r\n\\end{aligned}\r\n$$\r\nWe could evaluate this integral by multiplying and dividing the integrand by $\\sqrt{2-2 \\sin \\theta}$, or we could use a computer algebra system. In any event, we find that the length of the cardioid is $L=8$.\n"
+ },
+ {
+  "problem_text": "Estimate the volume of the solid that lies above the square $R=[0,2] \\times[0,2]$ and below the elliptic paraboloid $z=16-x^2-2 y^2$. Divide $R$ into four equal squares and choose the sample point to be the upper right corner of each square $R_{i j}$. ",
+  "answer_latex": " 34",
+  "answer_number": "34",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": "15.1.1 ",
+  "comment": " with solution",
+  "solution": "\nThe paraboloid is the graph of $f(x, y)=16-x^2-2 y^2$ and the area of each square is $\\Delta A=1$. Approximating the volume by the Riemann sum with $m=n=2$, we have\r\n$$\r\n\\begin{aligned}\r\nV & \\approx \\sum_{i=1}^2 \\sum_{j=1}^2 f\\left(x_i, y_j\\right) \\Delta A \\\\\r\n& =f(1,1) \\Delta A+f(1,2) \\Delta A+f(2,1) \\Delta A+f(2,2) \\Delta A \\\\\r\n& =13(1)+7(1)+10(1)+4(1)=34\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Find the average value of the function $f(x)=1+x^2$ on the interval $[-1,2]$.",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 6.5.1",
+  "comment": " with equation",
+  "solution": "With $a=-1$ and $b=2$ we have\r\n$$\r\n\\begin{aligned}\r\nf_{\\text {ave }} & =\\frac{1}{b-a} \\int_a^b f(x) d x=\\frac{1}{2-(-1)} \\int_{-1}^2\\left(1+x^2\\right) d x \\\\\r\n& =\\frac{1}{3}\\left[x+\\frac{x^3}{3}\\right]_{-1}^2=2\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Find the area of the region enclosed by the parabolas $y=x^2$ and $y=2 x-x^2$",
+  "answer_latex": " $\\frac{1}{3}$",
+  "answer_number": "0.333333333333333",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 6.1.2",
+  "comment": "with solution ",
+  "solution": "\nWe first find the points of intersection of the parabolas by solving their equations simultaneously. This gives $x^2=2 x-x^2$, or $2 x^2-2 x=0$. Thus $2 x(x-1)=0$, so $x=0$ or 1 . The points of intersection are $(0,0)$ and $(1,1)$. By analyzing these two equations, we can see that the top and bottom boundaries are\r\n$$\r\ny_T=2 x-x^2 \\quad \\text { and } \\quad y_B=x^2\r\n$$\r\nThe area of a typical rectangle is\r\n$$\r\n\\left(y_T-y_B\\right) \\Delta x=\\left(2 x-x^2-x^2\\right) \\Delta x\r\n$$\r\nand the region lies between $x=0$ and $x=1$. So the total area is\r\n$$\r\n\\begin{aligned}\r\nA & =\\int_0^1\\left(2 x-2 x^2\\right) d x=2 \\int_0^1\\left(x-x^2\\right) d x \\\\\r\n& =2\\left[\\frac{x^2}{2}-\\frac{x^3}{3}\\right]_0^1=2\\left(\\frac{1}{2}-\\frac{1}{3}\\right)=\\frac{1}{3}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "The region $\\mathscr{R}$ enclosed by the curves $y=x$ and $y=x^2$ is rotated about the $x$-axis. Find the volume of the resulting solid.",
+  "answer_latex": " $\\frac{2\\pi}{15}$",
+  "answer_number": "0.41887902047",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 6.2.4",
+  "comment": " with solution",
+  "solution": "\nThe curves $y=x$ and $y=x^2$ intersect at the points $(0,0)$ and $(1,1)$. A cross-section in the plane $P_x$ has the shape of a washer (an annular ring) with inner radius $x^2$ and outer radius $x$, so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle:\r\n$$\r\nA(x)=\\pi x^2-\\pi\\left(x^2\\right)^2=\\pi\\left(x^2-x^4\\right)\r\n$$\r\nTherefore we have\r\n$$\r\n\\begin{aligned}\r\nV & =\\int_0^1 A(x) d x=\\int_0^1 \\pi\\left(x^2-x^4\\right) d x \\\\\r\n& =\\pi\\left[\\frac{x^3}{3}-\\frac{x^5}{5}\\right]_0^1=\\frac{2 \\pi}{15}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Use Simpson's Rule with $n=10$ to approximate $\\int_1^2(1 / x) d x$.",
+  "answer_latex": " 0.693150",
+  "answer_number": "0.693150",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 7.7.4",
+  "comment": " with solution",
+  "solution": "Putting $f(x)=1 / x, n=10$, and $\\Delta x=0.1$ in Simpson's Rule, we obtain\r\n$$\r\n\\begin{aligned}\r\n\\int_1^2 \\frac{1}{x} d x & \\approx S_{10} \\\\\r\n& =\\frac{\\Delta x}{3}[f(1)+4 f(1.1)+2 f(1.2)+4 f(1.3)+\\cdots+2 f(1.8)+4 f(1.9)+f(2)] \\\\\r\n& =\\frac{0.1}{3}\\left(\\frac{1}{1}+\\frac{4}{1.1}+\\frac{2}{1.2}+\\frac{4}{1.3}+\\frac{2}{1.4}+\\frac{4}{1.5}+\\frac{2}{1.6}+\\frac{4}{1.7}+\\frac{2}{1.8}+\\frac{4}{1.9}+\\frac{1}{2}\\right) \\\\\r\n& \\approx 0.693150\r\n\\end{aligned}\r\n$$\r\n"
+ },
+ {
+  "problem_text": "Use the Midpoint Rule with $m=n=2$ to estimate the value of the integral $\\iint_R\\left(x-3 y^2\\right) d A$, where $R=\\{(x, y) \\mid 0 \\leqslant x \\leqslant 2,1 \\leqslant y \\leqslant 2\\}$.",
+  "answer_latex": " -11.875",
+  "answer_number": "-11.875",
+  "unit": " ",
+  "source": "calculus",
+  "problemid": " 15.1.3",
+  "comment": " ",
+  "solution": "\nIn using the Midpoint Rule with $m=n=2$, we evaluate $f(x, y)=x-3 y^2$ at the centers of the four subrectangles. So $\\bar{X}_1=\\frac{1}{2}, \\bar{X}_2=\\frac{3}{2}, \\bar{y}_1=\\frac{5}{4}$, and $\\bar{y}_2=\\frac{7}{4}$. The area of each subrectangle is $\\Delta A=\\frac{1}{2}$. Thus\r\n$$\r\n\\begin{aligned}\r\n\\iint_R\\left(x-3 y^2\\right) d A & \\approx \\sum_{i=1}^2 \\sum_{j=1}^2 f\\left(\\bar{x}_i, \\bar{y}_j\\right) \\Delta A \\\\\r\n& =f\\left(\\bar{x}_1, \\bar{y}_1\\right) \\Delta A+f\\left(\\bar{x}_1, \\bar{y}_2\\right) \\Delta A+f\\left(\\bar{x}_2, \\bar{y}_1\\right) \\Delta A+f\\left(\\bar{x}_2, \\bar{y}_2\\right) \\Delta A \\\\\r\n& =f\\left(\\frac{1}{2}, \\frac{5}{4}\\right) \\Delta A+f\\left(\\frac{1}{2}, \\frac{7}{4}\\right) \\Delta A+f\\left(\\frac{3}{2}, \\frac{5}{4}\\right) \\Delta A+f\\left(\\frac{3}{2}, \\frac{7}{4}\\right) \\Delta A \\\\\r\n& =\\left(-\\frac{67}{16}\\right) \\frac{1}{2}+\\left(-\\frac{139}{16}\\right) \\frac{1}{2}+\\left(-\\frac{51}{16}\\right) \\frac{1}{2}+\\left(-\\frac{123}{16}\\right) \\frac{1}{2} \\\\\r\n& =-\\frac{95}{8}=-11.875\r\n\\end{aligned}\r\n$$\r\nThus we have\r\n$$\r\n\\iint_R\\left(x-3 y^2\\right) d A \\approx-11.875\r\n$$\n"
+ },
+ {
+  "problem_text": "The base radius and height of a right circular cone are measured as $10 \\mathrm{~cm}$ and $25 \\mathrm{~cm}$, respectively, with a possible error in measurement of as much as $0.1 \\mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated volume of the cone.",
+  "answer_latex": " $20\\pi$",
+  "answer_number": "62.8318530718",
+  "unit": " $\\mathrm{~cm}^3$",
+  "source": "calculus",
+  "problemid": " 14.4.5",
+  "comment": "with solution ",
+  "solution": "The volume $V$ of a cone with base radius $r$ and height $h$ is $V=\\pi r^2 h / 3$. So the differential of $V$ is\r\n$$\r\nd V=\\frac{\\partial V}{\\partial r} d r+\\frac{\\partial V}{\\partial h} d h=\\frac{2 \\pi r h}{3} d r+\\frac{\\pi r^2}{3} d h\r\n$$\r\nSince each error is at most $0.1 \\mathrm{~cm}$, we have $|\\Delta r| \\leqslant 0.1,|\\Delta h| \\leqslant 0.1$. To estimate the largest error in the volume we take the largest error in the measurement of $r$ and of $h$. Therefore we take $d r=0.1$ and $d h=0.1$ along with $r=10, h=25$. This gives\r\n$$\r\nd V=\\frac{500 \\pi}{3}(0.1)+\\frac{100 \\pi}{3}(0.1)=20 \\pi\r\n$$\r\nThus the maximum error in the calculated volume is about $20 \\pi \\mathrm{cm}^3 \\approx 63 \\mathrm{~cm}^3$.\r\n"
+ },
+ {
+  "problem_text": "A force of $40 \\mathrm{~N}$ is required to hold a spring that has been stretched from its natural length of $10 \\mathrm{~cm}$ to a length of $15 \\mathrm{~cm}$. How much work is done in stretching the spring from $15 \\mathrm{~cm}$ to $18 \\mathrm{~cm}$ ?",
+  "answer_latex": " 1.56",
+  "answer_number": "1.56",
+  "unit": " $\\mathrm{~J}$",
+  "source": "calculus",
+  "problemid": " 6.4.3",
+  "comment": " with equation",
+  "solution": "According to Hooke's Law, the force required to hold the spring stretched $x$ meters beyond its natural length is $f(x)=k x$. When the spring is stretched from $10 \\mathrm{~cm}$ to $15 \\mathrm{~cm}$, the amount stretched is $5 \\mathrm{~cm}=0.05 \\mathrm{~m}$. This means that $f(0.05)=40$, so\r\n$$\r\n0.05 k=40 \\quad k=\\frac{40}{0.05}=800\r\n$$\r\nThus $f(x)=800 x$ and the work done in stretching the spring from $15 \\mathrm{~cm}$ to $18 \\mathrm{~cm}$ is\r\n$$\r\n\\begin{aligned}\r\nW & \\left.=\\int_{0.05}^{0.08} 800 x d x=800 \\frac{x^2}{2}\\right]_{0.05}^{0.08} \\\\\r\n& =400\\left[(0.08)^2-(0.05)^2\\right]=1.56 \\mathrm{~J}\r\n\\end{aligned}\r\n$$"
+ }
+]

chemmc.json

@@ -1 +1,344 @@
-[{"problem_text": "Calculate the de Broglie wavelength for (a) an electron with a kinetic energy of $100 \\mathrm{eV}$", "answer_latex": " 0.123", "answer_number": "0.123", "unit": "nm ", "source": "chemmc", "problemid": "1-38 ", "comment": " "}, {"problem_text": "The threshold wavelength for potassium metal is $564 \\mathrm{~nm}$. What is its work function? \r\n", "answer_latex": " 3.52", "answer_number": "3.52", "unit": "$10^{-19} \\mathrm{~J}$", "source": "chemmc", "problemid": " 1-18", "comment": " Only the first part, the work function is taken"}, {"problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=0}^{\\infty} \\frac{1}{3^n}\r\n$$", "answer_latex": " 3 / 2", "answer_number": "1.5", "unit": " ", "source": "chemmc", "problemid": "D-7 ", "comment": " Math Part D (after chapter 4)"}, {"problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{2^n}\r\n$$", "answer_latex": " 1", "answer_number": "1", "unit": " ", "source": "chemmc", "problemid": "D-17 ", "comment": " Math Part D (after chapter 4)"}, {"problem_text": "The relationship introduced in Problem $1-48$ has been interpreted to mean that a particle of mass $m\\left(E=m c^2\\right)$ can materialize from nothing provided that it returns to nothing within a time $\\Delta t \\leq h / m c^2$. Particles that last for time $\\Delta t$ or more are called real particles; particles that last less than time $\\Delta t$ are called virtual particles. The mass of the charged pion, a subatomic particle, is $2.5 \\times 10^{-28} \\mathrm{~kg}$. What is the minimum lifetime if the pion is to be considered a real particle?", "answer_latex": " 2.9", "answer_number": "2.9", "unit": "$10^{-23} \\mathrm{~s}$ ", "source": "chemmc", "problemid": "1-49 ", "comment": " "}, {"problem_text": "A household lightbulb is a blackbody radiator. Many lightbulbs use tungsten filaments that are heated by an electric current. What temperature is needed so that $\\lambda_{\\max }=550 \\mathrm{~nm}$ ?", "answer_latex": " 5300", "answer_number": "5300", "unit": " $\\mathrm{~K}$\r\n", "source": "chemmc", "problemid": " 1-17", "comment": " "}, {"problem_text": "Evaluate the series\r\n$$\r\nS=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}+\\cdots\r\n$$\r\n", "answer_latex": " 1", "answer_number": "1", "unit": " ", "source": "chemmc", "problemid": " D-6", "comment": " Math Part D (after chapter 4)"}, {"problem_text": "Through what potential must a proton initially at rest fall so that its de Broglie wavelength is $1.0 \\times 10^{-10} \\mathrm{~m}$ ?", "answer_latex": " 0.082", "answer_number": "0.082", "unit": "V ", "source": "chemmc", "problemid": "1-40 ", "comment": " "}, {"problem_text": "Example 5-3 shows that a Maclaurin expansion of a Morse potential leads to\r\n$$\r\nV(x)=D \\beta^2 x^2+\\cdots\r\n$$\r\nGiven that $D=7.31 \\times 10^{-19} \\mathrm{~J} \\cdot$ molecule ${ }^{-1}$ and $\\beta=1.81 \\times 10^{10} \\mathrm{~m}^{-1}$ for $\\mathrm{HCl}$, calculate the force constant of $\\mathrm{HCl}$.", "answer_latex": " 479", "answer_number": "479", "unit": "$\\mathrm{~N} \\cdot \\mathrm{m}^{-1}$ ", "source": "chemmc", "problemid": "5-9 ", "comment": " "}, {"problem_text": "A line in the Lyman series of hydrogen has a wavelength of $1.03 \\times 10^{-7} \\mathrm{~m}$. Find the original energy level of the electron.", "answer_latex": " 3", "answer_number": "3", "unit": " ", "source": "chemmc", "problemid": " 1-25", "comment": " no units"}, {"problem_text": "A helium-neon laser (used in supermarket scanners) emits light at $632.8 \\mathrm{~nm}$. Calculate the frequency of this light.", "answer_latex": " 4.738", "answer_number": "4.738", "unit": "$10^{14} \\mathrm{~Hz}$ ", "source": "chemmc", "problemid": " 1-15", "comment": " just the first part is taken: frequency of light"}, {"problem_text": "What is the uncertainty of the momentum of an electron if we know its position is somewhere in a $10 \\mathrm{pm}$ interval?", "answer_latex": " 6.6", "answer_number": " 6.6", "unit": " $10^{-23} \\mathrm{~kg} \\cdot \\mathrm{m} \\cdot \\mathrm{s}^{-1}$", "source": "chemmc", "problemid": "1-47 ", "comment": " discard the second part of the answer"}, {"problem_text": "Using the Bohr theory, calculate the ionization energy (in electron volts and in $\\mathrm{kJ} \\cdot \\mathrm{mol}^{-1}$ ) of singly ionized helium.", "answer_latex": " 54.394", "answer_number": "54.394", "unit": "$\\mathrm{eV}$ ", "source": "chemmc", "problemid": "1-34 ", "comment": " "}, {"problem_text": "When an excited nucleus decays, it emits a $\\gamma$ ray. The lifetime of an excited state of a nucleus is of the order of $10^{-12} \\mathrm{~s}$. What is the uncertainty in the energy of the $\\gamma$ ray produced?", "answer_latex": " 7", "answer_number": "7", "unit": "$10^{-22} \\mathrm{~J}$ ", "source": "chemmc", "problemid": "1-51 ", "comment": " "}, {"problem_text": "Calculate the wavelength and the energy of a photon associated with the series limit of the Lyman series.", "answer_latex": " 91.17", "answer_number": "91.17", "unit": "nm ", "source": "chemmc", "problemid": " 1-28", "comment": "only the first part of the question, the wavelength"}, {"problem_text": "Another application of the relationship given in Problem $1-48$ has to do with the excitedstate energies and lifetimes of atoms and molecules. If we know that the lifetime of an excited state is $10^{-9} \\mathrm{~s}$, then what is the uncertainty in the energy of this state?\r\n", "answer_latex": " 7", "answer_number": "7", "unit": " $10^{-25} \\mathrm{~J}$", "source": "chemmc", "problemid": " 1-50", "comment": " "}, {"problem_text": "One of the most powerful modern techniques for studying structure is neutron diffraction. This technique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by $v_{\\mathrm{n}}=\\left(3 k_{\\mathrm{B}} T / m\\right)^{1 / 2}$, where $m$ is the mass of a neutron, then what temperature is needed so that the neutrons have a de Broglie wavelength of $50 \\mathrm{pm}$ ?", "answer_latex": " 2500", "answer_number": "2500", "unit": "$\\mathrm{K}$ ", "source": "chemmc", "problemid": "1-42 ", "comment": " "}, {"problem_text": "The temperature of the fireball in a thermonuclear explosion can reach temperatures of approximately $10^7 \\mathrm{~K}$. What value of $\\lambda_{\\max }$ does this correspond to? ", "answer_latex": " 3", "answer_number": "3", "unit": " $10^{-10} \\mathrm{~m}$\r\n", "source": "chemmc", "problemid": "1-8 ", "comment": " "}, {"problem_text": "Show that l'H\u00f4pital's rule amounts to forming a Taylor expansion of both the numerator and the denominator. Evaluate the limit\r\n$$\r\n\\lim _{x \\rightarrow 0} \\frac{\\ln (1+x)-x}{x^2}\r\n$$\r\nboth ways.", "answer_latex": " -1/2", "answer_number": "-0.5", "unit": " ", "source": "chemmc", "problemid": "D-21 ", "comment": " Math Part D (after chapter 4)"}, {"problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{2^n}\r\n$$", "answer_latex": " 1/3", "answer_number": "0.3333333", "unit": " ", "source": "chemmc", "problemid": " D-8", "comment": " Math Part D (after chapter 4)"}, {"problem_text": "Calculate the percentage difference between $\\ln (1+x)$ and $x$ for $x=0.0050$", "answer_latex": " 0.249", "answer_number": "0.249", "unit": " %", "source": "chemmc", "problemid": " D-4", "comment": " Math Part D (after chapter 4)"}, {"problem_text": "Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00.", "answer_latex": " 7.00", "answer_number": "7.00", "unit": " ", "source": "chemmc", "problemid": "1-30 ", "comment": "no units "}, {"problem_text": "Two narrow slits are illuminated with red light of wavelength $694.3 \\mathrm{~nm}$ from a laser, producing a set of evenly placed bright bands on a screen located $3.00 \\mathrm{~m}$ beyond the slits. If the distance between the bands is $1.50 \\mathrm{~cm}$, then what is the distance between the slits?\r\n", "answer_latex": " 0.139", "answer_number": "0.139", "unit": "mm ", "source": "chemmc", "problemid": "1-45 ", "comment": " "}, {"problem_text": "Calculate the energy and wavelength associated with an $\\alpha$ particle that has fallen through a potential difference of $4.0 \\mathrm{~V}$. Take the mass of an $\\alpha$ particle to be $6.64 \\times 10^{-27} \\mathrm{~kg}$.", "answer_latex": " 1.3", "answer_number": "1.3", "unit": " $\r\n\\text { 1-41. } 1.3 \\times 10^{-18} \\mathrm{~J} / \\alpha \\text {-particle, }\r\n$", "source": "chemmc", "problemid": "1-41 ", "comment": " "}, {"problem_text": "Calculate the number of photons in a $2.00 \\mathrm{~mJ}$ light pulse at (a) $1.06 \\mu \\mathrm{m}$\r\n", "answer_latex": " 1.07", "answer_number": "1.07", "unit": " $10^{16}$ photons", "source": "chemmc", "problemid": " 1-13", "comment": " part (a) only"}, {"problem_text": "The force constant of ${ }^{35} \\mathrm{Cl}^{35} \\mathrm{Cl}$ is $319 \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$. Calculate the fundamental vibrational frequency", "answer_latex": " 556", "answer_number": "556", "unit": " $\\mathrm{~cm}^{-1}$", "source": "chemmc", "problemid": " 5-14", "comment": " "}, {"problem_text": "$$\r\n\\text {Calculate the energy of a photon for a wavelength of } 100 \\mathrm{pm} \\text { (about one atomic diameter). }\r\n$$\r\n", "answer_latex": " 2", "answer_number": "2", "unit": " $10^{-15} \\mathrm{~J}$", "source": "chemmc", "problemid": "1-11 ", "comment": " "}, {"problem_text": "A proton and a negatively charged $\\mu$ meson (called a muon) can form a short-lived species called a mesonic atom. The charge of a muon is the same as that on an electron and the mass of a muon is $207 m_{\\mathrm{e}}$. Assume that the Bohr theory can be applied to such a mesonic atom and calculate the ground-state energy, the radius of the first Bohr orbit, and the energy and frequency associated with the $n=1$ to $n=2$ transition in a mesonic atom.", "answer_latex": " 1.69", "answer_number": "1.69", "unit": " $10^{-28} \\mathrm{~kg}$", "source": "chemmc", "problemid": " 1-37", "comment": " only the ground state energy is there"}, {"problem_text": "$$\r\n\\beta=2 \\pi c \\tilde{\\omega}_{\\mathrm{obs}}\\left(\\frac{\\mu}{2 D}\\right)^{1 / 2}\r\n$$\r\nGiven that $\\tilde{\\omega}_{\\mathrm{obs}}=2886 \\mathrm{~cm}^{-1}$ and $D=440.2 \\mathrm{~kJ} \\cdot \\mathrm{mol}^{-1}$ for $\\mathrm{H}^{35} \\mathrm{Cl}$, calculate $\\beta$.", "answer_latex": " 1.81", "answer_number": "1.81", "unit": " $10^{10} \\mathrm{~m}^{-1}$", "source": "chemmc", "problemid": " 5-10", "comment": " "}, {"problem_text": "Two narrow slits separated by $0.10 \\mathrm{~mm}$ are illuminated by light of wavelength $600 \\mathrm{~nm}$. What is the angular position of the first maximum in the interference pattern? If a detector is located $2.00 \\mathrm{~m}$ beyond the slits, what is the distance between the central maximum and the first maximum?", "answer_latex": " 12", "answer_number": "12", "unit": " mm", "source": "chemmc", "problemid": "1-44 ", "comment": " "}, {"problem_text": "$$\r\n\\text { If we locate an electron to within } 20 \\mathrm{pm} \\text {, then what is the uncertainty in its speed? }\r\n$$", "answer_latex": " 3.7", "answer_number": "3.7", "unit": "$10^7 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "chemmc", "problemid": "1-46 ", "comment": " "}, {"problem_text": "The mean temperature of the earth's surface is $288 \\mathrm{~K}$. What is the maximum wavelength of the earth's blackbody radiation?", "answer_latex": " 1.01", "answer_number": "1.01", "unit": " 10^{-5} \\mathrm{~m}", "source": "chemmc", "problemid": " 1-14", "comment": " "}, {"problem_text": "The power output of a laser is measured in units of watts (W), where one watt is equal to one joule per second. $\\left(1 \\mathrm{~W}=1 \\mathrm{~J} \\cdot \\mathrm{s}^{-1}\\right.$.) What is the number of photons emitted per second by a $1.00 \\mathrm{~mW}$ nitrogen laser? The wavelength emitted by a nitrogen laser is $337 \\mathrm{~nm}$.", "answer_latex": " 1.70", "answer_number": "1.70", "unit": " $\r\n10^{15} \\text { photon } \\cdot \\mathrm{s}^{-1}\r\n$", "source": "chemmc", "problemid": " 1-16", "comment": " "}, {"problem_text": " Sirius, one of the hottest known stars, has approximately a blackbody spectrum with $\\lambda_{\\max }=260 \\mathrm{~nm}$. Estimate the surface temperature of Sirius.\r\n", "answer_latex": "11000", "answer_number": "11000", "unit": " $\\mathrm{~K}$\r\n", "source": "chemmc", "problemid": " 1-7", "comment": " "}, {"problem_text": "A ground-state hydrogen atom absorbs a photon of light that has a wavelength of $97.2 \\mathrm{~nm}$. It then gives off a photon that has a wavelength of $486 \\mathrm{~nm}$. What is the final state of the hydrogen atom?", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "chemmc", "problemid": " 1-26", "comment": " no units"}, {"problem_text": "It turns out that the solution of the Schr\u00f6dinger equation for the Morse potential can be expressed as\r\n$$\r\nG(v)=\\tilde{\\omega}_{\\mathrm{e}}\\left(v+\\frac{1}{2}\\right)-\\tilde{\\omega}_{\\mathrm{e}} \\tilde{x}_{\\mathrm{e}}\\left(v+\\frac{1}{2}\\right)^2\r\n$$\r\nChapter 5 / The Harmonic Oscillator and Vibrational Spectroscopy\r\nwhere\r\n$$\r\n\\tilde{x}_{\\mathrm{e}}=\\frac{h c \\tilde{\\omega}_{\\mathrm{e}}}{4 D}\r\n$$\r\nGiven that $\\tilde{\\omega}_{\\mathrm{e}}=2886 \\mathrm{~cm}^{-1}$ and $D=440.2 \\mathrm{~kJ} \\cdot \\mathrm{mol}^{-1}$ for $\\mathrm{H}^{35} \\mathrm{Cl}$, calculate $\\tilde{x}_{\\mathrm{e}}$ and $\\tilde{\\omega}_{\\mathrm{e}} \\tilde{x}_{\\mathrm{e}}$.", "answer_latex": " 0.01961", "answer_number": " 0.01961", "unit": " ", "source": "chemmc", "problemid": "5-12 ", "comment": "only first part taken of the question "}, {"problem_text": " In the infrared spectrum of $\\mathrm{H}^{127} \\mathrm{I}$, there is an intense line at $2309 \\mathrm{~cm}^{-1}$. Calculate the force constant of $\\mathrm{H}^{127} \\mathrm{I}$.", "answer_latex": "313", "answer_number": "313", "unit": " $ \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$", "source": "chemmc", "problemid": " 5-13", "comment": " "}, {"problem_text": "Calculate the percentage difference between $e^x$ and $1+x$ for $x=0.0050$", "answer_latex": " 1.25", "answer_number": "1.25", "unit": " $10^{-3} \\%$", "source": "chemmc", "problemid": "D-1", "comment": "Math Part D (after chapter 4)"}, {"problem_text": " Calculate (a) the wavelength and kinetic energy of an electron in a beam of electrons accelerated by a voltage increment of $100 \\mathrm{~V}$ ", "answer_latex": " 1.602", "answer_number": "1.602", "unit": " $10^{-17} \\mathrm{~J} \\cdot$ electron ${ }^{-1}$", "source": "chemmc", "problemid": "1-39 ", "comment": " "}]
+[
+ {
+  "problem_text": "Calculate the de Broglie wavelength for an electron with a kinetic energy of $100 \\mathrm{eV}$",
+  "answer_latex": " 0.123",
+  "answer_number": "0.123",
+  "unit": "nm ",
+  "source": "chemmc",
+  "problemid": "1-38 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "The threshold wavelength for potassium metal is $564 \\mathrm{~nm}$. What is its work function? \r\n",
+  "answer_latex": " 3.52",
+  "answer_number": "3.52",
+  "unit": "$10^{-19} \\mathrm{~J}$",
+  "source": "chemmc",
+  "problemid": " 1-18",
+  "comment": " Only the first part, the work function is taken"
+ },
+ {
+  "problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=0}^{\\infty} \\frac{1}{3^n}\r\n$$",
+  "answer_latex": " 3 / 2",
+  "answer_number": "1.5",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "D-7 ",
+  "comment": " Math Part D (after chapter 4)"
+ },
+ {
+  "problem_text": "The relationship introduced in Problem $1-48$ has been interpreted to mean that a particle of mass $m\\left(E=m c^2\\right)$ can materialize from nothing provided that it returns to nothing within a time $\\Delta t \\leq h / m c^2$. Particles that last for time $\\Delta t$ or more are called real particles; particles that last less than time $\\Delta t$ are called virtual particles. The mass of the charged pion, a subatomic particle, is $2.5 \\times 10^{-28} \\mathrm{~kg}$. What is the minimum lifetime if the pion is to be considered a real particle?",
+  "answer_latex": " 2.9",
+  "answer_number": "2.9",
+  "unit": "$10^{-23} \\mathrm{~s}$ ",
+  "source": "chemmc",
+  "problemid": "1-49 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "A household lightbulb is a blackbody radiator. Many lightbulbs use tungsten filaments that are heated by an electric current. What temperature is needed so that $\\lambda_{\\max }=550 \\mathrm{~nm}$ ?",
+  "answer_latex": " 5300",
+  "answer_number": "5300",
+  "unit": " $\\mathrm{~K}$\r\n",
+  "source": "chemmc",
+  "problemid": " 1-17",
+  "comment": " "
+ },
+ {
+  "problem_text": "Evaluate the series\r\n$$\r\nS=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}+\\cdots\r\n$$\r\n",
+  "answer_latex": " 1",
+  "answer_number": "1",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": " D-6",
+  "comment": " Math Part D (after chapter 4)"
+ },
+ {
+  "problem_text": "Through what potential must a proton initially at rest fall so that its de Broglie wavelength is $1.0 \\times 10^{-10} \\mathrm{~m}$ ?",
+  "answer_latex": " 0.082",
+  "answer_number": "0.082",
+  "unit": "V ",
+  "source": "chemmc",
+  "problemid": "1-40 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Example 5-3 shows that a Maclaurin expansion of a Morse potential leads to\r\n$$\r\nV(x)=D \\beta^2 x^2+\\cdots\r\n$$\r\nGiven that $D=7.31 \\times 10^{-19} \\mathrm{~J} \\cdot$ molecule ${ }^{-1}$ and $\\beta=1.81 \\times 10^{10} \\mathrm{~m}^{-1}$ for $\\mathrm{HCl}$, calculate the force constant of $\\mathrm{HCl}$.",
+  "answer_latex": " 479",
+  "answer_number": "479",
+  "unit": "$\\mathrm{~N} \\cdot \\mathrm{m}^{-1}$ ",
+  "source": "chemmc",
+  "problemid": "5-9 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "A line in the Lyman series of hydrogen has a wavelength of $1.03 \\times 10^{-7} \\mathrm{~m}$. Find the original energy level of the electron.",
+  "answer_latex": " 3",
+  "answer_number": "3",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": " 1-25",
+  "comment": " no units"
+ },
+ {
+  "problem_text": "A helium-neon laser (used in supermarket scanners) emits light at $632.8 \\mathrm{~nm}$. Calculate the frequency of this light.",
+  "answer_latex": " 4.738",
+  "answer_number": "4.738",
+  "unit": "$10^{14} \\mathrm{~Hz}$ ",
+  "source": "chemmc",
+  "problemid": " 1-15",
+  "comment": " just the first part is taken: frequency of light"
+ },
+ {
+  "problem_text": "What is the uncertainty of the momentum of an electron if we know its position is somewhere in a $10 \\mathrm{pm}$ interval?",
+  "answer_latex": " 6.6",
+  "answer_number": " 6.6",
+  "unit": " $10^{-23} \\mathrm{~kg} \\cdot \\mathrm{m} \\cdot \\mathrm{s}^{-1}$",
+  "source": "chemmc",
+  "problemid": "1-47 ",
+  "comment": " discard the second part of the answer"
+ },
+ {
+  "problem_text": "Using the Bohr theory, calculate the ionization energy (in electron volts and in $\\mathrm{kJ} \\cdot \\mathrm{mol}^{-1}$ ) of singly ionized helium.",
+  "answer_latex": " 54.394",
+  "answer_number": "54.394",
+  "unit": "$\\mathrm{eV}$ ",
+  "source": "chemmc",
+  "problemid": "1-34 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "When an excited nucleus decays, it emits a $\\gamma$ ray. The lifetime of an excited state of a nucleus is of the order of $10^{-12} \\mathrm{~s}$. What is the uncertainty in the energy of the $\\gamma$ ray produced?",
+  "answer_latex": " 7",
+  "answer_number": "7",
+  "unit": "$10^{-22} \\mathrm{~J}$ ",
+  "source": "chemmc",
+  "problemid": "1-51 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the wavelength and the energy of a photon associated with the series limit of the Lyman series.",
+  "answer_latex": " 91.17",
+  "answer_number": "91.17",
+  "unit": "nm ",
+  "source": "chemmc",
+  "problemid": " 1-28",
+  "comment": "only the first part of the question, the wavelength"
+ },
+ {
+  "problem_text": "Given a context information that there is also an uncertainty principle for energy and time:\n$$\n\\Delta E \\Delta t \\geq h\n$$, another application of the relationship has to do with the excitedstate energies and lifetimes of atoms and molecules. If we know that the lifetime of an excited state is $10^{-9} \\mathrm{~s}$, then what is the uncertainty in the energy of this state?",
+  "answer_latex": " 7",
+  "answer_number": "7",
+  "unit": " $10^{-25} \\mathrm{~J}$",
+  "source": "chemmc",
+  "problemid": " 1-50",
+  "comment": " "
+ },
+ {
+  "problem_text": "One of the most powerful modern techniques for studying structure is neutron diffraction. This technique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by $v_{\\mathrm{n}}=\\left(3 k_{\\mathrm{B}} T / m\\right)^{1 / 2}$, where $m$ is the mass of a neutron, then what temperature is needed so that the neutrons have a de Broglie wavelength of $50 \\mathrm{pm}$ ?",
+  "answer_latex": " 2500",
+  "answer_number": "2500",
+  "unit": "$\\mathrm{K}$ ",
+  "source": "chemmc",
+  "problemid": "1-42 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "The temperature of the fireball in a thermonuclear explosion can reach temperatures of approximately $10^7 \\mathrm{~K}$. What value of $\\lambda_{\\max }$ does this correspond to? ",
+  "answer_latex": " 3",
+  "answer_number": "3",
+  "unit": " $10^{-10} \\mathrm{~m}$\r\n",
+  "source": "chemmc",
+  "problemid": "1-8 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Show that l'H\u00f4pital's rule amounts to forming a Taylor expansion of both the numerator and the denominator. Evaluate the limit\r\n$$\r\n\\lim _{x \\rightarrow 0} \\frac{\\ln (1+x)-x}{x^2}\r\n$$\r\nboth ways and report the final result.",
+  "answer_latex": " -1/2",
+  "answer_number": "-0.5",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "D-21 ",
+  "comment": " Math Part D (after chapter 4)"
+ },
+ {
+  "problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{2^n}\r\n$$",
+  "answer_latex": " 1/3",
+  "answer_number": "0.3333333",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": " D-8",
+  "comment": " Math Part D (after chapter 4)"
+ },
+ {
+  "problem_text": "Calculate the percentage difference between $\\ln (1+x)$ and $x$ for $x=0.0050$",
+  "answer_latex": " 0.249",
+  "answer_number": "0.249",
+  "unit": " %",
+  "source": "chemmc",
+  "problemid": " D-4",
+  "comment": " Math Part D (after chapter 4)"
+ },
+ {
+  "problem_text": "Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00.",
+  "answer_latex": " 7.00",
+  "answer_number": "7.00",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "1-30 ",
+  "comment": "no units "
+ },
+ {
+  "problem_text": "Two narrow slits are illuminated with red light of wavelength $694.3 \\mathrm{~nm}$ from a laser, producing a set of evenly placed bright bands on a screen located $3.00 \\mathrm{~m}$ beyond the slits. If the distance between the bands is $1.50 \\mathrm{~cm}$, then what is the distance between the slits?\r\n",
+  "answer_latex": " 0.139",
+  "answer_number": "0.139",
+  "unit": "mm ",
+  "source": "chemmc",
+  "problemid": "1-45 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the energy associated with an $\\alpha$ particle that has fallen through a potential difference of $4.0 \\mathrm{~V}$. Take the mass of an $\\alpha$ particle to be $6.64 \\times 10^{-27} \\mathrm{~kg}$.",
+  "answer_latex": " 1.3",
+  "answer_number": "1.3",
+  "unit": "$10^{-18} \\mathrm{~J} / \\alpha \\text {-particle}$",
+  "source": "chemmc",
+  "problemid": "1-41 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the number of photons in a $2.00 \\mathrm{~mJ}$ light pulse at (a) $1.06 \\mu \\mathrm{m}$\r\n",
+  "answer_latex": " 1.07",
+  "answer_number": "1.07",
+  "unit": " $10^{16}$ photons",
+  "source": "chemmc",
+  "problemid": " 1-13",
+  "comment": " part (a) only"
+ },
+ {
+  "problem_text": "The force constant of ${ }^{35} \\mathrm{Cl}^{35} \\mathrm{Cl}$ is $319 \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$. Calculate the fundamental vibrational frequency",
+  "answer_latex": " 556",
+  "answer_number": "556",
+  "unit": " $\\mathrm{~cm}^{-1}$",
+  "source": "chemmc",
+  "problemid": " 5-14",
+  "comment": " "
+ },
+ {
+  "problem_text": "$$\r\n\\text {Calculate the energy of a photon for a wavelength of } 100 \\mathrm{pm} \\text { (about one atomic diameter). }\r\n$$\r\n",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " $10^{-15} \\mathrm{~J}$",
+  "source": "chemmc",
+  "problemid": "1-11 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "A proton and a negatively charged $\\mu$ meson (called a muon) can form a short-lived species called a mesonic atom. The charge of a muon is the same as that on an electron and the mass of a muon is $207 m_{\\mathrm{e}}$. Assume that the Bohr theory can be applied to such a mesonic atom and calculate the frequency associated with the $n=1$ to $n=2$ transition in a mesonic atom.",
+  "answer_latex": " 1.69",
+  "answer_number": "4.59",
+  "unit": "$10^{17} \\mathrm{~Hz}$",
+  "source": "chemmc",
+  "problemid": " 1-37",
+  "comment": " only the ground state energy is there"
+ },
+ {
+  "problem_text": "$$\r\n\\beta=2 \\pi c \\tilde{\\omega}_{\\mathrm{obs}}\\left(\\frac{\\mu}{2 D}\\right)^{1 / 2}\r\n$$\r\nGiven that $\\tilde{\\omega}_{\\mathrm{obs}}=2886 \\mathrm{~cm}^{-1}$ and $D=440.2 \\mathrm{~kJ} \\cdot \\mathrm{mol}^{-1}$ for $\\mathrm{H}^{35} \\mathrm{Cl}$, calculate $\\beta$.",
+  "answer_latex": " 1.81",
+  "answer_number": "1.81",
+  "unit": " $10^{10} \\mathrm{~m}^{-1}$",
+  "source": "chemmc",
+  "problemid": " 5-10",
+  "comment": " "
+ },
+ {
+  "problem_text": "Two narrow slits separated by $0.10 \\mathrm{~mm}$ are illuminated by light of wavelength $600 \\mathrm{~nm}$. If a detector is located $2.00 \\mathrm{~m}$ beyond the slits, what is the distance between the central maximum and the first maximum?",
+  "answer_latex": " 12",
+  "answer_number": "12",
+  "unit": " mm",
+  "source": "chemmc",
+  "problemid": "1-44 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "$$\r\n\\text { If we locate an electron to within } 20 \\mathrm{pm} \\text {, then what is the uncertainty in its speed? }\r\n$$",
+  "answer_latex": " 3.7",
+  "answer_number": "3.7",
+  "unit": "$10^7 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ",
+  "source": "chemmc",
+  "problemid": "1-46 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "The mean temperature of the earth's surface is $288 \\mathrm{~K}$. What is the maximum wavelength of the earth's blackbody radiation?",
+  "answer_latex": " 1.01",
+  "answer_number": "1.01",
+  "unit": " 10^{-5} \\mathrm{~m}",
+  "source": "chemmc",
+  "problemid": " 1-14",
+  "comment": " "
+ },
+ {
+  "problem_text": "The power output of a laser is measured in units of watts (W), where one watt is equal to one joule per second. $\\left(1 \\mathrm{~W}=1 \\mathrm{~J} \\cdot \\mathrm{s}^{-1}\\right.$.) What is the number of photons emitted per second by a $1.00 \\mathrm{~mW}$ nitrogen laser? The wavelength emitted by a nitrogen laser is $337 \\mathrm{~nm}$.",
+  "answer_latex": " 1.70",
+  "answer_number": "1.70",
+  "unit": " $\r\n10^{15} \\text { photon } \\cdot \\mathrm{s}^{-1}\r\n$",
+  "source": "chemmc",
+  "problemid": " 1-16",
+  "comment": " "
+ },
+ {
+  "problem_text": " Sirius, one of the hottest known stars, has approximately a blackbody spectrum with $\\lambda_{\\max }=260 \\mathrm{~nm}$. Estimate the surface temperature of Sirius.\r\n",
+  "answer_latex": "11000",
+  "answer_number": "11000",
+  "unit": " $\\mathrm{~K}$\r\n",
+  "source": "chemmc",
+  "problemid": " 1-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "A ground-state hydrogen atom absorbs a photon of light that has a wavelength of $97.2 \\mathrm{~nm}$. It then gives off a photon that has a wavelength of $486 \\mathrm{~nm}$. What is the final state of the hydrogen atom?",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": " 1-26",
+  "comment": " no units"
+ },
+ {
+  "problem_text": "It turns out that the solution of the Schr\u00f6dinger equation for the Morse potential can be expressed as\r\n$$\r\nG(v)=\\tilde{\\omega}_{\\mathrm{e}}\\left(v+\\frac{1}{2}\\right)-\\tilde{\\omega}_{\\mathrm{e}} \\tilde{x}_{\\mathrm{e}}\\left(v+\\frac{1}{2}\\right)^2\r\n$$\r\nThe Harmonic Oscillator and Vibrational Spectroscopy\r\nwhere\r\n$$\r\n\\tilde{x}_{\\mathrm{e}}=\\frac{h c \\tilde{\\omega}_{\\mathrm{e}}}{4 D}\r\n$$\r\nGiven that $\\tilde{\\omega}_{\\mathrm{e}}=2886 \\mathrm{~cm}^{-1}$ and $D=440.2 \\mathrm{~kJ} \\cdot \\mathrm{mol}^{-1}$ for $\\mathrm{H}^{35} \\mathrm{Cl}$, calculate $\\tilde{x}_{\\mathrm{e}}$.",
+  "answer_latex": " 0.01961",
+  "answer_number": " 0.01961",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "5-12 ",
+  "comment": "only first part taken of the question "
+ },
+ {
+  "problem_text": " In the infrared spectrum of $\\mathrm{H}^{127} \\mathrm{I}$, there is an intense line at $2309 \\mathrm{~cm}^{-1}$. Calculate the force constant of $\\mathrm{H}^{127} \\mathrm{I}$.",
+  "answer_latex": "313",
+  "answer_number": "313",
+  "unit": " $ \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$",
+  "source": "chemmc",
+  "problemid": " 5-13",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the percentage difference between $e^x$ and $1+x$ for $x=0.0050$",
+  "answer_latex": " 1.25",
+  "answer_number": "1.25",
+  "unit": " $10^{-3} \\%$",
+  "source": "chemmc",
+  "problemid": "D-1",
+  "comment": "Math Part D (after chapter 4)"
+ },
+ {
+  "problem_text": "Calculate the kinetic energy of an electron in a beam of electrons accelerated by a voltage increment of $100 \\mathrm{~V}$",
+  "answer_latex": " 1.602",
+  "answer_number": "1.602",
+  "unit": " $10^{-17} \\mathrm{~J} \\cdot$ electron ${ }^{-1}$",
+  "source": "chemmc",
+  "problemid": "1-39 ",
+  "comment": " "
+ }
+]

chemmc_sol.json

@@ -1 +1,92 @@
-[{"problem_text": "Given that the work function for sodium metal is $2.28 \\mathrm{eV}$, what is the threshold frequency $v_0$ for sodium?", "answer_latex": " 5.51", "answer_number": "5.51", "unit": "$10^{14}\\mathrm{~Hz}$", "source": "chemmc", "problemid": " 1.1_3", "comment": " ", "solution": "We must first convert $\\phi$ from electron volts to joules.\r\n$$\r\n\\begin{aligned}\r\n\\phi & =2.28 \\mathrm{eV}=(2.28 \\mathrm{eV})\\left(1.602 \\times 10^{-19} \\mathrm{~J} \\cdot \\mathrm{eV}^{-1}\\right) \\\\\r\n& =3.65 \\times 10^{-19} \\mathrm{~J}\r\n\\end{aligned}\r\n$$\r\nUsing Equation 1.11, we have\r\n$$\r\nv_0=\\frac{3.65 \\times 10^{-19} \\mathrm{~J}}{6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}}=5.51 \\times 10^{14} \\mathrm{~Hz}$$"}, {"problem_text": "Calculate the de Broglie wavelength of an electron traveling at $1.00 \\%$ of the speed of light.", "answer_latex": " 243", "answer_number": "243", "unit": " $\\mathrm{pm}$", "source": "chemmc", "problemid": " 1.1_11", "comment": " ", "solution": "The mass of an electron is $9.109 \\times 10^{-31} \\mathrm{~kg}$. One percent of the speed of light is\r\n$$\r\nv=(0.0100)\\left(2.998 \\times 10^8 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\\right)=2.998 \\times 10^6 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\r\n$$\r\nThe momentum of the electron is given by\r\n$$\r\n\\begin{aligned}\r\np=m_{\\mathrm{e}} v & =\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right)\\left(2.998 \\times 10^6 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\\right) \\\\\r\n& =2.73 \\times 10^{-24} \\mathrm{~kg} \\cdot \\mathrm{m} \\cdot \\mathrm{s}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe de Broglie wavelength of this electron is\r\n$$\r\n\\begin{aligned}\r\n\\lambda=\\frac{h}{p} & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}}{2.73 \\times 10^{-24} \\mathrm{~kg} \\cdot \\mathrm{m} \\cdot \\mathrm{s}^{-1}}=2.43 \\times 10^{-10} \\mathrm{~m} \\\\\r\n& =243 \\mathrm{pm}\r\n\\end{aligned}\r\n$$\r\nThis wavelength is of atomic dimensions.\r\n"}, {"problem_text": "Show that $u(\\theta, \\phi)=Y_1^1(\\theta, \\phi)$ given in Example E-2 satisfies the equation $\\nabla^2 u=\\frac{c}{r^2} u$, where $c$ is a constant. What is the value of $c$ ?", "answer_latex": "-2 ", "answer_number": "-2", "unit": " ", "source": "chemmc", "problemid": " E.E_4", "comment": " ", "solution": "Because $u(\\theta, \\phi)$ is independent of $r$, we start with\r\n$$\r\n\\nabla^2 u=\\frac{1}{r^2 \\sin \\theta} \\frac{\\partial}{\\partial \\theta}\\left(\\sin \\theta \\frac{\\partial u}{\\partial \\theta}\\right)+\\frac{1}{r^2 \\sin ^2 \\theta} \\frac{\\partial^2 u}{\\partial \\phi^2}\r\n$$\r\nSubstituting\r\n$$\r\nu(\\theta, \\phi)=-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} e^{i \\phi} \\sin \\theta\r\n$$\r\ninto $\\nabla^2 u$ gives\r\n$$\r\n\\begin{aligned}\r\n\\nabla^2 u & =-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2}\\left[\\frac{e^{i \\phi}}{r^2 \\sin \\theta}\\left(\\cos ^2 \\theta-\\sin ^2 \\theta\\right)-\\frac{\\sin \\theta}{r^2 \\sin ^2 \\theta} e^{i \\phi}\\right] \\\\\r\n& =-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} \\frac{e^{i \\phi}}{r^2}\\left(\\frac{1-2 \\sin ^2 \\theta}{\\sin \\theta}-\\frac{1}{\\sin \\theta}\\right) \\\\\r\n& =2\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} \\frac{e^{i \\phi} \\sin \\theta}{r^2}\r\n\\end{aligned}\r\n$$\r\nor $c=-2$."}, {"problem_text": "The wave function $\\Psi_2(1,2)$ given by Equation 9.39 is not normalized as it stands. Determine the normalization constant of $\\Psi_2(1,2)$ given that the \"1s\" parts are normalized.", "answer_latex": " $1 / \\sqrt{2}$", "answer_number": "0.70710678", "unit": " ", "source": "chemmc", "problemid": "9.9_4 ", "comment": " ", "solution": "We want to find the constant $c$ such that\r\n$$\r\nI=c^2\\left\\langle\\Psi_2(1,2) \\mid \\Psi_2(1,2)\\right\\rangle=1\r\n$$\r\nFirst notice that $\\Psi_2(1,2)$ can be factored into the product of a spatial part and a spin part:\r\n$$\r\n\\begin{aligned}\r\n\\Psi_2(1,2) & =1 s(1) 1 s(2)[\\alpha(1) \\beta(2)-\\alpha(2) \\beta(1)] \\\\\r\n& =1 s\\left(\\mathbf{r}_1\\right) 1 s\\left(\\mathbf{r}_2\\right)\\left[\\alpha\\left(\\sigma_1\\right) \\beta\\left(\\sigma_2\\right)-\\alpha\\left(\\sigma_2\\right) \\beta\\left(\\sigma_1\\right)\\right]\r\n\\end{aligned}\r\n$$\r\nThe normalization integral becomes the product of three integrals:\r\n$$\r\nI=c^2\\langle 1 s(1) \\mid 1 s(1)\\rangle\\langle 1 s(2) \\mid 1 s(2)\\rangle\\langle\\alpha(1) \\beta(1)-\\alpha(2) \\beta(1) \\mid \\alpha(1) \\beta(2)-\\alpha(2) \\beta(1)\\rangle\r\n$$\r\nThe spatial integrals are equal to 1 because we have taken the $1 s$ orbitals to be normalized. Now let's look at the spin integrals. When the two terms in the integrand of the spin integral are multiplied, we get four integrals. One of them is\r\n$$\r\n\\begin{aligned}\r\n\\iint \\alpha^*\\left(\\sigma_1\\right) \\beta^*\\left(\\sigma_2\\right) \\alpha\\left(\\sigma_1\\right) \\beta\\left(\\sigma_2\\right) d \\sigma_1 d \\sigma_2 & =\\langle\\alpha(1) \\beta(2) \\mid \\alpha(1) \\beta(2)\\rangle \\\\\r\n& =\\langle\\alpha(1) \\mid \\alpha(1)\\rangle\\langle\\beta(2) \\mid \\beta(2)\\rangle=1\r\n\\end{aligned}\r\n$$\r\nwhere once again we point out that integrating over $\\sigma_1$ and $\\sigma_2$ is purely symbolic; $\\sigma_1$ and $\\sigma_2$ are discrete variables. Another is\r\n$$\r\n\\langle\\alpha(1) \\beta(2) \\mid \\alpha(2) \\beta(1)\\rangle=\\langle\\alpha(1) \\mid \\beta(1)\\rangle\\langle\\beta(2) \\mid \\alpha(2)\\rangle=0\r\n$$\r\nThe other two are equal to 1 and 0 , and so\r\n$$\r\nI=c^2\\left\\langle\\Psi_2(1,2) \\mid \\Psi_2(1,2)\\right\\rangle=2 c^2=1\r\n$$\r\nor $c=1 / \\sqrt{2}$."}, {"problem_text": "Find the bonding and antibonding H\u00fcckel molecular orbitals for ethene.", "answer_latex": " $1 / \\sqrt{2}$", "answer_number": "0.70710678", "unit": " ", "source": "chemmc", "problemid": "11.11_11 ", "comment": " ", "solution": "The equations for $c_1$ and $c_2$ associated with Equation 11.7 are\r\n$$\r\nc_1(\\alpha-E)+c_2 \\beta=0 \\quad \\text { and } \\quad c_1 \\beta+c_2(\\alpha-E)=0\r\n$$\r\nFor $E=\\alpha+\\beta$, either equation yields $c_1=c_2$. Thus,\r\n$$\r\n\\psi_{\\mathrm{b}}=c_1\\left(2 p_{z 1}+2 p_{z 2}\\right)\r\n$$\r\n\r\nThe value of $c_1$ can be found by requiring that the wave function be normalized. The normalization condition on $\\psi_\\pi$ gives $c_1^2(1+2 S+1)=1$. Using the H\u00fcckel assumption that $S=0$, we find that $c_1=1 / \\sqrt{2}$.\r\n\r\nSubstituting $E=\\alpha-\\beta$ into either of the equations for $c_1$ and $c_2$ yields $c_1=-c_2$, or\r\n$$\r\n\\psi_{\\mathrm{a}}=c_1\\left(2 p_{z 1}-2 p_{z 2}\\right)\r\n$$\r\nThe normalization condition gives $c^2(1-2 S+1)=1$, or $c_1=1 / \\sqrt{2}$.\r\n"}, {"problem_text": "Using the explicit formulas for the Hermite polynomials given in Table 5.3, show that a $0 \\rightarrow 1$ vibrational transition is allowed and that a $0 \\rightarrow 2$ transition is forbidden.", "answer_latex": "0 ", "answer_number": "0", "unit": " ", "source": "chemmc", "problemid": "5.5_12 ", "comment": " ", "solution": "Letting $\\xi=\\alpha^{1 / 2} x$ in Table 5.3 , we have\r\n$$\r\n\\begin{aligned}\r\n& \\psi_0(\\xi)=\\left(\\frac{\\alpha}{\\pi}\\right)^{1 / 4} e^{-\\xi^2 / 2} \\\\\r\n& \\psi_1(\\xi)=\\sqrt{2}\\left(\\frac{\\alpha}{\\pi}\\right)^{1 / 4} \\xi e^{-\\xi^2 / 2} \\\\\r\n& \\psi_2(\\xi)=\\frac{1}{\\sqrt{2}}\\left(\\frac{\\alpha}{\\pi}\\right)^{1 / 4}\\left(2 \\xi^2-1\\right) e^{-\\xi^2 / 2}\r\n\\end{aligned}\r\n$$\r\nThe dipole transition moment is given by the integral\r\n$$\r\nI_{0 \\rightarrow v} \\propto \\int_{-\\infty}^{\\infty} \\psi_v(\\xi) \\xi \\psi_0(\\xi) d \\xi\r\n$$\r\nThe transition is allowed if $I_{0 \\rightarrow v} \\neq 0$ and is forbidden if $I_{0 \\rightarrow v}=0$. For $v=1$, we have\r\n$$\r\nI_{0 \\rightarrow 1} \\propto\\left(\\frac{2 \\alpha}{\\pi}\\right)^{1 / 2} \\int_{-\\infty}^{\\infty} \\xi^2 e^{-\\xi^2} d \\xi \\neq 0\r\n$$\r\nbecause the integrand is everywhere positive. For $v=2$,\r\n$$\r\nI_{0 \\rightarrow 2} \\propto\\left(\\frac{\\alpha}{2 \\pi}\\right)^{1 / 2} \\int_{-\\infty}^{\\infty}\\left(2 \\xi^3-\\xi\\right) e^{-\\xi^2} d \\xi=0\r\n$$\r\nbecause the integrand is an odd function and the limits go from $-\\infty$ to $+\\infty$.\r\n"}, {"problem_text": "To a good approximation, the microwave spectrum of $\\mathrm{H}^{35} \\mathrm{Cl}$ consists of a series of equally spaced lines, separated by $6.26 \\times 10^{11} \\mathrm{~Hz}$. Calculate the bond length of $\\mathrm{H}^{35} \\mathrm{Cl}$.", "answer_latex": " 129", "answer_number": "129", "unit": " $\\mathrm{pm}$", "source": "chemmc", "problemid": " 6.6_2", "comment": " ", "solution": "According to Equation 6.30, the spacing of the lines in the microwave spectrum of $\\mathrm{H}^{35} \\mathrm{Cl}$ is given by\r\n$$\r\n2 B=\\frac{h}{4 \\pi^2 I}\r\n$$\r\n\r\n$$\r\n\\frac{h}{4 \\pi^2 I}=6.26 \\times 10^{11} \\mathrm{~Hz}\r\n$$\r\nSolving this equation for $I$, we have\r\n$$\r\nI=\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}}{4 \\pi^2\\left(6.26 \\times 10^{11} \\mathrm{~s}^{-1}\\right)}=2.68 \\times 10^{-47} \\mathrm{~kg} \\cdot \\mathrm{m}^2\r\n$$\r\nThe reduced mass of $\\mathrm{H}^{35} \\mathrm{Cl}$ is\r\n$$\r\n\\mu=\\frac{(1.00 \\mathrm{amu})(35.0 \\mathrm{amu})}{36.0 \\mathrm{amu}}\\left(1.661 \\times 10^{-27} \\mathrm{~kg} \\cdot \\mathrm{amu}^{-1}\\right)=1.66 \\times 10^{-27} \\mathrm{~kg}\r\n$$\r\nUsing the fact that $I=\\mu l^2$, we obtain\r\n$$\r\nl=\\left(\\frac{2.68 \\times 10^{-47} \\mathrm{~kg} \\cdot \\mathrm{m}^2}{1.661 \\times 10^{-27} \\mathrm{~kg}}\\right)^{1 / 2}=1.29 \\times 10^{-10} \\mathrm{~m}=129 \\mathrm{pm}\r\n$$\r\nProblems 6-5 through 6-8 give other examples of the determination of bond lengths from microwave data."}, {"problem_text": "The unit of energy in atomic units is given by\r\n$$\r\n1 E_{\\mathrm{h}}=\\frac{m_{\\mathrm{e}} e^4}{16 \\pi^2 \\epsilon_0^2 \\hbar^2}\r\n$$\r\nExpress $1 E_{\\mathrm{h}}$ in units of joules (J), kilojoules per mole $\\left(\\mathrm{kJ} \\cdot \\mathrm{mol}^{-1}\\right)$, wave numbers $\\left(\\mathrm{cm}^{-1}\\right)$, and electron volts $(\\mathrm{eV})$.", "answer_latex": "27.211 ", "answer_number": "27.211", "unit": " $\\mathrm{eV}$", "source": "chemmc", "problemid": "9.9_1 ", "comment": " ", "solution": "To find $1 E_{\\mathrm{h}}$ expressed in joules, we substitute the SI values of $m_{\\mathrm{e}}, e$, $4 \\pi \\epsilon_0$, and $\\hbar$ into the above equation. Using these values from Table 9.1, we find\r\nAtomic and Molecular Calculations Are Expressed in Atomic Units\r\n$$\r\n\\begin{aligned}\r\n1 E_{\\mathrm{h}} & =\\frac{\\left(9.1094 \\times 10^{-31} \\mathrm{~kg}\\right)\\left(1.6022 \\times 10^{-19} \\mathrm{C}\\right)^4}{\\left(1.1127 \\times 10^{-10} \\mathrm{C}^2 \\cdot \\mathrm{J}^{-1} \\cdot \\mathrm{m}^{-1}\\right)^2\\left(1.0546 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}\\right)^2} \\\\\r\n& =4.3597 \\times 10^{-18} \\mathrm{~J}\r\n\\end{aligned}\r\n$$\r\nIf we multiply this result by the Avogadro constant, we obtain\r\n$$\r\n1 E_{\\mathrm{h}}=2625.5 \\mathrm{~kJ} \\cdot \\mathrm{mol}^{-1}\r\n$$\r\nTo express $1 E_{\\mathrm{h}}$ in wave numbers $\\left(\\mathrm{cm}^{-1}\\right)$, we use the fact that $1 E_{\\mathrm{h}}=4.3597 \\times 10^{-18} \\mathrm{~J}$ along with the equation\r\n$$\r\n\\begin{aligned}\r\n\\tilde{v} & =\\frac{1}{\\lambda}=\\frac{h v}{h c}=\\frac{E}{c h}=\\frac{4.3597 \\times 10^{-18} \\mathrm{~J}}{\\left(2.9979 \\times 10^8 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\\right)\\left(6.6261 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}\\right)} \\\\\r\n& =2.1947 \\times 10^7 \\mathrm{~m}^{-1}=2.1947 \\times 10^5 \\mathrm{~cm}^{-1}\r\n\\end{aligned}\r\n$$\r\nso that we can write\r\n$$\r\n1 E_{\\mathrm{h}}=2.1947 \\times 10^5 \\mathrm{~cm}^{-1}\r\n$$\r\nLast, to express $1 E_{\\mathrm{h}}$ in terms of electron volts, we use the conversion factor\r\n$$\r\n1 \\mathrm{eV}=1.6022 \\times 10^{-19} \\mathrm{~J}\r\n$$\r\nUsing the value of $1 E_{\\mathrm{h}}$ in joules obtained previously, we have\r\n$$\r\n\\begin{aligned}\r\n1 E_{\\mathrm{h}} & =\\left(4.3597 \\times 10^{-18} \\mathrm{~J}\\right)\\left(\\frac{1 \\mathrm{eV}}{1.6022 \\times 10^{-19} \\mathrm{~J}}\\right) \\\\\r\n& =27.211 \\mathrm{eV}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Calculate the probability that a particle in a one-dimensional box of length $a$ is found between 0 and $a / 2$.", "answer_latex": "$\\frac{1}{2}$", "answer_number": "0.5", "unit": " ", "source": "chemmc", "problemid": "3.3_6 ", "comment": " ", "solution": "The probability that the particle will be found between 0 and $a / 2$ is\r\n$$\r\n\\operatorname{Prob}(0 \\leq x \\leq a / 2)=\\int_0^{a / 2} \\psi^*(x) \\psi(x) d x=\\frac{2}{a} \\int_0^{a / 2} \\sin ^2 \\frac{n \\pi x}{a} d x\r\n$$\r\nIf we let $n \\pi x / a$ be $z$, then we find\r\n\r\n$$\r\n\\begin{aligned}\r\n\\operatorname{Prob}(0 \\leq x \\leq a / 2) & =\\frac{2}{n \\pi} \\int_0^{n \\pi / 2} \\sin ^2 z d z=\\frac{2}{n \\pi}\\left|\\frac{z}{2}-\\frac{\\sin 2 z}{4}\\right|_0^{n \\pi / 2} \\\\\r\n& =\\frac{2}{n \\pi}\\left(\\frac{n \\pi}{4}-\\frac{\\sin n \\pi}{4}\\right)=\\frac{1}{2} \\quad \\text { (for all } n \\text { ) }\r\n\\end{aligned}\r\n$$\r\nThus, the probability that the particle lies in one-half of the interval $0 \\leq x \\leq a$ is $\\frac{1}{2}$."}]
+[
+ {
+  "problem_text": "Given that the work function for sodium metal is $2.28 \\mathrm{eV}$, what is the threshold frequency $v_0$ for sodium?",
+  "answer_latex": " 5.51",
+  "answer_number": "5.51",
+  "unit": "$10^{14}\\mathrm{~Hz}$",
+  "source": "chemmc",
+  "problemid": " 1.1_3",
+  "comment": " ",
+  "solution": "\nWe must first convert $\\phi$ from electron volts to joules.\r\n$$\r\n\\begin{aligned}\r\n\\phi & =2.28 \\mathrm{eV}=(2.28 \\mathrm{eV})\\left(1.602 \\times 10^{-19} \\mathrm{~J} \\cdot \\mathrm{eV}^{-1}\\right) \\\\\r\n& =3.65 \\times 10^{-19} \\mathrm{~J}\r\n\\end{aligned}\r\n$$\r\nUsing Equation $h v_0=\\phi$, we have\r\n$$\r\nv_0=\\frac{3.65 \\times 10^{-19} \\mathrm{~J}}{6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}}=5.51 \\times 10^{14} \\mathrm{~Hz}$$\n"
+ },
+ {
+  "problem_text": "Calculate the de Broglie wavelength of an electron traveling at $1.00 \\%$ of the speed of light.",
+  "answer_latex": " 243",
+  "answer_number": "243",
+  "unit": " $\\mathrm{pm}$",
+  "source": "chemmc",
+  "problemid": " 1.1_11",
+  "comment": " ",
+  "solution": "The mass of an electron is $9.109 \\times 10^{-31} \\mathrm{~kg}$. One percent of the speed of light is\r\n$$\r\nv=(0.0100)\\left(2.998 \\times 10^8 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\\right)=2.998 \\times 10^6 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\r\n$$\r\nThe momentum of the electron is given by\r\n$$\r\n\\begin{aligned}\r\np=m_{\\mathrm{e}} v & =\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right)\\left(2.998 \\times 10^6 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\\right) \\\\\r\n& =2.73 \\times 10^{-24} \\mathrm{~kg} \\cdot \\mathrm{m} \\cdot \\mathrm{s}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe de Broglie wavelength of this electron is\r\n$$\r\n\\begin{aligned}\r\n\\lambda=\\frac{h}{p} & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}}{2.73 \\times 10^{-24} \\mathrm{~kg} \\cdot \\mathrm{m} \\cdot \\mathrm{s}^{-1}}=2.43 \\times 10^{-10} \\mathrm{~m} \\\\\r\n& =243 \\mathrm{pm}\r\n\\end{aligned}\r\n$$\r\nThis wavelength is of atomic dimensions.\r\n"
+ },
+ {
+  "problem_text": "Show that $u(\\theta, \\phi)=Y_1^1(\\theta, \\phi)$ given in Example $$\n\\begin{aligned}\n&Y_1^1(\\theta, \\phi)=-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} e^{i \\phi} \\sin \\theta\\\\\n&Y_1^{-1}(\\theta, \\phi)=\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} e^{-i \\phi} \\sin \\theta\n\\end{aligned}\n$$ satisfies the equation $\\nabla^2 u=\\frac{c}{r^2} u$, where $c$ is a constant. What is the value of $c$ ?",
+  "answer_latex": "-2 ",
+  "answer_number": "-2",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": " E.E_4",
+  "comment": " ",
+  "solution": "Because $u(\\theta, \\phi)$ is independent of $r$, we start with\r\n$$\r\n\\nabla^2 u=\\frac{1}{r^2 \\sin \\theta} \\frac{\\partial}{\\partial \\theta}\\left(\\sin \\theta \\frac{\\partial u}{\\partial \\theta}\\right)+\\frac{1}{r^2 \\sin ^2 \\theta} \\frac{\\partial^2 u}{\\partial \\phi^2}\r\n$$\r\nSubstituting\r\n$$\r\nu(\\theta, \\phi)=-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} e^{i \\phi} \\sin \\theta\r\n$$\r\ninto $\\nabla^2 u$ gives\r\n$$\r\n\\begin{aligned}\r\n\\nabla^2 u & =-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2}\\left[\\frac{e^{i \\phi}}{r^2 \\sin \\theta}\\left(\\cos ^2 \\theta-\\sin ^2 \\theta\\right)-\\frac{\\sin \\theta}{r^2 \\sin ^2 \\theta} e^{i \\phi}\\right] \\\\\r\n& =-\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} \\frac{e^{i \\phi}}{r^2}\\left(\\frac{1-2 \\sin ^2 \\theta}{\\sin \\theta}-\\frac{1}{\\sin \\theta}\\right) \\\\\r\n& =2\\left(\\frac{3}{8 \\pi}\\right)^{1 / 2} \\frac{e^{i \\phi} \\sin \\theta}{r^2}\r\n\\end{aligned}\r\n$$\r\nor $c=-2$."
+ },
+ {
+  "problem_text": "The wave function $\\Psi_2(1,2)$ given by Equation 9.39 is not normalized as it stands. Determine the normalization constant of $\\Psi_2(1,2)$ given that the \"1s\" parts are normalized.",
+  "answer_latex": " $1 / \\sqrt{2}$",
+  "answer_number": "0.70710678",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "9.9_4 ",
+  "comment": " ",
+  "solution": "We want to find the constant $c$ such that\r\n$$\r\nI=c^2\\left\\langle\\Psi_2(1,2) \\mid \\Psi_2(1,2)\\right\\rangle=1\r\n$$\r\nFirst notice that $\\Psi_2(1,2)$ can be factored into the product of a spatial part and a spin part:\r\n$$\r\n\\begin{aligned}\r\n\\Psi_2(1,2) & =1 s(1) 1 s(2)[\\alpha(1) \\beta(2)-\\alpha(2) \\beta(1)] \\\\\r\n& =1 s\\left(\\mathbf{r}_1\\right) 1 s\\left(\\mathbf{r}_2\\right)\\left[\\alpha\\left(\\sigma_1\\right) \\beta\\left(\\sigma_2\\right)-\\alpha\\left(\\sigma_2\\right) \\beta\\left(\\sigma_1\\right)\\right]\r\n\\end{aligned}\r\n$$\r\nThe normalization integral becomes the product of three integrals:\r\n$$\r\nI=c^2\\langle 1 s(1) \\mid 1 s(1)\\rangle\\langle 1 s(2) \\mid 1 s(2)\\rangle\\langle\\alpha(1) \\beta(1)-\\alpha(2) \\beta(1) \\mid \\alpha(1) \\beta(2)-\\alpha(2) \\beta(1)\\rangle\r\n$$\r\nThe spatial integrals are equal to 1 because we have taken the $1 s$ orbitals to be normalized. Now let's look at the spin integrals. When the two terms in the integrand of the spin integral are multiplied, we get four integrals. One of them is\r\n$$\r\n\\begin{aligned}\r\n\\iint \\alpha^*\\left(\\sigma_1\\right) \\beta^*\\left(\\sigma_2\\right) \\alpha\\left(\\sigma_1\\right) \\beta\\left(\\sigma_2\\right) d \\sigma_1 d \\sigma_2 & =\\langle\\alpha(1) \\beta(2) \\mid \\alpha(1) \\beta(2)\\rangle \\\\\r\n& =\\langle\\alpha(1) \\mid \\alpha(1)\\rangle\\langle\\beta(2) \\mid \\beta(2)\\rangle=1\r\n\\end{aligned}\r\n$$\r\nwhere once again we point out that integrating over $\\sigma_1$ and $\\sigma_2$ is purely symbolic; $\\sigma_1$ and $\\sigma_2$ are discrete variables. Another is\r\n$$\r\n\\langle\\alpha(1) \\beta(2) \\mid \\alpha(2) \\beta(1)\\rangle=\\langle\\alpha(1) \\mid \\beta(1)\\rangle\\langle\\beta(2) \\mid \\alpha(2)\\rangle=0\r\n$$\r\nThe other two are equal to 1 and 0 , and so\r\n$$\r\nI=c^2\\left\\langle\\Psi_2(1,2) \\mid \\Psi_2(1,2)\\right\\rangle=2 c^2=1\r\n$$\r\nor $c=1 / \\sqrt{2}$."
+ },
+ {
+  "problem_text": "Find the bonding and antibonding H\u00fcckel molecular orbitals for ethene.",
+  "answer_latex": " $1 / \\sqrt{2}$",
+  "answer_number": "0.70710678",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "11.11_11 ",
+  "comment": " ",
+  "solution": "\nThe equations for $c_1$ and $c_2$ associated with Equation $$\n\\left|\\begin{array}{ll}\nH_{11}-E S_{11} & H_{12}-E S_{12} \\\\\nH_{12}-E S_{12} & H_{22}-E S_{22}\n\\end{array}\\right|=0\n$$ are\r\n$$\r\nc_1(\\alpha-E)+c_2 \\beta=0 \\quad \\text { and } \\quad c_1 \\beta+c_2(\\alpha-E)=0\r\n$$\r\nFor $E=\\alpha+\\beta$, either equation yields $c_1=c_2$. Thus,\r\n$$\r\n\\psi_{\\mathrm{b}}=c_1\\left(2 p_{z 1}+2 p_{z 2}\\right)\r\n$$\r\n\r\nThe value of $c_1$ can be found by requiring that the wave function be normalized. The normalization condition on $\\psi_\\pi$ gives $c_1^2(1+2 S+1)=1$. Using the H\u00fcckel assumption that $S=0$, we find that $c_1=1 / \\sqrt{2}$.\r\n\r\nSubstituting $E=\\alpha-\\beta$ into either of the equations for $c_1$ and $c_2$ yields $c_1=-c_2$, or\r\n$$\r\n\\psi_{\\mathrm{a}}=c_1\\left(2 p_{z 1}-2 p_{z 2}\\right)\r\n$$\r\nThe normalization condition gives $c^2(1-2 S+1)=1$, or $c_1=1 / \\sqrt{2}$.\n"
+ },
+ {
+  "problem_text": "Using the explicit formulas for the Hermite polynomials given in Table 5.3 as below $$\n\\begin{array}{ll}\nH_0(\\xi)=1 & H_1(\\xi)=2 \\xi \\\\\nH_2(\\xi)=4 \\xi^2-2 & H_3(\\xi)=8 \\xi^3-12 \\xi \\\\\nH_4(\\xi)=16 \\xi^4-48 \\xi^2+12 & H_5(\\xi)=32 \\xi^5-160 \\xi^3+120 \\xi\n\\end{array}\n$$, show that a $0 \\rightarrow 1$ vibrational transition is allowed and that a $0 \\rightarrow 2$ transition is forbidden.",
+  "answer_latex": "0 ",
+  "answer_number": "0",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "5.5_12 ",
+  "comment": " ",
+  "solution": "Letting $\\xi=\\alpha^{1 / 2} x$ in Table 5.3 , we have\r\n$$\r\n\\begin{aligned}\r\n& \\psi_0(\\xi)=\\left(\\frac{\\alpha}{\\pi}\\right)^{1 / 4} e^{-\\xi^2 / 2} \\\\\r\n& \\psi_1(\\xi)=\\sqrt{2}\\left(\\frac{\\alpha}{\\pi}\\right)^{1 / 4} \\xi e^{-\\xi^2 / 2} \\\\\r\n& \\psi_2(\\xi)=\\frac{1}{\\sqrt{2}}\\left(\\frac{\\alpha}{\\pi}\\right)^{1 / 4}\\left(2 \\xi^2-1\\right) e^{-\\xi^2 / 2}\r\n\\end{aligned}\r\n$$\r\nThe dipole transition moment is given by the integral\r\n$$\r\nI_{0 \\rightarrow v} \\propto \\int_{-\\infty}^{\\infty} \\psi_v(\\xi) \\xi \\psi_0(\\xi) d \\xi\r\n$$\r\nThe transition is allowed if $I_{0 \\rightarrow v} \\neq 0$ and is forbidden if $I_{0 \\rightarrow v}=0$. For $v=1$, we have\r\n$$\r\nI_{0 \\rightarrow 1} \\propto\\left(\\frac{2 \\alpha}{\\pi}\\right)^{1 / 2} \\int_{-\\infty}^{\\infty} \\xi^2 e^{-\\xi^2} d \\xi \\neq 0\r\n$$\r\nbecause the integrand is everywhere positive. For $v=2$,\r\n$$\r\nI_{0 \\rightarrow 2} \\propto\\left(\\frac{\\alpha}{2 \\pi}\\right)^{1 / 2} \\int_{-\\infty}^{\\infty}\\left(2 \\xi^3-\\xi\\right) e^{-\\xi^2} d \\xi=0\r\n$$\r\nbecause the integrand is an odd function and the limits go from $-\\infty$ to $+\\infty$.\r\n"
+ },
+ {
+  "problem_text": "To a good approximation, the microwave spectrum of $\\mathrm{H}^{35} \\mathrm{Cl}$ consists of a series of equally spaced lines, separated by $6.26 \\times 10^{11} \\mathrm{~Hz}$. Calculate the bond length of $\\mathrm{H}^{35} \\mathrm{Cl}$.",
+  "answer_latex": " 129",
+  "answer_number": "129",
+  "unit": " $\\mathrm{pm}$",
+  "source": "chemmc",
+  "problemid": " 6.6_2",
+  "comment": " ",
+  "solution": "\nAccording to Equation $$\n\\begin{aligned}\n&v_{\\mathrm{obs}}=2 B(J+1) \\quad J=0,1,2, \\ldots\\\\\n&B=\\frac{h}{8 \\pi^2 I}\n\\end{aligned}\n$$, the spacing of the lines in the microwave spectrum of $\\mathrm{H}^{35} \\mathrm{Cl}$ is given by\r\n$$\r\n2 B=\\frac{h}{4 \\pi^2 I}\r\n$$\r\n\r\n$$\r\n\\frac{h}{4 \\pi^2 I}=6.26 \\times 10^{11} \\mathrm{~Hz}\r\n$$\r\nSolving this equation for $I$, we have\r\n$$\r\nI=\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}}{4 \\pi^2\\left(6.26 \\times 10^{11} \\mathrm{~s}^{-1}\\right)}=2.68 \\times 10^{-47} \\mathrm{~kg} \\cdot \\mathrm{m}^2\r\n$$\r\nThe reduced mass of $\\mathrm{H}^{35} \\mathrm{Cl}$ is\r\n$$\r\n\\mu=\\frac{(1.00 \\mathrm{amu})(35.0 \\mathrm{amu})}{36.0 \\mathrm{amu}}\\left(1.661 \\times 10^{-27} \\mathrm{~kg} \\cdot \\mathrm{amu}^{-1}\\right)=1.66 \\times 10^{-27} \\mathrm{~kg}\r\n$$\r\nUsing the fact that $I=\\mu l^2$, we obtain\r\n$$\r\nl=\\left(\\frac{2.68 \\times 10^{-47} \\mathrm{~kg} \\cdot \\mathrm{m}^2}{1.661 \\times 10^{-27} \\mathrm{~kg}}\\right)^{1 / 2}=1.29 \\times 10^{-10} \\mathrm{~m}=129 \\mathrm{pm}\r\n$$\n"
+ },
+ {
+  "problem_text": "The unit of energy in atomic units is given by\r\n$$\r\n1 E_{\\mathrm{h}}=\\frac{m_{\\mathrm{e}} e^4}{16 \\pi^2 \\epsilon_0^2 \\hbar^2}\r\n$$\r\nExpress $1 E_{\\mathrm{h}}$ in electron volts $(\\mathrm{eV})$.",
+  "answer_latex": "27.211 ",
+  "answer_number": "27.211",
+  "unit": " $\\mathrm{eV}$",
+  "source": "chemmc",
+  "problemid": "9.9_1 ",
+  "comment": " ",
+  "solution": "\nTo find $1 E_{\\mathrm{h}}$ expressed in joules, we substitute the SI values of $m_{\\mathrm{e}}, e$, $4 \\pi \\epsilon_0$, and $\\hbar$ into the above equation. Using these values from Table $$$\n\\begin{array}{lll}\n\\hline \\text { Property } & \\text { Atomic unit } & \\text { SI equivalent } \\\\\n\\hline \\text { Mass } & \\text { Mass of an electron, } m_{\\mathrm{e}} & 9.1094 \\times 10^{-31} \\mathrm{~kg} \\\\\n\\text { Charge } & \\text { Charge on a proton, } e & 1.6022 \\times 10^{-19} \\mathrm{C} \\\\\n\\text { Angular momentum } & \\text { Planck constant divided by } 2 \\pi, \\hbar & 1.0546 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s} \\\\\n\\text { Length } & \\text { Bohr radius, } a_0=\\frac{4 \\pi \\epsilon_0 \\hbar^2}{m_{\\mathrm{e}} e^2} & 5.2918 \\times 10^{-11} \\mathrm{~m} \\\\\n\\text { Energy } & \\frac{m_{\\mathrm{e}} e^4}{16 \\pi^2 \\epsilon_0^2 \\hbar^2}=\\frac{e^2}{4 \\pi \\epsilon_0 a_0}=E_{\\mathrm{h}} & 4.3597 \\times 10^{-18} \\mathrm{~J} \\\\\n\\text { Permittivity } & \\kappa_0=4 \\pi \\epsilon_0 & 1.1127 \\times 10^{-10} \\mathrm{C}^2 \\cdot \\mathrm{J}^{-1} \\cdot \\mathrm{m}^{-1} \\\\\n\\hline\n\\end{array}\n$$, we find\r\nAtomic and Molecular Calculations Are Expressed in Atomic Units\r\n$$\r\n\\begin{aligned}\r\n1 E_{\\mathrm{h}} & =\\frac{\\left(9.1094 \\times 10^{-31} \\mathrm{~kg}\\right)\\left(1.6022 \\times 10^{-19} \\mathrm{C}\\right)^4}{\\left(1.1127 \\times 10^{-10} \\mathrm{C}^2 \\cdot \\mathrm{J}^{-1} \\cdot \\mathrm{m}^{-1}\\right)^2\\left(1.0546 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}\\right)^2} \\\\\r\n& =4.3597 \\times 10^{-18} \\mathrm{~J}\r\n\\end{aligned}\r\n$$\r\nIf we multiply this result by the Avogadro constant, we obtain\r\n$$\r\n1 E_{\\mathrm{h}}=2625.5 \\mathrm{~kJ} \\cdot \\mathrm{mol}^{-1}\r\n$$\r\nTo express $1 E_{\\mathrm{h}}$ in wave numbers $\\left(\\mathrm{cm}^{-1}\\right)$, we use the fact that $1 E_{\\mathrm{h}}=4.3597 \\times 10^{-18} \\mathrm{~J}$ along with the equation\r\n$$\r\n\\begin{aligned}\r\n\\tilde{v} & =\\frac{1}{\\lambda}=\\frac{h v}{h c}=\\frac{E}{c h}=\\frac{4.3597 \\times 10^{-18} \\mathrm{~J}}{\\left(2.9979 \\times 10^8 \\mathrm{~m} \\cdot \\mathrm{s}^{-1}\\right)\\left(6.6261 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}\\right)} \\\\\r\n& =2.1947 \\times 10^7 \\mathrm{~m}^{-1}=2.1947 \\times 10^5 \\mathrm{~cm}^{-1}\r\n\\end{aligned}\r\n$$\r\nso that we can write\r\n$$\r\n1 E_{\\mathrm{h}}=2.1947 \\times 10^5 \\mathrm{~cm}^{-1}\r\n$$\r\nLast, to express $1 E_{\\mathrm{h}}$ in terms of electron volts, we use the conversion factor\r\n$$\r\n1 \\mathrm{eV}=1.6022 \\times 10^{-19} \\mathrm{~J}\r\n$$\r\nUsing the value of $1 E_{\\mathrm{h}}$ in joules obtained previously, we have\r\n$$\r\n\\begin{aligned}\r\n1 E_{\\mathrm{h}} & =\\left(4.3597 \\times 10^{-18} \\mathrm{~J}\\right)\\left(\\frac{1 \\mathrm{eV}}{1.6022 \\times 10^{-19} \\mathrm{~J}}\\right) \\\\\r\n& =27.211 \\mathrm{eV}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Calculate the probability that a particle in a one-dimensional box of length $a$ is found between 0 and $a / 2$.",
+  "answer_latex": "$\\frac{1}{2}$",
+  "answer_number": "0.5",
+  "unit": " ",
+  "source": "chemmc",
+  "problemid": "3.3_6 ",
+  "comment": " ",
+  "solution": "The probability that the particle will be found between 0 and $a / 2$ is\r\n$$\r\n\\operatorname{Prob}(0 \\leq x \\leq a / 2)=\\int_0^{a / 2} \\psi^*(x) \\psi(x) d x=\\frac{2}{a} \\int_0^{a / 2} \\sin ^2 \\frac{n \\pi x}{a} d x\r\n$$\r\nIf we let $n \\pi x / a$ be $z$, then we find\r\n\r\n$$\r\n\\begin{aligned}\r\n\\operatorname{Prob}(0 \\leq x \\leq a / 2) & =\\frac{2}{n \\pi} \\int_0^{n \\pi / 2} \\sin ^2 z d z=\\frac{2}{n \\pi}\\left|\\frac{z}{2}-\\frac{\\sin 2 z}{4}\\right|_0^{n \\pi / 2} \\\\\r\n& =\\frac{2}{n \\pi}\\left(\\frac{n \\pi}{4}-\\frac{\\sin n \\pi}{4}\\right)=\\frac{1}{2} \\quad \\text { (for all } n \\text { ) }\r\n\\end{aligned}\r\n$$\r\nThus, the probability that the particle lies in one-half of the interval $0 \\leq x \\leq a$ is $\\frac{1}{2}$."
+ }
+]

class.json

@@ -1 +1,631 @@
-[{"problem_text": " An automobile with a mass of $1000 \\mathrm{~kg}$, including passengers, settles $1.0 \\mathrm{~cm}$ closer to the road for every additional $100 \\mathrm{~kg}$ of passengers. It is driven with a constant horizontal component of speed $20 \\mathrm{~km} / \\mathrm{h}$ over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are $5.0 \\mathrm{~cm}$ and $20 \\mathrm{~cm}$, respectively. The distance between the front and back wheels is $2.4 \\mathrm{~m}$. Find the amplitude of oscillation of the automobile, assuming it moves vertically as an undamped driven harmonic oscillator. Neglect the mass of the wheels and springs and assume that the wheels are always in contact with the road.\r\n", "answer_latex": " -0.16", "answer_number": "-0.16", "unit": " $ \\mathrm{~mm}$", "source": "class", "problemid": " Problem 3.40", "comment": " "}, {"problem_text": "Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.", "answer_latex": " $2 \\sqrt{2} \\sin \\frac{\\pi}{2 \\sqrt{2}}$", "answer_number": "2.534324263", "unit": "", "source": "class", "problemid": " Problem 6.14", "comment": " "}, {"problem_text": "A simple pendulum of length $b$ and bob with mass $m$ is attached to a massless support moving vertically upward with constant acceleration $a$. Determine the period for small oscillations.\r\n", "answer_latex": " $2 \\pi$", "answer_number": "6.283185307", "unit": " $\\sqrt{\\frac{b}{a+g}}$", "source": "class", "problemid": " Problem 7.14", "comment": " "}, {"problem_text": "In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \\mathrm{~km} / \\mathrm{hr}$ and dropped the bales from a height of $80 \\mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \\mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?", "answer_latex": " 210", "answer_number": "210", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": "Problem 2.6 ", "comment": " "}, {"problem_text": "Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to $1 / e$ of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.\r\n", "answer_latex": " $\\frac{8 \\pi}{\\sqrt{64 \\pi^2+1}}$", "answer_number": "0.9992093669", "unit": "", "source": "class", "problemid": " Problem 3.44", "comment": " "}, {"problem_text": "What is the minimum escape velocity of a spacecraft from the moon?", "answer_latex": " 2380", "answer_number": "2380", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 8.28", "comment": " "}, {"problem_text": "Find the value of the integral $\\int_S \\mathbf{A} \\cdot d \\mathbf{a}$, where $\\mathbf{A}=x \\mathbf{i}-y \\mathbf{j}+z \\mathbf{k}$ and $S$ is the closed surface defined by the cylinder $c^2=x^2+y^2$. The top and bottom of the cylinder are at $z=d$ and 0 , respectively.", "answer_latex": " $\\pi$", "answer_number": "3.141592", "unit": " $c^2 d$", "source": "class", "problemid": " Question 1.36", "comment": " "}, {"problem_text": "A rocket has an initial mass of $7 \\times 10^4 \\mathrm{~kg}$ and on firing burns its fuel at a rate of 250 $\\mathrm{kg} / \\mathrm{s}$. The exhaust velocity is $2500 \\mathrm{~m} / \\mathrm{s}$. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off? What is wrong with the design of this rocket?\r\n", "answer_latex": "25", "answer_number": "25", "unit": "$\\mathrm{~s}$ ", "source": "class", "problemid": " Problem 9.60", "comment": " "}, {"problem_text": "A spacecraft of mass $10,000 \\mathrm{~kg}$ is parked in a circular orbit $200 \\mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\\tau=24 \\mathrm{hr}$ )?", "answer_latex": " 2.57", "answer_number": "2.57", "unit": "$10^{11} \\mathrm{~J}$ ", "source": "class", "problemid": " Problem 8.42", "comment": " "}, {"problem_text": "A uniformly solid sphere of mass $M$ and radius $R$ is fixed a distance $h$ above a thin infinite sheet of mass density $\\rho_5$ (mass/area). With what force does the sphere attract the sheet?", "answer_latex": " $2\\pi$", "answer_number": "6.283185307", "unit": " $\\rho_s G M$", "source": "class", "problemid": " Problem 5.16", "comment": " "}, {"problem_text": "Consider a thin rod of length $l$ and mass $m$ pivoted about one end. Calculate the moment of inertia. ", "answer_latex": " $\\frac{1}{3}$", "answer_number": "0.33333333", "unit": " $m l^2$", "source": "class", "problemid": " Problem 11.4", "comment": " "}, {"problem_text": "A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown $0.9 \\mathrm{~s}$ to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?\r\n", "answer_latex": "13.2", "answer_number": "13.2", "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 2.4", "comment": " "}, {"problem_text": "A billiard ball of initial velocity $u_1$ collides with another billiard ball (same mass) initially at rest. The first ball moves off at $\\psi=45^{\\circ}$. For an elastic collision, what are the velocities of both balls after the collision? ", "answer_latex": " $\\frac{1}{\\sqrt{2}}$", "answer_number": "0.7071067812", "unit": " $u_1$", "source": "class", "problemid": " Problem 9.34", "comment": " "}, {"problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \\mathrm{~km} / \\mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\\mathrm{LAB}$ angle $\\psi=10^{\\circ}$, what is the final speed of the deuteron?", "answer_latex": "14.44", "answer_number": "14.44", "unit": "$\\mathrm{~km} / \\mathrm{s}$", "source": "class", "problemid": " Problem 9.22", "comment": " "}, {"problem_text": "A mass $m$ moves in one dimension and is subject to a constant force $+F_0$ when $x<0$ and to a constant force $-F_0$ when $x>0$. Describe the motion by constructing a phase diagram. Calculate the period of the motion in terms of $m, F_0$, and the amplitude $A$ (disregard damping) .", "answer_latex": " 4", "answer_number": "4", "unit": " $\\sqrt{\\frac{2 m A}{F_0}}$", "source": "class", "problemid": " Problem 3.8", "comment": " "}, {"problem_text": "A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears the water splash $4.021 \\mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \\mathrm{~m} / \\mathrm{s}$, find the height of the building, neglecting air resistance.", "answer_latex": " 71", "answer_number": "71", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": " Problem 2.30", "comment": " "}, {"problem_text": "A thin disk of mass $M$ and radius $R$ lies in the $(x, y)$ plane with the $z$-axis passing through the center of the disk. Calculate the gravitational potential $\\Phi(z)$.", "answer_latex": " -2", "answer_number": "-2", "unit": " $\\frac{G M}{R^2}\\left(\\sqrt{z^2+R^2}-z\\right)$", "source": "class", "problemid": " Problem 5.20", "comment": " "}, {"problem_text": "A steel ball of velocity $5 \\mathrm{~m} / \\mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?", "answer_latex": " $4.3$", "answer_number": "4.3", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 9.42", "comment": " "}, {"problem_text": "Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \\mathrm{~cm}$ and the mass be $200 \\mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence. ", "answer_latex": " 35.2", "answer_number": "35.2", "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 2.18", "comment": " "}, {"problem_text": "A child slides a block of mass $2 \\mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\\mathrm{m} / \\mathrm{s}$ and the block hits a spring with spring constant $6 \\mathrm{~N} / \\mathrm{m}$, what is the maximum compression of the spring? ", "answer_latex": " 2.3", "answer_number": "2.3", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": " Problem 2.26", "comment": " "}, {"problem_text": "If the field vector is independent of the radial distance within a sphere, find the function describing the density $\\rho=\\rho(r)$ of the sphere.", "answer_latex": " $\\frac{1}{2 \\pi}$", "answer_number": "0.1591549431", "unit": " $\\frac{C}{G r}$", "source": "class", "problemid": " Problem 5.2", "comment": " "}, {"problem_text": "An Earth satellite has a perigee of $300 \\mathrm{~km}$ and an apogee of $3,500 \\mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\\circ}$ around Earth from perigee?", "answer_latex": "1590", "answer_number": "1590", "unit": "$\\mathrm{~km}$ ", "source": "class", "problemid": " Problem 8.24", "comment": " "}, {"problem_text": "Two masses $m_1=100 \\mathrm{~g}$ and $m_2=200 \\mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \\mathrm{~N} / \\mathrm{m}$. Find the frequency of oscillatory motion for this system.", "answer_latex": " 2.74", "answer_number": "2.74", "unit": "$\\mathrm{rad} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 3.6", "comment": " "}, {"problem_text": "A particle moves with $v=$ const. along the curve $r=k(1+\\cos \\theta)$ (a cardioid). Find $\\ddot{\\mathbf{r}} \\cdot \\mathbf{e}_r=\\mathbf{a} \\cdot \\mathbf{e}_r$.", "answer_latex": "$-\\frac{3}{4}$", "answer_number": "-0.75", "unit": " $\\frac{v^2}{k}$", "source": "class", "problemid": " Problem 1.26", "comment": " "}, {"problem_text": "Calculate the minimum $\\Delta v$ required to place a satellite already in Earth's heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth). Consider only the gravitational attraction of the Sun. ", "answer_latex": " 5275", "answer_number": "5275", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 8.38", "comment": " "}, {"problem_text": "Find the ratio of the radius $R$ to the height $H$ of a right-circular cylinder of fixed volume $V$ that minimizes the surface area $A$.", "answer_latex": " $\\frac{1}{2}$", "answer_number": "0.5", "unit": " $H$", "source": "class", "problemid": " Problem 6.10", "comment": " "}, {"problem_text": "Find the dimension of the parallelepiped of maximum volume circumscribed by a sphere of radius $R$.", "answer_latex": " $\\frac{2}{\\sqrt{3}}$", "answer_number": "1.154700538", "unit": "$R$", "source": "class", "problemid": " Problem 6.8", "comment": " "}, {"problem_text": "A potato of mass $0.5 \\mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. (a) Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \\mathrm{~s}^{-1}$. (b) Find the maximum height of the potato if it has the same value of $k$, but it is initially shot directly upward with a student-made potato gun with an initial velocity of $120 \\mathrm{~m} / \\mathrm{s}$.", "answer_latex": " 1000", "answer_number": "1000", "unit": "$\\mathrm{~m} / \\mathrm{s}$  ", "source": "class", "problemid": " Problem 2.54", "comment": " "}, {"problem_text": "A particle of mass $m$ and velocity $u_1$ makes a head-on collision with another particle of mass $2 m$ at rest. If the coefficient of restitution is such to make the loss of total kinetic energy a maximum, what are the velocities $v_1$ after the collision?", "answer_latex": "  $\\frac{1}{3}$", "answer_number": "0.33333333", "unit": "$u_1$", "source": "class", "problemid": " Problem 9.32", "comment": " "}, {"problem_text": "The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?", "answer_latex": " -2", "answer_number": "-2", "unit": "m ", "source": "class", "problemid": "Problem 1.40 ", "comment": " "}, {"problem_text": "Shot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\\circ} \\mathrm{N}$, and the lead fell a distance of $27 \\mathrm{~m}$. In what direction and how far did the lead bullets land from the direct vertical?", "answer_latex": "2.26", "answer_number": "2.26", "unit": " $\\mathrm{~mm}$", "source": "class", "problemid": "Problem 10.22", "comment": " "}, {"problem_text": "Perform an explicit calculation of the time average (i.e., the average over one complete period) of the potential energy for a particle moving in an elliptical orbit in a central inverse-square-law force field. Express the result in terms of the force constant of the field and the semimajor axis of the ellipse. ", "answer_latex": "-1", "answer_number": "-1", "unit": " $\\frac{k}{a}$", "source": "class", "problemid": " Problem 8.4", "comment": " "}, {"problem_text": "A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \\mathrm{dyne} / \\mathrm{cm}$. The mass is displaced $3 \\mathrm{~cm}$ and released from rest. Calculate the natural frequency $\\nu_0$.", "answer_latex": " 6.9", "answer_number": "6.9", "unit": " $10^{-2} \\mathrm{~s}^{-1}$", "source": "class", "problemid": " Problem 3.2", "comment": " "}, {"problem_text": " Find the center of mass of a uniformly solid cone of base diameter $2a$ and height  $h$", "answer_latex": " $\\frac{3}{4}$", "answer_number": "0.75", "unit": "$h$", "source": "class", "problemid": " Problem 9.2", "comment": " "}, {"problem_text": "A particle is projected with an initial velocity $v_0$ up a slope that makes an angle $\\alpha$ with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for $v_0=2.4 \\mathrm{~m} / \\mathrm{s}$ and $\\alpha=26^{\\circ}$.", "answer_latex": "2", "answer_number": "2", "unit": "$\\frac{v_0}{g \\sin \\alpha}$", "source": "class", "problemid": " Problem 2.16", "comment": " "}, {"problem_text": " Use the function described in Example 4.3, $x_{n+1}=\\alpha x_n\\left(1-x_n^2\\right)$ where $\\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Make a plot of $x_n$ versus $n$ for the two starting values and determine the lowest value of $n$ for which the two values diverge by more than $30 \\%$.", "answer_latex": " 30", "answer_number": "30", "unit": " ", "source": "class", "problemid": " Problem 4.14", "comment": " "}, {"problem_text": "A gun fires a projectile of mass $10 \\mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \\mathrm{~m} / \\mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \\mathrm{~m}$ away? Compare the results with those for the case of no retardation.", "answer_latex": " 17.4", "answer_number": "17.4", "unit": "$^{\\circ}$ ", "source": "class", "problemid": "Problem 2.20 ", "comment": " "}, {"problem_text": "A spacecraft is placed in orbit $200 \\mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth. ", "answer_latex": " 3.23", "answer_number": "3.23", "unit": " $ \\mathrm{~km} / \\mathrm{s}$", "source": "class", "problemid": " Problem 8.30", "comment": " "}, {"problem_text": "Find the value of the integral $\\int_S(\\nabla \\times \\mathbf{A}) \\cdot d \\mathbf{a}$ if the vector $\\mathbf{A}=y \\mathbf{i}+z \\mathbf{j}+x \\mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \\geq 0$.", "answer_latex": "$-\\pi$", "answer_number": "-3.141592", "unit": "", "source": "class", "problemid": "Problem 1.38", "comment": " "}, {"problem_text": "A skier weighing $90 \\mathrm{~kg}$ starts from rest down a hill inclined at $17^{\\circ}$. He skis $100 \\mathrm{~m}$ down the hill and then coasts for $70 \\mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. ", "answer_latex": "0.18", "answer_number": "0.18", "unit": " ", "source": "class", "problemid": " Problem 2.24", "comment": " "}, {"problem_text": "Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\\operatorname{Sun}$ is $\\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\\beta<1$, show that the time the comet spends within the orbit of Earth is given by\r\n$$\r\n\\sqrt{2(1-\\beta)} \\cdot(1+2 \\beta) / 3 \\pi \\times 1 \\text { year }\r\n$$\r\nIf the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?", "answer_latex": " 76", "answer_number": "76", "unit": "$ \\text { days }$ ", "source": "class", "problemid": " Problem 8.12", "comment": " "}, {"problem_text": "An automobile drag racer drives a car with acceleration $a$ and instantaneous velocity $v$. The tires (of radius $r_0$ ) are not slipping. For what initial velocity in the rotating system will the hockey puck appear to be subsequently motionless in the fixed system? ", "answer_latex": " 0.5", "answer_number": "0.5", "unit": " $\\omega R$", "source": "class", "problemid": " Problem 10.4", "comment": " "}, {"problem_text": " A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\\circ} \\mathrm{S}$. If the shells are fired at $37^{\\circ}$ elevation with a speed of $800 \\mathrm{~m} / \\mathrm{s}$, by how much do the shells miss their target and in what direction? Ignore air resistance.", "answer_latex": " 260", "answer_number": "260", "unit": " $\\mathrm{~m}$", "source": "class", "problemid": " Problem 10.18", "comment": " "}, {"problem_text": "Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?\r\n", "answer_latex": " 9", "answer_number": "9", "unit": "$10^7 \\mathrm{yr}$ ", "source": "class", "problemid": " Problem 8.46", "comment": " "}, {"problem_text": "To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \\mathrm{~m} / \\mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?", "answer_latex": "273", "answer_number": "273", "unit": " $\\mathrm{~s}$", "source": "class", "problemid": " Problem 9.62", "comment": " "}, {"problem_text": "In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\\mathbf{u}_1$ and $\\mathbf{u}_2=\\alpha \\mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\\alpha$ is positive. ", "answer_latex": " $3 \\pm 2 \\sqrt{2}$", "answer_number": "5.828427125", "unit": " ", "source": "class", "problemid": " Problem 9.36", "comment": " "}, {"problem_text": "Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \\mathrm{~m} / \\mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \\mathrm{~kg}$, including a pressurized tank of mass $10 \\mathrm{~kg}$. The tank includes only $2 \\mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \\mathrm{~m} / \\mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?", "answer_latex": "11", "answer_number": "11", "unit": "$ \\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 9.12", "comment": " "}]
+[
+ {
+  "problem_text": " An automobile with a mass of $1000 \\mathrm{~kg}$, including passengers, settles $1.0 \\mathrm{~cm}$ closer to the road for every additional $100 \\mathrm{~kg}$ of passengers. It is driven with a constant horizontal component of speed $20 \\mathrm{~km} / \\mathrm{h}$ over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are $5.0 \\mathrm{~cm}$ and $20 \\mathrm{~cm}$, respectively. The distance between the front and back wheels is $2.4 \\mathrm{~m}$. Find the amplitude of oscillation of the automobile, assuming it moves vertically as an undamped driven harmonic oscillator. Neglect the mass of the wheels and springs and assume that the wheels are always in contact with the road.\r\n",
+  "answer_latex": " -0.16",
+  "answer_number": "-0.16",
+  "unit": " $ \\mathrm{~mm}$",
+  "source": "class",
+  "problemid": " Problem 3.40",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.",
+  "answer_latex": " $2 \\sqrt{2} \\sin \\frac{\\pi}{2 \\sqrt{2}}$",
+  "answer_number": "2.534324263",
+  "unit": "",
+  "source": "class",
+  "problemid": " Problem 6.14",
+  "comment": " "
+ },
+ {
+  "problem_text": "In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \\mathrm{~km} / \\mathrm{hr}$ and dropped the bales from a height of $80 \\mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \\mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?",
+  "answer_latex": " 210",
+  "answer_number": "210",
+  "unit": "$\\mathrm{~m}$ ",
+  "source": "class",
+  "problemid": "Problem 2.6 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to $1 / e$ of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.\r\n",
+  "answer_latex": " $\\frac{8 \\pi}{\\sqrt{64 \\pi^2+1}}$",
+  "answer_number": "0.9992093669",
+  "unit": "",
+  "source": "class",
+  "problemid": " Problem 3.44",
+  "comment": " "
+ },
+ {
+  "problem_text": "What is the minimum escape velocity of a spacecraft from the moon?",
+  "answer_latex": " 2380",
+  "answer_number": "2380",
+  "unit": "$\\mathrm{~m} / \\mathrm{s}$ ",
+  "source": "class",
+  "problemid": " Problem 8.28",
+  "comment": " "
+ },
+ {
+  "problem_text": "A rocket has an initial mass of $7 \\times 10^4 \\mathrm{~kg}$ and on firing burns its fuel at a rate of 250 $\\mathrm{kg} / \\mathrm{s}$. The exhaust velocity is $2500 \\mathrm{~m} / \\mathrm{s}$. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off?",
+  "answer_latex": "25",
+  "answer_number": "25",
+  "unit": "$\\mathrm{~s}$ ",
+  "source": "class",
+  "problemid": " Problem 9.60",
+  "comment": " "
+ },
+ {
+  "problem_text": "A spacecraft of mass $10,000 \\mathrm{~kg}$ is parked in a circular orbit $200 \\mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\\tau=24 \\mathrm{hr}$ )?",
+  "answer_latex": " 2.57",
+  "answer_number": "2.57",
+  "unit": "$10^{11} \\mathrm{~J}$ ",
+  "source": "class",
+  "problemid": " Problem 8.42",
+  "comment": " "
+ },
+ {
+  "problem_text": "A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown $0.9 \\mathrm{~s}$ to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?\r\n",
+  "answer_latex": "13.2",
+  "answer_number": "13.2",
+  "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ",
+  "source": "class",
+  "problemid": " Problem 2.4",
+  "comment": " "
+ },
+ {
+  "problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \\mathrm{~km} / \\mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\\mathrm{LAB}$ angle $\\psi=10^{\\circ}$, what is the final speed of the deuteron?",
+  "answer_latex": "14.44",
+  "answer_number": "14.44",
+  "unit": "$\\mathrm{~km} / \\mathrm{s}$",
+  "source": "class",
+  "problemid": " Problem 9.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears the water splash $4.021 \\mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \\mathrm{~m} / \\mathrm{s}$, find the height of the building, neglecting air resistance.",
+  "answer_latex": " 71",
+  "answer_number": "71",
+  "unit": "$\\mathrm{~m}$ ",
+  "source": "class",
+  "problemid": " Problem 2.30",
+  "comment": " "
+ },
+ {
+  "problem_text": "A steel ball of velocity $5 \\mathrm{~m} / \\mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?",
+  "answer_latex": " $4.3$",
+  "answer_number": "4.3",
+  "unit": "$\\mathrm{~m} / \\mathrm{s}$ ",
+  "source": "class",
+  "problemid": " Problem 9.42",
+  "comment": " "
+ },
+ {
+  "problem_text": "Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \\mathrm{~cm}$ and the mass be $200 \\mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence. ",
+  "answer_latex": " 35.2",
+  "answer_number": "35.2",
+  "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ",
+  "source": "class",
+  "problemid": " Problem 2.18",
+  "comment": " "
+ },
+ {
+  "problem_text": "A child slides a block of mass $2 \\mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\\mathrm{m} / \\mathrm{s}$ and the block hits a spring with spring constant $6 \\mathrm{~N} / \\mathrm{m}$, what is the maximum compression of the spring? ",
+  "answer_latex": " 2.3",
+  "answer_number": "2.3",
+  "unit": "$\\mathrm{~m}$ ",
+  "source": "class",
+  "problemid": " Problem 2.26",
+  "comment": " "
+ },
+ {
+  "problem_text": "An Earth satellite has a perigee of $300 \\mathrm{~km}$ and an apogee of $3,500 \\mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\\circ}$ around Earth from perigee?",
+  "answer_latex": "1590",
+  "answer_number": "1590",
+  "unit": "$\\mathrm{~km}$ ",
+  "source": "class",
+  "problemid": " Problem 8.24",
+  "comment": " "
+ },
+ {
+  "problem_text": "Two masses $m_1=100 \\mathrm{~g}$ and $m_2=200 \\mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \\mathrm{~N} / \\mathrm{m}$. Find the frequency of oscillatory motion for this system.",
+  "answer_latex": " 2.74",
+  "answer_number": "2.74",
+  "unit": "$\\mathrm{rad} \\cdot \\mathrm{s}^{-1}$ ",
+  "source": "class",
+  "problemid": " Problem 3.6",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the minimum $\\Delta v$ required to place a satellite already in Earth's heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth). Consider only the gravitational attraction of the Sun. ",
+  "answer_latex": " 5275",
+  "answer_number": "5275",
+  "unit": "$\\mathrm{~m} / \\mathrm{s}$ ",
+  "source": "class",
+  "problemid": " Problem 8.38",
+  "comment": " "
+ },
+ {
+  "problem_text": "A potato of mass $0.5 \\mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \\mathrm{~s}^{-1}$. ",
+  "answer_latex": " 1000",
+  "answer_number": "1000",
+  "unit": "$\\mathrm{~m} / \\mathrm{s}$  ",
+  "source": "class",
+  "problemid": " Problem 2.54",
+  "comment": " "
+ },
+ {
+  "problem_text": "The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?",
+  "answer_latex": " -2",
+  "answer_number": "-2",
+  "unit": "m ",
+  "source": "class",
+  "problemid": "Problem 1.40 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Shot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\\circ} \\mathrm{N}$, and the lead fell a distance of $27 \\mathrm{~m}$. How far did the lead bullets land from the direct vertical?",
+  "answer_latex": "2.26",
+  "answer_number": "2.26",
+  "unit": " $\\mathrm{~mm}$",
+  "source": "class",
+  "problemid": "Problem 10.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \\mathrm{dyne} / \\mathrm{cm}$. The mass is displaced $3 \\mathrm{~cm}$ and released from rest. Calculate the natural frequency $\\nu_0$.",
+  "answer_latex": " 6.9",
+  "answer_number": "6.9",
+  "unit": " $10^{-2} \\mathrm{~s}^{-1}$",
+  "source": "class",
+  "problemid": " Problem 3.2",
+  "comment": " "
+ },
+ {
+  "problem_text": "Use the function described in Example 4.3, $x_{n+1}=\\alpha x_n\\left(1-x_n^2\\right)$ where $\\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Determine the lowest value of $n$ for which the two values diverge by more than $30 \\%$.",
+  "answer_latex": " 30",
+  "answer_number": "30",
+  "unit": " ",
+  "source": "class",
+  "problemid": " Problem 4.14",
+  "comment": " "
+ },
+ {
+  "problem_text": "A gun fires a projectile of mass $10 \\mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \\mathrm{~m} / \\mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \\mathrm{~m}$ away? Compare the results with those for the case of no retardation.",
+  "answer_latex": " 17.4",
+  "answer_number": "17.4",
+  "unit": "$^{\\circ}$ ",
+  "source": "class",
+  "problemid": "Problem 2.20 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "A spacecraft is placed in orbit $200 \\mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth. ",
+  "answer_latex": " 3.23",
+  "answer_number": "3.23",
+  "unit": " $ \\mathrm{~km} / \\mathrm{s}$",
+  "source": "class",
+  "problemid": " Problem 8.30",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the value of the integral $\\int_S(\\nabla \\times \\mathbf{A}) \\cdot d \\mathbf{a}$ if the vector $\\mathbf{A}=y \\mathbf{i}+z \\mathbf{j}+x \\mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \\geq 0$.",
+  "answer_latex": "$-\\pi$",
+  "answer_number": "-3.141592",
+  "unit": "",
+  "source": "class",
+  "problemid": "Problem 1.38",
+  "comment": " "
+ },
+ {
+  "problem_text": "A skier weighing $90 \\mathrm{~kg}$ starts from rest down a hill inclined at $17^{\\circ}$. He skis $100 \\mathrm{~m}$ down the hill and then coasts for $70 \\mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. ",
+  "answer_latex": "0.18",
+  "answer_number": "0.18",
+  "unit": " ",
+  "source": "class",
+  "problemid": " Problem 2.24",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\\operatorname{Sun}$ is $\\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\\beta<1$, show that the time the comet spends within the orbit of Earth is given by\r\n$$\r\n\\sqrt{2(1-\\beta)} \\cdot(1+2 \\beta) / 3 \\pi \\times 1 \\text { year }\r\n$$\r\nIf the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?",
+  "answer_latex": " 76",
+  "answer_number": "76",
+  "unit": "$ \\text { days }$ ",
+  "source": "class",
+  "problemid": " Problem 8.12",
+  "comment": " "
+ },
+ {
+  "problem_text": "A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\\circ} \\mathrm{S}$. If the shells are fired at $37^{\\circ}$ elevation with a speed of $800 \\mathrm{~m} / \\mathrm{s}$, by how much do the shells miss their target?",
+  "answer_latex": " 260",
+  "answer_number": "260",
+  "unit": " $\\mathrm{~m}$",
+  "source": "class",
+  "problemid": " Problem 10.18",
+  "comment": " "
+ },
+ {
+  "problem_text": "Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?\r\n",
+  "answer_latex": " 9",
+  "answer_number": "9",
+  "unit": "$10^7 \\mathrm{yr}$ ",
+  "source": "class",
+  "problemid": " Problem 8.46",
+  "comment": " "
+ },
+ {
+  "problem_text": "To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \\mathrm{~m} / \\mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?",
+  "answer_latex": "273",
+  "answer_number": "273",
+  "unit": " $\\mathrm{~s}$",
+  "source": "class",
+  "problemid": " Problem 9.62",
+  "comment": " "
+ },
+ {
+  "problem_text": "In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\\mathbf{u}_1$ and $\\mathbf{u}_2=\\alpha \\mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\\alpha$ is positive. ",
+  "answer_latex": " $3 \\pm 2 \\sqrt{2}$",
+  "answer_number": "5.828427125",
+  "unit": " ",
+  "source": "class",
+  "problemid": " Problem 9.36",
+  "comment": " "
+ },
+ {
+  "problem_text": "Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \\mathrm{~m} / \\mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \\mathrm{~kg}$, including a pressurized tank of mass $10 \\mathrm{~kg}$. The tank includes only $2 \\mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \\mathrm{~m} / \\mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?",
+  "answer_latex": "11",
+  "answer_number": "11",
+  "unit": "$ \\mathrm{~m} / \\mathrm{s}$ ",
+  "source": "class",
+  "problemid": " Problem 9.12",
+  "comment": " "
+ },
+ {
+  "problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9$ km / s collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a LAB angle $\\psi = 10^\\circ$, the final speed of the deuteron is $v_d = 14.44$ km / s and the final speed of the neutron is $v_n = 5.18$ km / s. Another set of solutions for the final speed is $v_d = 5.12$ km / s for the deuteron and $v_n = 19.79$ km / s for the neutron. What is the maximum possible scattering angle of the deuteron?",
+  "answer_latex": "$74.8^\\circ$, $5.2^\\circ$",
+  "answer_number": "30",
+  "unit": "$^\\circ$",
+  "source": "class",
+  "problemid": "9.22 B. ",
+  "comment": "",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A steel ball of velocity $5$ m/s strikes a smooth, heavy steel plate at an angle of $30^\\circ$ from the normal. If the coefficient of restitution is 0.8, at what angle from the normal does the steel ball bounce off the plate?",
+  "answer_latex": "$36^\\circ$",
+  "answer_number": "36",
+  "unit": "$^\\circ$",
+  "source": "class",
+  "problemid": "9.42 B. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A string is set into motion by being struck at a point $L/4$ from one end by a triangular hammer. The initial velocity is greatest at $x = L/4$ and decreases linearly to zero at $x = 0$ and $x = L/2$. The region $L/2 \\leq x \\leq L$ is initially undisturbed. Determine the subsequent motion of the string. How many decibels down from the fundamental are the second harmonics?'",
+  "answer_latex": "4.4, 13.3",
+  "answer_number": "4.4",
+  "unit": " dB",
+  "source": "class",
+  "problemid": "13.6 ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \\mathrm{~km} / \\mathrm{hr}$ and dropped the bales from a height of $80 \\mathrm{~m}$ above the flat range. To not hit the cattle, what is the largest time error she could make while pushing the bales out of the airplane? Ignore air resistance.",
+  "answer_latex": "0.68 ",
+  "answer_number": "0.68",
+  "unit": "seconds ",
+  "source": "class",
+  "problemid": " 2.6 B.",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A free neutron is unstable and decays into a proton and an electron. How much energy other than the rest energies of the proton and electron is available if a neutron at rest decays? (This is an example of nuclear beta decay. Another particle, called a neutrino-- actually an antineutrino $\\bar v$ is also produced.)",
+  "answer_latex": " 0.8",
+  "answer_number": "0.8",
+  "unit": "$MeV$ ",
+  "source": "class",
+  "problemid": " 14.30",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglecting air resistance and assuming that the acceleration of gravity is constant, the launched object can reach 3700 km above the surface of Earth. If the object has a radius of $20$ cm and the air resistance is proportional to the square of the object's speed with $c_w = 0.2$, assuming the density of air is constant, the maximum height reached is 890 km. Now also include the fact that the acceleration of gravity decreases as the object soars above Earth. Find the height reached.",
+  "answer_latex": "950 ",
+  "answer_number": "950",
+  "unit": "km ",
+  "source": "class",
+  "problemid": "9.64 C. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "Calculate the effective gravitational field vector $g$ at Earth's surface at the poles. Take account of the difference in the equatorial (6378 km) and polar (6357 km) radius as well as the centrifugal force. How well does the result agree with the difference calculated with the result $g = 9.780356[1 + 0.0052885sin^2\\lambda - 0.0000059 sin^2 (2\\lambda )]$ $m/s^2$ where $\\lambda$ is the latitude?",
+  "answer_latex": " 9.832",
+  "answer_number": "9.832",
+  "unit": "$m/s^2$ ",
+  "source": "class",
+  "problemid": " 10.20",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "In nuclear and particle physics, momentum is usually quoted in $MeV / c$ to facilitate calculations. Calculate the kinetic energy of an electron if it has a momentum of $1000$ $MeV/c$",
+  "answer_latex": "$T_{electron} = 999.5$, $T_{proton} = 433$",
+  "answer_number": "999.5",
+  "unit": " $MeV$",
+  "source": "class",
+  "problemid": " 14.32",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A skier weighing $90$ kg starts from rest down a hill inclined at $17^\\circ$. He skis 100 m down the hill and then coasts for 70 m along level snow until he stops. Given a coefficient of kinetic friction of $\\mu_k = 0.18$, what velocity does the skier have at the bottom of the hill?",
+  "answer_latex": "15.6 ",
+  "answer_number": "15.6",
+  "unit": "$m/s$ ",
+  "source": "class",
+  "problemid": " 2.24 B.",
+  "comment": "Uses answer from part A. (coefficient of kinetic friction) ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A rocket starts from rest in free space by emitting mass. At what fraction of the initial mass is the momentum a maximum?",
+  "answer_latex": "$e^{-1}$ ",
+  "answer_number": "0.367879",
+  "unit": " ",
+  "source": "class",
+  "problemid": " 9.54",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A particle moves in a plane elliptical orbit described by the position vector $r = 2b \\sin \\omega ti + b \\cos \\omega tj$ \r\n\r\nWhat is the angle between $v$ and $a$ at time $t = \\frac{\\pi}{2\\omega}$ ?",
+  "answer_latex": "$90^\\circ$",
+  "answer_number": "90",
+  "unit": "$^\\circ$",
+  "source": "class",
+  "problemid": " 1.10 B.",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "An Earth satellite has a perigee of $300$ km and an apogee of $3,500$ km above Earth's surface. How far is the satellite above Earth when it has moved halfway from perigee to apogee?",
+  "answer_latex": "1900 ",
+  "answer_number": "1900",
+  "unit": "$km$ ",
+  "source": "class",
+  "problemid": "8.24 (b) ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "In a typical model rocket (Estes Alpha III) the Estes C6 solid rocket engine provides a total impulse of $8.5$ N-s. Assume the total rocket mass at launch is $54$ g and that it has a rocket engine of mass $20$ g that burns evenly for $1.5$ s. The rocket diameter is $24$ mm. Assume a constant burn rate of the propellent mass ($11$ g), a rocket exhaust speed $800$ m/s, vertical ascent, and drag coefficient $c_w = 0.75$. Take into account the change of rocket mass with time and omit the effect of gravity. The rocket's speed at burn out is 131 m/s. How far has the rocket traveled at that moment?",
+  "answer_latex": " 108",
+  "answer_number": "108",
+  "unit": "m ",
+  "source": "class",
+  "problemid": "9.66 B. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9$ km / s collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a LAB angle $\\psi = 10^\\circ$, what is the final speed of the neutron?",
+  "answer_latex": " 5.18",
+  "answer_number": "5.18",
+  "unit": "km / s ",
+  "source": "class",
+  "problemid": "9.22 A. ",
+  "comment": "9.22 A, but only the neutron (not deuteron, which has already been annotated) ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglecting air resistance and assuming that the acceleration of gravity is constant, the launched object can reach 3700 km above the surface of Earth. If the object has a radius of $20$ cm and the air resistance is proportional to the square of the object's speed with $c_w = 0.2$, determine the maximum height reached. Assume the density of air is constant.",
+  "answer_latex": " 890",
+  "answer_number": "890",
+  "unit": "km ",
+  "source": "class",
+  "problemid": "9.64 B. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglecting air resistance and assuming that the acceleration of gravity is constant, the launched object can reach 3700 km above the surface of Earth. If the object has a radius of $20$ cm and the air resistance is proportional to the square of the object's speed with $c_w = 0.2$, assuming the density of air is constant, the maximum height reached is 890 km. Including the fact that the acceleration of gravity decreases as the object soars above Earth, the height reached is 950 km. Now add the effects of the decrease in air density with altitude to the calculation. We can very roughly represent the air density by $log_{10}(\\rho) = -0.05h + 0.11$ where $\\rho$ is the air density in $kg/m^3$ and $h$ is the altitude above Earth in km. Determine how high the object now goes.",
+  "answer_latex": "8900 ",
+  "answer_number": "8900",
+  "unit": "km ",
+  "source": "class",
+  "problemid": "9.64 D. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A racer attempting to break the land speed record rockets by two markers spaced $100$ m apart on the ground in a time of $0.4$ $\\mu s$ as measured by an observer on the ground. How far apart do the two markers appear to the racer? ",
+  "answer_latex": "55.3 ",
+  "answer_number": "55.3",
+  "unit": "$m$ ",
+  "source": "class",
+  "problemid": " 14.12",
+  "comment": "Omitted the last two questions / answers ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A billiard ball of intial velocity $u_1$ collides with another billard ball (same mass) initially at rest. The first ball moves off at $\\psi = 45^\\circ$. For an elastic collision, say the velocities of both balls after the collision is $v_1 = v_2 = \\frac{u_1}{\\sqrt(2)}$. At what LAB angle does the second ball emerge?",
+  "answer_latex": "45 ",
+  "answer_number": "45",
+  "unit": "$^\\circ$",
+  "source": "class",
+  "problemid": "9.34 B. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "Calculate the effective gravitational field vector $\\textbf{g}$ at Earth's surface at the equator. Take account of the difference in the equatorial (6378 km) and polar (6357 km) radius as well as the centrifugal force. ",
+  "answer_latex": " 9.780",
+  "answer_number": "9.780",
+  "unit": "$m/s^2$ ",
+  "source": "class",
+  "problemid": " 10.20 B.",
+  "comment": "Only calculating the equator, not the polar.",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "An astronaut travels to the nearest star system, 4 light years away, and returns at a speed $0.3c$. How much has the astronaut aged relative to those people remaining on Earth?",
+  "answer_latex": "25.4",
+  "answer_number": "25.4",
+  "unit": "years ",
+  "source": "class",
+  "problemid": " 14.20",
+  "comment": "Astronaut ages 25.4, people on Earth age 26.7 ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "In a typical model rocket (Estes Alpha III) the Estes C6 solid rocket engine provides a total impulse of $8.5$ N-s. Assume the total rocket mass at launch is $54$ g and that it has a rocket engine of mass $20$ g that burns evenly for $1.5$ s. The rocket diameter is $24$ mm. Assume a constant burn rate of the propellent mass ($11$ g), a rocket exhaust speed $800$ m/s, vertical ascent, and drag coefficient $c_w = 0.75$. Take into account the change of rocket mass with time and omit the effect of gravity. Find the rocket's speed at burn out.",
+  "answer_latex": "131 ",
+  "answer_number": "131",
+  "unit": "m/s ",
+  "source": "class",
+  "problemid": "9.66 A. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of $4000$ m/s. The total mass of the rocket is $10^5$ kg, with $90$% of its mass being fuel. The fuel burns quickly in $100$ s at a constant rate. For testing purposes, the rocket is launched vertically at rest from Earth's surface. Neglect air resistance and assume that the acceleration of gravity is constant. Determine how high the launched object can reach above the surface of Earth.",
+  "answer_latex": "3700 ",
+  "answer_number": "3700",
+  "unit": "km ",
+  "source": "class",
+  "problemid": "9.64 A. ",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_w = 0.5$, the softball radius be $5$ cm and the mass be $200$ g. Given a speed of 35.2 m/s, find the initial elevation angle that allows the ball to most easily clear the fence.",
+  "answer_latex": " $40.7^\\circ$",
+  "answer_number": "40.7",
+  "unit": "$^\\circ$",
+  "source": "class",
+  "problemid": " 2.18 B.",
+  "comment": "Used answer given in part A. for speed",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "true",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "Show that the small angular deviation of $\\epsilon$ of a plumb line from the true vertical (i.e., toward the center of Earth) at a point on Earth's surface at a latitude $\\lambda$ is $\\epsilon = \\frac{R\\omega^2sin\\lambda cos\\lambda}{g_0 - R\\omega^2 cos^2\\lambda}$ where R is the radius of Earth. What is the value (in seconds of arc) of the maximum deviation? Note that the entire denominator in the answer is actually the effective $g$, and $g_0$ denotes the pure gravitational component.",
+  "answer_latex": "6",
+  "answer_number": "6",
+  "unit": "min",
+  "source": "class",
+  "problemid": " 10.12",
+  "comment": " ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ },
+ {
+  "problem_text": "A potato of mass 0.5 kg moves under Earth's gravity with an air resistive force of -$kmv$. The terminal velocity of the potato when released from rest is $v = 1000$ m/s, with $k=0.01s^{-1}$. Find the maximum height of the potato if it has the same value of k, but it is initially shot directly upward with a student-made potato gun with an initial velocity of $120$ m/s.",
+  "answer_latex": "680 ",
+  "answer_number": "680",
+  "unit": "$m$ ",
+  "source": "class",
+  "problemid": " 2.54 B.",
+  "comment": "Used terminal velocity answer of 1000 m/s from part A. ",
+  "uid": "James",
+  "submit": "Submit",
+  "use_stored_img1": "true",
+  "use_stored_img2": "false",
+  "use_stored_img3": "true"
+ }
+]

class_sol.json

@@ -1 +1,72 @@
-[{"problem_text": "If the coefficient of static friction between the block and plane in the previous example is $\\mu_s=0.4$, at what angle $\\theta$ will the block start sliding if it is initially at rest?", "answer_latex": "22 ", "answer_number": "22", "unit": " $^{\\circ}$", "source": "class", "problemid": " 2.2", "comment": " ", "solution": "We need a new sketch to indicate the additional frictional force $f$ (see Figure 2-2b). The static frictional force has the approximate maximum value\r\n$$\r\nf_{\\max }=\\mu_s N\r\n$$\r\nand Equation 2.7 becomes, in component form, $y$-direction\r\n$$\r\n-F_g \\cos \\theta+N=0\r\n$$\r\n$x$-direction\r\n$$\r\n-f_s+F_g \\sin \\theta=m \\ddot{x}\r\n$$\r\nThe static frictional force $f_s$ will be some value $f_s \\leq f_{\\max }$ required to keep $\\ddot{x}=0$ -that is, to keep the block at rest. However, as the angle $\\theta$ of the plane increases, eventually the static frictional force will be unable to keep the block at rest. At that angle $\\theta^{\\prime}, f_s$ becomes\r\n$$\r\nf_s\\left(\\theta=\\theta^{\\prime}\\right)=f_{\\max }=\\mu_s N=\\mu_s F_g \\cos \\theta\r\n$$\r\nand\r\n$$\r\n\\begin{aligned}\r\nm \\ddot{x} & =F_g \\sin \\theta-f_{\\max } \\\\\r\nm \\ddot{x} & =F_g \\sin \\theta-\\mu_s F_g \\cos \\theta \\\\\r\n\\ddot{x} & =g\\left(\\sin \\theta-\\mu_s \\cos \\theta\\right)\r\n\\end{aligned}\r\n$$\r\nJust before the block starts to slide, the acceleration $\\ddot{x}=0$, so\r\n$$\r\n\\begin{aligned}\r\n\\sin \\theta-\\mu_s \\cos \\theta & =0 \\\\\r\n\\tan \\theta=\\mu_s & =0.4 \\\\\r\n\\theta=\\tan ^{-1}(0.4) & =22^{\\circ}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Halley's comet, which passed around the sun early in 1986, moves in a highly elliptical orbit with an eccentricity of 0.967 and a period of 76 years. Calculate its minimum  distances from the Sun.", "answer_latex": " 8.8", "answer_number": "8.8", "unit": " $10^{10} \\mathrm{m}$", "source": "class", "problemid": " 8.4", "comment": " ", "solution": "Equation 8.49 relates the period of motion with the semimajor axes. Because $m$ (Halley's comet) $\\ll m_{\\text {Sun }}$\r\n$$\r\n\\begin{aligned}\r\na & =\\left(\\frac{G m_{\\text {Sun }} \\tau^2}{4 \\pi^2}\\right)^{1 / 3} \\\\\r\n& =\\left[\\frac{\\left.\\left(6.67 \\times 10^{-11} \\frac{\\mathrm{Nm}^2}{\\mathrm{~kg}^2}\\right)\\left(1.99 \\times 10^{30} \\mathrm{~kg}\\right)\\left(76 \\mathrm{yr} \\frac{365 \\mathrm{day}}{\\mathrm{yr}} \\frac{24 \\mathrm{hr}}{\\mathrm{day}} \\frac{3600 \\mathrm{~s}}{\\mathrm{hr}}\\right)^2\\right]}{4 \\pi^2}\\right]^{1 / 3} \\\\\r\na & =2.68 \\times 10^{12} \\mathrm{m}\r\n\\end{aligned}\r\n$$\r\nUsing Equation 8.44 , we can determine $r_{\\min }$ and $r_{\\max }$\r\n$$\r\n\\begin{aligned}\r\n& r_{\\min }=2.68 \\times 10^{12} \\mathrm{~m}(1-0.967)=8.8 \\times 10^{10} \\mathrm{~m} \\\\\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Next, we treat projectile motion in two dimensions, first without considering air resistance. Let the muzzle velocity of the projectile be $v_0$ and the angle of elevation be $\\theta$ (Figure 2-7). Calculate the projectile's range.", "answer_latex": " 72", "answer_number": "72", "unit": " $\\mathrm{km}$", "source": "class", "problemid": " 2.6", "comment": " ", "solution": "Next, we treat projectile motion in two dimensions, first without considering air resistance. Let the muzzle velocity of the projectile be $v_0$ and the angle of elevation be $\\theta$ (Figure 2-7). Calculate the projectile's displacement, velocity, and range.\r\nSolution. Using $\\mathbf{F}=m \\mathrm{~g}$, the force components become\r\n$x$-direction\r\n$$\r\n0=m \\ddot{x}\r\n$$\r\ny-direction\r\n$-m g=m \\ddot{y}$\r\n$(2.31 b)$\r\n64\r\n2 / NEWTONIAN MECHANICS-SINGLE PARTICLE\r\nFIGURE 2-7 Example 2.6.\r\nNeglect the height of the gun, and assume $x=y=0$ at $t=0$. Then\r\n$$\r\n\\begin{aligned}\r\n& \\ddot{x}=0 \\\\\r\n& \\dot{x}=v_0 \\cos \\theta \\\\\r\n& x=v_0 t \\cos \\theta \\\\\r\n& y=-\\frac{-g t^2}{2}+v_0 t \\sin \\theta \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nand\r\n$$\r\n\\begin{aligned}\r\n& \\ddot{y}=-g \\\\\r\n& \\dot{y}=-g t+v_0 \\sin \\theta \\\\\r\n& y=\\frac{-g t^2}{2}+v_0 t \\sin \\theta\r\n\\end{aligned}\r\n$$\r\n\r\nWe can find the range by determining the value of $x$ when the projectile falls back to ground, that is, when $y=0$.\r\n$$\r\ny=t\\left(\\frac{-g t}{2}+v_0 \\sin \\theta\\right)=0\r\n$$\r\nOne value of $y=0$ occurs for $t=0$ and the other one for $t=T$.\r\n$$\r\n\\begin{aligned}\r\n\\frac{-g T}{2}+v_0 \\sin \\theta & =0 \\\\\r\nT & =\\frac{2 v_0 \\sin \\theta}{g}\r\n\\end{aligned}\r\n$$\r\n2.4 THE EQUATION OF MOTION FOR A PARTICLE\r\n65\r\nThe range $R$ is found from\r\n$$\r\n\\begin{aligned}\r\nx(t=T) & =\\text { range }=\\frac{2 v_0^2}{g} \\sin \\theta \\cos \\theta \\\\\r\nR & =\\text { range }=\\frac{v_0^2}{g} \\sin 2 \\theta\r\n\\end{aligned}\r\n$$\r\nNotice that the maximum range occurs for $\\theta=45^{\\circ}$.\r\nLet us use some actual numbers in these calculations. The Germans used a long-range gun named Big Bertha in World War I to bombard Paris. Its muzzle velocity was $1,450 \\mathrm{~m} / \\mathrm{s}$. Find its predicted range, maximum projectile height, and projectile time of flight if $\\theta=55^{\\circ}$. We have $v_0=1450 \\mathrm{~m} / \\mathrm{s}$ and $\\theta=55^{\\circ}$, so the range (from Equation 2.39) becomes\r\n$$\r\nR=\\frac{(1450 \\mathrm{~m} / \\mathrm{s})^2}{9.8 \\mathrm{~m} / \\mathrm{s}^2}\\left[\\sin \\left(110^{\\circ}\\right)\\right]=202 \\mathrm{~km}\r\n$$\r\nBig Bertha's actual range was $120 \\mathrm{~km}$. The difference is a result of the real effect of air resistance.\r\n\r\nTo find the maximum predicted height, we need to calculated $y$ for the time $T / 2$ where $T$ is the projectile time of flight:\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{(2)(1450 \\mathrm{~m} / \\mathrm{s})\\left(\\sin 55^{\\circ}\\right)}{9.8 \\mathrm{~m} / \\mathrm{s}^2}=242 \\mathrm{~s} \\\\\r\ny_{\\max }\\left(t=\\frac{T}{2}\\right) & =\\frac{-g T^2}{8}+\\frac{v_0 T}{2} \\sin \\theta \\\\\r\n& =\\frac{-(9.8 \\mathrm{~m} / \\mathrm{s})(242 \\mathrm{~s})^2}{8}+\\frac{(1450 \\mathrm{~m} / \\mathrm{s})(242 \\mathrm{~s}) \\sin \\left(55^{\\circ}\\right)}{2} \\\\\r\n& =72 \\mathrm{~km}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Calculate the time needed for a spacecraft to make a Hohmann transfer from Earth to Mars", "answer_latex": " 2.24", "answer_number": "2.24", "unit": " $10^7 \\mathrm{~s}$", "source": "class", "problemid": " 8.5", "comment": " ", "solution": "We need to insert the appropriate constants in Equation 8.58.\r\n$$\r\n\\begin{aligned}\r\n\\frac{m}{k} & =\\frac{m}{G m M_{\\text {Sun }}}=\\frac{1}{G M_{\\text {Sun }}} \\\\\r\n& =\\frac{1}{\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{s}^2 \\cdot \\mathrm{kg}\\right)\\left(1.99 \\times 10^{30} \\mathrm{~kg}\\right)} \\\\\r\n& =7.53 \\times 10^{-21} \\mathrm{~s}^2 / \\mathrm{m}^3\r\n\\end{aligned}\r\n$$\r\nBecause $k / m$ occurs so often in solar system calculations, we write it as well.\r\n$$\r\n\\begin{aligned}\r\n\\frac{k}{m} & =1.33 \\times 10^{20} \\mathrm{~m}^3 / \\mathrm{s}^2 \\\\\r\na_t & =\\frac{1}{2}\\left(r_{\\text {Earth-Sun }}+r_{\\text {Mars-Sun }}\\right) \\\\\r\n& =\\frac{1}{2}\\left(1.50 \\times 10^{11} \\mathrm{~m}+2.28 \\times 10^{11} \\mathrm{~m}\\right) \\\\\r\n& =1.89 \\times 10^{11} \\mathrm{~m} \\\\\r\nT_t & =\\pi\\left(7.53 \\times 10^{-21} \\mathrm{~s}^2 / \\mathrm{m}^3\\right)^{1 / 2}\\left(1.89 \\times 10^{11} \\mathrm{~m}\\right)^{3 / 2} \\\\\r\n& =2.24 \\times 10^7 \\mathrm{~s} \\\\\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Calculate the maximum height change in the ocean tides caused by the Moon.", "answer_latex": " 0.54", "answer_number": "0.54", "unit": "$\\mathrm{m}$", "source": "class", "problemid": " 5.5", "comment": " ", "solution": " We continue to use our simple model of the ocean surrounding Earth. Newton proposed a solution to this calculation by imagining that two wells be dug, one along the direction of high tide (our $x$-axis) and one along the direction of low tide (our $y$-axis). If the tidal height change we want to determine is $h$, then the difference in potential energy of mass $m$ due to the height difference is $m g h$. Let's calculate the difference in work if we move the mass $m$ from point $c$ in Figure 5-12 to the center of Earth and then to point $a$. This work $W$ done by gravity must equal the potential energy change $m g h$. The work $W$ is\r\n$$\r\nW=\\int_{r+\\delta_1}^0 F_{T_y} d y+\\int_0^{r+\\delta_2} F_{T_x} d x\r\n$$\r\nwhere we use the tidal forces $F_{T_y}$ and $F_{T x}$ of Equations 5.54. The small distances $\\delta_1$ and $\\delta_2$ are to account for the small variations from a spherical Earth, but these values are so small they can be henceforth neglected. The value for $W$ becomes\r\n$$\r\n\\begin{aligned}\r\nW & =\\frac{G m M_m}{D^3}\\left[\\int_r^0(-y) d y+\\int_0^r 2 x d x\\right] \\\\\r\n& =\\frac{G m M_m}{D^3}\\left(\\frac{r^2}{2}+r^2\\right)=\\frac{3 G m M_m r^2}{2 D^3}\r\n\\end{aligned}\r\n$$\r\nBecause this work is equal to $m g h$, we have\r\n$$\r\n\\begin{aligned}\r\nm g h & =\\frac{3 G m M_m r^2}{2 D^3} \\\\\r\nh & =\\frac{3 G M_m r^2}{2 g D^3}\r\n\\end{aligned}\r\n$$\r\nNote that the mass $m$ cancels, and the value of $h$ does not depend on $m$. Nor does it depend on the substance, so to the extent Earth is plastic, similar tidal effects should be (and are) observed for the surface land. If we insert the known values of the constants into Equation 5.55, we find\r\n$$\r\nh=\\frac{3\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{kg} \\cdot \\mathrm{s}^2\\right)\\left(7.350 \\times 10^{22} \\mathrm{~kg}\\right)\\left(6.37 \\times 10^6 \\mathrm{~m}\\right)^2}{2\\left(9.80 \\mathrm{~m} / \\mathrm{s}^2\\right)\\left(3.84 \\times 10^8 \\mathrm{~m}\\right)^3}=0.54 \\mathrm{~m}\r\n$$\r\n"}, {"problem_text": "A particle of mass $m$ starts at rest on top of a smooth fixed hemisphere of radius $a$. Determine the angle at which the particle leaves the hemisphere.", "answer_latex": " $\\cos ^{-1}\\left(\\frac{2}{3}\\right)$", "answer_number": "48.189685", "unit": "$^\\circ$", "source": "class", "problemid": " 7.10", "comment": " ", "solution": "Because we are considering the possibility of the particle leaving the hemisphere, we choose the generalized coordinates to be $r$ and $\\theta$. The constraint equation is\r\n$$\r\nf(r, \\theta)=r-a=0\r\n$$\r\nThe Lagrangian is determined from the kinetic and potential energies:\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{m}{2}\\left(\\dot{r}^2+r^2 \\dot{\\theta}^2\\right) \\\\\r\nU & =m g r \\cos \\theta \\\\\r\nL & =T-U \\\\\r\nL & =\\frac{m}{2}\\left(\\dot{r}^2+r^2 \\dot{\\theta}^2\\right)-m g r \\cos \\theta\r\n\\end{aligned}\r\n$$\r\nwhere the potential energy is zero at the bottom of the hemisphere. The Lagrange equations, Equation 7.65, are\r\n$$\r\n\\begin{aligned}\r\n& \\frac{\\partial L}{\\partial r}-\\frac{d}{d t} \\frac{\\partial L}{\\partial \\dot{r}}+\\lambda \\frac{\\partial f}{\\partial r}=0 \\\\\r\n& \\frac{\\partial L}{\\partial \\theta}-\\frac{d}{d t} \\frac{\\partial L}{\\partial \\dot{\\theta}}+\\lambda \\frac{\\partial f}{\\partial \\theta}=0\r\n\\end{aligned}\r\n$$\r\nPerforming the differentiations on Equation 7.80 gives\r\n$$\r\n\\frac{\\partial f}{\\partial r}=1, \\quad \\frac{\\partial f}{\\partial \\theta}=0\r\n$$\r\nEquations 7.82 and 7.83 become\r\n$$\r\n\\begin{gathered}\r\nm r \\dot{\\theta}^2-m g \\cos \\theta-m \\ddot{r}+\\lambda=0 \\\\\r\nm g r \\sin \\theta-m r^2 \\ddot{\\theta}-2 m r \\dot{r} \\dot{\\theta}=0\r\n\\end{gathered}\r\n$$\r\nNext, we apply the constraint $r=a$ to these equations of motion:\r\n$$\r\nr=a, \\quad \\dot{r}=0=\\ddot{r}\r\n$$\r\nEquations 7.85 and 7.86 then become\r\n$$\r\n\\begin{array}{r}\r\nm a \\dot{\\theta}^2-m g \\cos \\theta+\\lambda=0 \\\\\r\nm g a \\sin \\theta-m a^2 \\ddot{\\theta}=0\r\n\\end{array}\r\n$$\r\nFrom Equation 7.88 , we have\r\n$$\r\n\\ddot{\\theta}=\\frac{g}{a} \\sin \\theta\r\n$$\r\nWe can integrate Equation 7.89 to determine $\\dot{\\theta}^2$.\r\n$$\r\n\\ddot{\\theta}=\\frac{d}{d t} \\frac{d \\theta}{d t}=\\frac{d \\dot{\\theta}}{d t}=\\frac{d \\dot{\\theta}}{d \\theta} \\frac{d \\theta}{d t}=\\dot{\\theta} \\frac{d \\dot{\\theta}}{d \\theta}\r\n$$\r\nWe integrate Equation 7.89\r\n$$\r\n\\int \\dot{\\theta} d \\dot{\\theta}=\\frac{g}{a} \\int \\sin \\theta d \\theta\r\n$$\r\n254\r\n7 / HAMILTON'S PRINCIPLE-LAGRANGIAN AND HAMILTONIAN DYNAMICS\r\nwhich results in\r\n$$\r\n\\frac{\\dot{\\theta}^2}{2}=\\frac{-g}{a} \\cos \\theta+\\frac{g}{a}\r\n$$\r\nwhere the integration constant is $g / a$, because $\\dot{\\theta}=0$ at $t=0$ when $\\theta=0$. Substituting $\\dot{\\theta}^2$ from Equation 7.92 into Equation 7.87 gives, after solving for $\\lambda$,\r\n$$\r\n\\lambda=m g(3 \\cos \\theta-2)\r\n$$\r\nwhich is the force of constraint. The particle falls off the hemisphere at angle $\\theta_0$ when $\\lambda=0$.\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =0=m g\\left(3 \\cos \\theta_0-2\\right) \\\\\r\n\\theta_0 & =\\cos ^{-1}\\left(\\frac{2}{3}\\right)\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Consider the first stage of a Saturn $V$ rocket used for the Apollo moon program. The initial mass is $2.8 \\times 10^6 \\mathrm{~kg}$, and the mass of the first-stage fuel is $2.1 \\times 10^6$ kg. Assume a mean thrust of $37 \\times 10^6 \\mathrm{~N}$. The exhaust velocity is $2600 \\mathrm{~m} / \\mathrm{s}$. Calculate the final speed of the first stage at burnout. ", "answer_latex": " 2.16", "answer_number": "2.16", "unit": " $10^3 \\mathrm{~m} / \\mathrm{s}$", "source": "class", "problemid": " 9.12", "comment": " ", "solution": "From the thrust (Equation 9.157), we can determine the fuel burn rate:\r\n$$\r\n\\frac{d m}{d t}=\\frac{\\text { thrust }}{-u}=\\frac{37 \\times 10^6 \\mathrm{~N}}{-2600 \\mathrm{~m} / \\mathrm{s}}=-1.42 \\times 10^4 \\mathrm{~kg} / \\mathrm{s}\r\n$$\r\n9.11 ROCKET MOTION\r\n377\r\nThe final rocket mass is $\\left(2.8 \\times 10^6 \\mathrm{~kg}-2.1 \\times 10^6 \\mathrm{~kg}\\right)$ or $0.7 \\times 10^6 \\mathrm{~kg}$. We can determine the rocket speed at burnout $\\left(v_b\\right)$ using Equation 9.163.\r\n$$\r\n\\begin{aligned}\r\nv_b & =-\\frac{9.8 \\mathrm{~m} / \\mathrm{s}^2\\left(2.1 \\times 10^6 \\mathrm{~kg}\\right)}{1.42 \\times 10^4 \\mathrm{~kg} / \\mathrm{s}}+(2600 \\mathrm{~m} / \\mathrm{s}) \\ln \\left[\\frac{2.8 \\times 10^6 \\mathrm{~kg}}{0.7 \\times 10^6 \\mathrm{~kg}}\\right] \\\\\r\nv_b & =2.16 \\times 10^3 \\mathrm{~m} / \\mathrm{s}\r\n\\end{aligned}\r\n$$\r\n"}]
+[
+ {
+  "problem_text": "If the coefficient of static friction between the block and plane in the previous example is $\\mu_s=0.4$, at what angle $\\theta$ will the block start sliding if it is initially at rest?",
+  "answer_latex": "22 ",
+  "answer_number": "22",
+  "unit": " $^{\\circ}$",
+  "source": "class",
+  "problemid": " 2.2",
+  "comment": " ",
+  "solution": "\nThe static frictional force has the approximate maximum value$$f_{\\max }=\\mu_s N$$ Then we have, $y$-direction\r\n$$\r\n-F_g \\cos \\theta+N=0\r\n$$\r\n$x$-direction\r\n$$\r\n-f_s+F_g \\sin \\theta=m \\ddot{x}\r\n$$\r\nThe static frictional force $f_s$ will be some value $f_s \\leq f_{\\max }$ required to keep $\\ddot{x}=0$ -that is, to keep the block at rest. However, as the angle $\\theta$ of the plane increases, eventually the static frictional force will be unable to keep the block at rest. At that angle $\\theta^{\\prime}, f_s$ becomes\r\n$$\r\nf_s\\left(\\theta=\\theta^{\\prime}\\right)=f_{\\max }=\\mu_s N=\\mu_s F_g \\cos \\theta\r\n$$\r\nand\r\n$$\r\n\\begin{aligned}\r\nm \\ddot{x} & =F_g \\sin \\theta-f_{\\max } \\\\\r\nm \\ddot{x} & =F_g \\sin \\theta-\\mu_s F_g \\cos \\theta \\\\\r\n\\ddot{x} & =g\\left(\\sin \\theta-\\mu_s \\cos \\theta\\right)\r\n\\end{aligned}\r\n$$\r\nJust before the block starts to slide, the acceleration $\\ddot{x}=0$, so\r\n$$\r\n\\begin{aligned}\r\n\\sin \\theta-\\mu_s \\cos \\theta & =0 \\\\\r\n\\tan \\theta=\\mu_s & =0.4 \\\\\r\n\\theta=\\tan ^{-1}(0.4) & =22^{\\circ}\r\n\\end{aligned}\r\n$$\n\n\n"
+ },
+ {
+  "problem_text": "Halley's comet, which passed around the sun early in 1986, moves in a highly elliptical orbit with an eccentricity of 0.967 and a period of 76 years. Calculate its minimum  distances from the Sun.",
+  "answer_latex": " 8.8",
+  "answer_number": "8.8",
+  "unit": " $10^{10} \\mathrm{m}$",
+  "source": "class",
+  "problemid": " 8.4",
+  "comment": " ",
+  "solution": "\nThe Equation $\\tau^2=\\frac{4 \\pi^2 a^3}{G(m_1+m_2)} \\cong \\frac{4 \\pi^2 a^3}{G m_2}$ relates the period of motion with the semimajor axes. Because $m$ (Halley's comet) $\\ll m_{\\text {Sun }}$\r\n$$\r\n\\begin{aligned}\r\na & =\\left(\\frac{G m_{\\text {Sun }} \\tau^2}{4 \\pi^2}\\right)^{1 / 3} \\\\\r\n& =\\left[\\frac{\\left.\\left(6.67 \\times 10^{-11} \\frac{\\mathrm{Nm}^2}{\\mathrm{~kg}^2}\\right)\\left(1.99 \\times 10^{30} \\mathrm{~kg}\\right)\\left(76 \\mathrm{yr} \\frac{365 \\mathrm{day}}{\\mathrm{yr}} \\frac{24 \\mathrm{hr}}{\\mathrm{day}} \\frac{3600 \\mathrm{~s}}{\\mathrm{hr}}\\right)^2\\right]}{4 \\pi^2}\\right]^{1 / 3} \\\\\r\na & =2.68 \\times 10^{12} \\mathrm{m}\r\n\\end{aligned}\r\n$$\r\nUsing Equation $\\left.\\begin{array}{l}r_{\\min }=a(1-\\varepsilon)=\\frac{\\alpha}{1+\\varepsilon} \\ r_{\\max }=a(1+\\varepsilon)=\\frac{\\alpha}{1-\\varepsilon}\\end{array}\\right\\}$ , we can determine $r_{\\min }$ and $r_{\\max }$\r\n$$\r\n\\begin{aligned}\r\n& r_{\\min }=2.68 \\times 10^{12} \\mathrm{~m}(1-0.967)=8.8 \\times 10^{10} \\mathrm{~m} \\\\\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "We treat projectile motion in two dimensions, first without considering air resistance. Let the muzzle velocity of the projectile be $v_0$ and the angle of elevation be $\\theta$. The Germans used a long-range gun named Big Bertha in World War I to bombard Paris. Its muzzle velocity was $1,450 \\mathrm{~m} / \\mathrm{s}$. Find its predicted range of flight if $\\theta=55^{\\circ}$. ",
+  "answer_latex": " 72",
+  "answer_number": "202",
+  "unit": " $\\mathrm{km}$",
+  "source": "class",
+  "problemid": " 2.6",
+  "comment": " ",
+  "solution": "\nUsing $\\mathbf{F}=m \\mathrm{~g}$, the force components become\r\n$x$-direction\r\n$$\r\n0=m \\ddot{x}\r\n$$\r\ny-direction\r\n$-m g=m \\ddot{y}$ Neglect the height of the gun, and assume $x=y=0$ at $t=0$. Then\r\n$$\r\n\\begin{aligned}\r\n& \\ddot{x}=0 \\\\\r\n& \\dot{x}=v_0 \\cos \\theta \\\\\r\n& x=v_0 t \\cos \\theta \\\\\r\n& y=-\\frac{-g t^2}{2}+v_0 t \\sin \\theta \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nand\r\n$$\r\n\\begin{aligned}\r\n& \\ddot{y}=-g \\\\\r\n& \\dot{y}=-g t+v_0 \\sin \\theta \\\\\r\n& y=\\frac{-g t^2}{2}+v_0 t \\sin \\theta\r\n\\end{aligned}\r\n$$\r\n\r\nWe can find the range by determining the value of $x$ when the projectile falls back to ground, that is, when $y=0$.\r\n$$\r\ny=t\\left(\\frac{-g t}{2}+v_0 \\sin \\theta\\right)=0\r\n$$\r\nOne value of $y=0$ occurs for $t=0$ and the other one for $t=T$.\r\n$$\r\n\\begin{aligned}\r\n\\frac{-g T}{2}+v_0 \\sin \\theta & =0 \\\\\r\nT & =\\frac{2 v_0 \\sin \\theta}{g}\r\n\\end{aligned}\r\n$$The range $R$ is found from\r\n$$\r\n\\begin{aligned}\r\nx(t=T) & =\\text { range }=\\frac{2 v_0^2}{g} \\sin \\theta \\cos \\theta \\\\\r\nR & =\\text { range }=\\frac{v_0^2}{g} \\sin 2 \\theta\r\n\\end{aligned}$$We have $v_0=1450 \\mathrm{~m} / \\mathrm{s}$ and $\\theta=55^{\\circ}$, so the range becomes\r\n$$\r\nR=\\frac{(1450 \\mathrm{~m} / \\mathrm{s})^2}{9.8 \\mathrm{~m} / \\mathrm{s}^2}\\left[\\sin \\left(110^{\\circ}\\right)\\right]=202 \\mathrm{~km}$$. \n"
+ },
+ {
+  "problem_text": "Calculate the time needed for a spacecraft to make a Hohmann transfer from Earth to Mars",
+  "answer_latex": " 2.24",
+  "answer_number": "2.24",
+  "unit": " $10^7 \\mathrm{~s}$",
+  "source": "class",
+  "problemid": " 8.5",
+  "comment": " ",
+  "solution": "\n$$\r\n\\begin{aligned}\r\n\\frac{m}{k} & =\\frac{m}{G m M_{\\text {Sun }}}=\\frac{1}{G M_{\\text {Sun }}} \\\\\r\n& =\\frac{1}{\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{s}^2 \\cdot \\mathrm{kg}\\right)\\left(1.99 \\times 10^{30} \\mathrm{~kg}\\right)} \\\\\r\n& =7.53 \\times 10^{-21} \\mathrm{~s}^2 / \\mathrm{m}^3\r\n\\end{aligned}\r\n$$\r\nBecause $k / m$ occurs so often in solar system calculations, we write it as well.\r\n$$\r\n\\begin{aligned}\r\n\\frac{k}{m} & =1.33 \\times 10^{20} \\mathrm{~m}^3 / \\mathrm{s}^2 \\\\\r\na_t & =\\frac{1}{2}\\left(r_{\\text {Earth-Sun }}+r_{\\text {Mars-Sun }}\\right) \\\\\r\n& =\\frac{1}{2}\\left(1.50 \\times 10^{11} \\mathrm{~m}+2.28 \\times 10^{11} \\mathrm{~m}\\right) \\\\\r\n& =1.89 \\times 10^{11} \\mathrm{~m} \\\\\r\nT_t & =\\pi\\left(7.53 \\times 10^{-21} \\mathrm{~s}^2 / \\mathrm{m}^3\\right)^{1 / 2}\\left(1.89 \\times 10^{11} \\mathrm{~m}\\right)^{3 / 2} \\\\\r\n& =2.24 \\times 10^7 \\mathrm{~s} \\\\\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Calculate the maximum height change in the ocean tides caused by the Moon.",
+  "answer_latex": " 0.54",
+  "answer_number": "0.54",
+  "unit": "$\\mathrm{m}$",
+  "source": "class",
+  "problemid": " 5.5",
+  "comment": " ",
+  "solution": "\nWe continue to use our simple model of the ocean surrounding Earth. Newton proposed a solution to this calculation by imagining that two wells be dug, one along the direction of high tide (our $x$-axis) and one along the direction of low tide (our $y$-axis). If the tidal height change we want to determine is $h$, then the difference in potential energy of mass $m$ due to the height difference is $m g h$. Let's calculate the difference in work if we move the mass $m$ from point $c$ to the center of Earth and then to point $a$. This work $W$ done by gravity must equal the potential energy change $m g h$. The work $W$ is\r\n$$\r\nW=\\int_{r+\\delta_1}^0 F_{T_y} d y+\\int_0^{r+\\delta_2} F_{T_x} d x\r\n$$. The small distances $\\delta_1$ and $\\delta_2$ are to account for the small variations from a spherical Earth, but these values are so small they can be henceforth neglected. The value for $W$ becomes\r\n$$\r\n\\begin{aligned}\r\nW & =\\frac{G m M_m}{D^3}\\left[\\int_r^0(-y) d y+\\int_0^r 2 x d x\\right] \\\\\r\n& =\\frac{G m M_m}{D^3}\\left(\\frac{r^2}{2}+r^2\\right)=\\frac{3 G m M_m r^2}{2 D^3}\r\n\\end{aligned}\r\n$$\r\nBecause this work is equal to $m g h$, we have\r\n$$\r\n\\begin{aligned}\r\nm g h & =\\frac{3 G m M_m r^2}{2 D^3} \\\\\r\nh & =\\frac{3 G M_m r^2}{2 g D^3}\r\n\\end{aligned}\r\n$$\r\nNote that the mass $m$ cancels, and the value of $h$ does not depend on $m$. Nor does it depend on the substance, so to the extent Earth is plastic, similar tidal effects should be (and are) observed for the surface land. If we insert the known values of the constants into the Equation, we find\r\n$$\r\nh=\\frac{3\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{kg} \\cdot \\mathrm{s}^2\\right)\\left(7.350 \\times 10^{22} \\mathrm{~kg}\\right)\\left(6.37 \\times 10^6 \\mathrm{~m}\\right)^2}{2\\left(9.80 \\mathrm{~m} / \\mathrm{s}^2\\right)\\left(3.84 \\times 10^8 \\mathrm{~m}\\right)^3}=0.54 \\mathrm{~m}\r\n$$\n"
+ },
+ {
+  "problem_text": "A particle of mass $m$ starts at rest on top of a smooth fixed hemisphere of radius $a$. Determine the angle at which the particle leaves the hemisphere.",
+  "answer_latex": " $\\cos ^{-1}\\left(\\frac{2}{3}\\right)$",
+  "answer_number": "48.189685",
+  "unit": "$^\\circ$",
+  "source": "class",
+  "problemid": " 7.10",
+  "comment": " ",
+  "solution": "\nBecause we are considering the possibility of the particle leaving the hemisphere, we choose the generalized coordinates to be $r$ and $\\theta$. The constraint equation is\r\n$$\r\nf(r, \\theta)=r-a=0\r\n$$\r\nThe Lagrangian is determined from the kinetic and potential energies:\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{m}{2}\\left(\\dot{r}^2+r^2 \\dot{\\theta}^2\\right) \\\\\r\nU & =m g r \\cos \\theta \\\\\r\nL & =T-U \\\\\r\nL & =\\frac{m}{2}\\left(\\dot{r}^2+r^2 \\dot{\\theta}^2\\right)-m g r \\cos \\theta\r\n\\end{aligned}\r\n$$\r\nwhere the potential energy is zero at the bottom of the hemisphere. The Lagrange equations are\r\n$$\r\n\\begin{aligned}\r\n& \\frac{\\partial L}{\\partial r}-\\frac{d}{d t} \\frac{\\partial L}{\\partial \\dot{r}}+\\lambda \\frac{\\partial f}{\\partial r}=0 \\\\\r\n& \\frac{\\partial L}{\\partial \\theta}-\\frac{d}{d t} \\frac{\\partial L}{\\partial \\dot{\\theta}}+\\lambda \\frac{\\partial f}{\\partial \\theta}=0\r\n\\end{aligned}\r\n$$\r\nPerforming the differentiations gives\r\n$$\r\n\\frac{\\partial f}{\\partial r}=1, \\quad \\frac{\\partial f}{\\partial \\theta}=0\r\n$$Then equations become\r\n$$\r\n\\begin{gathered}\r\nm r \\dot{\\theta}^2-m g \\cos \\theta-m \\ddot{r}+\\lambda=0 \\\\\r\nm g r \\sin \\theta-m r^2 \\ddot{\\theta}-2 m r \\dot{r} \\dot{\\theta}=0\r\n\\end{gathered}\r\n$$\r\nNext, we apply the constraint $r=a$ to these equations of motion:\r\n$$\r\nr=a, \\quad \\dot{r}=0=\\ddot{r}\r\n$$Then $$\r\n\\begin{array}{r}\r\nm a \\dot{\\theta}^2-m g \\cos \\theta+\\lambda=0 \\\\\r\nm g a \\sin \\theta-m a^2 \\ddot{\\theta}=0\r\n\\end{array}\r\n$$We have\r\n$$\r\n\\ddot{\\theta}=\\frac{g}{a} \\sin \\theta\r\n$$\r\nWe can integrate Equation to determine $\\dot{\\theta}^2$.\r\n$$\r\n\\ddot{\\theta}=\\frac{d}{d t} \\frac{d \\theta}{d t}=\\frac{d \\dot{\\theta}}{d t}=\\frac{d \\dot{\\theta}}{d \\theta} \\frac{d \\theta}{d t}=\\dot{\\theta} \\frac{d \\dot{\\theta}}{d \\theta}\r\n$$\r\nWe integrate Equation $$\r\n\\int \\dot{\\theta} d \\dot{\\theta}=\\frac{g}{a} \\int \\sin \\theta d \\theta\r\n$$ which results in\r\n$$\r\n\\frac{\\dot{\\theta}^2}{2}=\\frac{-g}{a} \\cos \\theta+\\frac{g}{a}\r\n$$\r\nwhere the integration constant is $g / a$, because $\\dot{\\theta}=0$ at $t=0$ when $\\theta=0$. Substituting $\\dot{\\theta}^2$ from Equation, after solving for $\\lambda$,\r\n$$\r\n\\lambda=m g(3 \\cos \\theta-2)\r\n$$\r\nwhich is the force of constraint. The particle falls off the hemisphere at angle $\\theta_0$ when $\\lambda=0$.\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =0=m g\\left(3 \\cos \\theta_0-2\\right) \\\\\r\n\\theta_0 & =\\cos ^{-1}\\left(\\frac{2}{3}\\right)\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Consider the first stage of a Saturn $V$ rocket used for the Apollo moon program. The initial mass is $2.8 \\times 10^6 \\mathrm{~kg}$, and the mass of the first-stage fuel is $2.1 \\times 10^6$ kg. Assume a mean thrust of $37 \\times 10^6 \\mathrm{~N}$. The exhaust velocity is $2600 \\mathrm{~m} / \\mathrm{s}$. Calculate the final speed of the first stage at burnout. ",
+  "answer_latex": " 2.16",
+  "answer_number": "2.16",
+  "unit": " $10^3 \\mathrm{~m} / \\mathrm{s}$",
+  "source": "class",
+  "problemid": " 9.12",
+  "comment": " ",
+  "solution": "\nFrom the thrust, we can determine the fuel burn rate:$$\\frac{d m}{d t}=\\frac{\\text { thrust }}{-u}=\\frac{37 \\times 10^6 \\mathrm{~N}}{-2600 \\mathrm{~m} / \\mathrm{s}}=-1.42 \\times 10^4 \\mathrm{~kg} / \\mathrm{s}\r\n$$The final rocket mass is $\\left(2.8 \\times 10^6 \\mathrm{~kg}-2.1 \\times 10^6 \\mathrm{~kg}\\right)$ or $0.7 \\times 10^6 \\mathrm{~kg}$. We can determine the rocket speed at burnout $\\left(v_b\\right)$.\r\n$$\r\n\\begin{aligned}\r\nv_b & =-\\frac{9.8 \\mathrm{~m} / \\mathrm{s}^2\\left(2.1 \\times 10^6 \\mathrm{~kg}\\right)}{1.42 \\times 10^4 \\mathrm{~kg} / \\mathrm{s}}+(2600 \\mathrm{~m} / \\mathrm{s}) \\ln \\left[\\frac{2.8 \\times 10^6 \\mathrm{~kg}}{0.7 \\times 10^6 \\mathrm{~kg}}\\right] \\\\\r\nv_b & =2.16 \\times 10^3 \\mathrm{~m} / \\mathrm{s}\r\n\\end{aligned}\r\n$$\n"
+ }
+]

diff.json

@@ -1 +1,452 @@
-[{"problem_text": "Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.", "answer_latex": "7.25", "answer_number": "7.25", "unit": " %", "source": "diff", "problemid": " page 130-7", "comment": " "}, {"problem_text": "Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \\mathrm{~L}$ of a dye solution with a concentration of $1 \\mathrm{~g} / \\mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \\mathrm{~L} / \\mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \\%$ of its original value.", "answer_latex": " 460.5", "answer_number": "460.5", "unit": " min", "source": "diff", "problemid": " Page 59-1", "comment": " "}, {"problem_text": "A certain vibrating system satisfies the equation $u^{\\prime \\prime}+\\gamma u^{\\prime}+u=0$. Find the value of the damping coefficient $\\gamma$ for which the quasi period of the damped motion is $50 \\%$ greater than the period of the corresponding undamped motion.", "answer_latex": " 1.4907", "answer_number": "1.4907", "unit": " ", "source": "diff", "problemid": " page203-13", "comment": " "}, {"problem_text": "Find the value of $y_0$ for which the solution of the initial value problem\r\n$$\r\ny^{\\prime}-y=1+3 \\sin t, \\quad y(0)=y_0\r\n$$\r\nremains finite as $t \\rightarrow \\infty$", "answer_latex": " -2.5", "answer_number": " -2.5", "unit": " ", "source": "diff", "problemid": " Page 40-30", "comment": " "}, {"problem_text": "A certain spring-mass system satisfies the initial value problem\r\n$$\r\nu^{\\prime \\prime}+\\frac{1}{4} u^{\\prime}+u=k g(t), \\quad u(0)=0, \\quad u^{\\prime}(0)=0\r\n$$\r\nwhere $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.\r\nSuppose $k=2$. Find the time $\\tau$ after which $|u(t)|<0.1$ for all $t>\\tau$.", "answer_latex": " 25.6773", "answer_number": "25.6773", "unit": " ", "source": "diff", "problemid": " page336-16", "comment": " "}, {"problem_text": "Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.\r\nDetermine $T$ if $r=7 \\%$.", "answer_latex": " 9.90", "answer_number": "9.90", "unit": " year", "source": "diff", "problemid": " page 60-7", "comment": " "}, {"problem_text": "A mass weighing $2 \\mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position $u$ of the mass at any time $t$. Find the frequency of the motion.", "answer_latex": " $\\pi/4$", "answer_number": "0.7854", "unit": " s", "source": "diff", "problemid": " page202-5", "comment": " "}, {"problem_text": "If $\\mathbf{x}=\\left(\\begin{array}{c}2 \\\\ 3 i \\\\ 1-i\\end{array}\\right)$ and $\\mathbf{y}=\\left(\\begin{array}{c}-1+i \\\\ 2 \\\\ 3-i\\end{array}\\right)$, find $(\\mathbf{y}, \\mathbf{y})$.", "answer_latex": " 16", "answer_number": "16", "unit": " ", "source": "diff", "problemid": " page372-8", "comment": " "}, {"problem_text": "15. Consider the initial value problem\r\n$$\r\n4 y^{\\prime \\prime}+12 y^{\\prime}+9 y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=-4 .\r\n$$\r\nDetermine where the solution has the value zero.", "answer_latex": " 0.4", "answer_number": " 0.4", "unit": " ", "source": "diff", "problemid": " page172-15", "comment": " "}, {"problem_text": "A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?", "answer_latex": " 258.14", "answer_number": " 258.14", "unit": " $", "source": "diff", "problemid": " page131-9", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=k \\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0\r\n$$\r\nwhere $k$ is the magnitude of an impulse at $t=1$ and $\\gamma$ is the damping coefficient (or resistance).\r\nLet $\\gamma=\\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.", "answer_latex": " 2.8108", "answer_number": "2.8108", "unit": " ", "source": "diff", "problemid": " page344-15", "comment": " "}, {"problem_text": "If a series circuit has a capacitor of $C=0.8 \\times 10^{-6} \\mathrm{~F}$ and an inductor of $L=0.2 \\mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.", "answer_latex": " 1000", "answer_number": "1000", "unit": " $\\Omega$", "source": "diff", "problemid": " page203-18", "comment": " "}, {"problem_text": "If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\\prime \\prime}+2 y^{\\prime}+t e^t y=0$ and if $W\\left(y_1, y_2\\right)(1)=2$, find the value of $W\\left(y_1, y_2\\right)(5)$.", "answer_latex": " 2/25", "answer_number": "0.08", "unit": " ", "source": "diff", "problemid": " page156-34", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\n5 u^{\\prime \\prime}+2 u^{\\prime}+7 u=0, \\quad u(0)=2, \\quad u^{\\prime}(0)=1\r\n$$\r\nFind the smallest $T$ such that $|u(t)| \\leq 0.1$ for all $t>T$.", "answer_latex": "14.5115", "answer_number": "14.5115", "unit": " ", "source": "diff", "problemid": " page163-24", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}=t y(4-y) / 3, \\quad y(0)=y_0\r\n$$\r\nSuppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.", "answer_latex": " 3.29527", "answer_number": "3.29527", "unit": " ", "source": "diff", "problemid": "Page 49 27 ", "comment": " "}, {"problem_text": "25. Consider the initial value problem\r\n$$\r\n2 y^{\\prime \\prime}+3 y^{\\prime}-2 y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=-\\beta,\r\n$$\r\nwhere $\\beta>0$.\r\nFind the smallest value of $\\beta$ for which the solution has no minimum point.", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "diff", "problemid": " page144-25", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}=t y(4-y) /(1+t), \\quad y(0)=y_0>0 .\r\n$$\r\nIf $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.", "answer_latex": " 2.84367", "answer_number": "2.84367", "unit": " ", "source": "diff", "problemid": " Page 49 28", "comment": " "}, {"problem_text": "28. A mass of $0.25 \\mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\\mathrm{m} / \\mathrm{s}$.\r\nIf the mass is to attain a velocity of no more than $10 \\mathrm{~m} / \\mathrm{s}$, find the maximum height from which it can be dropped.", "answer_latex": " 13.45", "answer_number": " 13.45", "unit": " m", "source": "diff", "problemid": "page66-28", "comment": " "}, {"problem_text": "A home buyer can afford to spend no more than $\\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \\%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.\r\nDetermine the maximum amount that this buyer can afford to borrow.", "answer_latex": " 89,034.79", "answer_number": "89,034.79", "unit": " $", "source": "diff", "problemid": " page 61-10", "comment": " "}, {"problem_text": "A spring is stretched 6 in by a mass that weighs $8 \\mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \\mathrm{lb} \\cdot \\mathrm{s} / \\mathrm{ft}$ and is acted on by an external force of $4 \\cos 2 t \\mathrm{lb}$.\r\nIf the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.", "answer_latex": " 4", "answer_number": "4", "unit": " slugs", "source": "diff", "problemid": " page216-11", "comment": " "}, {"problem_text": "A recent college graduate borrows $\\$ 100,000$ at an interest rate of $9 \\%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.\r\nAssuming that this payment schedule can be maintained, when will the loan be fully paid?", "answer_latex": " 135.36", "answer_number": " 135.36", "unit": " months", "source": "diff", "problemid": " page61-11", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{1}{4} y=3+2 \\cos 2 t, \\quad y(0)=0\r\n$$\r\nDetermine the value of $t$ for which the solution first intersects the line $y=12$.", "answer_latex": " 10.065778", "answer_number": "10.065778", "unit": " ", "source": "diff", "problemid": " Page 40 29", "comment": " "}, {"problem_text": "An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \\$25 per month. Find the balance in the account after 3 years.", "answer_latex": " 2283.63", "answer_number": "2283.63", "unit": " $", "source": "diff", "problemid": " page 131-8", "comment": " "}, {"problem_text": "A mass of $0.25 \\mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\\mathrm{m} / \\mathrm{s}$.\r\nIf the mass is dropped from a height of $30 \\mathrm{~m}$, find its velocity when it hits the ground.", "answer_latex": " 11.58", "answer_number": " 11.58", "unit": " m/s", "source": "diff", "problemid": " page 66-28", "comment": " "}, {"problem_text": "A mass of $100 \\mathrm{~g}$ stretches a spring $5 \\mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \\mathrm{~cm} / \\mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.", "answer_latex": " $\\pi/14$", "answer_number": "0.2244", "unit": " s", "source": "diff", "problemid": " page202-6", "comment": " "}, {"problem_text": "Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.\r\nConsider a water tank in the form of a right circular cylinder that is $3 \\mathrm{~m}$ high above the outlet. The radius of the tank is $1 \\mathrm{~m}$ and the radius of the circular outlet is $0.1 \\mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.", "answer_latex": " 130.41", "answer_number": "130.41", "unit": " s", "source": "diff", "problemid": "page 60-6", "comment": " "}, {"problem_text": "Solve the initial value problem $y^{\\prime \\prime}-y^{\\prime}-2 y=0, y(0)=\\alpha, y^{\\prime}(0)=2$. Then find $\\alpha$ so that the solution approaches zero as $t \\rightarrow \\infty$.", "answer_latex": " \u22122", "answer_number": "\u22122", "unit": " ", "source": "diff", "problemid": " page144-21", "comment": " "}, {"problem_text": "If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\\prime \\prime}-2 y^{\\prime}+(3+t) y=0$ and if $W\\left(y_1, y_2\\right)(2)=3$, find the value of $W\\left(y_1, y_2\\right)(4)$.", "answer_latex": " 4.946", "answer_number": "4.946", "unit": " ", "source": "diff", "problemid": " page156-35", "comment": " "}, {"problem_text": " Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.", "answer_latex": " 672.4", "answer_number": " 672.4", "unit": " Year", "source": "diff", "problemid": " Page 17 14", "comment": " "}, {"problem_text": "A tank originally contains $100 \\mathrm{gal}$ of fresh water. Then water containing $\\frac{1}{2} \\mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \\mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \\mathrm{~min}$.", "answer_latex": " 7.42", "answer_number": " 7.42", "unit": " lb", "source": "diff", "problemid": "Page 60-3 ", "comment": " "}, {"problem_text": "A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.\r\nIf $r=7.5 \\%$, determine $k$ so that $\\$ 1$ million will be available for retirement in 40 years.", "answer_latex": " 3930", "answer_number": "3930", "unit": " $", "source": "diff", "problemid": " page 60-8", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+2 a y^{\\prime}+\\left(a^2+1\\right) y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=0 .\r\n$$\r\nFor $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.", "answer_latex": "1.8763", "answer_number": "1.8763", "unit": " ", "source": "diff", "problemid": " page164-26", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=\\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0,\r\n$$\r\nwhere $\\gamma$ is the damping coefficient (or resistance).\r\nFind the time $t_1$ at which the solution attains its maximum value.", "answer_latex": " 2.3613", "answer_number": "2.3613", "unit": " ", "source": "diff", "problemid": " page344-14", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{2}{3} y=1-\\frac{1}{2} t, \\quad y(0)=y_0 .\r\n$$\r\nFind the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.", "answer_latex": " \u22121.642876", "answer_number": "\u22121.642876", "unit": " ", "source": "diff", "problemid": " Page 40 28", "comment": " "}, {"problem_text": "A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \\mathrm{mg}$ of thorium-234 decays to $82.04 \\mathrm{mg}$ in 1 week, determine the decay rate $r$.", "answer_latex": " 0.02828", "answer_number": "0.02828", "unit": " $\\text{day}^{-1}$", "source": "diff", "problemid": " Section 1.2, page 15 12. (a)", "comment": " "}, {"problem_text": "Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\\circ} \\mathrm{F}$ when freshly poured, and $1 \\mathrm{~min}$ later has cooled to $190^{\\circ} \\mathrm{F}$ in a room at $70^{\\circ} \\mathrm{F}$, determine when the coffee reaches a temperature of $150^{\\circ} \\mathrm{F}$.", "answer_latex": " 6.07", "answer_number": " 6.07", "unit": " min", "source": "diff", "problemid": " page62-16", "comment": " "}, {"problem_text": "Solve the initial value problem $4 y^{\\prime \\prime}-y=0, y(0)=2, y^{\\prime}(0)=\\beta$. Then find $\\beta$ so that the solution approaches zero as $t \\rightarrow \\infty$.", "answer_latex": " -1", "answer_number": " -1", "unit": " ", "source": "diff", "problemid": " page144-22", "comment": " "}, {"problem_text": "Consider the initial value problem (see Example 5)\r\n$$\r\ny^{\\prime \\prime}+5 y^{\\prime}+6 y=0, \\quad y(0)=2, \\quad y^{\\prime}(0)=\\beta\r\n$$\r\nwhere $\\beta>0$.\r\nDetermine the smallest value of $\\beta$ for which $y_m \\geq 4$.", "answer_latex": " 16.3923", "answer_number": "16.3923", "unit": " ", "source": "diff", "problemid": " page145-26", "comment": " "}, {"problem_text": "A home buyer can afford to spend no more than $\\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \\%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.\r\nDetermine the total interest paid during the term of the mortgage.", "answer_latex": " 102,965.21", "answer_number": "102,965.21", "unit": " $", "source": "diff", "problemid": " page61-10", "comment": " "}, {"problem_text": "Find the fundamental period of the given function:\r\n$$f(x)=\\left\\{\\begin{array}{ll}(-1)^n, & 2 n-1 \\leq x<2 n, \\\\ 1, & 2 n \\leq x<2 n+1 ;\\end{array} \\quad n=0, \\pm 1, \\pm 2, \\ldots\\right.$$", "answer_latex": " 4", "answer_number": "4", "unit": " ", "source": "diff", "problemid": " page593-8", "comment": " "}, {"problem_text": "A homebuyer wishes to finance the purchase with a \\$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \\$900?", "answer_latex": " 9.73", "answer_number": " 9.73", "unit": " %", "source": "diff", "problemid": " page131-13", "comment": " "}, {"problem_text": "A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?", "answer_latex": "804.62", "answer_number": "804.62", "unit": "$", "source": "diff", "problemid": " page131-10", "comment": " "}, {"problem_text": "Let a metallic rod $20 \\mathrm{~cm}$ long be heated to a uniform temperature of $100^{\\circ} \\mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\\circ} \\mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \\mathrm{~s}$ if the bar is made of silver.", "answer_latex": " 35.91", "answer_number": " 35.91", "unit": " ${ }^{\\circ} \\mathrm{C}$", "source": "diff", "problemid": " page619-18", "comment": " "}, {"problem_text": "Find $\\gamma$ so that the solution of the initial value problem $x^2 y^{\\prime \\prime}-2 y=0, y(1)=1, y^{\\prime}(1)=\\gamma$ is bounded as $x \\rightarrow 0$.", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "diff", "problemid": " page277-37", "comment": " "}, {"problem_text": "A tank contains 100 gal of water and $50 \\mathrm{oz}$ of salt. Water containing a salt concentration of $\\frac{1}{4}\\left(1+\\frac{1}{2} \\sin t\\right) \\mathrm{oz} / \\mathrm{gal}$ flows into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, and the mixture in the tank flows out at the same rate.\r\nThe long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?", "answer_latex": " 0.24995", "answer_number": "0.24995", "unit": " ", "source": "diff", "problemid": " Page 60-5", "comment": " "}, {"problem_text": "A mass weighing $8 \\mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\\gamma$. Determine the value of $\\gamma$ for which the system is critically damped; be sure to give the units for $\\gamma$", "answer_latex": "8", "answer_number": "8", "unit": " $\\mathrm{lb} \\cdot \\mathrm{s} / \\mathrm{ft}$", "source": "diff", "problemid": " page203-17", "comment": " "}, {"problem_text": "Your swimming pool containing 60,000 gal of water has been contaminated by $5 \\mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \\mathrm{gal} / \\mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \\mathrm{~g} / \\mathrm{gal}$.", "answer_latex": " 7.136", "answer_number": "7.136", "unit": " hour", "source": "diff", "problemid": " Page 18 19", "comment": " "}, {"problem_text": "For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.  If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.", "answer_latex": "$\\frac{2}{49}$", "answer_number": "0.0408", "unit": " ", "source": "diff", "problemid": " 1 25(c)", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\n3 u^{\\prime \\prime}-u^{\\prime}+2 u=0, \\quad u(0)=2, \\quad u^{\\prime}(0)=0\r\n$$\r\nFor $t>0$ find the first time at which $|u(t)|=10$.", "answer_latex": " 10.7598", "answer_number": " 10.7598", "unit": " ", "source": "diff", "problemid": " page163-23", "comment": " "}, {"problem_text": "Consider the initial value problem\r\n$$\r\n9 y^{\\prime \\prime}+12 y^{\\prime}+4 y=0, \\quad y(0)=a>0, \\quad y^{\\prime}(0)=-1\r\n$$\r\nFind the critical value of $a$ that separates solutions that become negative from those that are always positive.", "answer_latex": " 1.5", "answer_number": "1.5", "unit": " ", "source": "diff", "problemid": " page172-18", "comment": " "}]
+[
+ {
+  "problem_text": "Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.",
+  "answer_latex": "7.25",
+  "answer_number": "7.25",
+  "unit": " %",
+  "source": "diff",
+  "problemid": " page 130-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \\mathrm{~L}$ of a dye solution with a concentration of $1 \\mathrm{~g} / \\mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \\mathrm{~L} / \\mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \\%$ of its original value.",
+  "answer_latex": " 460.5",
+  "answer_number": "460.5",
+  "unit": " min",
+  "source": "diff",
+  "problemid": " Page 59-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "A certain vibrating system satisfies the equation $u^{\\prime \\prime}+\\gamma u^{\\prime}+u=0$. Find the value of the damping coefficient $\\gamma$ for which the quasi period of the damped motion is $50 \\%$ greater than the period of the corresponding undamped motion.",
+  "answer_latex": " 1.4907",
+  "answer_number": "1.4907",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page203-13",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the value of $y_0$ for which the solution of the initial value problem\r\n$$\r\ny^{\\prime}-y=1+3 \\sin t, \\quad y(0)=y_0\r\n$$\r\nremains finite as $t \\rightarrow \\infty$",
+  "answer_latex": " -2.5",
+  "answer_number": " -2.5",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " Page 40-30",
+  "comment": " "
+ },
+ {
+  "problem_text": "A certain spring-mass system satisfies the initial value problem\r\n$$\r\nu^{\\prime \\prime}+\\frac{1}{4} u^{\\prime}+u=k g(t), \\quad u(0)=0, \\quad u^{\\prime}(0)=0\r\n$$\r\nwhere $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.\r\nSuppose $k=2$. Find the time $\\tau$ after which $|u(t)|<0.1$ for all $t>\\tau$.",
+  "answer_latex": " 25.6773",
+  "answer_number": "25.6773",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page336-16",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.\r\nDetermine $T$ if $r=7 \\%$.",
+  "answer_latex": " 9.90",
+  "answer_number": "9.90",
+  "unit": " year",
+  "source": "diff",
+  "problemid": " page 60-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mass weighing $2 \\mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, by determining the position $u$ of the mass at any time $t$, find the frequency of the motion",
+  "answer_latex": " $\\pi/4$",
+  "answer_number": "0.7854",
+  "unit": " s",
+  "source": "diff",
+  "problemid": " page202-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "If $\\mathbf{x}=\\left(\\begin{array}{c}2 \\\\ 3 i \\\\ 1-i\\end{array}\\right)$ and $\\mathbf{y}=\\left(\\begin{array}{c}-1+i \\\\ 2 \\\\ 3-i\\end{array}\\right)$, find $(\\mathbf{y}, \\mathbf{y})$.",
+  "answer_latex": " 16",
+  "answer_number": "16",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page372-8",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\n4 y^{\\prime \\prime}+12 y^{\\prime}+9 y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=-4 .\r\n$$\r\nDetermine where the solution has the value zero.",
+  "answer_latex": " 0.4",
+  "answer_number": " 0.4",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page172-15",
+  "comment": " "
+ },
+ {
+  "problem_text": "A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?",
+  "answer_latex": " 258.14",
+  "answer_number": " 258.14",
+  "unit": " $",
+  "source": "diff",
+  "problemid": " page131-9",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=k \\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0\r\n$$\r\nwhere $k$ is the magnitude of an impulse at $t=1$ and $\\gamma$ is the damping coefficient (or resistance).\r\nLet $\\gamma=\\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.",
+  "answer_latex": " 2.8108",
+  "answer_number": "2.8108",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page344-15",
+  "comment": " "
+ },
+ {
+  "problem_text": "If a series circuit has a capacitor of $C=0.8 \\times 10^{-6} \\mathrm{~F}$ and an inductor of $L=0.2 \\mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.",
+  "answer_latex": " 1000",
+  "answer_number": "1000",
+  "unit": " $\\Omega$",
+  "source": "diff",
+  "problemid": " page203-18",
+  "comment": " "
+ },
+ {
+  "problem_text": "If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\\prime \\prime}+2 y^{\\prime}+t e^t y=0$ and if $W\\left(y_1, y_2\\right)(1)=2$, find the value of $W\\left(y_1, y_2\\right)(5)$.",
+  "answer_latex": " 2/25",
+  "answer_number": "0.08",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page156-34",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\n5 u^{\\prime \\prime}+2 u^{\\prime}+7 u=0, \\quad u(0)=2, \\quad u^{\\prime}(0)=1\r\n$$\r\nFind the smallest $T$ such that $|u(t)| \\leq 0.1$ for all $t>T$.",
+  "answer_latex": "14.5115",
+  "answer_number": "14.5115",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page163-24",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}=t y(4-y) / 3, \\quad y(0)=y_0\r\n$$\r\nSuppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.",
+  "answer_latex": " 3.29527",
+  "answer_number": "3.29527",
+  "unit": " ",
+  "source": "diff",
+  "problemid": "Page 49 27 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\n2 y^{\\prime \\prime}+3 y^{\\prime}-2 y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=-\\beta,\r\n$$\r\nwhere $\\beta>0$.\r\nFind the smallest value of $\\beta$ for which the solution has no minimum point.",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page144-25",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}=t y(4-y) /(1+t), \\quad y(0)=y_0>0 .\r\n$$\r\nIf $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.",
+  "answer_latex": " 2.84367",
+  "answer_number": "2.84367",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " Page 49 28",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mass of $0.25 \\mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\\mathrm{m} / \\mathrm{s}$.\r\nIf the mass is to attain a velocity of no more than $10 \\mathrm{~m} / \\mathrm{s}$, find the maximum height from which it can be dropped.",
+  "answer_latex": " 13.45",
+  "answer_number": " 13.45",
+  "unit": " m",
+  "source": "diff",
+  "problemid": "page66-28",
+  "comment": " "
+ },
+ {
+  "problem_text": "A home buyer can afford to spend no more than $\\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \\%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.\r\nDetermine the maximum amount that this buyer can afford to borrow.",
+  "answer_latex": " 89,034.79",
+  "answer_number": "89,034.79",
+  "unit": " $",
+  "source": "diff",
+  "problemid": " page 61-10",
+  "comment": " "
+ },
+ {
+  "problem_text": "A spring is stretched 6 in by a mass that weighs $8 \\mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \\mathrm{lb} \\cdot \\mathrm{s} / \\mathrm{ft}$ and is acted on by an external force of $4 \\cos 2 t \\mathrm{lb}$.\r\nIf the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.",
+  "answer_latex": " 4",
+  "answer_number": "4",
+  "unit": " slugs",
+  "source": "diff",
+  "problemid": " page216-11",
+  "comment": " "
+ },
+ {
+  "problem_text": "A recent college graduate borrows $\\$ 100,000$ at an interest rate of $9 \\%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.\r\nAssuming that this payment schedule can be maintained, when will the loan be fully paid?",
+  "answer_latex": " 135.36",
+  "answer_number": " 135.36",
+  "unit": " months",
+  "source": "diff",
+  "problemid": " page61-11",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{1}{4} y=3+2 \\cos 2 t, \\quad y(0)=0\r\n$$\r\nDetermine the value of $t$ for which the solution first intersects the line $y=12$.",
+  "answer_latex": " 10.065778",
+  "answer_number": "10.065778",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " Page 40 29",
+  "comment": " "
+ },
+ {
+  "problem_text": "An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \\$25 per month. Find the balance in the account after 3 years.",
+  "answer_latex": " 2283.63",
+  "answer_number": "2283.63",
+  "unit": " $",
+  "source": "diff",
+  "problemid": " page 131-8",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mass of $0.25 \\mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\\mathrm{m} / \\mathrm{s}$.\r\nIf the mass is dropped from a height of $30 \\mathrm{~m}$, find its velocity when it hits the ground.",
+  "answer_latex": " 11.58",
+  "answer_number": " 11.58",
+  "unit": " m/s",
+  "source": "diff",
+  "problemid": " page 66-28",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mass of $100 \\mathrm{~g}$ stretches a spring $5 \\mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \\mathrm{~cm} / \\mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.",
+  "answer_latex": " $\\pi/14$",
+  "answer_number": "0.2244",
+  "unit": " s",
+  "source": "diff",
+  "problemid": " page202-6",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.\r\nConsider a water tank in the form of a right circular cylinder that is $3 \\mathrm{~m}$ high above the outlet. The radius of the tank is $1 \\mathrm{~m}$ and the radius of the circular outlet is $0.1 \\mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.",
+  "answer_latex": " 130.41",
+  "answer_number": "130.41",
+  "unit": " s",
+  "source": "diff",
+  "problemid": "page 60-6",
+  "comment": " "
+ },
+ {
+  "problem_text": "Solve the initial value problem $y^{\\prime \\prime}-y^{\\prime}-2 y=0, y(0)=\\alpha, y^{\\prime}(0)=2$. Then find $\\alpha$ so that the solution approaches zero as $t \\rightarrow \\infty$.",
+  "answer_latex": " \u22122",
+  "answer_number": "\u22122",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page144-21",
+  "comment": " "
+ },
+ {
+  "problem_text": "If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\\prime \\prime}-2 y^{\\prime}+(3+t) y=0$ and if $W\\left(y_1, y_2\\right)(2)=3$, find the value of $W\\left(y_1, y_2\\right)(4)$.",
+  "answer_latex": " 4.946",
+  "answer_number": "4.946",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page156-35",
+  "comment": " "
+ },
+ {
+  "problem_text": " Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.",
+  "answer_latex": " 672.4",
+  "answer_number": " 672.4",
+  "unit": " Year",
+  "source": "diff",
+  "problemid": " Page 17 14",
+  "comment": " "
+ },
+ {
+  "problem_text": "A tank originally contains $100 \\mathrm{gal}$ of fresh water. Then water containing $\\frac{1}{2} \\mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \\mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \\mathrm{~min}$.",
+  "answer_latex": " 7.42",
+  "answer_number": " 7.42",
+  "unit": " lb",
+  "source": "diff",
+  "problemid": "Page 60-3 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.\r\nIf $r=7.5 \\%$, determine $k$ so that $\\$ 1$ million will be available for retirement in 40 years.",
+  "answer_latex": " 3930",
+  "answer_number": "3930",
+  "unit": " $",
+  "source": "diff",
+  "problemid": " page 60-8",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+2 a y^{\\prime}+\\left(a^2+1\\right) y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=0 .\r\n$$\r\nFor $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.",
+  "answer_latex": "1.8763",
+  "answer_number": "1.8763",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page164-26",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=\\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0,\r\n$$\r\nwhere $\\gamma$ is the damping coefficient (or resistance).\r\nFind the time $t_1$ at which the solution attains its maximum value.",
+  "answer_latex": " 2.3613",
+  "answer_number": "2.3613",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page344-14",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{2}{3} y=1-\\frac{1}{2} t, \\quad y(0)=y_0 .\r\n$$\r\nFind the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.",
+  "answer_latex": " \u22121.642876",
+  "answer_number": "\u22121.642876",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " Page 40 28",
+  "comment": " "
+ },
+ {
+  "problem_text": "A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \\mathrm{mg}$ of thorium-234 decays to $82.04 \\mathrm{mg}$ in 1 week, determine the decay rate $r$.",
+  "answer_latex": " 0.02828",
+  "answer_number": "0.02828",
+  "unit": " $\\text{day}^{-1}$",
+  "source": "diff",
+  "problemid": " Section 1.2, page 15 12. (a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\\circ} \\mathrm{F}$ when freshly poured, and $1 \\mathrm{~min}$ later has cooled to $190^{\\circ} \\mathrm{F}$ in a room at $70^{\\circ} \\mathrm{F}$, determine when the coffee reaches a temperature of $150^{\\circ} \\mathrm{F}$.",
+  "answer_latex": " 6.07",
+  "answer_number": " 6.07",
+  "unit": " min",
+  "source": "diff",
+  "problemid": " page62-16",
+  "comment": " "
+ },
+ {
+  "problem_text": "Solve the initial value problem $4 y^{\\prime \\prime}-y=0, y(0)=2, y^{\\prime}(0)=\\beta$. Then find $\\beta$ so that the solution approaches zero as $t \\rightarrow \\infty$.",
+  "answer_latex": " -1",
+  "answer_number": " -1",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page144-22",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem (see Example 5)\r\n$$\r\ny^{\\prime \\prime}+5 y^{\\prime}+6 y=0, \\quad y(0)=2, \\quad y^{\\prime}(0)=\\beta\r\n$$\r\nwhere $\\beta>0$.\r\nDetermine the smallest value of $\\beta$ for which $y_m \\geq 4$.",
+  "answer_latex": " 16.3923",
+  "answer_number": "16.3923",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page145-26",
+  "comment": " "
+ },
+ {
+  "problem_text": "A home buyer can afford to spend no more than $\\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \\%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.\r\nDetermine the total interest paid during the term of the mortgage.",
+  "answer_latex": " 102,965.21",
+  "answer_number": "102,965.21",
+  "unit": " $",
+  "source": "diff",
+  "problemid": " page61-10",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find the fundamental period of the given function:\r\n$$f(x)=\\left\\{\\begin{array}{ll}(-1)^n, & 2 n-1 \\leq x<2 n, \\\\ 1, & 2 n \\leq x<2 n+1 ;\\end{array} \\quad n=0, \\pm 1, \\pm 2, \\ldots\\right.$$",
+  "answer_latex": " 4",
+  "answer_number": "4",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page593-8",
+  "comment": " "
+ },
+ {
+  "problem_text": "A homebuyer wishes to finance the purchase with a \\$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \\$900?",
+  "answer_latex": " 9.73",
+  "answer_number": " 9.73",
+  "unit": " %",
+  "source": "diff",
+  "problemid": " page131-13",
+  "comment": " "
+ },
+ {
+  "problem_text": "A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?",
+  "answer_latex": "804.62",
+  "answer_number": "804.62",
+  "unit": "$",
+  "source": "diff",
+  "problemid": " page131-10",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let a metallic rod $20 \\mathrm{~cm}$ long be heated to a uniform temperature of $100^{\\circ} \\mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\\circ} \\mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \\mathrm{~s}$ if the bar is made of silver.",
+  "answer_latex": " 35.91",
+  "answer_number": " 35.91",
+  "unit": " ${ }^{\\circ} \\mathrm{C}$",
+  "source": "diff",
+  "problemid": " page619-18",
+  "comment": " "
+ },
+ {
+  "problem_text": "Find $\\gamma$ so that the solution of the initial value problem $x^2 y^{\\prime \\prime}-2 y=0, y(1)=1, y^{\\prime}(1)=\\gamma$ is bounded as $x \\rightarrow 0$.",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page277-37",
+  "comment": " "
+ },
+ {
+  "problem_text": "A tank contains 100 gal of water and $50 \\mathrm{oz}$ of salt. Water containing a salt concentration of $\\frac{1}{4}\\left(1+\\frac{1}{2} \\sin t\\right) \\mathrm{oz} / \\mathrm{gal}$ flows into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, and the mixture in the tank flows out at the same rate.\r\nThe long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?",
+  "answer_latex": " 0.24995",
+  "answer_number": "0.24995",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " Page 60-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mass weighing $8 \\mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\\gamma$. Determine the value of $\\gamma$ for which the system is critically damped; be sure to give the units for $\\gamma$",
+  "answer_latex": "8",
+  "answer_number": "8",
+  "unit": " $\\mathrm{lb} \\cdot \\mathrm{s} / \\mathrm{ft}$",
+  "source": "diff",
+  "problemid": " page203-17",
+  "comment": " "
+ },
+ {
+  "problem_text": "Your swimming pool containing 60,000 gal of water has been contaminated by $5 \\mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \\mathrm{gal} / \\mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \\mathrm{~g} / \\mathrm{gal}$.",
+  "answer_latex": " 7.136",
+  "answer_number": "7.136",
+  "unit": " hour",
+  "source": "diff",
+  "problemid": " Page 18 19",
+  "comment": " "
+ },
+ {
+  "problem_text": "For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.  If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.",
+  "answer_latex": "$\\frac{2}{49}$",
+  "answer_number": "0.0408",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " 1 25(c)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\n3 u^{\\prime \\prime}-u^{\\prime}+2 u=0, \\quad u(0)=2, \\quad u^{\\prime}(0)=0\r\n$$\r\nFor $t>0$ find the first time at which $|u(t)|=10$.",
+  "answer_latex": " 10.7598",
+  "answer_number": " 10.7598",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page163-23",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the initial value problem\r\n$$\r\n9 y^{\\prime \\prime}+12 y^{\\prime}+4 y=0, \\quad y(0)=a>0, \\quad y^{\\prime}(0)=-1\r\n$$\r\nFind the critical value of $a$ that separates solutions that become negative from those that are always positive.",
+  "answer_latex": " 1.5",
+  "answer_number": "1.5",
+  "unit": " ",
+  "source": "diff",
+  "problemid": " page172-18",
+  "comment": " "
+ }
+]

diff_sol.json

@@ -1 +1,52 @@
-[{"problem_text": "For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \\%$, what will be the balance in the IRA at age 65 ?", "answer_latex": " 588313", "answer_number": "588313", "unit": " $\\$$", "source": "diff", "problemid": " 2.3.2", "comment": " ", "solution": "We have $S_0=0, r=0.08$, and $k=\\$ 2000$, and we wish to determine $S(40)$. From Eq. (16) we have\r\n$$\r\nS(40)=(25,000)\\left(e^{3.2}-1\\right)=\\$ 588,313\r\n$$\r\n"}, {"problem_text": "Suppose that a mass weighing $10 \\mathrm{lb}$ stretches a spring $2 \\mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \\mathrm{ft} / \\mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion.", "answer_latex": "0.18162 ", "answer_number": "0.18162", "unit": " $\\mathrm{ft}$", "source": "diff", "problemid": " 3.7.2", "comment": " ", "solution": "The spring constant is $k=10 \\mathrm{lb} / 2 \\mathrm{in} .=60 \\mathrm{lb} / \\mathrm{ft}$, and the mass is $m=w / g=10 / 32 \\mathrm{lb} \\cdot \\mathrm{s}^2 / \\mathrm{ft}$. Hence the equation of motion reduces to\r\n$$\r\nu^{\\prime \\prime}+192 u=0\r\n$$\r\nand the general solution is\r\n$$\r\nu=A \\cos (8 \\sqrt{3} t)+B \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe solution satisfying the initial conditions $u(0)=1 / 6 \\mathrm{ft}$ and $u^{\\prime}(0)=-1 \\mathrm{ft} / \\mathrm{s}$ is\r\n$$\r\nu=\\frac{1}{6} \\cos (8 \\sqrt{3} t)-\\frac{1}{8 \\sqrt{3}} \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe natural frequency is $\\omega_0=\\sqrt{192} \\cong 13.856 \\mathrm{rad} / \\mathrm{s}$, so the period is $T=2 \\pi / \\omega_0 \\cong 0.45345 \\mathrm{~s}$. The amplitude $R$ and phase $\\delta$ are found from Eqs. (17). We have\r\n$$\r\nR^2=\\frac{1}{36}+\\frac{1}{192}=\\frac{19}{576}, \\quad \\text { so } \\quad R \\cong 0.18162 \\mathrm{ft}\r\n$$"}, {"problem_text": "At time $t=0$ a tank contains $Q_0 \\mathrm{lb}$ of salt dissolved in 100 gal of water; see Figure 2.3.1. Assume that water containing $\\frac{1}{4} \\mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \\mathrm{gal} / \\mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \\%$ of $Q_L$. ", "answer_latex": "$(\\ln 50) / 0.03$", "answer_number": "130.400766848", "unit": " $\\mathrm{~min}$", "source": "diff", "problemid": " 2.3.1", "comment": " ", "solution": "We assume that salt is neither created nor destroyed in the tank. Therefore variations in the amount of salt are due solely to the flows in and out of the tank. More precisely, the rate of change of salt in the tank, $d Q / d t$, is equal to the rate at which salt is flowing in minus the rate at which it is flowing out. In symbols,\r\n$$\r\n\\frac{d Q}{d t}=\\text { rate in }- \\text { rate out }\r\n$$\r\nThe rate at which salt enters the tank is the concentration $\\frac{1}{4} \\mathrm{lb} / \\mathrm{gal}$ times the flow rate $r \\mathrm{gal} / \\mathrm{min}$, or $(r / 4) \\mathrm{lb} / \\mathrm{min}$. To find the rate at which salt leaves the tankl we need to multiply the concentration of salt in the tank by the rate of outflow, $r \\mathrm{gal} / \\mathrm{min}$. Since the rates of flow in and out are equal, the volume of water in the tank remains constant at $100 \\mathrm{gal}$, and since the mixture is \"well-stirred,\" the concentration throughout the tank is the same, namely, $[Q(t) / 100] \\mathrm{lb} / \\mathrm{gal}$.\r\nith First Order Equations\r\n53\r\nTherefore the rate at which salt leaves the tank is $[r Q(t) / 100] \\mathrm{lb} / \\mathrm{min}$. Thus the differential equation governing this process is\r\n$$\r\n\\frac{d Q}{d t}=\\frac{r}{4}-\\frac{r Q}{100}\r\n$$\r\nThe initial condition is\r\n$$\r\nQ(0)=Q_0\r\n$$\r\nUpon thinking about the problem physically, we might anticipate that eventually the mixture originally in the tank will be essentially replaced by the mixture flowing in, whose concentration is $\\frac{1}{4} \\mathrm{lb} / \\mathrm{gal}$. Consequently, we might expect that ultimately the amount of salt in the tank would be very close to $25 \\mathrm{lb}$. We can also find the limiting amount $Q_L=25$ by setting $d Q / d t$ equal to zero in Eq. (2) and solving the resulting algebraic equation for $Q$.\r\n\r\nTo solve the initial value problem (2), (3) analytically, note that Eq. (2) is both linear and separable. Rewriting it in the standard form for a linear equation, we have\r\n$$\r\n\\frac{d Q}{d t}+\\frac{r Q}{100}=\\frac{r}{4}\r\n$$\r\nThus the integrating factor is $e^{r t / 100}$ and the general solution is\r\n$$\r\nQ(t)=25+c e^{-r t / 100}\r\n$$\r\nwhere $c$ is an arbitrary constant. To satisfy the initial condition (3), we must choose $c=Q_0-25$. Therefore the solution of the initial value problem (2),(3) is\r\n$$\r\nQ(t)=25+\\left(Q_0-25\\right) e^{-r t / 100}\r\n$$\r\nor\r\n$$\r\nQ(t)=25\\left(1-e^{-r t / 100}\\right)+Q_0 e^{-r t / 100}\r\n$$\r\nFrom Eq. (6) or (7), you can see that $Q(t) \\rightarrow 25$ (lb) as $t \\rightarrow \\infty$, so the limiting value $Q_L$ is 25 , confirming our physical intuition. Further, $Q(t)$ approaches the limit more rapidly as $r$ increases. In interpreting the solution (7), note that the second term on the right side is the portion of the original salt that remains at time $t$, while the first term gives the amount of salt in the tank due to the action of the flow processes. Plots of the solution for $r=3$ and for several values of $Q_0$ are shown in Figure 2.3.2.\r\nNow suppose that $r=3$ and $Q_0=2 Q_L=50$; then Eq. (6) becomes\r\n$$\r\nQ(t)=25+25 e^{-0.03 t}\r\n$$\r\nSince $2 \\%$ of 25 is 0.5 , we wish to find the time $T$ at which $Q(t)$ has the value 25.5. Substituting $t=T$ and $Q=25.5$ in Eq. (8) and solving for $T$, we obtain\r\n$$\r\nT=(\\ln 50) / 0.03 \\cong 130.400766848(\\mathrm{~min}) .\r\n$$\r\n"}, {"problem_text": "Suppose that a mass weighing $10 \\mathrm{lb}$ stretches a spring $2 \\mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \\mathrm{ft} / \\mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion.", "answer_latex": " $-\\arctan (\\sqrt{3} / 4)$", "answer_number": "-0.40864", "unit": " $\\mathrm{rad}$", "source": "diff", "problemid": " 3.7.2", "comment": " ", "solution": "The spring constant is $k=10 \\mathrm{lb} / 2 \\mathrm{in} .=60 \\mathrm{lb} / \\mathrm{ft}$, and the mass is $m=w / g=10 / 32 \\mathrm{lb} \\cdot \\mathrm{s}^2 / \\mathrm{ft}$. Hence the equation of motion reduces to\r\n$$\r\nu^{\\prime \\prime}+192 u=0\r\n$$\r\nand the general solution is\r\n$$\r\nu=A \\cos (8 \\sqrt{3} t)+B \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe solution satisfying the initial conditions $u(0)=1 / 6 \\mathrm{ft}$ and $u^{\\prime}(0)=-1 \\mathrm{ft} / \\mathrm{s}$ is\r\n$$\r\nu=\\frac{1}{6} \\cos (8 \\sqrt{3} t)-\\frac{1}{8 \\sqrt{3}} \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe natural frequency is $\\omega_0=\\sqrt{192} \\cong 13.856 \\mathrm{rad} / \\mathrm{s}$, so the period is $T=2 \\pi / \\omega_0 \\cong 0.45345 \\mathrm{~s}$. The amplitude $R$ and phase $\\delta$ are found from Eqs. (17). We have\r\n$$\r\nR^2=\\frac{1}{36}+\\frac{1}{192}=\\frac{19}{576}, \\quad \\text { so } \\quad R \\cong 0.18162 \\mathrm{ft}\r\n$$\r\nThe second of Eqs. (17) yields $\\tan \\delta=-\\sqrt{3} / 4$. There are two solutions of this equation, one in the second quadrant and one in the fourth. In the present problem $\\cos \\delta>0$ and $\\sin \\delta<0$, so $\\delta$ is in the fourth quadrant, namely,\r\n$$\r\n\\delta=-\\arctan (\\sqrt{3} / 4) \\cong-0.40864 \\mathrm{rad}\r\n$$\r\n"}, {"problem_text": "The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \\times 10^6 \\mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later. ", "answer_latex": " 46.7", "answer_number": "46.7", "unit": " $10^6 \\mathrm{~kg}$", "source": "diff", "problemid": "2.5.1 ", "comment": " ", "solution": "It is convenient to scale the solution (11) to the carrying capacity $K$; thus we write Eq. (11) in the form\r\n$$\r\n\\frac{y}{K}=\\frac{y_0 / K}{\\left(y_0 / K\\right)+\\left[1-\\left(y_0 / K\\right)\\right] e^{-r t}}\r\n$$\r\nUsing the data given in the problem, we find that\r\n$$\r\n\\frac{y(2)}{K}=\\frac{0.25}{0.25+0.75 e^{-1.42}} \\cong 0.5797 .\r\n$$\r\nConsequently, $y(2) \\cong 46.7 \\times 10^6 \\mathrm{~kg}$.\r\n"}]
+[
+ {
+  "problem_text": "For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \\%$, what will be the balance in the IRA at age 65 ?",
+  "answer_latex": " 588313",
+  "answer_number": "588313",
+  "unit": " $\\$$",
+  "source": "diff",
+  "problemid": " 2.3.2",
+  "comment": " ",
+  "solution": "\nWe have $S_0=0, r=0.08$, and $k=\\$ 2000$, and we wish to determine $S(40)$. From Eq. $$\nS(t)=S_0 e^{r t}+(k / r)\\left(e^{r t}-1\\right)\n$$ we have\r\n$$\r\nS(40)=(25,000)\\left(e^{3.2}-1\\right)=\\$ 588,313\r\n$$\n"
+ },
+ {
+  "problem_text": "Suppose that a mass weighing $10 \\mathrm{lb}$ stretches a spring $2 \\mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \\mathrm{ft} / \\mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion.",
+  "answer_latex": "0.18162 ",
+  "answer_number": "0.18162",
+  "unit": " $\\mathrm{ft}$",
+  "source": "diff",
+  "problemid": " 3.7.2",
+  "comment": " ",
+  "solution": "\nThe spring constant is $k=10 \\mathrm{lb} / 2 \\mathrm{in} .=60 \\mathrm{lb} / \\mathrm{ft}$, and the mass is $m=w / g=10 / 32 \\mathrm{lb} \\cdot \\mathrm{s}^2 / \\mathrm{ft}$. Hence the equation of motion reduces to\r\n$$\r\nu^{\\prime \\prime}+192 u=0\r\n$$\r\nand the general solution is\r\n$$\r\nu=A \\cos (8 \\sqrt{3} t)+B \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe solution satisfying the initial conditions $u(0)=1 / 6 \\mathrm{ft}$ and $u^{\\prime}(0)=-1 \\mathrm{ft} / \\mathrm{s}$ is\r\n$$\r\nu=\\frac{1}{6} \\cos (8 \\sqrt{3} t)-\\frac{1}{8 \\sqrt{3}} \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe natural frequency is $\\omega_0=\\sqrt{192} \\cong 13.856 \\mathrm{rad} / \\mathrm{s}$, so the period is $T=2 \\pi / \\omega_0 \\cong 0.45345 \\mathrm{~s}$. The amplitude $R$ and phase $\\delta$ are found from Eqs. $$\nR=\\sqrt{A^2+B^2}, \\quad \\tan \\delta=B / A\n$$. We have\r\n$$\r\nR^2=\\frac{1}{36}+\\frac{1}{192}=\\frac{19}{576}, \\quad \\text { so } \\quad R \\cong 0.18162 \\mathrm{ft}\r\n$$\n"
+ },
+ {
+  "problem_text": "At time $t=0$ a tank contains $Q_0 \\mathrm{lb}$ of salt dissolved in 100 gal of water. Assume that water containing $\\frac{1}{4} \\mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \\mathrm{gal} / \\mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \\%$ of $Q_L$.",
+  "answer_latex": "$(\\ln 50) / 0.03$",
+  "answer_number": "130.400766848",
+  "unit": " $\\mathrm{~min}$",
+  "source": "diff",
+  "problemid": " 2.3.1",
+  "comment": " ",
+  "solution": "\nWe assume that salt is neither created nor destroyed in the tank. Therefore variations in the amount of salt are due solely to the flows in and out of the tank. More precisely, the rate of change of salt in the tank, $d Q / d t$, is equal to the rate at which salt is flowing in minus the rate at which it is flowing out. In symbols,\r\n$$\r\n\\frac{d Q}{d t}=\\text { rate in }- \\text { rate out }\r\n$$\r\nThe rate at which salt enters the tank is the concentration $\\frac{1}{4} \\mathrm{lb} / \\mathrm{gal}$ times the flow rate $r \\mathrm{gal} / \\mathrm{min}$, or $(r / 4) \\mathrm{lb} / \\mathrm{min}$. To find the rate at which salt leaves the tankl we need to multiply the concentration of salt in the tank by the rate of outflow, $r \\mathrm{gal} / \\mathrm{min}$. Since the rates of flow in and out are equal, the volume of water in the tank remains constant at $100 \\mathrm{gal}$, and since the mixture is \"well-stirred,\" the concentration throughout the tank is the same, namely, $[Q(t) / 100] \\mathrm{lb} / \\mathrm{gal}$. Therefore the rate at which salt leaves the tank is $[r Q(t) / 100] \\mathrm{lb} / \\mathrm{min}$. Thus the differential equation governing this process is\r\n$$\r\n\\frac{d Q}{d t}=\\frac{r}{4}-\\frac{r Q}{100}\r\n$$\r\nThe initial condition is\r\n$$\r\nQ(0)=Q_0\r\n$$\r\nUpon thinking about the problem physically, we might anticipate that eventually the mixture originally in the tank will be essentially replaced by the mixture flowing in, whose concentration is $\\frac{1}{4} \\mathrm{lb} / \\mathrm{gal}$. Consequently, we might expect that ultimately the amount of salt in the tank would be very close to $25 \\mathrm{lb}$. We can also find the limiting amount $Q_L=25$ by setting $d Q / d t$ equal to zero in the equation and solving the resulting algebraic equation for $Q$. Rewriting the above equation in the standard form for a linear equation, we have\r\n$$\r\n\\frac{d Q}{d t}+\\frac{r Q}{100}=\\frac{r}{4}\r\n$$\r\nThus the integrating factor is $e^{r t / 100}$ and the general solution is\r\n$$\r\nQ(t)=25+c e^{-r t / 100}\r\n$$\r\nwhere $c$ is an arbitrary constant. To satisfy the initial condition, we must choose $c=Q_0-25$. Therefore the solution of the initial value problem is\r\n$$\r\nQ(t)=25+\\left(Q_0-25\\right) e^{-r t / 100}\r\n$$\r\nor\r\n$$\r\nQ(t)=25\\left(1-e^{-r t / 100}\\right)+Q_0 e^{-r t / 100}\r\n$$\r\nFrom Eq., you can see that $Q(t) \\rightarrow 25$ (lb) as $t \\rightarrow \\infty$, so the limiting value $Q_L$ is 25 , confirming our physical intuition. Further, $Q(t)$ approaches the limit more rapidly as $r$ increases. In interpreting the solution, note that the second term on the right side is the portion of the original salt that remains at time $t$, while the first term gives the amount of salt in the tank due to the action of the flow processes. Now suppose that $r=3$ and $Q_0=2 Q_L=50$; then Eq. becomes\r\n$$\r\nQ(t)=25+25 e^{-0.03 t}\r\n$$\r\nSince $2 \\%$ of 25 is 0.5 , we wish to find the time $T$ at which $Q(t)$ has the value 25.5. Substituting $t=T$ and $Q=25.5$ in Eq. (8) and solving for $T$, we obtain\r\n$$\r\nT=(\\ln 50) / 0.03 \\cong 130.400766848(\\mathrm{~min}) .\r\n$$\n"
+ },
+ {
+  "problem_text": "Suppose that a mass weighing $10 \\mathrm{lb}$ stretches a spring $2 \\mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \\mathrm{ft} / \\mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion.",
+  "answer_latex": " $-\\arctan (\\sqrt{3} / 4)$",
+  "answer_number": "-0.40864",
+  "unit": " $\\mathrm{rad}$",
+  "source": "diff",
+  "problemid": " 3.7.2",
+  "comment": " ",
+  "solution": "\nThe spring constant is $k=10 \\mathrm{lb} / 2 \\mathrm{in} .=60 \\mathrm{lb} / \\mathrm{ft}$, and the mass is $m=w / g=10 / 32 \\mathrm{lb} \\cdot \\mathrm{s}^2 / \\mathrm{ft}$. Hence the equation of motion reduces to\r\n$$\r\nu^{\\prime \\prime}+192 u=0\r\n$$\r\nand the general solution is\r\n$$\r\nu=A \\cos (8 \\sqrt{3} t)+B \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe solution satisfying the initial conditions $u(0)=1 / 6 \\mathrm{ft}$ and $u^{\\prime}(0)=-1 \\mathrm{ft} / \\mathrm{s}$ is\r\n$$\r\nu=\\frac{1}{6} \\cos (8 \\sqrt{3} t)-\\frac{1}{8 \\sqrt{3}} \\sin (8 \\sqrt{3} t)\r\n$$\r\nThe natural frequency is $\\omega_0=\\sqrt{192} \\cong 13.856 \\mathrm{rad} / \\mathrm{s}$, so the period is $T=2 \\pi / \\omega_0 \\cong 0.45345 \\mathrm{~s}$. The amplitude $R$ and phase $\\delta$ are found from Eqs. (17) $$R=\\sqrt{A^2+B^2}, \\quad \\tan \\delta=B / A$$. We have\r\n$$\r\nR^2=\\frac{1}{36}+\\frac{1}{192}=\\frac{19}{576}, \\quad \\text { so } \\quad R \\cong 0.18162 \\mathrm{ft}\r\n$$\r\nThe second of Eqs. (17) yields $\\tan \\delta=-\\sqrt{3} / 4$. There are two solutions of this equation, one in the second quadrant and one in the fourth. In the present problem $\\cos \\delta>0$ and $\\sin \\delta<0$, so $\\delta$ is in the fourth quadrant, namely,\r\n$$\r\n\\delta=-\\arctan (\\sqrt{3} / 4) \\cong-0.40864 \\mathrm{rad}\r\n$$"
+ },
+ {
+  "problem_text": "The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \\times 10^6 \\mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later. ",
+  "answer_latex": " 46.7",
+  "answer_number": "46.7",
+  "unit": " $10^6 \\mathrm{~kg}$",
+  "source": "diff",
+  "problemid": "2.5.1 ",
+  "comment": " ",
+  "solution": "\nIt is convenient to scale the solution (11) $$y=\\frac{y_0 K}{y_0+\\left(K-y_0\\right) e^{-r t}}\r\n$$ to the carrying capacity $K$; thus we write Eq. (11) in the form\r\n$$\r\n\\frac{y}{K}=\\frac{y_0 / K}{\\left(y_0 / K\\right)+\\left[1-\\left(y_0 / K\\right)\\right] e^{-r t}}\r\n$$\r\nUsing the data given in the problem, we find that\r\n$$\r\n\\frac{y(2)}{K}=\\frac{0.25}{0.25+0.75 e^{-1.42}} \\cong 0.5797 .\r\n$$\r\nConsequently, $y(2) \\cong 46.7 \\times 10^6 \\mathrm{~kg}$.\n"
+ }
+]

fund.json

@@ -1 +1,704 @@
-[{"problem_text": "In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\\vec{a}, 2.0 \\mathrm{~km}$ due east (directly toward the east); (b) $\\vec{b}, 2.0 \\mathrm{~km} 30^{\\circ}$ north of east (at an angle of $30^{\\circ}$ toward the north from due east); (c) $\\vec{c}, 1.0 \\mathrm{~km}$ due west. Alternatively, you may substitute either $-\\vec{b}$ for $\\vec{b}$ or $-\\vec{c}$ for $\\vec{c}$. What is the greatest distance you can be from base camp at the end of the third displacement? (We are not concerned about the direction.)", "answer_latex": " 4.8", "answer_number": "4.8", "unit": " m", "source": "fund", "problemid": " 3.01", "comment": " "}, {"problem_text": "\"Top gun\" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function.\r\nThere are several warning signs. When the centripetal acceleration is $2 g$ or $3 g$, the pilot feels heavy. At about $4 g$, the pilot's vision switches to black and white and narrows to \"tunnel vision.\" If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious - a condition known as $g$-LOC for \" $g$-induced loss of consciousness.\"\r\n\r\nWhat is the magnitude of the acceleration, in $g$ units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of $\\vec{v}_i=(400 \\hat{\\mathrm{i}}+500 \\hat{\\mathrm{j}}) \\mathrm{m} / \\mathrm{s}$ and $24.0 \\mathrm{~s}$ later leaves the turn with a velocity of $\\vec{v}_f=(-400 \\hat{\\mathrm{i}}-500 \\hat{\\mathrm{j}}) \\mathrm{m} / \\mathrm{s}$ ?", "answer_latex": "83.81", "answer_number": "83.81", "unit": " $\\mathrm{m} / \\mathrm{s}^2$", "source": "fund", "problemid": " 4.06", "comment": " "}, {"problem_text": "The world\u2019s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball?", "answer_latex": "2", "answer_number": "2", "unit": " $10^6$ m", "source": "fund", "problemid": " 1.01", "comment": " "}, {"problem_text": "You drive a beat-up pickup truck along a straight road for $8.4 \\mathrm{~km}$ at $70 \\mathrm{~km} / \\mathrm{h}$, at which point the truck runs out of gasoline and stops. Over the next $30 \\mathrm{~min}$, you walk another $2.0 \\mathrm{~km}$ farther along the road to a gasoline station.\r\nWhat is your overall displacement from the beginning of your drive to your arrival at the station?", "answer_latex": " 10.4", "answer_number": "10.4", "unit": " km", "source": "fund", "problemid": " 2.01", "comment": " "}, {"problem_text": "A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms of the void ratio $e$ for a sample of the ground:\r\n$$\r\ne=\\frac{V_{\\text {voids }}}{V_{\\text {grains }}} .\r\n$$\r\nHere, $V_{\\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\\rho_{\\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\\rho_{\\mathrm{SiO}_2}=2.600 \\times 10^3 \\mathrm{~kg} / \\mathrm{m}^3$.", "answer_latex": "1.4", "answer_number": "1.4", "unit": " $10^3 \\mathrm{~kg} / \\mathrm{m}^3$", "source": "fund", "problemid": "1.02 ", "comment": " "}, {"problem_text": "What is the angle $\\phi$ between $\\vec{a}=3.0 \\hat{\\mathrm{i}}-4.0 \\hat{\\mathrm{j}}$ and $\\vec{b}=$ $-2.0 \\hat{\\mathrm{i}}+3.0 \\hat{\\mathrm{k}}$ ? (Caution: Although many of the following steps can be bypassed with a vector-capable calculator, you will learn more about scalar products if, at least here, you use these steps.)", "answer_latex": "109", "answer_number": "109", "unit": " $^{\\circ}$", "source": "fund", "problemid": " 3.05", "comment": " "}, {"problem_text": "During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\\vec{d}=(-3.0 \\mathrm{~m}) \\hat{\\mathrm{i}}$ while a steady wind pushes against the crate with a force $\\vec{F}=(2.0 \\mathrm{~N}) \\hat{\\mathrm{i}}+(-6.0 \\mathrm{~N}) \\hat{\\mathrm{j}}$. The situation and coordinate axes are shown in Figure. If the crate has a kinetic energy of $10 \\mathrm{~J}$ at the beginning of displacement $\\vec{d}$, what is its kinetic energy at the end of $\\vec{d}$ ?", "answer_latex": " 4.0", "answer_number": "4.0", "unit": " J", "source": "fund", "problemid": "  7.03", "comment": " "}, {"problem_text": "When the force on an object depends on the position of the object, we cannot find the work done by it on the object by simply multiplying the force by the displacement. The reason is that there is no one value for the force-it changes. So, we must find the work in tiny little displacements and then add up all the work results. We effectively say, \"Yes, the force varies over any given tiny little displacement, but the variation is so small we can approximate the force as being constant during the displacement.\" Sure, it is not precise, but if we make the displacements infinitesimal, then our error becomes infinitesimal and the result becomes precise. But, to add an infinite number of work contributions by hand would take us forever, longer than a semester. So, we add them up via an integration, which allows us to do all this in minutes (much less than a semester).\r\n\r\nForce $\\vec{F}=\\left(3 x^2 \\mathrm{~N}\\right) \\hat{\\mathrm{i}}+(4 \\mathrm{~N}) \\hat{\\mathrm{j}}$, with $x$ in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates $(2 \\mathrm{~m}, 3 \\mathrm{~m})$ to $(3 \\mathrm{~m}, 0 \\mathrm{~m})$ ? Does the speed of the particle increase, decrease, or remain the same?", "answer_latex": " 7.0", "answer_number": " 7.0", "unit": " J", "source": "fund", "problemid": " 7.08", "comment": " "}, {"problem_text": " The charges of an electron and a positron are $-e$ and $+e$. The mass of each is $9.11 \\times 10^{-31} \\mathrm{~kg}$. What is the ratio of the electrical force to the gravitational force between an electron and a positron?\r\n", "answer_latex": " $4.16$", "answer_number": "4.16", "unit": "$10^{42}$", "source": "fund", "problemid": " Question 21.75", "comment": " ", "solution": "$4.16 \\times 10^{42}$"}, {"problem_text": "In Fig. 21-26, particle 1 of charge $+q$ and particle 2 of charge $+4.00 q$ are held at separation $L=9.00 \\mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_3$ is to be located such that the three particles remain in place when released, what must be the $x$ coordinate of particle 3?\r\n", "answer_latex": " $3.00$", "answer_number": "3.00", "unit": "$\\mathrm{~cm}$", "source": "fund", "problemid": " Question 21.19", "comment": " ", "solution": "$3.00 \\mathrm{~cm}$"}, {"problem_text": "Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_1=2.1 \\times 10^{-8} \\mathrm{C}$ is at position $x=20 \\mathrm{~cm}$ and particle 2 of charge $q_2=-4.00 q_1$ is at position $x=70 \\mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?\r\n", "answer_latex": " $-30$", "answer_number": "-30", "unit": " $\\mathrm{~cm}$", "source": "fund", "problemid": " Question 22.11", "comment": " ", "solution": "$-30 \\mathrm{~cm}$"}, {"problem_text": "The volume charge density of a solid nonconducting sphere of radius $R=5.60 \\mathrm{~cm}$ varies with radial distance $r$ as given by $\\rho=$ $\\left(14.1 \\mathrm{pC} / \\mathrm{m}^3\\right) r / R$. What is the sphere's total charge?", "answer_latex": " $7.78$", "answer_number": "7.78", "unit": "$\\mathrm{fC} $ ", "source": "fund", "problemid": " Question 23.53", "comment": " ", "solution": "$7.78 \\mathrm{fC} $"}, {"problem_text": "Two charged concentric spher-\r\nical shells have radii $10.0 \\mathrm{~cm}$ and $15.0 \\mathrm{~cm}$. The charge on the inner shell is $4.00 \\times 10^{-8} \\mathrm{C}$, and that on the outer shell is $2.00 \\times 10^{-8} \\mathrm{C}$. Find the electric field at $r=12.0 \\mathrm{~cm}$.", "answer_latex": " $2.50$", "answer_number": "2.50", "unit": "$10^4 \\mathrm{~N} / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 23.45", "comment": " ", "solution": "$2.50 \\times 10^4 \\mathrm{~N} / \\mathrm{C}$"}, {"problem_text": "Assume that a honeybee is a sphere of diameter 1.000 $\\mathrm{cm}$ with a charge of $+45.0 \\mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \\mu \\mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \\mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \\mathrm{pC}$ on the far side. What is the magnitude of the net electrostatic force on the grain due to the bee? ", "answer_latex": " $2.6$", "answer_number": "2.6", "unit": "$10^{-10} \\mathrm{~N}$ ", "source": "fund", "problemid": " Question 22.51", "comment": " ", "solution": "$2.6 \\times 10^{-10} \\mathrm{~N}$"}, {"problem_text": "In the radioactive decay of Eq. 21-13, $\\mathrm{a}^{238} \\mathrm{U}$ nucleus transforms to ${ }^{234} \\mathrm{Th}$ and an ejected ${ }^4 \\mathrm{He}$. (These are nuclei, not atoms, and thus electrons are not involved.) When the separation between ${ }^{234} \\mathrm{Th}$ and ${ }^4 \\mathrm{He}$ is $9.0 \\times 10^{-15} \\mathrm{~m}$, what are the magnitudes of the electrostatic force between them?\r\n", "answer_latex": " $5.1$", "answer_number": "5.1", "unit": " $10^2 \\mathrm{~N}$", "source": "fund", "problemid": " Question 21.69", "comment": " ", "solution": "$5.1 \\times 10^2 \\mathrm{~N}$"}, {"problem_text": "The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \\hat{\\mathrm{i}}+3.0 \\hat{\\mathrm{j}}) \\mathrm{N} / \\mathrm{C}$ at the point $(3.0,3.0) \\mathrm{cm}$ and $100 \\hat{\\mathrm{i}} \\mathrm{N} / \\mathrm{C}$ at the point $(2.0,0) \\mathrm{cm}$. What is the $x$ coordinate of the particle?", "answer_latex": " $-1.0$", "answer_number": "-1.0", "unit": "$\\mathrm{~cm}$ ", "source": "fund", "problemid": " Question 22.73", "comment": " ", "solution": "$-1.0 \\mathrm{~cm}$"}, {"problem_text": "A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude $E=K r^4$, directed radially outward from the center of the sphere. Here $r$ is the radial distance from that center, and $K$ is a constant. What is the volume density $\\rho$ of the charge distribution?", "answer_latex": " $6$", "answer_number": "6", "unit": " $K \\varepsilon_0 r^3$", "source": "fund", "problemid": " Question 23.55", "comment": " ", "solution": "$6 K \\varepsilon_0 r^3$"}, {"problem_text": "Two particles, each with a charge of magnitude $12 \\mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \\mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive?", "answer_latex": " $47$", "answer_number": "47", "unit": "$\\mathrm{~N} / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 22.69", "comment": " ", "solution": "$47 \\mathrm{~N} / \\mathrm{C} $"}, {"problem_text": "How much work is required to turn an electric dipole $180^{\\circ}$ in a uniform electric field of magnitude $E=46.0 \\mathrm{~N} / \\mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \\times$ $10^{-25} \\mathrm{C} \\cdot \\mathrm{m}$ and the initial angle is $64^{\\circ} ?$\r\n", "answer_latex": " $1.22$", "answer_number": "1.22", "unit": "$10^{-23} \\mathrm{~J}$ ", "source": "fund", "problemid": " Question 22.59", "comment": " ", "solution": "$1.22 \\times 10^{-23} \\mathrm{~J}$"}, {"problem_text": "We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \\%$. With what force would two copper coins, placed $1.0 \\mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \\times 10^{22}$ copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) What do you conclude?", "answer_latex": " $1.7$", "answer_number": "1.7", "unit": "$10^8 \\mathrm{~N}$ ", "source": "fund", "problemid": " Question 21.57", "comment": " ", "solution": "$1.7 \\times 10^8 \\mathrm{~N}$"}, {"problem_text": "What must be the distance between point charge $q_1=$ $26.0 \\mu \\mathrm{C}$ and point charge $q_2=-47.0 \\mu \\mathrm{C}$ for the electrostatic force between them to have a magnitude of $5.70 \\mathrm{~N}$ ?\r\n", "answer_latex": " 1.39", "answer_number": "1.39", "unit": " m", "source": "fund", "problemid": " Question 21.3", "comment": " ", "solution": "1.39"}, {"problem_text": "Three charged particles form a triangle: particle 1 with charge $Q_1=80.0 \\mathrm{nC}$ is at $x y$ coordinates $(0,3.00 \\mathrm{~mm})$, particle 2 with charge $Q_2$ is at $(0,-3.00 \\mathrm{~mm})$, and particle 3 with charge $q=18.0$ $\\mathrm{nC}$ is at $(4.00 \\mathrm{~mm}, 0)$. In unit-vector notation, what is the electrostatic force on particle 3 due to the other two particles if $Q_2$ is equal to $80.0 \\mathrm{nC}$?\r\n", "answer_latex": "$(0.829)$", "answer_number": "0.829", "unit": " $\\mathrm{~N} \\hat{\\mathrm{i}}$", "source": "fund", "problemid": " Question 21.61", "comment": " ", "solution": "$(0.829 \\mathrm{~N}) \\hat{\\mathrm{i}}$"}, {"problem_text": "A particle of charge $-q_1$ is at the origin of an $x$ axis. (a) At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \\mathrm{~mm}$ on the axis? ", "answer_latex": " $6.0$", "answer_number": "6.0", "unit": "$\\mathrm{~mm}$ ", "source": "fund", "problemid": " Question 22.77", "comment": " ", "solution": "$6.0 \\mathrm{~mm}$"}, {"problem_text": "An electron is released from rest in a uniform electric field of magnitude $2.00 \\times 10^4 \\mathrm{~N} / \\mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.)", "answer_latex": " $3.51$", "answer_number": "3.51", "unit": "$10^{15} \\mathrm{~m} / \\mathrm{s}^2$ ", "source": "fund", "problemid": " Question 22.43", "comment": " ", "solution": "$ 3.51 \\times 10^{15} \\mathrm{~m} / \\mathrm{s}^2$"}, {"problem_text": "Identify $\\mathrm{X}$ in the following nuclear reactions: (a) ${ }^1 \\mathrm{H}+$ ${ }^9 \\mathrm{Be} \\rightarrow \\mathrm{X}+\\mathrm{n} ;$ (b) ${ }^{12} \\mathrm{C}+{ }^1 \\mathrm{H} \\rightarrow \\mathrm{X} ;$ (c) ${ }^{15} \\mathrm{~N}+{ }^1 \\mathrm{H} \\rightarrow{ }^4 \\mathrm{He}+\\mathrm{X}$. Appendix F will help.", "answer_latex": " ${ }^9 \\mathrm{~B}$", "answer_number": "", "unit": " ", "source": "fund", "problemid": " Question 21.37", "comment": " ", "solution": "${ }^9 \\mathrm{~B}$"}, {"problem_text": "The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \\mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons?", "answer_latex": "$3.07$", "answer_number": "3.07", "unit": "$10^{21} \\mathrm{~N} / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 22.3", "comment": " ", "solution": "$3.07 \\times 10^{21} \\mathrm{~N} / \\mathrm{C}$"}, {"problem_text": "A nonconducting spherical shell, with an inner radius of $4.0 \\mathrm{~cm}$ and an outer radius of $6.0 \\mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell $\\rho=b / r$, where $r$ is the distance in meters from the center of the shell and $b=3.0 \\mu \\mathrm{C} / \\mathrm{m}^2$. What is the net charge in the shell?\r\n", "answer_latex": " $3.8$", "answer_number": "3.8", "unit": " $10^{-8} \\mathrm{C}$", "source": "fund", "problemid": " Question 21.21", "comment": " ", "solution": "$3.8 \\times 10^{-8} \\mathrm{C}$"}, {"problem_text": "An electron is shot directly\r\nFigure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \\times 10^{-6} \\mathrm{C} / \\mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \\times 10^{-17} \\mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?", "answer_latex": " $0.44$", "answer_number": "0.44", "unit": "$\\mathrm{~mm}$ ", "source": "fund", "problemid": " Question 23.41", "comment": " ", "solution": "$0.44 \\mathrm{~mm}$"}, {"problem_text": "A square metal plate of edge length $8.0 \\mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \\times 10^{-6} \\mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \\mathrm{~mm}$ from the center) by assuming that the charge is spread uniformly over the two faces of the plate. ", "answer_latex": "$5.4$", "answer_number": "5.4", "unit": "$10^7 \\mathrm{~N} / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 23.37", "comment": " ", "solution": "$5.3 \\times 10^7 \\mathrm{~N} / \\mathrm{C}$"}, {"problem_text": "A neutron consists of one \"up\" quark of charge $+2 e / 3$ and two \"down\" quarks each having charge $-e / 3$. If we assume that the down quarks are $2.6 \\times 10^{-15} \\mathrm{~m}$ apart inside the neutron, what is the magnitude of the electrostatic force between them?\r\n", "answer_latex": "$3.8$", "answer_number": "3.8", "unit": "$N$ ", "source": "fund", "problemid": " Question 21.49", "comment": " ", "solution": "$3.8 N$"}, {"problem_text": "In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \\mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit?", "answer_latex": " $2.19$", "answer_number": "2.19", "unit": "$10^6 \\mathrm{~m} / \\mathrm{s}$ ", "source": "fund", "problemid": " Question 21.73", "comment": " ", "solution": "$2.19 \\times 10^6 \\mathrm{~m} / \\mathrm{s}$"}, {"problem_text": "At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \\mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?", "answer_latex": " $0.346$", "answer_number": "0.346", "unit": "$\\mathrm{~m}$ ", "source": "fund", "problemid": " Question 22.35", "comment": " ", "solution": "$0.346 \\mathrm{~m}$"}, {"problem_text": "Of the charge $Q$ on a tiny sphere, a fraction $\\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles.  What value of $\\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres? ", "answer_latex": " $0.5$", "answer_number": "0.5", "unit": " ", "source": "fund", "problemid": " Question 21.55", "comment": " ", "solution": "$0.5$"}, {"problem_text": "In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \\times 10^{-13} \\mathrm{C} / \\mathrm{m}^2$. What is the magnitude of the electron's acceleration when it reaches radial distances $r=0.500 R$?", "answer_latex": " $0$", "answer_number": "0", "unit": " ", "source": "fund", "problemid": " Question 21.71", "comment": " ", "solution": "$0$"}, {"problem_text": "A particle of charge $+3.00 \\times 10^{-6} \\mathrm{C}$ is $12.0 \\mathrm{~cm}$ distant from a second particle of charge $-1.50 \\times 10^{-6} \\mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles.", "answer_latex": "2.81 ", "answer_number": "2.81", "unit": "N ", "source": "fund", "problemid": " Question 21.5", "comment": " ", "solution": "2.81"}, {"problem_text": "A charged particle produces an electric field with a magnitude of $2.0 \\mathrm{~N} / \\mathrm{C}$ at a point that is $50 \\mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?\r\n", "answer_latex": " $56$", "answer_number": "56", "unit": "$\\mathrm{pC}$ ", "source": "fund", "problemid": " Question 22.5", "comment": " ", "solution": "$ 56 \\mathrm{pC}$"}, {"problem_text": "In Millikan's experiment, an oil drop of radius $1.64 \\mu \\mathrm{m}$ and density $0.851 \\mathrm{~g} / \\mathrm{cm}^3$ is suspended in chamber C (Fig. 22-16) when a downward electric field of $1.92 \\times 10^5 \\mathrm{~N} / \\mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$.", "answer_latex": " $-5$", "answer_number": "-5", "unit": "$e$ ", "source": "fund", "problemid": " Question 22.39", "comment": " ", "solution": "$-5 e$"}, {"problem_text": "The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \\mu \\mathrm{C}, x_1=3.5 \\mathrm{~cm}, y_1=0.50 \\mathrm{~cm}$, and $q_2=-4.0 \\mu \\mathrm{C}, x_2=-2.0 \\mathrm{~cm}, y_2=1.5 \\mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1.", "answer_latex": "$35$", "answer_number": "35", "unit": "$\\mathrm{~N}$", "source": "fund", "problemid": " Question 21.15", "comment": " ", "solution": "$35$"}, {"problem_text": "An electron on the axis of an electric dipole is $25 \\mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \\times 10^{-29} \\mathrm{C} \\cdot \\mathrm{m}$ ? Assume that $25 \\mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole.", "answer_latex": " $6.6$", "answer_number": "6.6", "unit": "$10^{-15} \\mathrm{~N}$ ", "source": "fund", "problemid": " Question 22.45", "comment": " ", "solution": "$ 6.6 \\times 10^{-15} \\mathrm{~N}$"}, {"problem_text": "Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?", "answer_latex": "$122$", "answer_number": "122", "unit": " $\\mathrm{~mA}$", "source": "fund", "problemid": " Question 21.31", "comment": " ", "solution": "$122 \\mathrm{~mA}$"}, {"problem_text": "An electric field $\\vec{E}$ with an average magnitude of about $150 \\mathrm{~N} / \\mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to \"float\" a sulfur sphere weighing $4.4 \\mathrm{~N}$ in this field by charging the sphere. What charge (both sign and magnitude) must be used?", "answer_latex": " $-0.029$", "answer_number": "-0.029", "unit": " $C$", "source": "fund", "problemid": " Question 22.81", "comment": " ", "solution": "$-0.029 C$"}, {"problem_text": "Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \\mathrm{~N}$ when their center-to-center separation is $50.0 \\mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.0360 \\mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them?", "answer_latex": " $-1.00 \\mu \\mathrm{C}$", "answer_number": "-1.00", "unit": "$ \\mu \\mathrm{C}$", "source": "fund", "problemid": " Question 21.9", "comment": " ", "solution": "$-1.00 \\mu \\mathrm{C}$"}, {"problem_text": "A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \\mathrm{~cm}$ away, in a time $1.5 \\times 10^{-8} \\mathrm{~s}$. What is the speed of the electron as it strikes the second plate? ", "answer_latex": " $2.7$", "answer_number": "2.7", "unit": " $10^6$", "source": "fund", "problemid": " Question 22.55", "comment": " ", "solution": "$2.7 \\times 10^6$"}, {"problem_text": " Two point charges of $30 \\mathrm{nC}$ and $-40 \\mathrm{nC}$ are held fixed on an $x$ axis, at the origin and at $x=72 \\mathrm{~cm}$, respectively. A particle with a charge of $42 \\mu \\mathrm{C}$ is released from rest at $x=28 \\mathrm{~cm}$. If the initial acceleration of the particle has a magnitude of $100 \\mathrm{~km} / \\mathrm{s}^2$, what is the particle's mass?", "answer_latex": " $2.2$", "answer_number": "2.2", "unit": " $10^{-6} \\mathrm{~kg}$", "source": "fund", "problemid": " Question 21.63", "comment": " ", "solution": "$2.2 \\times 10^{-6} \\mathrm{~kg}$"}, {"problem_text": "In Fig. 21-26, particle 1 of charge $-5.00 q$ and particle 2 of charge $+2.00 q$ are held at separation $L$ on an $x$ axis. If particle 3 of unknown charge $q_3$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the $x$ coordinate of particle 3?", "answer_latex": " $2.72$", "answer_number": "2.72", "unit": " $L$", "source": "fund", "problemid": " Question 21.67", "comment": " ", "solution": "$2.72 L$"}, {"problem_text": "An isolated conductor has net charge $+10 \\times 10^{-6} \\mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \\times 10^{-6} \\mathrm{C}$. What is the charge on the cavity wall?", "answer_latex": " $-3.0$", "answer_number": "-3.0", "unit": "$10^{-6} \\mathrm{C} $ ", "source": "fund", "problemid": " Question 23.21", "comment": " ", "solution": "$-3.0 \\times 10^{-6} \\mathrm{C} $"}, {"problem_text": "Point charges of $+6.0 \\mu \\mathrm{C}$ and $-4.0 \\mu \\mathrm{C}$ are placed on an $x$ axis, at $x=8.0 \\mathrm{~m}$ and $x=16 \\mathrm{~m}$, respectively. What charge must be placed at $x=24 \\mathrm{~m}$ so that any charge placed at the origin would experience no electrostatic force?\r\n", "answer_latex": " $-45$", "answer_number": "-45", "unit": " $\\mu \\mathrm{C}$", "source": "fund", "problemid": " Question 21.47", "comment": " ", "solution": "$-45 \\mu \\mathrm{C}$"}, {"problem_text": "The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of $300 \\mathrm{~m}$ the field has magnitude $60.0 \\mathrm{~N} / \\mathrm{C}$; at an altitude of $200 \\mathrm{~m}$, the magnitude is $100 \\mathrm{~N} / \\mathrm{C}$. Find the net amount of charge contained in a cube $100 \\mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \\mathrm{~m}$.", "answer_latex": " $3.54$", "answer_number": "3.54", "unit": "$\\mu \\mathrm{C}$ ", "source": "fund", "problemid": " Question 23.13", "comment": " ", "solution": "$3.54 \\mu \\mathrm{C}$"}, {"problem_text": "What would be the magnitude of the electrostatic force between two 1.00 C point charges separated by a distance of $1.00 \\mathrm{~m}$  if such point charges existed (they do not) and this configuration could be set up?", "answer_latex": " $8.99$", "answer_number": "8.99", "unit": " $10^9 \\mathrm{~N}$", "source": "fund", "problemid": " Question 21.53", "comment": " ", "solution": "$8.99 \\times 10^9 \\mathrm{~N}$"}, {"problem_text": "An electric dipole consisting of charges of magnitude $1.50 \\mathrm{nC}$ separated by $6.20 \\mu \\mathrm{m}$ is in an electric field of strength 1100 $\\mathrm{N} / \\mathrm{C}$. What is the magnitude of the electric dipole moment?", "answer_latex": " $9.30$", "answer_number": "9.30", "unit": "$10^{-15} \\mathrm{C} \\cdot \\mathrm{m}$ ", "source": "fund", "problemid": " Question 22.57", "comment": " ", "solution": "$9.30 \\times 10^{-15} \\mathrm{C} \\cdot \\mathrm{m}$"}, {"problem_text": "What equal positive charges would have to be placed on Earth and on the Moon to neutralize their gravitational attraction?", "answer_latex": " $5.7$", "answer_number": "5.7", "unit": "$10^{13} \\mathrm{C}$", "source": "fund", "problemid": " Question 21.41", "comment": " ", "solution": "$5.7 \\times 10^{13} \\mathrm{C}$"}, {"problem_text": "The initial charges on the three identical metal spheres in Fig. 21-24 are the following: sphere $A, Q$; sphere $B,-Q / 4$; and sphere $C, Q / 2$, where $Q=2.00 \\times 10^{-14}$ C. Spheres $A$ and $B$ are fixed in place, with a center-to-center separation of $d=1.20 \\mathrm{~m}$, which is much larger than the spheres. Sphere $C$ is touched first to sphere $A$ and then to sphere $B$ and is then removed. What then is the magnitude of the electrostatic force between spheres $A$ and $B$ ?\r\n", "answer_latex": " $4.68$", "answer_number": "4.68", "unit": " $10^{-19} \\mathrm{~N}$", "source": "fund", "problemid": " Question 21.65", "comment": " ", "solution": "$4.68 \\times 10^{-19} \\mathrm{~N}$"}, {"problem_text": "A $10.0 \\mathrm{~g}$ block with a charge of $+8.00 \\times 10^{-5} \\mathrm{C}$ is placed in an electric field $\\vec{E}=(3000 \\hat{\\mathrm{i}}-600 \\hat{\\mathrm{j}}) \\mathrm{N} / \\mathrm{C}$. What is the magnitude of the electrostatic force on the block?", "answer_latex": " $0.245$", "answer_number": "0.245", "unit": "$\\mathrm{~N}$ ", "source": "fund", "problemid": " Question 22.49", "comment": " ", "solution": "$0.245 \\mathrm{~N}$"}, {"problem_text": "Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \\mathrm{~cm}$. The charge per unit length is $5.0 \\times 10^{-6} \\mathrm{C} / \\mathrm{m}$ on the inner shell and $-7.0 \\times 10^{-6} \\mathrm{C} / \\mathrm{m}$ on the outer shell. What is the magnitude $E$ of the electric field at radial distance $r=4.0 \\mathrm{~cm}$?", "answer_latex": " $2.3$", "answer_number": "2.3", "unit": "$10^6 \\mathrm{~N} / \\mathrm{C} $ ", "source": "fund", "problemid": " Question 23.31", "comment": " ", "solution": "$2.3 \\times 10^6 \\mathrm{~N} / \\mathrm{C} $"}, {"problem_text": "A particle of charge $1.8 \\mu \\mathrm{C}$ is at the center of a Gaussian cube $55 \\mathrm{~cm}$ on edge. What is the net electric flux through the surface?", "answer_latex": " $2.0$", "answer_number": "2.0", "unit": "$10^5 \\mathrm{~N} \\cdot \\mathrm{m}^2 / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 23.7", "comment": " ", "solution": "$2.0 \\times 10^5 \\mathrm{~N} \\cdot \\mathrm{m}^2 / \\mathrm{C}$"}, {"problem_text": "The drum of a photocopying machine has a length of $42 \\mathrm{~cm}$ and a diameter of $12 \\mathrm{~cm}$. The electric field just above the drum's surface is $2.3 \\times 10^5 \\mathrm{~N} / \\mathrm{C}$. What is the total charge on the drum? ", "answer_latex": " $0.32$", "answer_number": "0.32", "unit": "$\\mu C$ ", "source": "fund", "problemid": " Question 23.23", "comment": " ", "solution": "$0.32 \\mu C$"}, {"problem_text": "A spherical water drop $1.20 \\mu \\mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \\mathrm{~N} / \\mathrm{C}$. What is the magnitude of the gravitational force on the drop?", "answer_latex": " $8.87$", "answer_number": "8.87", "unit": "$10^{-15} \\mathrm{~N} $ ", "source": "fund", "problemid": " Question 22.63", "comment": " ", "solution": "$8.87 \\times 10^{-15} \\mathrm{~N} $"}, {"problem_text": "How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \\times 10^{-7} \\mathrm{C}$ ?", "answer_latex": " $6.3$", "answer_number": "6.3", "unit": "$10^{11}$", "source": "fund", "problemid": "Question 21.25 ", "comment": " ", "solution": "$6.3 \\times 10^{11}$"}, {"problem_text": "An unknown charge sits on a conducting solid sphere of radius $10 \\mathrm{~cm}$. If the electric field $15 \\mathrm{~cm}$ from the center of the sphere has the magnitude $3.0 \\times 10^3 \\mathrm{~N} / \\mathrm{C}$ and is directed radially inward, what is the net charge on the sphere?\r\n", "answer_latex": " $-7.5$", "answer_number": "-7.5", "unit": "$\\mathrm{nC}$ ", "source": "fund", "problemid": " Question 23.47", "comment": " ", "solution": "$-7.5 \\mathrm{nC}$"}, {"problem_text": "Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite $1.3 \\mathrm{~m}$ in diameter accumulates $2.4 \\mu \\mathrm{C}$ of charge in one orbital revolution. Find the resulting surface charge density. ", "answer_latex": " $4.5$", "answer_number": "4.5", "unit": "$10^{-7} \\mathrm{C} / \\mathrm{m}^2 $ ", "source": "fund", "problemid": " Question 23.19", "comment": " ", "solution": "$4.5 \\times 10^{-7} \\mathrm{C} / \\mathrm{m}^2 $"}, {"problem_text": " A charge of $20 \\mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \\mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \\mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc?", "answer_latex": "$38$", "answer_number": "38", "unit": "$\\mathrm{~N} / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 22.71", "comment": " ", "solution": "$38 \\mathrm{~N} / \\mathrm{C}$"}, {"problem_text": "Calculate the number of coulombs of positive charge in 250 $\\mathrm{cm}^3$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.)", "answer_latex": " $1.3$", "answer_number": "1.3", "unit": "$10^7 \\mathrm{C}$", "source": "fund", "problemid": " Question 21.33", "comment": " ", "solution": "$1.3 \\times 10^7 \\mathrm{C}$"}, {"problem_text": "A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \\times 10^{-9} \\mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \\times 10^{-6} \\mathrm{~N}$ when placed in this field. What is the magnitude of the electric field? ", "answer_latex": " $1.5$", "answer_number": "1.5", "unit": "$10^3 \\mathrm{~N} / \\mathrm{C}$ ", "source": "fund", "problemid": " Question 22.41", "comment": " ", "solution": "$1.5 \\times 10^3 \\mathrm{~N} / \\mathrm{C}$"}, {"problem_text": " An electric dipole with dipole moment\r\n$$\r\n\\vec{p}=(3.00 \\hat{\\mathrm{i}}+4.00 \\hat{\\mathrm{j}})\\left(1.24 \\times 10^{-30} \\mathrm{C} \\cdot \\mathrm{m}\\right)\r\n$$\r\nis in an electric field $\\vec{E}=(4000 \\mathrm{~N} / \\mathrm{C}) \\hat{\\mathrm{i}}$. What is the potential energy of the electric dipole? (b) What is the torque acting on it?", "answer_latex": " $-1.49$", "answer_number": "-1.49", "unit": "$10^{-26} \\mathrm{~J} $ ", "source": "fund", "problemid": " Question 22.83", "comment": " ", "solution": "$-1.49 \\times 10^{-26} \\mathrm{~J} $"}, {"problem_text": "What is the total charge in coulombs of $75.0 \\mathrm{~kg}$ of electrons?", "answer_latex": " $-1.32$", "answer_number": "-1.32", "unit": " $10^{13} \\mathrm{C}$", "source": "fund", "problemid": " Question 21.59", "comment": " ", "solution": "$-1.32 \\times 10^{13} \\mathrm{C}$"}, {"problem_text": "A uniformly charged conducting sphere of $1.2 \\mathrm{~m}$ diameter has surface charge density $8.1 \\mu \\mathrm{C} / \\mathrm{m}^2$. Find the net charge on the sphere.", "answer_latex": " $37$", "answer_number": "37", "unit": "$\\mu \\mathrm{C}$ ", "source": "fund", "problemid": " Question 23.17", "comment": " ", "solution": "$37 \\mu \\mathrm{C}$"}, {"problem_text": "The magnitude of the electrostatic force between two identical ions that are separated by a distance of $5.0 \\times 10^{-10} \\mathrm{~m}$ is $3.7 \\times 10^{-9}$ N.  What is the charge of each ion? ", "answer_latex": "$3.2$", "answer_number": "3.2", "unit": " $10^{-19} \\mathrm{C}$", "source": "fund", "problemid": " Question 21.27", "comment": " ", "solution": "$3.2 \\times 10^{-19} \\mathrm{C}$"}, {"problem_text": "How many megacoulombs of positive charge are in $1.00 \\mathrm{~mol}$ of neutral molecular-hydrogen gas $\\left(\\mathrm{H}_2\\right)$ ?", "answer_latex": " $0.19$", "answer_number": "0.19", "unit": "$\\mathrm{MC}$", "source": "fund", "problemid": " Question 21.45", "comment": " ", "solution": "$0.19 \\mathrm{MC}$"}, {"problem_text": "A charge (uniform linear density $=9.0 \\mathrm{nC} / \\mathrm{m}$ ) lies on a string that is stretched along an $x$ axis from $x=0$ to $x=3.0 \\mathrm{~m}$. Determine the magnitude of the electric field at $x=4.0 \\mathrm{~m}$ on the $x$ axis.", "answer_latex": " $61$", "answer_number": "61", "unit": " $\\mathrm{~N} / \\mathrm{C}$", "source": "fund", "problemid": " Question 22.67", "comment": " ", "solution": "$61 \\mathrm{~N} / \\mathrm{C}$"}, {"problem_text": "A long, straight wire has fixed negative charge with a linear charge density of magnitude $3.6 \\mathrm{nC} / \\mathrm{m}$. The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius $1.5 \\mathrm{~cm}$. The shell is to have positive charge on its outside surface with a surface charge density $\\sigma$ that makes the net external electric field zero. Calculate $\\sigma$.\r\n", "answer_latex": " $3.8$", "answer_number": "3.8", "unit": "$10^{-8} \\mathrm{C} / \\mathrm{m}^2$ ", "source": "fund", "problemid": " Question 23.27", "comment": " ", "solution": "$3.8 \\times 10^{-8} \\mathrm{C} / \\mathrm{m}^2$"}, {"problem_text": "Beams of high-speed protons can be produced in \"guns\" using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun's electric field were $2.00 \\times 10^4 \\mathrm{~N} / \\mathrm{C}$ ? (b) What speed would the proton attain if the field accelerated the proton through a distance of $1.00 \\mathrm{~cm}$ ?", "answer_latex": "$1.92$", "answer_number": "1.92", "unit": "$10^{12} \\mathrm{~m} / \\mathrm{s}^2 $ ", "source": "fund", "problemid": " Question 22.47", "comment": " ", "solution": "$1.92 \\times 10^{12} \\mathrm{~m} / \\mathrm{s}^2 $"}, {"problem_text": "An infinite line of charge produces a field of magnitude $4.5 \\times$ $10^4 \\mathrm{~N} / \\mathrm{C}$ at distance $2.0 \\mathrm{~m}$. Find the linear charge density.\r\n", "answer_latex": " $5.0$", "answer_number": "5.0", "unit": "$\\mu \\mathrm{C} / \\mathrm{m}$", "source": "fund", "problemid": " Question 23.25", "comment": " ", "solution": "$5.0 \\mu \\mathrm{C} / \\mathrm{m}$"}, {"problem_text": "A charged nonconducting rod, with a length of $2.00 \\mathrm{~m}$ and a cross-sectional area of $4.00 \\mathrm{~cm}^2$, lies along the positive side of an $x$ axis with one end at the origin. The volume charge density $\\rho$ is charge per unit volume in coulombs per cubic meter. How many excess electrons are on the rod if $\\rho$ is uniform, with a value of $-4.00 \\mu \\mathrm{C} / \\mathrm{m}^3$?\r\n", "answer_latex": " $2.00$", "answer_number": "2.00", "unit": "$10^{10} \\text { electrons; }$", "source": "fund", "problemid": " Question 21.51", "comment": " ", "solution": "$2.00 \\times 10^{10} \\text { electrons; }$"}]
+[
+ {
+  "problem_text": "In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\\vec{a}, 2.0 \\mathrm{~km}$ due east (directly toward the east); (b) $\\vec{b}, 2.0 \\mathrm{~km} 30^{\\circ}$ north of east (at an angle of $30^{\\circ}$ toward the north from due east); (c) $\\vec{c}, 1.0 \\mathrm{~km}$ due west. Alternatively, you may substitute either $-\\vec{b}$ for $\\vec{b}$ or $-\\vec{c}$ for $\\vec{c}$. What is the greatest distance you can be from base camp at the end of the third displacement? (We are not concerned about the direction.)",
+  "answer_latex": " 4.8",
+  "answer_number": "4.8",
+  "unit": " m",
+  "source": "fund",
+  "problemid": " 3.01",
+  "comment": " "
+ },
+ {
+  "problem_text": "\"Top gun\" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function.\r\nThere are several warning signs. When the centripetal acceleration is $2 g$ or $3 g$, the pilot feels heavy. At about $4 g$, the pilot's vision switches to black and white and narrows to \"tunnel vision.\" If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious - a condition known as $g$-LOC for \" $g$-induced loss of consciousness.\"\r\n\r\nWhat is the magnitude of the acceleration, in $g$ units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of $\\vec{v}_i=(400 \\hat{\\mathrm{i}}+500 \\hat{\\mathrm{j}}) \\mathrm{m} / \\mathrm{s}$ and $24.0 \\mathrm{~s}$ later leaves the turn with a velocity of $\\vec{v}_f=(-400 \\hat{\\mathrm{i}}-500 \\hat{\\mathrm{j}}) \\mathrm{m} / \\mathrm{s}$ ?",
+  "answer_latex": "83.81",
+  "answer_number": "83.81",
+  "unit": " $\\mathrm{m} / \\mathrm{s}^2$",
+  "source": "fund",
+  "problemid": " 4.06",
+  "comment": " "
+ },
+ {
+  "problem_text": "The world\u2019s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball?",
+  "answer_latex": "2",
+  "answer_number": "2",
+  "unit": " $10^6$ m",
+  "source": "fund",
+  "problemid": " 1.01",
+  "comment": " "
+ },
+ {
+  "problem_text": "You drive a beat-up pickup truck along a straight road for $8.4 \\mathrm{~km}$ at $70 \\mathrm{~km} / \\mathrm{h}$, at which point the truck runs out of gasoline and stops. Over the next $30 \\mathrm{~min}$, you walk another $2.0 \\mathrm{~km}$ farther along the road to a gasoline station.\r\nWhat is your overall displacement from the beginning of your drive to your arrival at the station?",
+  "answer_latex": " 10.4",
+  "answer_number": "10.4",
+  "unit": " km",
+  "source": "fund",
+  "problemid": " 2.01",
+  "comment": " "
+ },
+ {
+  "problem_text": "A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms of the void ratio $e$ for a sample of the ground:\r\n$$\r\ne=\\frac{V_{\\text {voids }}}{V_{\\text {grains }}} .\r\n$$\r\nHere, $V_{\\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\\rho_{\\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\\rho_{\\mathrm{SiO}_2}=2.600 \\times 10^3 \\mathrm{~kg} / \\mathrm{m}^3$.",
+  "answer_latex": "1.4",
+  "answer_number": "1.4",
+  "unit": " $10^3 \\mathrm{~kg} / \\mathrm{m}^3$",
+  "source": "fund",
+  "problemid": "1.02 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "What is the angle $\\phi$ between $\\vec{a}=3.0 \\hat{\\mathrm{i}}-4.0 \\hat{\\mathrm{j}}$ and $\\vec{b}=$ $-2.0 \\hat{\\mathrm{i}}+3.0 \\hat{\\mathrm{k}}$ ?",
+  "answer_latex": "109",
+  "answer_number": "109",
+  "unit": " $^{\\circ}$",
+  "source": "fund",
+  "problemid": " 3.05",
+  "comment": " "
+ },
+ {
+  "problem_text": "During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\\vec{d}=(-3.0 \\mathrm{~m}) \\hat{\\mathrm{i}}$ while a steady wind pushes against the crate with a force $\\vec{F}=(2.0 \\mathrm{~N}) \\hat{\\mathrm{i}}+(-6.0 \\mathrm{~N}) \\hat{\\mathrm{j}}$. If the crate has a kinetic energy of $10 \\mathrm{~J}$ at the beginning of displacement $\\vec{d}$, what is its kinetic energy at the end of $\\vec{d}$ ?",
+  "answer_latex": " 4.0",
+  "answer_number": "4.0",
+  "unit": " J",
+  "source": "fund",
+  "problemid": "  7.03",
+  "comment": " "
+ },
+ {
+  "problem_text": "When the force on an object depends on the position of the object, we cannot find the work done by it on the object by simply multiplying the force by the displacement. The reason is that there is no one value for the force-it changes. So, we must find the work in tiny little displacements and then add up all the work results. We effectively say, \"Yes, the force varies over any given tiny little displacement, but the variation is so small we can approximate the force as being constant during the displacement.\" Sure, it is not precise, but if we make the displacements infinitesimal, then our error becomes infinitesimal and the result becomes precise. But, to add an infinite number of work contributions by hand would take us forever, longer than a semester. So, we add them up via an integration, which allows us to do all this in minutes (much less than a semester).\r\n\r\nForce $\\vec{F}=\\left(3 x^2 \\mathrm{~N}\\right) \\hat{\\mathrm{i}}+(4 \\mathrm{~N}) \\hat{\\mathrm{j}}$, with $x$ in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates $(2 \\mathrm{~m}, 3 \\mathrm{~m})$ to $(3 \\mathrm{~m}, 0 \\mathrm{~m})$ ?",
+  "answer_latex": " 7.0",
+  "answer_number": " 7.0",
+  "unit": " J",
+  "source": "fund",
+  "problemid": " 7.08",
+  "comment": " "
+ },
+ {
+  "problem_text": " The charges of an electron and a positron are $-e$ and $+e$. The mass of each is $9.11 \\times 10^{-31} \\mathrm{~kg}$. What is the ratio of the electrical force to the gravitational force between an electron and a positron?\r\n",
+  "answer_latex": " $4.16$",
+  "answer_number": "4.16",
+  "unit": "$10^{42}$",
+  "source": "fund",
+  "problemid": " Question 21.75",
+  "comment": " ",
+  "solution": "$4.16 \\times 10^{42}$"
+ },
+ {
+  "problem_text": "Particle 1 of charge $+q$ and particle 2 of charge $+4.00 q$ are held at separation $L=9.00 \\mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_3$ is to be located such that the three particles remain in place when released, what must be the $x$ coordinate of particle 3?",
+  "answer_latex": " $3.00$",
+  "answer_number": "3.00",
+  "unit": "$\\mathrm{~cm}$",
+  "source": "fund",
+  "problemid": " Question 21.19",
+  "comment": " ",
+  "solution": "$3.00 \\mathrm{~cm}$"
+ },
+ {
+  "problem_text": "Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_1=2.1 \\times 10^{-8} \\mathrm{C}$ is at position $x=20 \\mathrm{~cm}$ and particle 2 of charge $q_2=-4.00 q_1$ is at position $x=70 \\mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?\r\n",
+  "answer_latex": " $-30$",
+  "answer_number": "-30",
+  "unit": " $\\mathrm{~cm}$",
+  "source": "fund",
+  "problemid": " Question 22.11",
+  "comment": " ",
+  "solution": "$-30 \\mathrm{~cm}$"
+ },
+ {
+  "problem_text": "The volume charge density of a solid nonconducting sphere of radius $R=5.60 \\mathrm{~cm}$ varies with radial distance $r$ as given by $\\rho=$ $\\left(14.1 \\mathrm{pC} / \\mathrm{m}^3\\right) r / R$. What is the sphere's total charge?",
+  "answer_latex": " $7.78$",
+  "answer_number": "7.78",
+  "unit": "$\\mathrm{fC} $ ",
+  "source": "fund",
+  "problemid": " Question 23.53",
+  "comment": " ",
+  "solution": "$7.78 \\mathrm{fC} $"
+ },
+ {
+  "problem_text": "Two charged concentric spherical shells have radii $10.0 \\mathrm{~cm}$ and $15.0 \\mathrm{~cm}$. The charge on the inner shell is $4.00 \\times 10^{-8} \\mathrm{C}$, and that on the outer shell is $2.00 \\times 10^{-8} \\mathrm{C}$. Find the electric field at $r=12.0 \\mathrm{~cm}$.",
+  "answer_latex": " $2.50$",
+  "answer_number": "2.50",
+  "unit": "$10^4 \\mathrm{~N} / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 23.45",
+  "comment": " ",
+  "solution": "$2.50 \\times 10^4 \\mathrm{~N} / \\mathrm{C}$"
+ },
+ {
+  "problem_text": "Assume that a honeybee is a sphere of diameter 1.000 $\\mathrm{cm}$ with a charge of $+45.0 \\mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \\mu \\mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \\mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \\mathrm{pC}$ on the far side. What is the magnitude of the net electrostatic force on the grain due to the bee? ",
+  "answer_latex": " $2.6$",
+  "answer_number": "2.6",
+  "unit": "$10^{-10} \\mathrm{~N}$ ",
+  "source": "fund",
+  "problemid": " Question 22.51",
+  "comment": " ",
+  "solution": "$2.6 \\times 10^{-10} \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "In the radioactive decay of Eq. 21-13, $\\mathrm{a}^{238} \\mathrm{U}$ nucleus transforms to ${ }^{234} \\mathrm{Th}$ and an ejected ${ }^4 \\mathrm{He}$. (These are nuclei, not atoms, and thus electrons are not involved.) When the separation between ${ }^{234} \\mathrm{Th}$ and ${ }^4 \\mathrm{He}$ is $9.0 \\times 10^{-15} \\mathrm{~m}$, what are the magnitudes of the electrostatic force between them?\r\n",
+  "answer_latex": " $5.1$",
+  "answer_number": "5.1",
+  "unit": " $10^2 \\mathrm{~N}$",
+  "source": "fund",
+  "problemid": " Question 21.69",
+  "comment": " ",
+  "solution": "$5.1 \\times 10^2 \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \\hat{\\mathrm{i}}+3.0 \\hat{\\mathrm{j}}) \\mathrm{N} / \\mathrm{C}$ at the point $(3.0,3.0) \\mathrm{cm}$ and $100 \\hat{\\mathrm{i}} \\mathrm{N} / \\mathrm{C}$ at the point $(2.0,0) \\mathrm{cm}$. What is the $x$ coordinate of the particle?",
+  "answer_latex": " $-1.0$",
+  "answer_number": "-1.0",
+  "unit": "$\\mathrm{~cm}$ ",
+  "source": "fund",
+  "problemid": " Question 22.73",
+  "comment": " ",
+  "solution": "$-1.0 \\mathrm{~cm}$"
+ },
+ {
+  "problem_text": "Two particles, each with a charge of magnitude $12 \\mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \\mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive?",
+  "answer_latex": " $47$",
+  "answer_number": "47",
+  "unit": "$\\mathrm{~N} / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 22.69",
+  "comment": " ",
+  "solution": "$47 \\mathrm{~N} / \\mathrm{C} $"
+ },
+ {
+  "problem_text": "How much work is required to turn an electric dipole $180^{\\circ}$ in a uniform electric field of magnitude $E=46.0 \\mathrm{~N} / \\mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \\times$ $10^{-25} \\mathrm{C} \\cdot \\mathrm{m}$ and the initial angle is $64^{\\circ} ?$\r\n",
+  "answer_latex": " $1.22$",
+  "answer_number": "1.22",
+  "unit": "$10^{-23} \\mathrm{~J}$ ",
+  "source": "fund",
+  "problemid": " Question 22.59",
+  "comment": " ",
+  "solution": "$1.22 \\times 10^{-23} \\mathrm{~J}$"
+ },
+ {
+  "problem_text": "We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \\%$. With what force would two copper coins, placed $1.0 \\mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \\times 10^{22}$ copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.)",
+  "answer_latex": " $1.7$",
+  "answer_number": "1.7",
+  "unit": "$10^8 \\mathrm{~N}$ ",
+  "source": "fund",
+  "problemid": " Question 21.57",
+  "comment": " ",
+  "solution": "$1.7 \\times 10^8 \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "What must be the distance between point charge $q_1=$ $26.0 \\mu \\mathrm{C}$ and point charge $q_2=-47.0 \\mu \\mathrm{C}$ for the electrostatic force between them to have a magnitude of $5.70 \\mathrm{~N}$ ?\r\n",
+  "answer_latex": " 1.39",
+  "answer_number": "1.39",
+  "unit": " m",
+  "source": "fund",
+  "problemid": " Question 21.3",
+  "comment": " ",
+  "solution": "1.39"
+ },
+ {
+  "problem_text": "Three charged particles form a triangle: particle 1 with charge $Q_1=80.0 \\mathrm{nC}$ is at $x y$ coordinates $(0,3.00 \\mathrm{~mm})$, particle 2 with charge $Q_2$ is at $(0,-3.00 \\mathrm{~mm})$, and particle 3 with charge $q=18.0$ $\\mathrm{nC}$ is at $(4.00 \\mathrm{~mm}, 0)$. In unit-vector notation, what is the electrostatic force on particle 3 due to the other two particles if $Q_2$ is equal to $80.0 \\mathrm{nC}$?\r\n",
+  "answer_latex": "$(0.829)$",
+  "answer_number": "0.829",
+  "unit": " $\\mathrm{~N} \\hat{\\mathrm{i}}$",
+  "source": "fund",
+  "problemid": " Question 21.61",
+  "comment": " ",
+  "solution": "$(0.829 \\mathrm{~N}) \\hat{\\mathrm{i}}$"
+ },
+ {
+  "problem_text": "A particle of charge $-q_1$ is at the origin of an $x$ axis. At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \\mathrm{~mm}$ on the axis? ",
+  "answer_latex": " $6.0$",
+  "answer_number": "6.0",
+  "unit": "$\\mathrm{~mm}$ ",
+  "source": "fund",
+  "problemid": " Question 22.77",
+  "comment": " ",
+  "solution": "$6.0 \\mathrm{~mm}$"
+ },
+ {
+  "problem_text": "An electron is released from rest in a uniform electric field of magnitude $2.00 \\times 10^4 \\mathrm{~N} / \\mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.)",
+  "answer_latex": " $3.51$",
+  "answer_number": "3.51",
+  "unit": "$10^{15} \\mathrm{~m} / \\mathrm{s}^2$ ",
+  "source": "fund",
+  "problemid": " Question 22.43",
+  "comment": " ",
+  "solution": "$ 3.51 \\times 10^{15} \\mathrm{~m} / \\mathrm{s}^2$"
+ },
+ {
+  "problem_text": "The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \\mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons?",
+  "answer_latex": "$3.07$",
+  "answer_number": "3.07",
+  "unit": "$10^{21} \\mathrm{~N} / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 22.3",
+  "comment": " ",
+  "solution": "$3.07 \\times 10^{21} \\mathrm{~N} / \\mathrm{C}$"
+ },
+ {
+  "problem_text": "A nonconducting spherical shell, with an inner radius of $4.0 \\mathrm{~cm}$ and an outer radius of $6.0 \\mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell $\\rho=b / r$, where $r$ is the distance in meters from the center of the shell and $b=3.0 \\mu \\mathrm{C} / \\mathrm{m}^2$. What is the net charge in the shell?\r\n",
+  "answer_latex": " $3.8$",
+  "answer_number": "3.8",
+  "unit": " $10^{-8} \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.21",
+  "comment": " ",
+  "solution": "$3.8 \\times 10^{-8} \\mathrm{C}$"
+ },
+ {
+  "problem_text": "An electron is shot directly\r\nFigure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \\times 10^{-6} \\mathrm{C} / \\mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \\times 10^{-17} \\mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?",
+  "answer_latex": " $0.44$",
+  "answer_number": "0.44",
+  "unit": "$\\mathrm{~mm}$ ",
+  "source": "fund",
+  "problemid": " Question 23.41",
+  "comment": " ",
+  "solution": "$0.44 \\mathrm{~mm}$"
+ },
+ {
+  "problem_text": "A square metal plate of edge length $8.0 \\mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \\times 10^{-6} \\mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \\mathrm{~mm}$ from the center) by assuming that the charge is spread uniformly over the two faces of the plate. ",
+  "answer_latex": "$5.4$",
+  "answer_number": "5.4",
+  "unit": "$10^7 \\mathrm{~N} / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 23.37",
+  "comment": " ",
+  "solution": "$5.3 \\times 10^7 \\mathrm{~N} / \\mathrm{C}$"
+ },
+ {
+  "problem_text": "A neutron consists of one \"up\" quark of charge $+2 e / 3$ and two \"down\" quarks each having charge $-e / 3$. If we assume that the down quarks are $2.6 \\times 10^{-15} \\mathrm{~m}$ apart inside the neutron, what is the magnitude of the electrostatic force between them?\r\n",
+  "answer_latex": "$3.8$",
+  "answer_number": "3.8",
+  "unit": "$N$ ",
+  "source": "fund",
+  "problemid": " Question 21.49",
+  "comment": " ",
+  "solution": "$3.8 N$"
+ },
+ {
+  "problem_text": "In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \\mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit?",
+  "answer_latex": " $2.19$",
+  "answer_number": "2.19",
+  "unit": "$10^6 \\mathrm{~m} / \\mathrm{s}$ ",
+  "source": "fund",
+  "problemid": " Question 21.73",
+  "comment": " ",
+  "solution": "$2.19 \\times 10^6 \\mathrm{~m} / \\mathrm{s}$"
+ },
+ {
+  "problem_text": "At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \\mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?",
+  "answer_latex": " $0.346$",
+  "answer_number": "0.346",
+  "unit": "$\\mathrm{~m}$ ",
+  "source": "fund",
+  "problemid": " Question 22.35",
+  "comment": " ",
+  "solution": "$0.346 \\mathrm{~m}$"
+ },
+ {
+  "problem_text": "Of the charge $Q$ on a tiny sphere, a fraction $\\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles.  What value of $\\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres? ",
+  "answer_latex": " $0.5$",
+  "answer_number": "0.5",
+  "unit": " ",
+  "source": "fund",
+  "problemid": " Question 21.55",
+  "comment": " ",
+  "solution": "$0.5$"
+ },
+ {
+  "problem_text": "In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \\times 10^{-13} \\mathrm{C} / \\mathrm{m}^2$. What is the magnitude of the electron's acceleration when it reaches radial distances $r=0.500 R$?",
+  "answer_latex": " $0$",
+  "answer_number": "0",
+  "unit": " ",
+  "source": "fund",
+  "problemid": " Question 21.71",
+  "comment": " ",
+  "solution": "$0$"
+ },
+ {
+  "problem_text": "A particle of charge $+3.00 \\times 10^{-6} \\mathrm{C}$ is $12.0 \\mathrm{~cm}$ distant from a second particle of charge $-1.50 \\times 10^{-6} \\mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles.",
+  "answer_latex": "2.81 ",
+  "answer_number": "2.81",
+  "unit": "N ",
+  "source": "fund",
+  "problemid": " Question 21.5",
+  "comment": " ",
+  "solution": "2.81"
+ },
+ {
+  "problem_text": "A charged particle produces an electric field with a magnitude of $2.0 \\mathrm{~N} / \\mathrm{C}$ at a point that is $50 \\mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?\r\n",
+  "answer_latex": " $56$",
+  "answer_number": "56",
+  "unit": "$\\mathrm{pC}$ ",
+  "source": "fund",
+  "problemid": " Question 22.5",
+  "comment": " ",
+  "solution": "$ 56 \\mathrm{pC}$"
+ },
+ {
+  "problem_text": "In Millikan's experiment, an oil drop of radius $1.64 \\mu \\mathrm{m}$ and density $0.851 \\mathrm{~g} / \\mathrm{cm}^3$ is suspended in chamber C when a downward electric field of $1.92 \\times 10^5 \\mathrm{~N} / \\mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$.",
+  "answer_latex": " $-5$",
+  "answer_number": "-5",
+  "unit": "$e$ ",
+  "source": "fund",
+  "problemid": " Question 22.39",
+  "comment": " ",
+  "solution": "$-5 e$"
+ },
+ {
+  "problem_text": "The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \\mu \\mathrm{C}, x_1=3.5 \\mathrm{~cm}, y_1=0.50 \\mathrm{~cm}$, and $q_2=-4.0 \\mu \\mathrm{C}, x_2=-2.0 \\mathrm{~cm}, y_2=1.5 \\mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1.",
+  "answer_latex": "$35$",
+  "answer_number": "35",
+  "unit": "$\\mathrm{~N}$",
+  "source": "fund",
+  "problemid": " Question 21.15",
+  "comment": " ",
+  "solution": "$35$"
+ },
+ {
+  "problem_text": "An electron on the axis of an electric dipole is $25 \\mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \\times 10^{-29} \\mathrm{C} \\cdot \\mathrm{m}$ ? Assume that $25 \\mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole.",
+  "answer_latex": " $6.6$",
+  "answer_number": "6.6",
+  "unit": "$10^{-15} \\mathrm{~N}$ ",
+  "source": "fund",
+  "problemid": " Question 22.45",
+  "comment": " ",
+  "solution": "$ 6.6 \\times 10^{-15} \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?",
+  "answer_latex": "$122$",
+  "answer_number": "122",
+  "unit": " $\\mathrm{~mA}$",
+  "source": "fund",
+  "problemid": " Question 21.31",
+  "comment": " ",
+  "solution": "$122 \\mathrm{~mA}$"
+ },
+ {
+  "problem_text": "An electric field $\\vec{E}$ with an average magnitude of about $150 \\mathrm{~N} / \\mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to \"float\" a sulfur sphere weighing $4.4 \\mathrm{~N}$ in this field by charging the sphere. What charge (both sign and magnitude) must be used?",
+  "answer_latex": " $-0.029$",
+  "answer_number": "-0.029",
+  "unit": " $C$",
+  "source": "fund",
+  "problemid": " Question 22.81",
+  "comment": " ",
+  "solution": "$-0.029 C$"
+ },
+ {
+  "problem_text": "Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \\mathrm{~N}$ when their center-to-center separation is $50.0 \\mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.0360 \\mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was the negative charge on one of them?",
+  "answer_latex": " $-1.00 \\mu \\mathrm{C}$",
+  "answer_number": "-1.00",
+  "unit": "$ \\mu \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.9",
+  "comment": " ",
+  "solution": "$-1.00 \\mu \\mathrm{C}$"
+ },
+ {
+  "problem_text": "A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \\mathrm{~cm}$ away, in a time $1.5 \\times 10^{-8} \\mathrm{~s}$. What is the speed of the electron as it strikes the second plate? ",
+  "answer_latex": " $2.7$",
+  "answer_number": "2.7",
+  "unit": " $10^6$",
+  "source": "fund",
+  "problemid": " Question 22.55",
+  "comment": " ",
+  "solution": "$2.7 \\times 10^6$"
+ },
+ {
+  "problem_text": " Two point charges of $30 \\mathrm{nC}$ and $-40 \\mathrm{nC}$ are held fixed on an $x$ axis, at the origin and at $x=72 \\mathrm{~cm}$, respectively. A particle with a charge of $42 \\mu \\mathrm{C}$ is released from rest at $x=28 \\mathrm{~cm}$. If the initial acceleration of the particle has a magnitude of $100 \\mathrm{~km} / \\mathrm{s}^2$, what is the particle's mass?",
+  "answer_latex": " $2.2$",
+  "answer_number": "2.2",
+  "unit": " $10^{-6} \\mathrm{~kg}$",
+  "source": "fund",
+  "problemid": " Question 21.63",
+  "comment": " ",
+  "solution": "$2.2 \\times 10^{-6} \\mathrm{~kg}$"
+ },
+ {
+  "problem_text": "In Fig. 21-26, particle 1 of charge $-5.00 q$ and particle 2 of charge $+2.00 q$ are held at separation $L$ on an $x$ axis. If particle 3 of unknown charge $q_3$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the $x$ coordinate of particle 3?",
+  "answer_latex": " $2.72$",
+  "answer_number": "2.72",
+  "unit": " $L$",
+  "source": "fund",
+  "problemid": " Question 21.67",
+  "comment": " ",
+  "solution": "$2.72 L$"
+ },
+ {
+  "problem_text": "An isolated conductor has net charge $+10 \\times 10^{-6} \\mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \\times 10^{-6} \\mathrm{C}$. What is the charge on the cavity wall?",
+  "answer_latex": " $-3.0$",
+  "answer_number": "-3.0",
+  "unit": "$10^{-6} \\mathrm{C} $ ",
+  "source": "fund",
+  "problemid": " Question 23.21",
+  "comment": " ",
+  "solution": "$-3.0 \\times 10^{-6} \\mathrm{C} $"
+ },
+ {
+  "problem_text": "Point charges of $+6.0 \\mu \\mathrm{C}$ and $-4.0 \\mu \\mathrm{C}$ are placed on an $x$ axis, at $x=8.0 \\mathrm{~m}$ and $x=16 \\mathrm{~m}$, respectively. What charge must be placed at $x=24 \\mathrm{~m}$ so that any charge placed at the origin would experience no electrostatic force?\r\n",
+  "answer_latex": " $-45$",
+  "answer_number": "-45",
+  "unit": " $\\mu \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.47",
+  "comment": " ",
+  "solution": "$-45 \\mu \\mathrm{C}$"
+ },
+ {
+  "problem_text": "The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of $300 \\mathrm{~m}$ the field has magnitude $60.0 \\mathrm{~N} / \\mathrm{C}$; at an altitude of $200 \\mathrm{~m}$, the magnitude is $100 \\mathrm{~N} / \\mathrm{C}$. Find the net amount of charge contained in a cube $100 \\mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \\mathrm{~m}$.",
+  "answer_latex": " $3.54$",
+  "answer_number": "3.54",
+  "unit": "$\\mu \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 23.13",
+  "comment": " ",
+  "solution": "$3.54 \\mu \\mathrm{C}$"
+ },
+ {
+  "problem_text": "What would be the magnitude of the electrostatic force between two 1.00 C point charges separated by a distance of $1.00 \\mathrm{~m}$  if such point charges existed (they do not) and this configuration could be set up?",
+  "answer_latex": " $8.99$",
+  "answer_number": "8.99",
+  "unit": " $10^9 \\mathrm{~N}$",
+  "source": "fund",
+  "problemid": " Question 21.53",
+  "comment": " ",
+  "solution": "$8.99 \\times 10^9 \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "An electric dipole consisting of charges of magnitude $1.50 \\mathrm{nC}$ separated by $6.20 \\mu \\mathrm{m}$ is in an electric field of strength 1100 $\\mathrm{N} / \\mathrm{C}$. What is the magnitude of the electric dipole moment?",
+  "answer_latex": " $9.30$",
+  "answer_number": "9.30",
+  "unit": "$10^{-15} \\mathrm{C} \\cdot \\mathrm{m}$ ",
+  "source": "fund",
+  "problemid": " Question 22.57",
+  "comment": " ",
+  "solution": "$9.30 \\times 10^{-15} \\mathrm{C} \\cdot \\mathrm{m}$"
+ },
+ {
+  "problem_text": "What equal positive charges would have to be placed on Earth and on the Moon to neutralize their gravitational attraction?",
+  "answer_latex": " $5.7$",
+  "answer_number": "5.7",
+  "unit": "$10^{13} \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.41",
+  "comment": " ",
+  "solution": "$5.7 \\times 10^{13} \\mathrm{C}$"
+ },
+ {
+  "problem_text": "The initial charges on the three identical metal spheres in Fig. 21-24 are the following: sphere $A, Q$; sphere $B,-Q / 4$; and sphere $C, Q / 2$, where $Q=2.00 \\times 10^{-14}$ C. Spheres $A$ and $B$ are fixed in place, with a center-to-center separation of $d=1.20 \\mathrm{~m}$, which is much larger than the spheres. Sphere $C$ is touched first to sphere $A$ and then to sphere $B$ and is then removed. What then is the magnitude of the electrostatic force between spheres $A$ and $B$ ?\r\n",
+  "answer_latex": " $4.68$",
+  "answer_number": "4.68",
+  "unit": " $10^{-19} \\mathrm{~N}$",
+  "source": "fund",
+  "problemid": " Question 21.65",
+  "comment": " ",
+  "solution": "$4.68 \\times 10^{-19} \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "A $10.0 \\mathrm{~g}$ block with a charge of $+8.00 \\times 10^{-5} \\mathrm{C}$ is placed in an electric field $\\vec{E}=(3000 \\hat{\\mathrm{i}}-600 \\hat{\\mathrm{j}}) \\mathrm{N} / \\mathrm{C}$. What is the magnitude of the electrostatic force on the block?",
+  "answer_latex": " $0.245$",
+  "answer_number": "0.245",
+  "unit": "$\\mathrm{~N}$ ",
+  "source": "fund",
+  "problemid": " Question 22.49",
+  "comment": " ",
+  "solution": "$0.245 \\mathrm{~N}$"
+ },
+ {
+  "problem_text": "Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \\mathrm{~cm}$. The charge per unit length is $5.0 \\times 10^{-6} \\mathrm{C} / \\mathrm{m}$ on the inner shell and $-7.0 \\times 10^{-6} \\mathrm{C} / \\mathrm{m}$ on the outer shell. What is the magnitude $E$ of the electric field at radial distance $r=4.0 \\mathrm{~cm}$?",
+  "answer_latex": " $2.3$",
+  "answer_number": "2.3",
+  "unit": "$10^6 \\mathrm{~N} / \\mathrm{C} $ ",
+  "source": "fund",
+  "problemid": " Question 23.31",
+  "comment": " ",
+  "solution": "$2.3 \\times 10^6 \\mathrm{~N} / \\mathrm{C} $"
+ },
+ {
+  "problem_text": "A particle of charge $1.8 \\mu \\mathrm{C}$ is at the center of a Gaussian cube $55 \\mathrm{~cm}$ on edge. What is the net electric flux through the surface?",
+  "answer_latex": " $2.0$",
+  "answer_number": "2.0",
+  "unit": "$10^5 \\mathrm{~N} \\cdot \\mathrm{m}^2 / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 23.7",
+  "comment": " ",
+  "solution": "$2.0 \\times 10^5 \\mathrm{~N} \\cdot \\mathrm{m}^2 / \\mathrm{C}$"
+ },
+ {
+  "problem_text": "The drum of a photocopying machine has a length of $42 \\mathrm{~cm}$ and a diameter of $12 \\mathrm{~cm}$. The electric field just above the drum's surface is $2.3 \\times 10^5 \\mathrm{~N} / \\mathrm{C}$. What is the total charge on the drum? ",
+  "answer_latex": " $0.32$",
+  "answer_number": "0.32",
+  "unit": "$\\mu C$ ",
+  "source": "fund",
+  "problemid": " Question 23.23",
+  "comment": " ",
+  "solution": "$0.32 \\mu C$"
+ },
+ {
+  "problem_text": "A spherical water drop $1.20 \\mu \\mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \\mathrm{~N} / \\mathrm{C}$. What is the magnitude of the gravitational force on the drop?",
+  "answer_latex": " $8.87$",
+  "answer_number": "8.87",
+  "unit": "$10^{-15} \\mathrm{~N} $ ",
+  "source": "fund",
+  "problemid": " Question 22.63",
+  "comment": " ",
+  "solution": "$8.87 \\times 10^{-15} \\mathrm{~N} $"
+ },
+ {
+  "problem_text": "How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \\times 10^{-7} \\mathrm{C}$ ?",
+  "answer_latex": " $6.3$",
+  "answer_number": "6.3",
+  "unit": "$10^{11}$",
+  "source": "fund",
+  "problemid": "Question 21.25 ",
+  "comment": " ",
+  "solution": "$6.3 \\times 10^{11}$"
+ },
+ {
+  "problem_text": "An unknown charge sits on a conducting solid sphere of radius $10 \\mathrm{~cm}$. If the electric field $15 \\mathrm{~cm}$ from the center of the sphere has the magnitude $3.0 \\times 10^3 \\mathrm{~N} / \\mathrm{C}$ and is directed radially inward, what is the net charge on the sphere?\r\n",
+  "answer_latex": " $-7.5$",
+  "answer_number": "-7.5",
+  "unit": "$\\mathrm{nC}$ ",
+  "source": "fund",
+  "problemid": " Question 23.47",
+  "comment": " ",
+  "solution": "$-7.5 \\mathrm{nC}$"
+ },
+ {
+  "problem_text": "Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite $1.3 \\mathrm{~m}$ in diameter accumulates $2.4 \\mu \\mathrm{C}$ of charge in one orbital revolution. Find the resulting surface charge density. ",
+  "answer_latex": " $4.5$",
+  "answer_number": "4.5",
+  "unit": "$10^{-7} \\mathrm{C} / \\mathrm{m}^2 $ ",
+  "source": "fund",
+  "problemid": " Question 23.19",
+  "comment": " ",
+  "solution": "$4.5 \\times 10^{-7} \\mathrm{C} / \\mathrm{m}^2 $"
+ },
+ {
+  "problem_text": " A charge of $20 \\mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \\mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \\mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc?",
+  "answer_latex": "$38$",
+  "answer_number": "38",
+  "unit": "$\\mathrm{~N} / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 22.71",
+  "comment": " ",
+  "solution": "$38 \\mathrm{~N} / \\mathrm{C}$"
+ },
+ {
+  "problem_text": "Calculate the number of coulombs of positive charge in 250 $\\mathrm{cm}^3$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.)",
+  "answer_latex": " $1.3$",
+  "answer_number": "1.3",
+  "unit": "$10^7 \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.33",
+  "comment": " ",
+  "solution": "$1.3 \\times 10^7 \\mathrm{C}$"
+ },
+ {
+  "problem_text": "A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \\times 10^{-9} \\mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \\times 10^{-6} \\mathrm{~N}$ when placed in this field. What is the magnitude of the electric field? ",
+  "answer_latex": " $1.5$",
+  "answer_number": "1.5",
+  "unit": "$10^3 \\mathrm{~N} / \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 22.41",
+  "comment": " ",
+  "solution": "$1.5 \\times 10^3 \\mathrm{~N} / \\mathrm{C}$"
+ },
+ {
+  "problem_text": " An electric dipole with dipole moment\r\n$$\r\n\\vec{p}=(3.00 \\hat{\\mathrm{i}}+4.00 \\hat{\\mathrm{j}})\\left(1.24 \\times 10^{-30} \\mathrm{C} \\cdot \\mathrm{m}\\right)\r\n$$\r\nis in an electric field $\\vec{E}=(4000 \\mathrm{~N} / \\mathrm{C}) \\hat{\\mathrm{i}}$. What is the potential energy of the electric dipole?",
+  "answer_latex": " $-1.49$",
+  "answer_number": "-1.49",
+  "unit": "$10^{-26} \\mathrm{~J} $ ",
+  "source": "fund",
+  "problemid": " Question 22.83",
+  "comment": " ",
+  "solution": "$-1.49 \\times 10^{-26} \\mathrm{~J} $"
+ },
+ {
+  "problem_text": "What is the total charge in coulombs of $75.0 \\mathrm{~kg}$ of electrons?",
+  "answer_latex": " $-1.32$",
+  "answer_number": "-1.32",
+  "unit": " $10^{13} \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.59",
+  "comment": " ",
+  "solution": "$-1.32 \\times 10^{13} \\mathrm{C}$"
+ },
+ {
+  "problem_text": "A uniformly charged conducting sphere of $1.2 \\mathrm{~m}$ diameter has surface charge density $8.1 \\mu \\mathrm{C} / \\mathrm{m}^2$. Find the net charge on the sphere.",
+  "answer_latex": " $37$",
+  "answer_number": "37",
+  "unit": "$\\mu \\mathrm{C}$ ",
+  "source": "fund",
+  "problemid": " Question 23.17",
+  "comment": " ",
+  "solution": "$37 \\mu \\mathrm{C}$"
+ },
+ {
+  "problem_text": "The magnitude of the electrostatic force between two identical ions that are separated by a distance of $5.0 \\times 10^{-10} \\mathrm{~m}$ is $3.7 \\times 10^{-9}$ N.  What is the charge of each ion? ",
+  "answer_latex": "$3.2$",
+  "answer_number": "3.2",
+  "unit": " $10^{-19} \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 21.27",
+  "comment": " ",
+  "solution": "$3.2 \\times 10^{-19} \\mathrm{C}$"
+ },
+ {
+  "problem_text": "How many megacoulombs of positive charge are in $1.00 \\mathrm{~mol}$ of neutral molecular-hydrogen gas $\\left(\\mathrm{H}_2\\right)$ ?",
+  "answer_latex": " $0.19$",
+  "answer_number": "0.19",
+  "unit": "$\\mathrm{MC}$",
+  "source": "fund",
+  "problemid": " Question 21.45",
+  "comment": " ",
+  "solution": "$0.19 \\mathrm{MC}$"
+ },
+ {
+  "problem_text": "A charge (uniform linear density $=9.0 \\mathrm{nC} / \\mathrm{m}$ ) lies on a string that is stretched along an $x$ axis from $x=0$ to $x=3.0 \\mathrm{~m}$. Determine the magnitude of the electric field at $x=4.0 \\mathrm{~m}$ on the $x$ axis.",
+  "answer_latex": " $61$",
+  "answer_number": "61",
+  "unit": " $\\mathrm{~N} / \\mathrm{C}$",
+  "source": "fund",
+  "problemid": " Question 22.67",
+  "comment": " ",
+  "solution": "$61 \\mathrm{~N} / \\mathrm{C}$"
+ },
+ {
+  "problem_text": "A long, straight wire has fixed negative charge with a linear charge density of magnitude $3.6 \\mathrm{nC} / \\mathrm{m}$. The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius $1.5 \\mathrm{~cm}$. The shell is to have positive charge on its outside surface with a surface charge density $\\sigma$ that makes the net external electric field zero. Calculate $\\sigma$.\r\n",
+  "answer_latex": " $3.8$",
+  "answer_number": "3.8",
+  "unit": "$10^{-8} \\mathrm{C} / \\mathrm{m}^2$ ",
+  "source": "fund",
+  "problemid": " Question 23.27",
+  "comment": " ",
+  "solution": "$3.8 \\times 10^{-8} \\mathrm{C} / \\mathrm{m}^2$"
+ },
+ {
+  "problem_text": "Beams of high-speed protons can be produced in \"guns\" using electric fields to accelerate the protons. What acceleration would a proton experience if the gun's electric field were $2.00 \\times 10^4 \\mathrm{~N} / \\mathrm{C}$ ?",
+  "answer_latex": "$1.92$",
+  "answer_number": "1.92",
+  "unit": "$10^{12} \\mathrm{~m} / \\mathrm{s}^2 $ ",
+  "source": "fund",
+  "problemid": " Question 22.47",
+  "comment": " ",
+  "solution": "$1.92 \\times 10^{12} \\mathrm{~m} / \\mathrm{s}^2 $"
+ },
+ {
+  "problem_text": "An infinite line of charge produces a field of magnitude $4.5 \\times$ $10^4 \\mathrm{~N} / \\mathrm{C}$ at distance $2.0 \\mathrm{~m}$. Find the linear charge density.\r\n",
+  "answer_latex": " $5.0$",
+  "answer_number": "5.0",
+  "unit": "$\\mu \\mathrm{C} / \\mathrm{m}$",
+  "source": "fund",
+  "problemid": " Question 23.25",
+  "comment": " ",
+  "solution": "$5.0 \\mu \\mathrm{C} / \\mathrm{m}$"
+ },
+ {
+  "problem_text": "A charged nonconducting rod, with a length of $2.00 \\mathrm{~m}$ and a cross-sectional area of $4.00 \\mathrm{~cm}^2$, lies along the positive side of an $x$ axis with one end at the origin. The volume charge density $\\rho$ is charge per unit volume in coulombs per cubic meter. How many excess electrons are on the rod if $\\rho$ is uniform, with a value of $-4.00 \\mu \\mathrm{C} / \\mathrm{m}^3$?\r\n",
+  "answer_latex": " $2.00$",
+  "answer_number": "2.00",
+  "unit": "$10^{10} \\text { electrons; }$",
+  "source": "fund",
+  "problemid": " Question 21.51",
+  "comment": " ",
+  "solution": "$2.00 \\times 10^{10} \\text { electrons; }$"
+ }
+]

fund_sol.json

@@ -1 +1,102 @@
-[{"problem_text": "A $2.00 \\mathrm{~kg}$ particle moves along an $x$ axis in one-dimensional motion while a conservative force along that axis acts on it. The potential energy $U(x)$ associated with the force is plotted in Fig. 8-10a. That is, if the particle were placed at any position between $x=0$ and $x=7.00 \\mathrm{~m}$, it would have the plotted value of $U$. At $x=6.5 \\mathrm{~m}$, the particle has velocity $\\vec{v}_0=(-4.00 \\mathrm{~m} / \\mathrm{s}) \\hat{\\mathrm{i}}$\r\nFrom Figure, determine the particle's speed at $x_1=4.5 \\mathrm{~m}$.", "answer_latex": "3.0", "answer_number": "3.0", "unit": " m/s", "source": "fund", "problemid": " 8.04", "comment": " ", "solution": "\n(1) The particle's kinetic energy is given by Eq. 7-1 $\\left(K=\\frac{1}{2} m v^2\\right)$. (2) Because only a conservative force acts on the particle, the mechanical energy $E_{\\mathrm{mec}}(=K+U)$ is conserved as the particle moves. (3) Therefore, on a plot of $U(x)$ such as Fig. 8-10a, the kinetic energy is equal to the difference between $E_{\\mathrm{mec}}$ and $U$.\nCalculations: At $x=6.5 \\mathrm{~m}$, the particle has kinetic energy\n$$\n\\begin{aligned}\nK_0 & =\\frac{1}{2} m v_0^2=\\frac{1}{2}(2.00 \\mathrm{~kg})(4.00 \\mathrm{~m} / \\mathrm{s})^2 \\\\\n& =16.0 \\mathrm{~J} .\n\\end{aligned}\n$$\nBecause the potential energy there is $U=0$, the mechanical energy is\n$$\nE_{\\text {mec }}=K_0+U_0=16.0 \\mathrm{~J}+0=16.0 \\mathrm{~J} .\n$$\nThis value for $E_{\\mathrm{mec}}$ is plotted as a horizontal line in Fig. 8-10a. From that figure we see that at $x=4.5 \\mathrm{~m}$, the potential energy is $U_1=7.0 \\mathrm{~J}$. The kinetic energy $K_1$ is the difference between $E_{\\text {mec }}$ and $U_1$ :\n$$\nK_1=E_{\\text {mec }}-U_1=16.0 \\mathrm{~J}-7.0 \\mathrm{~J}=9.0 \\mathrm{~J} .\n$$\nBecause $K_1=\\frac{1}{2} m v_1^2$, we find\n$$\nv_1=3.0 \\mathrm{~m} / \\mathrm{s}\n$$\n"}, {"problem_text": "A playful astronaut releases a bowling ball, of mass $m=$ $7.20 \\mathrm{~kg}$, into circular orbit about Earth at an altitude $h$ of $350 \\mathrm{~km}$.\r\nWhat is the mechanical energy $E$ of the ball in its orbit?", "answer_latex": " -214", "answer_number": "-214", "unit": " MJ", "source": "fund", "problemid": " 13.05", "comment": " ", "solution": "We can get $E$ from the orbital energy, given by Eq. $13-40$ ( $E=-G M m / 2 r)$ ), if we first find the orbital radius $r$. (It is not simply the given altitude.)\nCalculations: The orbital radius must be\n$$\nr=R+h=6370 \\mathrm{~km}+350 \\mathrm{~km}=6.72 \\times 10^6 \\mathrm{~m},\n$$\nin which $R$ is the radius of Earth. Then, from Eq. 13-40 with Earth mass $M=5.98 \\times 10^{24} \\mathrm{~kg}$, the mechanical energy is\n$$\n\\begin{aligned}\nE & =-\\frac{G M m}{2 r} \\\\\n& =-\\frac{\\left(6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^2 / \\mathrm{kg}^2\\right)\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)(7.20 \\mathrm{~kg})}{(2)\\left(6.72 \\times 10^6 \\mathrm{~m}\\right)} \\\\\n& =-2.14 \\times 10^8 \\mathrm{~J}=-214 \\mathrm{MJ} .\n\\end{aligned}\n$$"}, {"problem_text": "Let the disk in Figure start from rest at time $t=0$ and also let the tension in the massless cord be $6.0 \\mathrm{~N}$ and the angular acceleration of the disk be $-24 \\mathrm{rad} / \\mathrm{s}^2$. What is its rotational kinetic energy $K$ at $t=2.5 \\mathrm{~s}$ ?", "answer_latex": " 90", "answer_number": "90", "unit": " J", "source": "fund", "problemid": " 10.11", "comment": " ", "solution": "We can find $K$ with Eq. $10-34\\left(K=\\frac{1}{2} I \\omega^2\\right)$. We already know that $I=\\frac{1}{2} M R^2$, but we do not yet know $\\omega$ at $t=2.5 \\mathrm{~s}$. However, because the angular acceleration $\\alpha$ has the constant value of $-24 \\mathrm{rad} / \\mathrm{s}^2$, we can apply the equations for constant angular acceleration in Table 10-1.\n\nCalculations: Because we want $\\omega$ and know $\\alpha$ and $\\omega_0(=0)$, we use Eq. 10-12:\n$$\n\\omega=\\omega_0+\\alpha t=0+\\alpha t=\\alpha t .\n$$\nSubstituting $\\omega=\\alpha t$ and $I=\\frac{1}{2} M R^2$ into Eq. 10-34, we find\n$$\n\\begin{aligned}\nK & =\\frac{1}{2} I \\omega^2=\\frac{1}{2}\\left(\\frac{1}{2} M R^2\\right)(\\alpha t)^2=\\frac{1}{4} M(R \\alpha t)^2 \\\\\n& =\\frac{1}{4}(2.5 \\mathrm{~kg})\\left[(0.20 \\mathrm{~m})\\left(-24 \\mathrm{rad} / \\mathrm{s}^2\\right)(2.5 \\mathrm{~s})\\right]^2 \\\\\n& =90 \\mathrm{~J} .\n\\end{aligned}\n$$\n"}, {"problem_text": "A food shipper pushes a wood crate of cabbage heads (total mass $m=14 \\mathrm{~kg}$ ) across a concrete floor with a constant horizontal force $\\vec{F}$ of magnitude $40 \\mathrm{~N}$. In a straight-line displacement of magnitude $d=0.50 \\mathrm{~m}$, the speed of the crate decreases from $v_0=0.60 \\mathrm{~m} / \\mathrm{s}$ to $v=0.20 \\mathrm{~m} / \\mathrm{s}$. What is the increase $\\Delta E_{\\text {th }}$ in the thermal energy of the crate and floor?", "answer_latex": " 22.2", "answer_number": "22.2", "unit": " J", "source": "fund", "problemid": " 8.05", "comment": " ", "solution": "\nWe can relate $\\Delta E_{\\text {th }}$ to the work $W$ done by $\\vec{F}$ with the energy statement of Eq. 8-33 for a system that involves friction:\n$$\nW=\\Delta E_{\\text {mec }}+\\Delta E_{\\text {th }} .\n$$\nCalculations: We know the value of $W$ from (a). The change $\\Delta E_{\\text {mec }}$ in the crate's mechanical energy is just the change in its kinetic energy because no potential energy changes occur, so we have\n$$\n\\Delta E_{\\mathrm{mec}}=\\Delta K=\\frac{1}{2} m v^2-\\frac{1}{2} m v_0^2 .\n$$\nSubstituting this into Eq. 8-34 and solving for $\\Delta E_{\\mathrm{th}}$, we find\n$$\n\\begin{aligned}\n\\Delta E_{\\mathrm{th}} & =W-\\left(\\frac{1}{2} m v^2-\\frac{1}{2} m v_0^2\\right)=W-\\frac{1}{2} m\\left(v^2-v_0^2\\right) \\\\\n& =20 \\mathrm{~J}-\\frac{1}{2}(14 \\mathrm{~kg})\\left[(0.20 \\mathrm{~m} / \\mathrm{s})^2-(0.60 \\mathrm{~m} / \\mathrm{s})^2\\right] \\\\\n& =22.2 \\mathrm{~J} \\approx 22 \\mathrm{~J} .\n\\end{aligned}\n$$\n"}, {"problem_text": "While you are operating a Rotor (a large, vertical, rotating cylinder found in amusement parks), you spot a passenger in acute distress and decrease the angular velocity of the cylinder from $3.40 \\mathrm{rad} / \\mathrm{s}$ to $2.00 \\mathrm{rad} / \\mathrm{s}$ in $20.0 \\mathrm{rev}$, at constant angular acceleration. (The passenger is obviously more of a \"translation person\" than a \"rotation person.\")\r\nWhat is the constant angular acceleration during this decrease in angular speed?", "answer_latex": "-0.0301", "answer_number": "-0.0301", "unit": " $\\mathrm{rad} / \\mathrm{s}^2$", "source": "fund", "problemid": " 10.04", "comment": " ", "solution": "Because the cylinder's angular acceleration is constant, we can relate it to the angular velocity and angular displacement via the basic equations for constant angular acceleration (Eqs. 10-12 and 10-13).\n\nCalculations: Let's first do a quick check to see if we can solve the basic equations. The initial angular velocity is $\\omega_0=3.40$\n$\\mathrm{rad} / \\mathrm{s}$, the angular displacement is $\\theta-\\theta_0=20.0 \\mathrm{rev}$, and the angular velocity at the end of that displacement is $\\omega=2.00$ $\\mathrm{rad} / \\mathrm{s}$. In addition to the angular acceleration $\\alpha$ that we want, both basic equations also contain time $t$, which we do not necessarily want.\n\nTo eliminate the unknown $t$, we use Eq. 10-12 to write\n$$\nt=\\frac{\\omega-\\omega_0}{\\alpha}\n$$\nwhich we then substitute into Eq. 10-13 to write\n$$\n\\theta-\\theta_0=\\omega_0\\left(\\frac{\\omega-\\omega_0}{\\alpha}\\right)+\\frac{1}{2} \\alpha\\left(\\frac{\\omega-\\omega_0}{\\alpha}\\right)^2 .\n$$\nSolving for $\\alpha$, substituting known data, and converting 20 rev to $125.7 \\mathrm{rad}$, we find\n$$\n\\begin{aligned}\n\\alpha & =\\frac{\\omega^2-\\omega_0^2}{2\\left(\\theta-\\theta_0\\right)}=\\frac{(2.00 \\mathrm{rad} / \\mathrm{s})^2-(3.40 \\mathrm{rad} / \\mathrm{s})^2}{2(125.7 \\mathrm{rad})} \\\\\n& =-0.0301 \\mathrm{rad} / \\mathrm{s}^2\n\\end{aligned}\n$$\n"}, {"problem_text": "A living room has floor dimensions of $3.5 \\mathrm{~m}$ and $4.2 \\mathrm{~m}$ and a height of $2.4 \\mathrm{~m}$.\r\nWhat does the air in the room weigh when the air pressure is $1.0 \\mathrm{~atm}$ ?", "answer_latex": " 418", "answer_number": "418", "unit": " N", "source": "fund", "problemid": " 14.01", "comment": " ", "solution": "(1) The air's weight is equal to $m g$, where $m$ is its mass.\n(2) Mass $m$ is related to the air density $\\rho$ and the air volume $V$ by Eq.14-2 $(\\rho=m / V)$.\n\nCalculation: Putting the two ideas together and taking the density of air at 1.0 atm from Table 14-1, we find\n$$\n\\begin{aligned}\nm g & =(\\rho V) g \\\\\n& =\\left(1.21 \\mathrm{~kg} / \\mathrm{m}^3\\right)(3.5 \\mathrm{~m} \\times 4.2 \\mathrm{~m} \\times 2.4 \\mathrm{~m})\\left(9.8 \\mathrm{~m} / \\mathrm{s}^2\\right) \\\\\n& =418 \\mathrm{~N} \n\\end{aligned}\n$$"}, {"problem_text": "An astronaut whose height $h$ is $1.70 \\mathrm{~m}$ floats \"feet down\" in an orbiting space shuttle at distance $r=6.77 \\times 10^6 \\mathrm{~m}$ away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head?", "answer_latex": "-4.37 ", "answer_number": "-4.37 ", "unit": " $10 ^ {-6}m/s^2$", "source": "fund", "problemid": " 13.02", "comment": " ", "solution": "We can approximate Earth as a uniform sphere of mass $M_E$. Then, from Eq. 13-11, the gravitational acceleration at any distance $r$ from the center of Earth is\n$$\na_g=\\frac{G M_E}{r^2}\n$$\nWe might simply apply this equation twice, first with $r=$ $6.77 \\times 10^6 \\mathrm{~m}$ for the location of the feet and then with $r=6.77 \\times 10^6 \\mathrm{~m}+1.70 \\mathrm{~m}$ for the location of the head. However, a calculator may give us the same value for $a_g$ twice, and thus a difference of zero, because $h$ is so much smaller than $r$. Here's a more promising approach: Because we have a differential change $d r$ in $r$ between the astronaut's feet and head, we should differentiate Eq. 13-15 with respect to $r$.\nCalculations: The differentiation gives us\n$$\nd a_g=-2 \\frac{G M_E}{r^3} d r\n$$\nwhere $d a_g$ is the differential change in the gravitational acceleration due to the differential change $d r$ in $r$. For the astronaut, $d r=h$ and $r=6.77 \\times 10^6 \\mathrm{~m}$. Substituting data into Eq.13-16, we find\n$$\n\\begin{aligned}\nd a_g & =-2 \\frac{\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{kg} \\cdot \\mathrm{s}^2\\right)\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)}{\\left(6.77 \\times 10^6 \\mathrm{~m}\\right)^3}(1.70 \\mathrm{~m}) \\\\\n& =-4.37 \\times 10^{-6} \\mathrm{~m} / \\mathrm{s}^2, \\quad \\text { (Answer) }\n\\end{aligned}\n$$"}, {"problem_text": "If the particles in a system all move together, the com moves with them-no trouble there. But what happens when they move in different directions with different accelerations? Here is an example.\r\n\r\nThe three particles in Figure are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are $F_1=6.0 \\mathrm{~N}, F_2=12 \\mathrm{~N}$, and $F_3=14 \\mathrm{~N}$. What is the acceleration of the center of mass of the system?", "answer_latex": " 1.16", "answer_number": " 1.16", "unit": " $m/s^2$", "source": "fund", "problemid": " 9.03", "comment": " ", "solution": "The position of the center of mass is marked by a dot in the figure. We can treat the center of mass as if it were a real particle, with a mass equal to the system's total mass $M=16 \\mathrm{~kg}$. We can also treat the three external forces as if they act at the center of mass (Fig. 9-7b).\n\nCalculations: We can now apply Newton's second law $\\left(\\vec{F}_{\\text {net }}=m \\vec{a}\\right)$ to the center of mass, writing\n$$\n\\vec{F}_{\\text {net }}=M \\vec{a}_{\\mathrm{com}}\n$$\nor\n$$\n\\begin{aligned}\n& \\vec{F}_1+\\vec{F}_2+\\vec{F}_3=M \\vec{a}_{\\mathrm{com}} \\\\\n& \\vec{a}_{\\mathrm{com}}=\\frac{\\vec{F}_1+\\vec{F}_2+\\vec{F}_3}{M} .\n\\end{aligned}\n$$\nEquation 9-20 tells us that the acceleration $\\vec{a}_{\\text {com }}$ of the center of mass is in the same direction as the net external force $\\vec{F}_{\\text {net }}$ on the system (Fig. 9-7b). Because the particles are initially at rest, the center of mass must also be at rest. As the center of mass then begins to accelerate, it must move off in the common direction of $\\vec{a}_{\\text {com }}$ and $\\vec{F}_{\\text {net }}$.\n\nWe can evaluate the right side of Eq. 9-21 directly on a vector-capable calculator, or we can rewrite Eq. 9-21 in component form, find the components of $\\vec{a}_{\\text {com }}$, and then find $\\vec{a}_{\\text {com }}$. Along the $x$ axis, we have\n$$\n\\begin{aligned}\na_{\\mathrm{com}, x} & =\\frac{F_{1 x}+F_{2 x}+F_{3 x}}{M} \\\\\n& =\\frac{-6.0 \\mathrm{~N}+(12 \\mathrm{~N}) \\cos 45^{\\circ}+14 \\mathrm{~N}}{16 \\mathrm{~kg}}=1.03 \\mathrm{~m} / \\mathrm{s}^2 .\n\\end{aligned}\n$$\nAlong the $y$ axis, we have\n$$\n\\begin{aligned}\na_{\\mathrm{com}, y} & =\\frac{F_{1 y}+F_{2 y}+F_{3 y}}{M} \\\\\n& =\\frac{0+(12 \\mathrm{~N}) \\sin 45^{\\circ}+0}{16 \\mathrm{~kg}}=0.530 \\mathrm{~m} / \\mathrm{s}^2 .\n\\end{aligned}\n$$\nFrom these components, we find that $\\vec{a}_{\\mathrm{com}}$ has the magnitude\n$$\n\\begin{aligned}\na_{\\mathrm{com}} & =\\sqrt{\\left(a_{\\mathrm{com}, x}\\right)^2+\\left(a_{\\text {com }, y}\\right)^2} \\\\\n& =1.16 \\mathrm{~m} / \\mathrm{s}^2 \n\\end{aligned}\n$$\n"}, {"problem_text": "An asteroid, headed directly toward Earth, has a speed of $12 \\mathrm{~km} / \\mathrm{s}$ relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed $v_f$ when it reaches Earth's surface.", "answer_latex": "1.60", "answer_number": "1.60", "unit": " $10^4 \\mathrm{~m} / \\mathrm{s}$", "source": "fund", "problemid": " 13.03", "comment": " ", "solution": "Because we are to neglect the effects of the atmosphere on the asteroid, the mechanical energy of the asteroid-Earth system is conserved during the fall. Thus, the final mechanical energy (when the asteroid reaches Earth's surface) is equal to the initial mechanical energy. With kinetic energy $K$ and gravitational potential energy $U$, we can write this as\n$$\nK_f+U_f=K_i+U_i\n$$\nAlso, if we assume the system is isolated, the system's linear momentum must be conserved during the fall. Therefore, the momentum change of the asteroid and that of Earth must be equal in magnitude and opposite in sign. However, because Earth's mass is so much greater than the asteroid's mass, the change in Earth's speed is negligible relative to the change in the asteroid's speed. So, the change in Earth's kinetic energy is also negligible. Thus, we can assume that the kinetic energies in Eq.13-29 are those of the asteroid alone.\n\nCalculations: Let $m$ represent the asteroid's mass and $M$ represent Earth's mass $\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)$. The asteroid is ini-\ntially at distance $10 R_E$ and finally at distance $R_E$, where $R_E$ is Earth's radius $\\left(6.37 \\times 10^6 \\mathrm{~m}\\right)$. Substituting Eq. 13-21 for $U$ and $\\frac{1}{2} m v^2$ for $K$, we rewrite Eq. 13-29 as\n$$\n\\frac{1}{2} m v_f^2-\\frac{G M m}{R_E}=\\frac{1}{2} m v_i^2-\\frac{G M m}{10 R_E} .\n$$\nRearranging and substituting known values, we find\n$$\n\\begin{aligned}\nv_f^2= & v_i^2+\\frac{2 G M}{R_E}\\left(1-\\frac{1}{10}\\right) \\\\\n= & \\left(12 \\times 10^3 \\mathrm{~m} / \\mathrm{s}\\right)^2 \\\\\n& +\\frac{2\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{kg} \\cdot \\mathrm{s}^2\\right)\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)}{6.37 \\times 10^6 \\mathrm{~m}} 0.9 \\\\\n= & 2.567 \\times 10^8 \\mathrm{~m}^2 / \\mathrm{s}^2,\n\\end{aligned}\n$$\nand\n$$\nv_f=1.60 \\times 10^4 \\mathrm{~m} / \\mathrm{s}\n$$"}, {"problem_text": "The huge advantage of using the conservation of energy instead of Newton's laws of motion is that we can jump from the initial state to the final state without considering all the intermediate motion. Here is an example. In Figure, a child of mass $m$ is released from rest at the top of a water slide, at height $h=8.5 \\mathrm{~m}$ above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.", "answer_latex": " 13", "answer_number": "13", "unit": " m/s", "source": "fund", "problemid": "  8.03", "comment": " ", "solution": "Let the mechanical energy be $E_{\\text {mec }, t}$ when the child is at the top of the slide and $E_{\\mathrm{mec}, b}$ when she is at the bottom. Then the conservation principle tells us\n$$\nE_{\\mathrm{mec}, b}=E_{\\mathrm{mec}, t} .\n$$\n8-3 READING A POTENTIAL ENERGY CURVE\n187\nTo show both kinds of mechanical energy, we have\n$$\n\\begin{aligned}\nK_b+U_b & =K_t+U_t, \\\\\n\\frac{1}{2} m v_b^2+m g y_b & =\\frac{1}{2} m v_t^2+m g y_t .\n\\end{aligned}\n$$\nDividing by $m$ and rearranging yield\n$$\nv_b^2=v_t^2+2 g\\left(y_t-y_b\\right)\n$$\nPutting $v_t=0$ and $y_t-y_b=h$ leads to\n$$\n\\begin{aligned}\nv_b & =\\sqrt{2 g h}=\\sqrt{(2)\\left(9.8 \\mathrm{~m} / \\mathrm{s}^2\\right)(8.5 \\mathrm{~m})} \\\\\n& =13 \\mathrm{~m} / \\mathrm{s} .\n\\end{aligned}\n$$"}]
+[
+ {
+  "problem_text": "A $2.00 \\mathrm{~kg}$ particle moves along an $x$ axis in one-dimensional motion while a conservative force along that axis acts on it. The potential energy $U(x)$ is 0 when $x = 6.5 \\mathrm{~m} $ and is $7 \\mathrm{~J}$ when $x = 4.5 \\mathrm{~m} $. At $x=6.5 \\mathrm{~m}$, the particle has velocity $\\vec{v}_0=(-4.00 \\mathrm{~m} / \\mathrm{s}) \\hat{\\mathrm{i}}$ Determine the particle's speed at $x_1=4.5 \\mathrm{~m}$.",
+  "answer_latex": "3.0",
+  "answer_number": "3.0",
+  "unit": " m/s",
+  "source": "fund",
+  "problemid": " 8.04",
+  "comment": " ",
+  "solution": "\nAt $x=6.5 \\mathrm{~m}$, the particle has kinetic energy\n$$\n\\begin{aligned}\nK_0 & =\\frac{1}{2} m v_0^2=\\frac{1}{2}(2.00 \\mathrm{~kg})(4.00 \\mathrm{~m} / \\mathrm{s})^2 \\\\\n& =16.0 \\mathrm{~J} .\n\\end{aligned}\n$$\nBecause the potential energy there is $U=0$, the mechanical energy is\n$$\nE_{\\text {mec }}=K_0+U_0=16.0 \\mathrm{~J}+0=16.0 \\mathrm{~J} .\n$$The kinetic energy $K_1$ is the difference between $E_{\\text {mec }}$ and $U_1$ :\n$$\nK_1=E_{\\text {mec }}-U_1=16.0 \\mathrm{~J}-7.0 \\mathrm{~J}=9.0 \\mathrm{~J} .\n$$\nBecause $K_1=\\frac{1}{2} m v_1^2$, we find\n$$\nv_1=3.0 \\mathrm{~m} / \\mathrm{s}\n$$\n\n\n"
+ },
+ {
+  "problem_text": "A playful astronaut releases a bowling ball, of mass $m=$ $7.20 \\mathrm{~kg}$, into circular orbit about Earth at an altitude $h$ of $350 \\mathrm{~km}$.\r\nWhat is the mechanical energy $E$ of the ball in its orbit?",
+  "answer_latex": " -214",
+  "answer_number": "-214",
+  "unit": " MJ",
+  "source": "fund",
+  "problemid": " 13.05",
+  "comment": " ",
+  "solution": "We can get $E$ from the orbital energy, given by Eq. $13-40$ ( $E=-G M m / 2 r)$ ), if we first find the orbital radius $r$. (It is not simply the given altitude.)\nCalculations: The orbital radius must be\n$$\nr=R+h=6370 \\mathrm{~km}+350 \\mathrm{~km}=6.72 \\times 10^6 \\mathrm{~m},\n$$\nin which $R$ is the radius of Earth. Then, from Eq. 13-40 with Earth mass $M=5.98 \\times 10^{24} \\mathrm{~kg}$, the mechanical energy is\n$$\n\\begin{aligned}\nE & =-\\frac{G M m}{2 r} \\\\\n& =-\\frac{\\left(6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^2 / \\mathrm{kg}^2\\right)\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)(7.20 \\mathrm{~kg})}{(2)\\left(6.72 \\times 10^6 \\mathrm{~m}\\right)} \\\\\n& =-2.14 \\times 10^8 \\mathrm{~J}=-214 \\mathrm{MJ} .\n\\end{aligned}\n$$"
+ },
+ {
+  "problem_text": "There is a disk mounted on a fixed horizontal axle. A block hangs from a massless cord that is wrapped around the rim of the disk. The cord does not slip and there is no friction at the axle. Let the disk start from rest at time $t=0$ and also let the tension in the massless cord be $6.0 \\mathrm{~N}$ and the angular acceleration of the disk be $-24 \\mathrm{rad} / \\mathrm{s}^2$. What is its rotational kinetic energy $K$ at $t=2.5 \\mathrm{~s}$ ?",
+  "answer_latex": " 90",
+  "answer_number": "90",
+  "unit": " J",
+  "source": "fund",
+  "problemid": " 10.11",
+  "comment": " ",
+  "solution": "\nWe can find $K$ with Eq. $10-34\\left(K=\\frac{1}{2} I \\omega^2\\right)$. We already know that $I=\\frac{1}{2} M R^2$, but we do not yet know $\\omega$ at $t=2.5 \\mathrm{~s}$. However, because the angular acceleration $\\alpha$ has the constant value of $-24 \\mathrm{rad} / \\mathrm{s}^2$, we can apply the equations for constant angular acceleration.\n\nCalculations: Because we want $\\omega$ and know $\\alpha$ and $\\omega_0(=0)$, we use Eq. 10-12:\n$$\n\\omega=\\omega_0+\\alpha t=0+\\alpha t=\\alpha t .\n$$\nSubstituting $\\omega=\\alpha t$ and $I=\\frac{1}{2} M R^2$ into Eq. 10-34, we find\n$$\n\\begin{aligned}\nK & =\\frac{1}{2} I \\omega^2=\\frac{1}{2}\\left(\\frac{1}{2} M R^2\\right)(\\alpha t)^2=\\frac{1}{4} M(R \\alpha t)^2 \\\\\n& =\\frac{1}{4}(2.5 \\mathrm{~kg})\\left[(0.20 \\mathrm{~m})\\left(-24 \\mathrm{rad} / \\mathrm{s}^2\\right)(2.5 \\mathrm{~s})\\right]^2 \\\\\n& =90 \\mathrm{~J} .\n\\end{aligned}\n$$\n\n"
+ },
+ {
+  "problem_text": "A food shipper pushes a wood crate of cabbage heads (total mass $m=14 \\mathrm{~kg}$ ) across a concrete floor with a constant horizontal force $\\vec{F}$ of magnitude $40 \\mathrm{~N}$. In a straight-line displacement of magnitude $d=0.50 \\mathrm{~m}$, the speed of the crate decreases from $v_0=0.60 \\mathrm{~m} / \\mathrm{s}$ to $v=0.20 \\mathrm{~m} / \\mathrm{s}$. What is the increase $\\Delta E_{\\text {th }}$ in the thermal energy of the crate and floor?",
+  "answer_latex": " 22.2",
+  "answer_number": "22.2",
+  "unit": " J",
+  "source": "fund",
+  "problemid": " 8.05",
+  "comment": " ",
+  "solution": "\nWe can relate $\\Delta E_{\\text{th }}$ to the work $W$ done by $\\vec{F}$ with the energy statement for a system that involves friction:\n$$\nW=\\Delta E_{\\text {mec }}+\\Delta E_{\\text {th }} .\n$$\nCalculations: We know the value of $W=F d \\cos \\phi=(40 \\mathrm{~N})(0.50 \\mathrm{~m}) \\cos 0^{\\circ}=20 \\mathrm{~J}$. The change $\\Delta E_{\\text {mec }}$ in the crate's mechanical energy is just the change in its kinetic energy because no potential energy changes occur, so we have\n$$\n\\Delta E_{\\mathrm{mec}}=\\Delta K=\\frac{1}{2} m v^2-\\frac{1}{2} m v_0^2 .\n$$\nSubstituting this into above equation and solving for $\\Delta E_{\\mathrm{th}}$, we find\n$$\n\\begin{aligned}\n\\Delta E_{\\mathrm{th}} & =W-\\left(\\frac{1}{2} m v^2-\\frac{1}{2} m v_0^2\\right)=W-\\frac{1}{2} m\\left(v^2-v_0^2\\right) \\\\\n& =20 \\mathrm{~J}-\\frac{1}{2}(14 \\mathrm{~kg})\\left[(0.20 \\mathrm{~m} / \\mathrm{s})^2-(0.60 \\mathrm{~m} / \\mathrm{s})^2\\right] \\\\\n& =22.2 \\mathrm{~J}.\n\\end{aligned}\n$$\n"
+ },
+ {
+  "problem_text": "While you are operating a Rotor (a large, vertical, rotating cylinder found in amusement parks), you spot a passenger in acute distress and decrease the angular velocity of the cylinder from $3.40 \\mathrm{rad} / \\mathrm{s}$ to $2.00 \\mathrm{rad} / \\mathrm{s}$ in $20.0 \\mathrm{rev}$, at constant angular acceleration. (The passenger is obviously more of a \"translation person\" than a \"rotation person.\")\r\nWhat is the constant angular acceleration during this decrease in angular speed?",
+  "answer_latex": "-0.0301",
+  "answer_number": "-0.0301",
+  "unit": " $\\mathrm{rad} / \\mathrm{s}^2$",
+  "source": "fund",
+  "problemid": " 10.04",
+  "comment": " ",
+  "solution": "\nBecause the cylinder's angular acceleration is constant, we can relate it to the angular velocity and angular displacement via the basic equations for constant angular acceleration.\n\nCalculations: Let's first do a quick check to see if we can solve the basic equations. The initial angular velocity is $\\omega_0=3.40$\n$\\mathrm{rad} / \\mathrm{s}$, the angular displacement is $\\theta-\\theta_0=20.0 \\mathrm{rev}$, and the angular velocity at the end of that displacement is $\\omega=2.00$ $\\mathrm{rad} / \\mathrm{s}$. In addition to the angular acceleration $\\alpha$ that we want, both basic equations also contain time $t$, which we do not necessarily want.\n\nTo eliminate the unknown $t$, we use $$\\omega  =\\omega_0+\\alpha t$$ to write\n$$\nt=\\frac{\\omega-\\omega_0}{\\alpha}\n$$\nwhich we then substitute into equation $$\\theta-\\theta_0 =\\omega_0 t+\\frac{1}{2} \\alpha t^2 $$ to write\n$$\n\\theta-\\theta_0=\\omega_0\\left(\\frac{\\omega-\\omega_0}{\\alpha}\\right)+\\frac{1}{2} \\alpha\\left(\\frac{\\omega-\\omega_0}{\\alpha}\\right)^2 .\n$$\nSolving for $\\alpha$, substituting known data, and converting 20 rev to $125.7 \\mathrm{rad}$, we find\n$$\n\\begin{aligned}\n\\alpha & =\\frac{\\omega^2-\\omega_0^2}{2\\left(\\theta-\\theta_0\\right)}=\\frac{(2.00 \\mathrm{rad} / \\mathrm{s})^2-(3.40 \\mathrm{rad} / \\mathrm{s})^2}{2(125.7 \\mathrm{rad})} \\\\\n& =-0.0301 \\mathrm{rad} / \\mathrm{s}^2\n\\end{aligned}\n$$\n\n"
+ },
+ {
+  "problem_text": "A living room has floor dimensions of $3.5 \\mathrm{~m}$ and $4.2 \\mathrm{~m}$ and a height of $2.4 \\mathrm{~m}$.\r\nWhat does the air in the room weigh when the air pressure is $1.0 \\mathrm{~atm}$ ?",
+  "answer_latex": " 418",
+  "answer_number": "418",
+  "unit": " N",
+  "source": "fund",
+  "problemid": " 14.01",
+  "comment": " ",
+  "solution": "\n(1) The air's weight is equal to $m g$, where $m$ is its mass.\n(2) Mass $m$ is related to the air density $\\rho$ and the air volume $V$ by Eq $(\\rho=m / V)$.\n\nCalculation: Putting the two ideas together and taking the density of air at 1.0 atm, we find\n$$\n\\begin{aligned}\nm g & =(\\rho V) g \\\\\n& =\\left(1.21 \\mathrm{~kg} / \\mathrm{m}^3\\right)(3.5 \\mathrm{~m} \\times 4.2 \\mathrm{~m} \\times 2.4 \\mathrm{~m})\\left(9.8 \\mathrm{~m} / \\mathrm{s}^2\\right) \\\\\n& =418 \\mathrm{~N} \n\\end{aligned}\n$$\n"
+ },
+ {
+  "problem_text": "An astronaut whose height $h$ is $1.70 \\mathrm{~m}$ floats \"feet down\" in an orbiting space shuttle at distance $r=6.77 \\times 10^6 \\mathrm{~m}$ away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head?",
+  "answer_latex": "-4.37 ",
+  "answer_number": "-4.37 ",
+  "unit": " $10 ^ {-6}m/s^2$",
+  "source": "fund",
+  "problemid": " 13.02",
+  "comment": " ",
+  "solution": "\nWe can approximate Earth as a uniform sphere of mass $M_E$. Then the gravitational acceleration at any distance $r$ from the center of Earth is\n$$\na_g=\\frac{G M_E}{r^2}\n$$\nWe might simply apply this equation twice, first with $r=$ $6.77 \\times 10^6 \\mathrm{~m}$ for the location of the feet and then with $r=6.77 \\times 10^6 \\mathrm{~m}+1.70 \\mathrm{~m}$ for the location of the head. However, a calculator may give us the same value for $a_g$ twice, and thus a difference of zero, because $h$ is so much smaller than $r$. Here's a more promising approach: Because we have a differential change $d r$ in $r$ between the astronaut's feet and head, we should differentiate above equation with respect to $r$.\nCalculations: The differentiation gives us\n$$\nd a_g=-2 \\frac{G M_E}{r^3} d r\n$$\nwhere $d a_g$ is the differential change in the gravitational acceleration due to the differential change $d r$ in $r$. For the astronaut, $d r=h$ and $r=6.77 \\times 10^6 \\mathrm{~m}$. Substituting data into equation, we find\n$$\n\\begin{aligned}\nd a_g & =-2 \\frac{\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{kg} \\cdot \\mathrm{s}^2\\right)\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)}{\\left(6.77 \\times 10^6 \\mathrm{~m}\\right)^3}(1.70 \\mathrm{~m}) \\\\\n& =-4.37 \\times 10^{-6} \\mathrm{~m} / \\mathrm{s}^2, \\quad \\text { (Answer) }\n\\end{aligned}\n$$\n"
+ },
+ {
+  "problem_text": "If the particles in a system all move together, the com moves with them-no trouble there. But what happens when they move in different directions with different accelerations? Here is an example.\r\n\r\nThe three particles in Figure are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are $F_1=6.0 \\mathrm{~N}, F_2=12 \\mathrm{~N}$, and $F_3=14 \\mathrm{~N}$. What is the acceleration of the center of mass of the system?",
+  "answer_latex": " 1.16",
+  "answer_number": " 1.16",
+  "unit": " $m/s^2$",
+  "source": "fund",
+  "problemid": " 9.03",
+  "comment": " ",
+  "solution": "\nWe can treat the center of mass as if it were a real particle, with a mass equal to the system's total mass $M=16 \\mathrm{~kg}$. We can also treat the three external forces as if they act at the center of mass.\n\nCalculations: We can now apply Newton's second law $\\left(\\vec{F}_{\\text {net }}=m \\vec{a}\\right)$ to the center of mass, writing\n$$\n\\vec{F}_{\\text {net }}=M \\vec{a}_{\\mathrm{com}}\n$$\nor\n$$\n\\begin{aligned}\n& \\vec{F}_1+\\vec{F}_2+\\vec{F}_3=M \\vec{a}_{\\mathrm{com}} \\\\\n& \\vec{a}_{\\mathrm{com}}=\\frac{\\vec{F}_1+\\vec{F}_2+\\vec{F}_3}{M} .\n\\end{aligned}\n$$\nEquation tells us that the acceleration $\\vec{a}_{\\text {com }}$ of the center of mass is in the same direction as the net external force $\\vec{F}_{\\text {net }}$ on the system. Because the particles are initially at rest, the center of mass must also be at rest. As the center of mass then begins to accelerate, it must move off in the common direction of $\\vec{a}_{\\text {com }}$ and $\\vec{F}_{\\text {net }}$.\n\nWe can evaluate the right side of the above equation directly on a vector-capable calculator, or we can rewrite the equation in component form, find the components of $\\vec{a}_{\\text {com }}$, and then find $\\vec{a}_{\\text {com }}$. Along the $x$ axis, we have\n$$\n\\begin{aligned}\na_{\\mathrm{com}, x} & =\\frac{F_{1 x}+F_{2 x}+F_{3 x}}{M} \\\\\n& =\\frac{-6.0 \\mathrm{~N}+(12 \\mathrm{~N}) \\cos 45^{\\circ}+14 \\mathrm{~N}}{16 \\mathrm{~kg}}=1.03 \\mathrm{~m} / \\mathrm{s}^2 .\n\\end{aligned}\n$$\nAlong the $y$ axis, we have\n$$\n\\begin{aligned}\na_{\\mathrm{com}, y} & =\\frac{F_{1 y}+F_{2 y}+F_{3 y}}{M} \\\\\n& =\\frac{0+(12 \\mathrm{~N}) \\sin 45^{\\circ}+0}{16 \\mathrm{~kg}}=0.530 \\mathrm{~m} / \\mathrm{s}^2 .\n\\end{aligned}\n$$\nFrom these components, we find that $\\vec{a}_{\\mathrm{com}}$ has the magnitude\n$$\n\\begin{aligned}\na_{\\mathrm{com}} & =\\sqrt{\\left(a_{\\mathrm{com}, x}\\right)^2+\\left(a_{\\text {com }, y}\\right)^2} \\\\\n& =1.16 \\mathrm{~m} / \\mathrm{s}^2 \n\\end{aligned}\n$$\n"
+ },
+ {
+  "problem_text": "An asteroid, headed directly toward Earth, has a speed of $12 \\mathrm{~km} / \\mathrm{s}$ relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed $v_f$ when it reaches Earth's surface.",
+  "answer_latex": "1.60",
+  "answer_number": "1.60",
+  "unit": " $10^4 \\mathrm{~m} / \\mathrm{s}$",
+  "source": "fund",
+  "problemid": " 13.03",
+  "comment": " ",
+  "solution": "\nBecause we are to neglect the effects of the atmosphere on the asteroid, the mechanical energy of the asteroid-Earth system is conserved during the fall. Thus, the final mechanical energy (when the asteroid reaches Earth's surface) is equal to the initial mechanical energy. With kinetic energy $K$ and gravitational potential energy $U$, we can write this as\n$$\nK_f+U_f=K_i+U_i\n$$\nAlso, if we assume the system is isolated, the system's linear momentum must be conserved during the fall. Therefore, the momentum change of the asteroid and that of Earth must be equal in magnitude and opposite in sign. However, because Earth's mass is so much greater than the asteroid's mass, the change in Earth's speed is negligible relative to the change in the asteroid's speed. So, the change in Earth's kinetic energy is also negligible. Thus, we can assume that the kinetic energies in above equation are those of the asteroid alone.\n\nCalculations: Let $m$ represent the asteroid's mass and $M$ represent Earth's mass $\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)$. The asteroid is ini-\ntially at distance $10 R_E$ and finally at distance $R_E$, where $R_E$ is Earth's radius $\\left(6.37 \\times 10^6 \\mathrm{~m}\\right)$. Substituting $-\\frac{G M m}{r}$ for $U$ and $\\frac{1}{2} m v^2$ for $K$, we rewrite above equation as\n$$\n\\frac{1}{2} m v_f^2-\\frac{G M m}{R_E}=\\frac{1}{2} m v_i^2-\\frac{G M m}{10 R_E} .\n$$\nRearranging and substituting known values, we find\n$$\n\\begin{aligned}\nv_f^2= & v_i^2+\\frac{2 G M}{R_E}\\left(1-\\frac{1}{10}\\right) \\\\\n= & \\left(12 \\times 10^3 \\mathrm{~m} / \\mathrm{s}\\right)^2 \\\\\n& +\\frac{2\\left(6.67 \\times 10^{-11} \\mathrm{~m}^3 / \\mathrm{kg} \\cdot \\mathrm{s}^2\\right)\\left(5.98 \\times 10^{24} \\mathrm{~kg}\\right)}{6.37 \\times 10^6 \\mathrm{~m}} 0.9 \\\\\n= & 2.567 \\times 10^8 \\mathrm{~m}^2 / \\mathrm{s}^2,\n\\end{aligned}\n$$\nand\n$$\nv_f=1.60 \\times 10^4 \\mathrm{~m} / \\mathrm{s}\n$$\n"
+ },
+ {
+  "problem_text": "The huge advantage of using the conservation of energy instead of Newton's laws of motion is that we can jump from the initial state to the final state without considering all the intermediate motion. Here is an example. In Figure, a child of mass $m$ is released from rest at the top of a water slide, at height $h=8.5 \\mathrm{~m}$ above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.",
+  "answer_latex": " 13",
+  "answer_number": "13",
+  "unit": " m/s",
+  "source": "fund",
+  "problemid": "  8.03",
+  "comment": " ",
+  "solution": "\nLet the mechanical energy be $E_{\\text {mec }, t}$ when the child is at the top of the slide and $E_{\\mathrm{mec}, b}$ when she is at the bottom. Then the conservation principle tells us\n$$\nE_{\\mathrm{mec}, b}=E_{\\mathrm{mec}, t} .\n$$\nTo show both kinds of mechanical energy, we have\n$$\n\\begin{aligned}\nK_b+U_b & =K_t+U_t, \\\\\n\\frac{1}{2} m v_b^2+m g y_b & =\\frac{1}{2} m v_t^2+m g y_t .\n\\end{aligned}\n$$\nDividing by $m$ and rearranging yield\n$$\nv_b^2=v_t^2+2 g\\left(y_t-y_b\\right)\n$$\nPutting $v_t=0$ and $y_t-y_b=h$ leads to\n$$\n\\begin{aligned}\nv_b & =\\sqrt{2 g h}=\\sqrt{(2)\\left(9.8 \\mathrm{~m} / \\mathrm{s}^2\\right)(8.5 \\mathrm{~m})} \\\\\n& =13 \\mathrm{~m} / \\mathrm{s} .\n\\end{aligned}\n$$\n"
+ }
+]

matter.json

@@ -1 +1,425 @@
-[{"problem_text": "Radiation from an X-ray source consists of two components of wavelengths $154.433 \\mathrm{pm}$ and $154.051 \\mathrm{pm}$. Calculate the difference in glancing angles $(2 \\theta)$ of the diffraction lines arising from the two components in a diffraction pattern from planes of separation $77.8 \\mathrm{pm}$.", "answer_latex": " 2.14", "answer_number": "2.14", "unit": " ${\\circ}$", "source": "matter", "problemid": " 37.7(a)", "comment": " "}, {"problem_text": "A chemical reaction takes place in a container of cross-sectional area $50 \\mathrm{~cm}^2$. As a result of the reaction, a piston is pushed out through $15 \\mathrm{~cm}$ against an external pressure of $1.0 \\mathrm{~atm}$. Calculate the work done by the system.", "answer_latex": " -75", "answer_number": "-75", "unit": " $\\mathrm{~J}$", "source": "matter", "problemid": " 55.4(a)", "comment": " "}, {"problem_text": "A mixture of water and ethanol is prepared with a mole fraction of water of 0.60 . If a small change in the mixture composition results in an increase in the chemical potential of water by $0.25 \\mathrm{~J} \\mathrm{~mol}^{-1}$, by how much will the chemical potential of ethanol change?", "answer_latex": " -0.38", "answer_number": "-0.38", "unit": " $\\mathrm{~J} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 69.2(a)", "comment": " "}, {"problem_text": "The enthalpy of fusion of mercury is $2.292 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, and its normal freezing point is $234.3 \\mathrm{~K}$ with a change in molar volume of $+0.517 \\mathrm{~cm}^3 \\mathrm{~mol}^{-1}$ on melting. At what temperature will the bottom of a column of mercury (density $13.6 \\mathrm{~g} \\mathrm{~cm}^{-3}$ ) of height $10.0 \\mathrm{~m}$ be expected to freeze?", "answer_latex": " 234.4", "answer_number": "234.4", "unit": " $ \\mathrm{~K}$ ", "source": "matter", "problemid": " 69.3", "comment": " problem"}, {"problem_text": "Suppose a nanostructure is modelled by an electron confined to a rectangular region with sides of lengths $L_1=1.0 \\mathrm{~nm}$ and $L_2=2.0 \\mathrm{~nm}$ and is subjected to thermal motion with a typical energy equal to $k T$, where $k$ is Boltzmann's constant. How low should the temperature be for the thermal energy to be comparable to the zero-point energy\uff1f", "answer_latex": " 5.5", "answer_number": "5.5", "unit": " $10^3 \\mathrm{~K}$", "source": "matter", "problemid": " 11.3(a)", "comment": " "}, {"problem_text": "Calculate the change in Gibbs energy of $35 \\mathrm{~g}$ of ethanol (mass density $0.789 \\mathrm{~g} \\mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from 1 atm to 3000 atm.", "answer_latex": " 12", "answer_number": "12", "unit": " $\\mathrm{~kJ}$", "source": "matter", "problemid": " 66.5(a)", "comment": " "}, {"problem_text": "The promotion of an electron from the valence band into the conduction band in pure $\\mathrm{TIO}_2$ by light absorption requires a wavelength of less than $350 \\mathrm{~nm}$. Calculate the energy gap in electronvolts between the valence and conduction bands.", "answer_latex": " 3.54", "answer_number": "3.54", "unit": " $\\mathrm{eV}$", "source": "matter", "problemid": " 39.1(a)", "comment": " "}, {"problem_text": "Although the crystallization of large biological molecules may not be as readily accomplished as that of small molecules, their crystal lattices are no different. Tobacco seed globulin forms face-centred cubic crystals with unit cell dimension of $12.3 \\mathrm{~nm}$ and a density of $1.287 \\mathrm{~g} \\mathrm{~cm}^{-3}$. Determine its molar mass.", "answer_latex": "3.61", "answer_number": "3.61", "unit": " $10^5 \\mathrm{~g} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 37.1", "comment": " problem"}, {"problem_text": "An electron is accelerated in an electron microscope from rest through a potential difference $\\Delta \\phi=100 \\mathrm{kV}$ and acquires an energy of $e \\Delta \\phi$. What is its final speed?", "answer_latex": " 1.88", "answer_number": "1.88", "unit": " $10^8 \\mathrm{~m} \\mathrm{~s}^{-1}$", "source": "matter", "problemid": " 2.7(a)", "comment": " "}, {"problem_text": "The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature. By how much does the standard molar enthalpy of $\\mathrm{SO}_2(\\mathrm{~g})$ increase when the temperature is raised from $298.15 \\mathrm{~K}$ to $1500 \\mathrm{~K}$ ?", "answer_latex": " 62.2", "answer_number": "62.2", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 56.1", "comment": " problem"}, {"problem_text": "Suppose that the normalized wavefunction for an electron in a carbon nanotube of length $L=10.0 \\mathrm{~nm}$ is: $\\psi=(2 / L)^{1 / 2} \\sin (\\pi x / L)$. Calculate the probability that the electron is between $x=4.95 \\mathrm{~nm}$ and $5.05 \\mathrm{~nm}$.", "answer_latex": " 0.020", "answer_number": "0.020", "unit": " ", "source": "matter", "problemid": " 5.1(a)", "comment": " problem"}, {"problem_text": "A sample of the sugar D-ribose of mass $0.727 \\mathrm{~g}$ was placed in a calorimeter and then ignited in the presence of excess oxygen. The temperature rose by $0.910 \\mathrm{~K}$. In a separate experiment in the same calorimeter, the combustion of $0.825 \\mathrm{~g}$ of benzoic acid, for which the internal energy of combustion is $-3251 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, gave a temperature rise of $1.940 \\mathrm{~K}$. Calculate the enthalpy of formation of D-ribose.", "answer_latex": " -1270", "answer_number": "-1270", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 57.1", "comment": " problem"}, {"problem_text": "An electron confined to a metallic nanoparticle is modelled as a particle in a one-dimensional box of length $L$. If the electron is in the state $n=1$, calculate the probability of finding it in the following regions: $0 \\leq x \\leq \\frac{1}{2} L$.", "answer_latex": " $\\frac{1}{2}$", "answer_number": "0.5", "unit": " ", "source": "matter", "problemid": " 9.3(a)", "comment": " problem"}, {"problem_text": "The carbon-carbon bond length in diamond is $154.45 \\mathrm{pm}$. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is $3.516 \\mathrm{~g} \\mathrm{~cm}^{-3}$.", "answer_latex": " 7.654", "answer_number": "7.654", "unit": " $\\mathrm{~g} \\mathrm{~cm}^{-3}$", "source": "matter", "problemid": " 38.3", "comment": " problem"}, {"problem_text": "A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of seawater in the visible region is $6.2 \\times 10^{-3} \\mathrm{dm}^3 \\mathrm{~mol}^{-1} \\mathrm{~cm}^{-1}$, calculate the depth at which a diver will experience half the surface intensity of light.", "answer_latex": " 0.87", "answer_number": "0.87", "unit": " $\\mathrm{~m}$", "source": "matter", "problemid": " 40.7(a)", "comment": " "}, {"problem_text": "Calculate the molar energy required to reverse the direction of an $\\mathrm{H}_2 \\mathrm{O}$ molecule located $100 \\mathrm{pm}$ from a $\\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \\mathrm{D}$.", "answer_latex": " 1.07", "answer_number": "1.07", "unit": " $10^3 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 35.1(a)", "comment": " "}, {"problem_text": "In an industrial process, nitrogen is heated to $500 \\mathrm{~K}$ at a constant volume of $1.000 \\mathrm{~m}^3$. The gas enters the container at $300 \\mathrm{~K}$ and $100 \\mathrm{~atm}$. The mass of the gas is $92.4 \\mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \\mathrm{~K}$. For nitrogen, $a=1.39 \\mathrm{dm}^6 \\mathrm{~atm} \\mathrm{~mol}^{-2}, b=0.0391 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$.", "answer_latex": " 140", "answer_number": "140", "unit": " $\\mathrm{~atm}$", "source": "matter", "problemid": " 36.6(a)", "comment": " "}, {"problem_text": "The chemical shift of the $\\mathrm{CH}_3$ protons in acetaldehyde (ethanal) is $\\delta=2.20$ and that of the $\\mathrm{CHO}$ proton is 9.80 . What is the difference in local magnetic field between the two regions of the molecule when the applied field is $1.5 \\mathrm{~T}$", "answer_latex": " 11", "answer_number": "11", "unit": " $\\mu \\mathrm{T}$", "source": "matter", "problemid": " 48.2(a)", "comment": " "}, {"problem_text": "Suppose that the junction between two semiconductors can be represented by a barrier of height $2.0 \\mathrm{eV}$ and length $100 \\mathrm{pm}$. Calculate the transmission probability of an electron with energy $1.5 \\mathrm{eV}$.", "answer_latex": " 0.8", "answer_number": "0.8", "unit": " ", "source": "matter", "problemid": " 10.1(a)", "comment": " "}, {"problem_text": "The diffusion coefficient of a particular kind of t-RNA molecule is $D=1.0 \\times 10^{-11} \\mathrm{~m}^2 \\mathrm{~s}^{-1}$ in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance $1.0 \\mu \\mathrm{m}$, corresponding to the radius of the cell?", "answer_latex": " 1.7", "answer_number": "1.7", "unit": " $10^{-2} \\mathrm{~s}$", "source": "matter", "problemid": " 81.11", "comment": " problem"}, {"problem_text": "At what pressure does the mean free path of argon at $20^{\\circ} \\mathrm{C}$ become comparable to the diameter of a $100 \\mathrm{~cm}^3$ vessel that contains it? Take $\\sigma=0.36 \\mathrm{~nm}^2$", "answer_latex": " 0.195", "answer_number": "0.195", "unit": " $\\mathrm{Pa}$", "source": "matter", "problemid": " 78.6(a)", "comment": " "}, {"problem_text": "The equilibrium pressure of $\\mathrm{O}_2$ over solid silver and silver oxide, $\\mathrm{Ag}_2 \\mathrm{O}$, at $298 \\mathrm{~K}$ is $11.85 \\mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\\mathrm{Ag}_2 \\mathrm{O}(\\mathrm{s})$ at $298 \\mathrm{~K}$.", "answer_latex": " -11.2", "answer_number": "-11.2", "unit": " $\\mathrm{~kJ}   \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 73.4(a)", "comment": " "}, {"problem_text": "When alkali metals dissolve in liquid ammonia, their atoms each lose an electron and give rise to a deep-blue solution that contains unpaired electrons occupying cavities in the solvent. These 'metal-ammonia solutions' have a maximum absorption at $1500 \\mathrm{~nm}$. Supposing that the absorption is due to the excitation of an electron in a spherical square well from its ground state to the next-higher state (see the preceding problem for information), what is the radius of the cavity?", "answer_latex": " 0.69", "answer_number": "0.69", "unit": " $\\mathrm{~nm}$", "source": "matter", "problemid": " 11.5", "comment": " problem"}, {"problem_text": "Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \\mathrm{pm}$ ? ", "answer_latex": " 7.27", "answer_number": "7.27", "unit": " $10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$", "source": "matter", "problemid": " 4.8(a)", "comment": " "}, {"problem_text": "Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \\mathrm{pm}$ ? ", "answer_latex": " 7.27", "answer_number": "7.27", "unit": " $10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$", "source": "matter", "problemid": " 4.8(a)", "comment": " "}, {"problem_text": "Nelson, et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the $\\mathrm{H}$ atoms in $\\mathrm{NH}_3$ formed hydrogen bonds, but found none. For example, they found that the complex of $\\mathrm{NH}_3$ and $\\mathrm{CO}_2$ has the carbon atom nearest the nitrogen (299 pm away): the $\\mathrm{CO}_2$ molecule is at right angles to the $\\mathrm{C}-\\mathrm{N}$ 'bond', and the $\\mathrm{H}$ atoms of $\\mathrm{NH}_3$ are pointing away from the $\\mathrm{CO}_2$. The magnitude of the permanent dipole moment of this complex is reported as $1.77 \\mathrm{D}$. If the $\\mathrm{N}$ and $\\mathrm{C}$ atoms are the centres of the negative and positive charge distributions, respectively, what is the magnitude of those partial charges (as multiples of $e$ )?", "answer_latex": " 0.123", "answer_number": "0.123", "unit": " ", "source": "matter", "problemid": " 34.5", "comment": " problem"}, {"problem_text": "The NOF molecule is an asymmetric rotor with rotational constants $3.1752 \\mathrm{~cm}^{-1}, 0.3951 \\mathrm{~cm}^{-1}$, and $0.3505 \\mathrm{~cm}^{-1}$. Calculate the rotational partition function of the molecule at $25^{\\circ} \\mathrm{C}$.", "answer_latex": " 7.97", "answer_number": "7.97", "unit": "$10^3$", "source": "matter", "problemid": " 52.4(a)", "comment": " "}, {"problem_text": "Suppose that $2.5 \\mathrm{mmol} \\mathrm{N}_2$ (g) occupies $42 \\mathrm{~cm}^3$ at $300 \\mathrm{~K}$ and expands isothermally to $600 \\mathrm{~cm}^3$. Calculate $\\Delta G$ for the process.", "answer_latex": " -17", "answer_number": "-17", "unit": " $\\mathrm{~J}$ ", "source": "matter", "problemid": " 66.1(a)", "comment": " "}, {"problem_text": "Calculate the standard potential of the $\\mathrm{Ce}^{4+} / \\mathrm{Ce}$ couple from the values for the $\\mathrm{Ce}^{3+} / \\mathrm{Ce}$ and $\\mathrm{Ce}^{4+} / \\mathrm{Ce}^{3+}$ couples.\r\n", "answer_latex": " -1.46", "answer_number": "-1.46", "unit": " $\\mathrm{V}$", "source": "matter", "problemid": " 77.1(a)", "comment": " "}, {"problem_text": "An effusion cell has a circular hole of diameter $1.50 \\mathrm{~mm}$. If the molar mass of the solid in the cell is $300 \\mathrm{~g} \\mathrm{~mol}^{-1}$ and its vapour pressure is $0.735 \\mathrm{~Pa}$ at $500 \\mathrm{~K}$, by how much will the mass of the solid decrease in a period of $1.00 \\mathrm{~h}$ ?", "answer_latex": " 16", "answer_number": "16", "unit": " $\\mathrm{mg}$", "source": "matter", "problemid": " 78.11(a)", "comment": " "}, {"problem_text": "The speed of a certain proton is $6.1 \\times 10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated?", "answer_latex": " 52", "answer_number": "52", "unit": " $\\mathrm{pm}$", "source": "matter", "problemid": " 8.1(a)", "comment": " "}, {"problem_text": "It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of $1.0 \\mathrm{kT}$ ?", "answer_latex": " 2.8", "answer_number": "2.8", "unit": " $10^{13} \\mathrm{~Hz}$", "source": "matter", "problemid": " 50.1", "comment": " problem"}, {"problem_text": "A particle of mass $1.0 \\mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \\mathrm{~m} \\mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \\mathrm{~s}$. Ignore air resistance?", "answer_latex": "48", "answer_number": "48", "unit": " $\\mathrm{~mJ}$", "source": "matter", "problemid": " 2.1(a)", "comment": " "}, {"problem_text": "A particle of mass $1.0 \\mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \\mathrm{~m} \\mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \\mathrm{~s}$. Ignore air resistance?", "answer_latex": "48", "answer_number": "48", "unit": " $\\mathrm{~mJ}$", "source": "matter", "problemid": " 2.1(a)", "comment": " "}, {"problem_text": "The flux of visible photons reaching Earth from the North Star is about $4 \\times 10^3 \\mathrm{~mm}^{-2} \\mathrm{~s}^{-1}$. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about $40 \\mathrm{~mm}^2$ and the response time of the eye is about $0.1 \\mathrm{~s}$. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused onto the retina in $0.1 \\mathrm{~s}$ ?", "answer_latex": " 4.4", "answer_number": "4.4", "unit": " $10^3$", "source": "matter", "problemid": " 40.3", "comment": " problem"}, {"problem_text": "When ultraviolet radiation of wavelength $58.4 \\mathrm{~nm}$ from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of $1.59 \\times 10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$. Calculate the ionization energy of krypton.\r\n", "answer_latex": " 14", "answer_number": "14", "unit": " $\\mathrm{eV}$", "source": "matter", "problemid": " 17.2(a)", "comment": " "}, {"problem_text": " If $125 \\mathrm{~cm}^3$ of hydrogen gas effuses through a small hole in 135 seconds, how long will it take the same volume of oxygen gas to effuse under the same temperature and pressure?", "answer_latex": " 537", "answer_number": "537", "unit": " $\\mathrm{s}$", "source": "matter", "problemid": " 78.10(a)", "comment": " "}, {"problem_text": "The vibrational wavenumber of $\\mathrm{Br}_2$ is $323.2 \\mathrm{~cm}^{-1}$. Evaluate the vibrational partition function explicitly (without approximation) and plot its value as a function of temperature. At what temperature is the value within 5 per cent of the value calculated from the approximate formula?", "answer_latex": " 4500", "answer_number": "4500", "unit": "$\\mathrm{~K}$", "source": "matter", "problemid": " 52.10(a)", "comment": " "}, {"problem_text": "A thermodynamic study of $\\mathrm{DyCl}_3$ (E.H.P. Cordfunke, et al., J. Chem. Thermodynamics 28, 1387 (1996)) determined its standard enthalpy of formation from the following information\r\n(1) $\\mathrm{DyCl}_3(\\mathrm{~s}) \\rightarrow \\mathrm{DyCl}_3(\\mathrm{aq}$, in $4.0 \\mathrm{M} \\mathrm{HCl}) \\quad \\Delta_{\\mathrm{r}} H^{\\ominus}=-180.06 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\n(2) $\\mathrm{Dy}(\\mathrm{s})+3 \\mathrm{HCl}(\\mathrm{aq}, 4.0 \\mathrm{~m}) \\rightarrow \\mathrm{DyCl}_3(\\mathrm{aq}$, in $4.0 \\mathrm{M} \\mathrm{HCl}(\\mathrm{aq}))+\\frac{3}{2} \\mathrm{H}_2(\\mathrm{~g})$ $\\Delta_{\\mathrm{r}} H^{\\ominus}=-699.43 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\n(3) $\\frac{1}{2} \\mathrm{H}_2(\\mathrm{~g})+\\frac{1}{2} \\mathrm{Cl}_2(\\mathrm{~g}) \\rightarrow \\mathrm{HCl}(\\mathrm{aq}, 4.0 \\mathrm{M}) \\quad \\Delta_{\\mathrm{r}} H^{\\ominus}=-158.31 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\nDetermine $\\Delta_{\\mathrm{f}} H^{\\ominus}\\left(\\mathrm{DyCl}_3, \\mathrm{~s}\\right)$ from these data.", "answer_latex": " -994.3", "answer_number": "-994.3", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 57.5", "comment": " problem"}, {"problem_text": "Calculate $\\Delta_{\\mathrm{r}} G^{\\ominus}(375 \\mathrm{~K})$ for the reaction $2 \\mathrm{CO}(\\mathrm{g})+\\mathrm{O}_2(\\mathrm{~g}) \\rightarrow 2 \\mathrm{CO}_2(\\mathrm{~g})$ from the values of $\\Delta_{\\mathrm{r}} G^{\\ominus}(298 \\mathrm{~K})$ : and $\\Delta_{\\mathrm{r}} H^{\\ominus}(298 \\mathrm{~K})$, and the GibbsHelmholtz equation.", "answer_latex": "-501", "answer_number": "-501", "unit": "$\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 66.1", "comment": " problem"}, {"problem_text": "The vapour pressure of benzene is $53.3 \\mathrm{kPa}$ at $60.6^{\\circ} \\mathrm{C}$, but it fell to $51.5 \\mathrm{kPa}$ when $19.0 \\mathrm{~g}$ of an non-volatile organic compound was dissolved in $500 \\mathrm{~g}$ of benzene. Calculate the molar mass of the compound.", "answer_latex": " 85", "answer_number": "85", "unit": " $\\mathrm{~g} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 70.8(a)", "comment": " "}, {"problem_text": "J.G. Dojahn, et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. The ground state of $\\mathrm{F}_2^{-}$is ${ }^2 \\sum_{\\mathrm{u}}^{+}$with a fundamental vibrational wavenumber of $450.0 \\mathrm{~cm}^{-1}$ and equilibrium internuclear distance of $190.0 \\mathrm{pm}$. The first two excited states are at 1.609 and $1.702 \\mathrm{eV}$ above the ground state. Compute the standard molar entropy of $\\mathrm{F}_2^{-}$at $298 \\mathrm{~K}$.", "answer_latex": " 199.4", "answer_number": "199.4", "unit": " $\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$", "source": "matter", "problemid": " 60.3", "comment": " problem"}, {"problem_text": "The duration of a $90^{\\circ}$ or $180^{\\circ}$ pulse depends on the strength of the $\\mathscr{B}_1$ field. If a $180^{\\circ}$ pulse requires $12.5 \\mu \\mathrm{s}$, what is the strength of the $\\mathscr{B}_1$ field? ", "answer_latex": " 5.9", "answer_number": "5.9", "unit": " $10^{-4} \\mathrm{~T}$", "source": "matter", "problemid": " 49.1(a)", "comment": " "}, {"problem_text": "In 1976 it was mistakenly believed that the first of the 'superheavy' elements had been discovered in a sample of mica. Its atomic number  was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)", "answer_latex": " 0.42", "answer_number": "0.42", "unit": " $\\mathrm{pm}$", "source": "matter", "problemid": " 18.1", "comment": " problem"}, {"problem_text": "The ground level of $\\mathrm{Cl}$ is ${ }^2 \\mathrm{P}_{3 / 2}$ and a ${ }^2 \\mathrm{P}_{1 / 2}$ level lies $881 \\mathrm{~cm}^{-1}$ above it. Calculate the electronic contribution to the molar Gibbs energy of $\\mathrm{Cl}$ atoms at  $500 \\mathrm{~K}$.", "answer_latex": " -6.42", "answer_number": "-6.42", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 64.5(a)", "comment": " "}, {"problem_text": "Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately $0.92 \\mathrm{~g} \\mathrm{~cm}^{-3}$ and that of liquid water is $1.00 \\mathrm{~g} \\mathrm{~cm}^{-3}$.", "answer_latex": " 272.8", "answer_number": "272.8", "unit": " $\\mathrm{K}$", "source": "matter", "problemid": " 69.9(a)", "comment": " "}, {"problem_text": "What is the temperature of a two-level system of energy separation equivalent to $400 \\mathrm{~cm}^{-1}$ when the population of the upper state is one-third that of the lower state?", "answer_latex": " 524", "answer_number": "524", "unit": " $ \\mathrm{~K}$", "source": "matter", "problemid": " 51.4(a)", "comment": " "}, {"problem_text": "At $300 \\mathrm{~K}$ and $20 \\mathrm{~atm}$, the compression factor of a gas is 0.86 . Calculate the volume occupied by $8.2 \\mathrm{mmol}$ of the gas under these conditions.", "answer_latex": " 8.7", "answer_number": "8.7", "unit": " $\\mathrm{~cm}^3$", "source": "matter", "problemid": " 36.3(a)", "comment": " problem"}, {"problem_text": "A very crude model of the buckminsterfullerene molecule $\\left(\\mathrm{C}_{60}\\right)$ is to treat it as a collection of electrons in a cube with sides of length equal to the mean diameter of the molecule $(0.7 \\mathrm{~nm})$. Suppose that only the $\\pi$ electrons of the carbon atoms contribute, and predict the wavelength of the first excitation of $\\mathrm{C}_{60}$. (The actual value is $730 \\mathrm{~nm}$.)", "answer_latex": " 1.6", "answer_number": "1.6", "unit": " $\\mu \\mathrm{m}$", "source": "matter", "problemid": " 11.3", "comment": " problem"}]
+[
+ {
+  "problem_text": "Radiation from an X-ray source consists of two components of wavelengths $154.433 \\mathrm{pm}$ and $154.051 \\mathrm{pm}$. Calculate the difference in glancing angles $(2 \\theta)$ of the diffraction lines arising from the two components in a diffraction pattern from planes of separation $77.8 \\mathrm{pm}$.",
+  "answer_latex": " 2.14",
+  "answer_number": "2.14",
+  "unit": " ${\\circ}$",
+  "source": "matter",
+  "problemid": " 37.7(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "A chemical reaction takes place in a container of cross-sectional area $50 \\mathrm{~cm}^2$. As a result of the reaction, a piston is pushed out through $15 \\mathrm{~cm}$ against an external pressure of $1.0 \\mathrm{~atm}$. Calculate the work done by the system.",
+  "answer_latex": " -75",
+  "answer_number": "-75",
+  "unit": " $\\mathrm{~J}$",
+  "source": "matter",
+  "problemid": " 55.4(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mixture of water and ethanol is prepared with a mole fraction of water of 0.60 . If a small change in the mixture composition results in an increase in the chemical potential of water by $0.25 \\mathrm{~J} \\mathrm{~mol}^{-1}$, by how much will the chemical potential of ethanol change?",
+  "answer_latex": " -0.38",
+  "answer_number": "-0.38",
+  "unit": " $\\mathrm{~J} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 69.2(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The enthalpy of fusion of mercury is $2.292 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, and its normal freezing point is $234.3 \\mathrm{~K}$ with a change in molar volume of $+0.517 \\mathrm{~cm}^3 \\mathrm{~mol}^{-1}$ on melting. At what temperature will the bottom of a column of mercury (density $13.6 \\mathrm{~g} \\mathrm{~cm}^{-3}$ ) of height $10.0 \\mathrm{~m}$ be expected to freeze?",
+  "answer_latex": " 234.4",
+  "answer_number": "234.4",
+  "unit": " $ \\mathrm{~K}$ ",
+  "source": "matter",
+  "problemid": " 69.3",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "Suppose a nanostructure is modelled by an electron confined to a rectangular region with sides of lengths $L_1=1.0 \\mathrm{~nm}$ and $L_2=2.0 \\mathrm{~nm}$ and is subjected to thermal motion with a typical energy equal to $k T$, where $k$ is Boltzmann's constant. How low should the temperature be for the thermal energy to be comparable to the zero-point energy\uff1f",
+  "answer_latex": " 5.5",
+  "answer_number": "5.5",
+  "unit": " $10^3 \\mathrm{~K}$",
+  "source": "matter",
+  "problemid": " 11.3(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the change in Gibbs energy of $35 \\mathrm{~g}$ of ethanol (mass density $0.789 \\mathrm{~g} \\mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from 1 atm to 3000 atm.",
+  "answer_latex": " 12",
+  "answer_number": "12",
+  "unit": " $\\mathrm{~kJ}$",
+  "source": "matter",
+  "problemid": " 66.5(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The promotion of an electron from the valence band into the conduction band in pure $\\mathrm{TIO}_2$ by light absorption requires a wavelength of less than $350 \\mathrm{~nm}$. Calculate the energy gap in electronvolts between the valence and conduction bands.",
+  "answer_latex": " 3.54",
+  "answer_number": "3.54",
+  "unit": " $\\mathrm{eV}$",
+  "source": "matter",
+  "problemid": " 39.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Although the crystallization of large biological molecules may not be as readily accomplished as that of small molecules, their crystal lattices are no different. Tobacco seed globulin forms face-centred cubic crystals with unit cell dimension of $12.3 \\mathrm{~nm}$ and a density of $1.287 \\mathrm{~g} \\mathrm{~cm}^{-3}$. Determine its molar mass.",
+  "answer_latex": "3.61",
+  "answer_number": "3.61",
+  "unit": " $10^5 \\mathrm{~g} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 37.1",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "An electron is accelerated in an electron microscope from rest through a potential difference $\\Delta \\phi=100 \\mathrm{kV}$ and acquires an energy of $e \\Delta \\phi$. What is its final speed?",
+  "answer_latex": " 1.88",
+  "answer_number": "1.88",
+  "unit": " $10^8 \\mathrm{~m} \\mathrm{~s}^{-1}$",
+  "source": "matter",
+  "problemid": " 2.7(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature. By how much does the standard molar enthalpy of $\\mathrm{SO}_2(\\mathrm{~g})$ increase when the temperature is raised from $298.15 \\mathrm{~K}$ to $1500 \\mathrm{~K}$ ?",
+  "answer_latex": " 62.2",
+  "answer_number": "62.2",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 56.1",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "Suppose that the normalized wavefunction for an electron in a carbon nanotube of length $L=10.0 \\mathrm{~nm}$ is: $\\psi=(2 / L)^{1 / 2} \\sin (\\pi x / L)$. Calculate the probability that the electron is between $x=4.95 \\mathrm{~nm}$ and $5.05 \\mathrm{~nm}$.",
+  "answer_latex": " 0.020",
+  "answer_number": "0.020",
+  "unit": " ",
+  "source": "matter",
+  "problemid": " 5.1(a)",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "A sample of the sugar D-ribose of mass $0.727 \\mathrm{~g}$ was placed in a calorimeter and then ignited in the presence of excess oxygen. The temperature rose by $0.910 \\mathrm{~K}$. In a separate experiment in the same calorimeter, the combustion of $0.825 \\mathrm{~g}$ of benzoic acid, for which the internal energy of combustion is $-3251 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, gave a temperature rise of $1.940 \\mathrm{~K}$. Calculate the enthalpy of formation of D-ribose.",
+  "answer_latex": " -1270",
+  "answer_number": "-1270",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 57.1",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "An electron confined to a metallic nanoparticle is modelled as a particle in a one-dimensional box of length $L$. If the electron is in the state $n=1$, calculate the probability of finding it in the following regions: $0 \\leq x \\leq \\frac{1}{2} L$.",
+  "answer_latex": " $\\frac{1}{2}$",
+  "answer_number": "0.5",
+  "unit": " ",
+  "source": "matter",
+  "problemid": " 9.3(a)",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "The carbon-carbon bond length in diamond is $154.45 \\mathrm{pm}$. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is $3.516 \\mathrm{~g} \\mathrm{~cm}^{-3}$.",
+  "answer_latex": " 7.654",
+  "answer_number": "7.654",
+  "unit": " $\\mathrm{~g} \\mathrm{~cm}^{-3}$",
+  "source": "matter",
+  "problemid": " 38.3",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of seawater in the visible region is $6.2 \\times 10^{-3} \\mathrm{dm}^3 \\mathrm{~mol}^{-1} \\mathrm{~cm}^{-1}$, calculate the depth at which a diver will experience half the surface intensity of light.",
+  "answer_latex": " 0.87",
+  "answer_number": "0.87",
+  "unit": " $\\mathrm{~m}$",
+  "source": "matter",
+  "problemid": " 40.7(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the molar energy required to reverse the direction of an $\\mathrm{H}_2 \\mathrm{O}$ molecule located $100 \\mathrm{pm}$ from a $\\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \\mathrm{D}$.",
+  "answer_latex": " 1.07",
+  "answer_number": "1.07",
+  "unit": " $10^3 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 35.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "In an industrial process, nitrogen is heated to $500 \\mathrm{~K}$ at a constant volume of $1.000 \\mathrm{~m}^3$. The gas enters the container at $300 \\mathrm{~K}$ and $100 \\mathrm{~atm}$. The mass of the gas is $92.4 \\mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \\mathrm{~K}$. For nitrogen, $a=1.39 \\mathrm{dm}^6 \\mathrm{~atm} \\mathrm{~mol}^{-2}, b=0.0391 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$.",
+  "answer_latex": " 140",
+  "answer_number": "140",
+  "unit": " $\\mathrm{~atm}$",
+  "source": "matter",
+  "problemid": " 36.6(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The chemical shift of the $\\mathrm{CH}_3$ protons in acetaldehyde (ethanal) is $\\delta=2.20$ and that of the $\\mathrm{CHO}$ proton is 9.80 . What is the difference in local magnetic field between the two regions of the molecule when the applied field is $1.5 \\mathrm{~T}$",
+  "answer_latex": " 11",
+  "answer_number": "11",
+  "unit": " $\\mu \\mathrm{T}$",
+  "source": "matter",
+  "problemid": " 48.2(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that the junction between two semiconductors can be represented by a barrier of height $2.0 \\mathrm{eV}$ and length $100 \\mathrm{pm}$. Calculate the transmission probability of an electron with energy $1.5 \\mathrm{eV}$.",
+  "answer_latex": " 0.8",
+  "answer_number": "0.8",
+  "unit": " ",
+  "source": "matter",
+  "problemid": " 10.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The diffusion coefficient of a particular kind of t-RNA molecule is $D=1.0 \\times 10^{-11} \\mathrm{~m}^2 \\mathrm{~s}^{-1}$ in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance $1.0 \\mu \\mathrm{m}$, corresponding to the radius of the cell?",
+  "answer_latex": " 1.7",
+  "answer_number": "1.7",
+  "unit": " $10^{-2} \\mathrm{~s}$",
+  "source": "matter",
+  "problemid": " 81.11",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "At what pressure does the mean free path of argon at $20^{\\circ} \\mathrm{C}$ become comparable to the diameter of a $100 \\mathrm{~cm}^3$ vessel that contains it? Take $\\sigma=0.36 \\mathrm{~nm}^2$",
+  "answer_latex": " 0.195",
+  "answer_number": "0.195",
+  "unit": " $\\mathrm{Pa}$",
+  "source": "matter",
+  "problemid": " 78.6(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The equilibrium pressure of $\\mathrm{O}_2$ over solid silver and silver oxide, $\\mathrm{Ag}_2 \\mathrm{O}$, at $298 \\mathrm{~K}$ is $11.85 \\mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\\mathrm{Ag}_2 \\mathrm{O}(\\mathrm{s})$ at $298 \\mathrm{~K}$.",
+  "answer_latex": " -11.2",
+  "answer_number": "-11.2",
+  "unit": " $\\mathrm{~kJ}   \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 73.4(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "When alkali metals dissolve in liquid ammonia, their atoms each lose an electron and give rise to a deep-blue solution that contains unpaired electrons occupying cavities in the solvent. These 'metal-ammonia solutions' have a maximum absorption at $1500 \\mathrm{~nm}$. Supposing that the absorption is due to the excitation of an electron in a spherical square well from its ground state to the next-higher state (see the preceding problem for information), what is the radius of the cavity?",
+  "answer_latex": " 0.69",
+  "answer_number": "0.69",
+  "unit": " $\\mathrm{~nm}$",
+  "source": "matter",
+  "problemid": " 11.5",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \\mathrm{pm}$ ? ",
+  "answer_latex": " 7.27",
+  "answer_number": "7.27",
+  "unit": " $10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$",
+  "source": "matter",
+  "problemid": " 4.8(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Nelson, et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the $\\mathrm{H}$ atoms in $\\mathrm{NH}_3$ formed hydrogen bonds, but found none. For example, they found that the complex of $\\mathrm{NH}_3$ and $\\mathrm{CO}_2$ has the carbon atom nearest the nitrogen (299 pm away): the $\\mathrm{CO}_2$ molecule is at right angles to the $\\mathrm{C}-\\mathrm{N}$ 'bond', and the $\\mathrm{H}$ atoms of $\\mathrm{NH}_3$ are pointing away from the $\\mathrm{CO}_2$. The magnitude of the permanent dipole moment of this complex is reported as $1.77 \\mathrm{D}$. If the $\\mathrm{N}$ and $\\mathrm{C}$ atoms are the centres of the negative and positive charge distributions, respectively, what is the magnitude of those partial charges (as multiples of $e$ )?",
+  "answer_latex": " 0.123",
+  "answer_number": "0.123",
+  "unit": " ",
+  "source": "matter",
+  "problemid": " 34.5",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "The NOF molecule is an asymmetric rotor with rotational constants $3.1752 \\mathrm{~cm}^{-1}, 0.3951 \\mathrm{~cm}^{-1}$, and $0.3505 \\mathrm{~cm}^{-1}$. Calculate the rotational partition function of the molecule at $25^{\\circ} \\mathrm{C}$.",
+  "answer_latex": " 7.97",
+  "answer_number": "7.97",
+  "unit": "$10^3$",
+  "source": "matter",
+  "problemid": " 52.4(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that $2.5 \\mathrm{mmol} \\mathrm{N}_2$ (g) occupies $42 \\mathrm{~cm}^3$ at $300 \\mathrm{~K}$ and expands isothermally to $600 \\mathrm{~cm}^3$. Calculate $\\Delta G$ for the process.",
+  "answer_latex": " -17",
+  "answer_number": "-17",
+  "unit": " $\\mathrm{~J}$ ",
+  "source": "matter",
+  "problemid": " 66.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the standard potential of the $\\mathrm{Ce}^{4+} / \\mathrm{Ce}$ couple from the values for the $\\mathrm{Ce}^{3+} / \\mathrm{Ce}$ and $\\mathrm{Ce}^{4+} / \\mathrm{Ce}^{3+}$ couples.\r\n",
+  "answer_latex": " -1.46",
+  "answer_number": "-1.46",
+  "unit": " $\\mathrm{V}$",
+  "source": "matter",
+  "problemid": " 77.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "An effusion cell has a circular hole of diameter $1.50 \\mathrm{~mm}$. If the molar mass of the solid in the cell is $300 \\mathrm{~g} \\mathrm{~mol}^{-1}$ and its vapour pressure is $0.735 \\mathrm{~Pa}$ at $500 \\mathrm{~K}$, by how much will the mass of the solid decrease in a period of $1.00 \\mathrm{~h}$ ?",
+  "answer_latex": " 16",
+  "answer_number": "16",
+  "unit": " $\\mathrm{mg}$",
+  "source": "matter",
+  "problemid": " 78.11(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The speed of a certain proton is $6.1 \\times 10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated?",
+  "answer_latex": " 52",
+  "answer_number": "52",
+  "unit": " $\\mathrm{pm}$",
+  "source": "matter",
+  "problemid": " 8.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of $1.0 \\mathrm{kT}$ ?",
+  "answer_latex": " 2.8",
+  "answer_number": "2.8",
+  "unit": " $10^{13} \\mathrm{~Hz}$",
+  "source": "matter",
+  "problemid": " 50.1",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "A particle of mass $1.0 \\mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \\mathrm{~m} \\mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \\mathrm{~s}$. Ignore air resistance?",
+  "answer_latex": "48",
+  "answer_number": "48",
+  "unit": " $\\mathrm{~mJ}$",
+  "source": "matter",
+  "problemid": " 2.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The flux of visible photons reaching Earth from the North Star is about $4 \\times 10^3 \\mathrm{~mm}^{-2} \\mathrm{~s}^{-1}$. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about $40 \\mathrm{~mm}^2$ and the response time of the eye is about $0.1 \\mathrm{~s}$. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused onto the retina in $0.1 \\mathrm{~s}$ ?",
+  "answer_latex": " 4.4",
+  "answer_number": "4.4",
+  "unit": " $10^3$",
+  "source": "matter",
+  "problemid": " 40.3",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "When ultraviolet radiation of wavelength $58.4 \\mathrm{~nm}$ from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of $1.59 \\times 10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$. Calculate the ionization energy of krypton.\r\n",
+  "answer_latex": " 14",
+  "answer_number": "14",
+  "unit": " $\\mathrm{eV}$",
+  "source": "matter",
+  "problemid": " 17.2(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": " If $125 \\mathrm{~cm}^3$ of hydrogen gas effuses through a small hole in 135 seconds, how long will it take the same volume of oxygen gas to effuse under the same temperature and pressure?",
+  "answer_latex": " 537",
+  "answer_number": "537",
+  "unit": " $\\mathrm{s}$",
+  "source": "matter",
+  "problemid": " 78.10(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "The vibrational wavenumber of $\\mathrm{Br}_2$ is $323.2 \\mathrm{~cm}^{-1}$. By evaluating the vibrational partition function explicitly (without approximation), at what temperature is the value within 5 per cent of the value calculated from the approximate formula?",
+  "answer_latex": " 4500",
+  "answer_number": "4500",
+  "unit": "$\\mathrm{~K}$",
+  "source": "matter",
+  "problemid": " 52.10(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "A thermodynamic study of $\\mathrm{DyCl}_3$ (E.H.P. Cordfunke, et al., J. Chem. Thermodynamics 28, 1387 (1996)) determined its standard enthalpy of formation from the following information\r\n(1) $\\mathrm{DyCl}_3(\\mathrm{~s}) \\rightarrow \\mathrm{DyCl}_3(\\mathrm{aq}$, in $4.0 \\mathrm{M} \\mathrm{HCl}) \\quad \\Delta_{\\mathrm{r}} H^{\\ominus}=-180.06 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\n(2) $\\mathrm{Dy}(\\mathrm{s})+3 \\mathrm{HCl}(\\mathrm{aq}, 4.0 \\mathrm{~m}) \\rightarrow \\mathrm{DyCl}_3(\\mathrm{aq}$, in $4.0 \\mathrm{M} \\mathrm{HCl}(\\mathrm{aq}))+\\frac{3}{2} \\mathrm{H}_2(\\mathrm{~g})$ $\\Delta_{\\mathrm{r}} H^{\\ominus}=-699.43 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\n(3) $\\frac{1}{2} \\mathrm{H}_2(\\mathrm{~g})+\\frac{1}{2} \\mathrm{Cl}_2(\\mathrm{~g}) \\rightarrow \\mathrm{HCl}(\\mathrm{aq}, 4.0 \\mathrm{M}) \\quad \\Delta_{\\mathrm{r}} H^{\\ominus}=-158.31 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\nDetermine $\\Delta_{\\mathrm{f}} H^{\\ominus}\\left(\\mathrm{DyCl}_3, \\mathrm{~s}\\right)$ from these data.",
+  "answer_latex": " -994.3",
+  "answer_number": "-994.3",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 57.5",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "Calculate $\\Delta_{\\mathrm{r}} G^{\\ominus}(375 \\mathrm{~K})$ for the reaction $2 \\mathrm{CO}(\\mathrm{g})+\\mathrm{O}_2(\\mathrm{~g}) \\rightarrow 2 \\mathrm{CO}_2(\\mathrm{~g})$ from the values of $\\Delta_{\\mathrm{r}} G^{\\ominus}(298 \\mathrm{~K})$ : and $\\Delta_{\\mathrm{r}} H^{\\ominus}(298 \\mathrm{~K})$, and the GibbsHelmholtz equation.",
+  "answer_latex": "-501",
+  "answer_number": "-501",
+  "unit": "$\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 66.1",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "The vapour pressure of benzene is $53.3 \\mathrm{kPa}$ at $60.6^{\\circ} \\mathrm{C}$, but it fell to $51.5 \\mathrm{kPa}$ when $19.0 \\mathrm{~g}$ of an non-volatile organic compound was dissolved in $500 \\mathrm{~g}$ of benzene. Calculate the molar mass of the compound.",
+  "answer_latex": " 85",
+  "answer_number": "85",
+  "unit": " $\\mathrm{~g} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 70.8(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "J.G. Dojahn, et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. The ground state of $\\mathrm{F}_2^{-}$ is ${ }^2 \\sum_{\\mathrm{u}}^{+}$ with a fundamental vibrational wavenumber of $450.0 \\mathrm{~cm}^{-1}$ and equilibrium internuclear distance of $190.0 \\mathrm{pm}$. The first two excited states are at 1.609 and $1.702 \\mathrm{eV}$ above the ground state. Compute the standard molar entropy of $\\mathrm{F}_2^{-}$ at $ 298 \\mathrm{~K}$.",
+  "answer_latex": " 199.4",
+  "answer_number": "199.4",
+  "unit": " $\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$",
+  "source": "matter",
+  "problemid": " 60.3",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "The duration of a $90^{\\circ}$ or $180^{\\circ}$ pulse depends on the strength of the $\\mathscr{B}_1$ field. If a $180^{\\circ}$ pulse requires $12.5 \\mu \\mathrm{s}$, what is the strength of the $\\mathscr{B}_1$ field? ",
+  "answer_latex": " 5.9",
+  "answer_number": "5.9",
+  "unit": " $10^{-4} \\mathrm{~T}$",
+  "source": "matter",
+  "problemid": " 49.1(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "In 1976 it was mistakenly believed that the first of the 'superheavy' elements had been discovered in a sample of mica. Its atomic number  was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)",
+  "answer_latex": " 0.42",
+  "answer_number": "0.42",
+  "unit": " $\\mathrm{pm}$",
+  "source": "matter",
+  "problemid": " 18.1",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "The ground level of $\\mathrm{Cl}$ is ${ }^2 \\mathrm{P}_{3 / 2}$ and a ${ }^2 \\mathrm{P}_{1 / 2}$ level lies $881 \\mathrm{~cm}^{-1}$ above it. Calculate the electronic contribution to the molar Gibbs energy of $\\mathrm{Cl}$ atoms at  $500 \\mathrm{~K}$.",
+  "answer_latex": " -6.42",
+  "answer_number": "-6.42",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 64.5(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately $0.92 \\mathrm{~g} \\mathrm{~cm}^{-3}$ and that of liquid water is $1.00 \\mathrm{~g} \\mathrm{~cm}^{-3}$.",
+  "answer_latex": " 272.8",
+  "answer_number": "272.8",
+  "unit": " $\\mathrm{K}$",
+  "source": "matter",
+  "problemid": " 69.9(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "What is the temperature of a two-level system of energy separation equivalent to $400 \\mathrm{~cm}^{-1}$ when the population of the upper state is one-third that of the lower state?",
+  "answer_latex": " 524",
+  "answer_number": "524",
+  "unit": " $ \\mathrm{~K}$",
+  "source": "matter",
+  "problemid": " 51.4(a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "At $300 \\mathrm{~K}$ and $20 \\mathrm{~atm}$, the compression factor of a gas is 0.86 . Calculate the volume occupied by $8.2 \\mathrm{mmol}$ of the gas under these conditions.",
+  "answer_latex": " 8.7",
+  "answer_number": "8.7",
+  "unit": " $\\mathrm{~cm}^3$",
+  "source": "matter",
+  "problemid": " 36.3(a)",
+  "comment": " problem"
+ },
+ {
+  "problem_text": "A very crude model of the buckminsterfullerene molecule $\\left(\\mathrm{C}_{60}\\right)$ is to treat it as a collection of electrons in a cube with sides of length equal to the mean diameter of the molecule $(0.7 \\mathrm{~nm})$. Suppose that only the $\\pi$ electrons of the carbon atoms contribute, and predict the wavelength of the first excitation of $\\mathrm{C}_{60}$. (The actual value is $730 \\mathrm{~nm}$.)",
+  "answer_latex": " 1.6",
+  "answer_number": "1.6",
+  "unit": " $\\mu \\mathrm{m}$",
+  "source": "matter",
+  "problemid": " 11.3",
+  "comment": " problem"
+ }
+]

matter_sol.json

@@ -1 +1,102 @@
-[{"problem_text": "Calculating the maximum wavelength capable of photoejection\r\nA photon of radiation of wavelength $305 \\mathrm{~nm}$ ejects an electron from a metal with a kinetic energy of $1.77 \\mathrm{eV}$. Calculate the maximum wavelength of radiation capable of ejecting an electron from the metal.", "answer_latex": " 540", "answer_number": "540", "unit": "$\\mathrm{~nm}$ ", "source": "matter", "problemid": " 4.1", "comment": " with solution", "solution": "From the expression for the work function $\\Phi=h \\nu-E_{\\mathrm{k}}$ the minimum frequency for photoejection is\r\n$$\r\n\\nu_{\\min }=\\frac{\\Phi}{h}=\\frac{h v-E_{\\mathrm{k}}}{h} \\stackrel{\\nu=c \\mid \\lambda}{=} \\frac{c}{\\lambda}-\\frac{E_{\\mathrm{k}}}{h}\r\n$$\r\nThe maximum wavelength is therefore\r\n$$\r\n\\lambda_{\\max }=\\frac{c}{v_{\\min }}=\\frac{c}{c / \\lambda-E_{\\mathrm{k}} / h}=\\frac{1}{1 / \\lambda-E_{\\mathrm{k}} / h c}\r\n$$\r\nNow we substitute the data. The kinetic energy of the electron is\r\n$$\r\n\\begin{aligned}\r\n& E_k=1.77 \\mathrm{eV} \\times\\left(1.602 \\times 10^{-19} \\mathrm{JeV}^{-1}\\right)=2.83 \\ldots \\times 10^{-19} \\mathrm{~J} \\\\\r\n& \\frac{E_k}{h c}=\\frac{2.83 \\ldots \\times 10^{-19} \\mathrm{~J}}{\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right) \\times\\left(2.998 \\times 10^8 \\mathrm{~m} \\mathrm{~s}^{-1}\\right)}=1.42 \\ldots \\times 10^6 \\mathrm{~m}^{-1}\r\n\\end{aligned}\r\n$$\r\nTherefore, with\r\n$$\r\n\\begin{aligned}\r\n& 1 / \\lambda=1 / 305 \\mathrm{~nm}=3.27 \\ldots \\times 10^6 \\mathrm{~m}^{-1}, \\\\\r\n& \\lambda_{\\max }=\\frac{1}{\\left(3.27 \\ldots \\times 10^6 \\mathrm{~m}^{-1}\\right)-\\left(1.42 \\ldots \\times 10^6 \\mathrm{~m}^{-1}\\right)}=5.40 \\times 10^{-7} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\nor $540 \\mathrm{~nm}$.\r\n"}, {"problem_text": "Estimate the molar volume of $\\mathrm{CO}_2$ at $500 \\mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.", "answer_latex": " 0.366", "answer_number": "0.366", "unit": "$\\mathrm{dm}^3 \\mathrm{~mol}^{-1}$", "source": "matter", "problemid": " 36.1", "comment": " with solution", "solution": "According to Table 36.3, $a=3.610 \\mathrm{dm}^6$ atm $\\mathrm{mol}^{-2}$ and $b=4.29 \\times 10^{-2} \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. Under the stated conditions, $R T / p=0.410 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. The coefficients in the equation for $V_{\\mathrm{m}}$ are therefore\r\n$$\r\n\\begin{aligned}\r\nb+R T / p & =0.453 \\mathrm{dm}^3 \\mathrm{~mol}^{-1} \\\\\r\na / p & =3.61 \\times 10^{-2}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^2 \\\\\r\na b / p & =1.55 \\times 10^{-3}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^3\r\n\\end{aligned}\r\n$$\r\nTherefore, on writing $x=V_{\\mathrm{m}} /\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)$, the equation to solve is\r\n$$\r\nx^3-0.453 x^2+\\left(3.61 \\times 10^{-2}\\right) x-\\left(1.55 \\times 10^{-3}\\right)=0\r\n$$\r\nThe acceptable root is $x=0.366$, which implies that $V_{\\mathrm{m}}=0.366$ $\\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. The molar volume of a perfect gas under these conditions is $0.410 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$.\r\n"}, {"problem_text": "The single electron in a certain excited state of a hydrogenic $\\mathrm{He}^{+}$ion $(Z=2)$ is described by the wavefunction $R_{3,2}(r) \\times$ $Y_{2,-1}(\\theta, \\phi)$. What is the energy of its electron?", "answer_latex": " -6.04697", "answer_number": " -6.04697", "unit": "$ \\mathrm{eV}$ ", "source": "matter", "problemid": " 17.1", "comment": " with solution", "solution": "Replacing $\\mu$ by $m_{\\mathrm{e}}$ and using $\\hbar=h / 2 \\pi$, we can write the expression for the energy (eqn 17.7) as\r\n$$\r\nE_n=-\\frac{Z^2 m_e e^4}{8 \\varepsilon_0^2 h^2 n^2}=-\\frac{Z^2 h c \\tilde{R}_{\\infty}}{n^2}\r\n$$\r\nwith\r\n$$\r\n\\begin{aligned}\r\n& \\times \\underbrace{2.997926 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}}_c \\\\\r\n& =109737 \\mathrm{~cm}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nand\r\n$$\r\n\\begin{aligned}\r\nh c \\tilde{R}_{\\infty}= & \\left(6.62608 \\times 10^{-34} \\mathrm{Js}\\right) \\times\\left(2.997926 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right) \\\\\r\n& \\times\\left(109737 \\mathrm{~cm}^{-1}\\right) \\\\\r\n= & 2.17987 \\times 10^{-18} \\mathrm{~J}\r\n\\end{aligned}\r\n$$\r\nTherefore, for $n=3$, the energy is\r\n$$\r\n\\begin{aligned}\r\n& E_3=-\\frac{\\overbrace{4}^{Z^2} \\times \\overbrace{2.17987 \\times 10^{-18} \\mathrm{~J}}^{h c \\tilde{R}_{\\infty}}}{\\underset{\\tilde{n}^2}{9}} \\\\\r\n& =-9.68831 \\times 10^{-19} \\mathrm{~J} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nor $-0.968831 \\mathrm{aJ}$ (a, for atto, is the prefix that denotes $10^{-18}$ ). In some applications it is useful to express the energy in electronvolts $\\left(1 \\mathrm{eV}=1.602176 \\times 10^{-19} \\mathrm{~J}\\right)$; in this case, $E_3=-6.04697 \\mathrm{eV}$"}, {"problem_text": "Calculate the typical wavelength of neutrons after reaching thermal equilibrium with their surroundings at $373 \\mathrm{~K}$. For simplicity, assume that the particles are travelling in one dimension.", "answer_latex": " 226", "answer_number": "226", "unit": "$\\mathrm{pm}$", "source": "matter", "problemid": " 37.4", "comment": " with solution", "solution": "From the equipartition principle, we know that the mean translational kinetic energy of a neutron at a temperature $T$ travelling in the $x$-direction is $E_{\\mathrm{k}}=\\frac{1}{2} k T$. The kinetic energy is also equal to $p^2 / 2 m$, where $p$ is the momentum of the neutron and $m$ is its mass. Hence, $p=(m k T)^{1 / 2}$. It follows from the de Broglie relation $\\lambda=h / p$ that the neutron's wavelength is\r\n$$\r\n\\lambda=\\frac{h}{(m k T)^{1 / 2}}\r\n$$\r\nTherefore, at $373 \\mathrm{~K}$,\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left\\{\\left(1.675 \\times 10^{-27} \\mathrm{~kg}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\times(373 \\mathrm{~K})\\right\\}^{1 / 2}} \\\\\r\n& =\\frac{6.626 \\times 10^{-34}}{\\left(1.675 \\times 10^{-27} \\times 1.381 \\times 10^{-23} \\times 373\\right)^{1 / 2}} \\frac{\\mathrm{kg} \\mathrm{m}^2 \\mathrm{~s}^{-1}}{\\left(\\mathrm{~kg}^2 \\mathrm{~m}^2 \\mathrm{~s}^{-2}\\right)^{1 / 2}} \\\\\r\n& =2.26 \\times 10^{-10} \\mathrm{~m}=226 \\mathrm{pm}\r\n\\end{aligned}\r\n$$\r\nwhere we have used $1 \\mathrm{~J}=1 \\mathrm{~kg} \\mathrm{~m}^2 \\mathrm{~s}^{-2}$."}, {"problem_text": "Using the perfect gas equation\r\nCalculate the pressure in kilopascals exerted by $1.25 \\mathrm{~g}$ of nitrogen gas in a flask of volume $250 \\mathrm{~cm}^3$ at $20^{\\circ} \\mathrm{C}$.", "answer_latex": " 435", "answer_number": "435", "unit": " $\\mathrm{kPa}$", "source": "matter", "problemid": " 1.1", "comment": " with equation", "solution": "The amount of $\\mathrm{N}_2$ molecules (of molar mass $28.02 \\mathrm{~g}$ $\\mathrm{mol}^{-1}$ ) present is\r\n$$\r\nn\\left(\\mathrm{~N}_2\\right)=\\frac{m}{M\\left(\\mathrm{~N}_2\\right)}=\\frac{1.25 \\mathrm{~g}}{28.02 \\mathrm{~g} \\mathrm{~mol}^{-1}}=\\frac{1.25}{28.02} \\mathrm{~mol}\r\n$$\r\nThe temperature of the sample is\r\n$$\r\nT / K=20+273.15, \\text { so } T=(20+273.15) \\mathrm{K}\r\n$$\r\nTherefore, after rewriting eqn 1.5 as $p=n R T / V$,\r\n$$\r\n\\begin{aligned}\r\np & =\\frac{\\overbrace{(1.25 / 28.02) \\mathrm{mol}}^n \\times \\overbrace{\\left(8.3145 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right)}^R \\times \\overbrace{(20+273.15) \\mathrm{K}}^{\\left(2.50 \\times 10^{-4}\\right) \\mathrm{m}^3}}{\\underbrace{R}_V} \\\\\r\n& =\\frac{(1.25 / 28.02) \\times(8.3145) \\times(20+273.15)}{2.50 \\times 10^{-4}} \\frac{\\mathrm{J}}{\\mathrm{m}^3} \\\\\r\n1 \\mathrm{Jm}^{-3} & =1 \\mathrm{~Pa} \\\\\r\n& \\stackrel{\\infty}{=} 4.35 \\times 10^5 \\mathrm{~Pa}=435 \\mathrm{kPa}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "Determine the energies and degeneracies of the lowest four energy levels of an ${ }^1 \\mathrm{H}^{35} \\mathrm{Cl}$ molecule freely rotating in three dimensions. What is the frequency of the transition between the lowest two rotational levels? The moment of inertia of an ${ }^1 \\mathrm{H}^{35} \\mathrm{Cl}$ molecule is $2.6422 \\times 10^{-47} \\mathrm{~kg} \\mathrm{~m}^2$.\r\n", "answer_latex": " 635.7", "answer_number": "635.7", "unit": " $\\mathrm{GHz}$", "source": "matter", "problemid": " 14.1", "comment": " with solution", "solution": "First, note that\r\n$$\r\n\\frac{\\hbar^2}{2 I}=\\frac{\\left(1.055 \\times 10^{-34} \\mathrm{Js}^2\\right.}{2 \\times\\left(2.6422 \\times 10^{-47} \\mathrm{~kg} \\mathrm{~m}^2\\right)}=2.106 \\ldots \\times 10^{-22} \\mathrm{~J}\r\n$$\r\nor $0.2106 \\ldots$ zJ. We now draw up the following table, where the molar energies are obtained by multiplying the individual energies by Avogadro's constant:\r\n\\begin{tabular}{llll}\r\n\\hline$J$ & $E / z J$ & $E /\\left(\\mathrm{J} \\mathrm{mol}^{-1}\\right)$ & Degeneracy \\\\\r\n\\hline 0 & 0 & 0 & 1 \\\\\r\n1 & 0.4212 & 253.6 & 3 \\\\\r\n2 & 1.264 & 760.9 & 5 \\\\\r\n3 & 2.527 & 1522 & 7 \\\\\r\n\\hline\r\n\\end{tabular}\r\n\r\nThe energy separation between the two lowest rotational energy levels $\\left(J=0\\right.$ and 1 ) is $4.212 \\times 10^{-22} \\mathrm{~J}$, which corresponds to a photon frequency of\r\n$$\r\n\\nu=\\frac{\\Delta E}{h}=\\frac{4.212 \\times 10^{-22} \\mathrm{~J}}{6.626 \\times 10^{-34} \\mathrm{Js}}=6.357 \\times 10^{11} \\mathrm{~s}^{-1}=635.7 \\mathrm{GHz}\r\n$$"}, {"problem_text": "Using the Planck distribution\r\nCompare the energy output of a black-body radiator (such as an incandescent lamp) at two different wavelengths by calculating the ratio of the energy output at $450 \\mathrm{~nm}$ (blue light) to that at $700 \\mathrm{~nm}$ (red light) at $298 \\mathrm{~K}$.\r\n", "answer_latex": " 2.10", "answer_number": "2.10", "unit": " $10^{-16}$", "source": "matter", "problemid": " 4.1", "comment": " with solution", "solution": "At a temperature $T$, the ratio of the spectral density of states at a wavelength $\\lambda_1$ to that at $\\lambda_2$ is given by\r\n$$\r\n\\frac{\\rho\\left(\\lambda_1, T\\right)}{\\rho\\left(\\lambda_2, T\\right)}=\\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^5 \\times \\frac{\\left(\\mathrm{e}^{h c / \\lambda_2 k T}-1\\right)}{\\left(\\mathrm{e}^{h c / \\lambda_1 k T}-1\\right)}\r\n$$\r\nInsert the data and evaluate this ratio.\r\nAnswer With $\\lambda_1=450 \\mathrm{~nm}$ and $\\lambda_2=700 \\mathrm{~nm}$,\r\n$$\r\n\\begin{aligned}\r\n\\frac{h c}{\\lambda_1 k T} & =\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right) \\times\\left(2.998 \\times 10^8 \\mathrm{~m} \\mathrm{~s}^{-1}\\right)}{\\left(450 \\times 10^{-9} \\mathrm{~m}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\times(298 \\mathrm{~K})}=107.2 \\ldots \\\\\r\n\\frac{h c}{\\lambda_2 k T} & =\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right) \\times\\left(2.998 \\times 10^8 \\mathrm{~m} \\mathrm{~s}^{-1}\\right)}{\\left(700 \\times 10^{-9} \\mathrm{~m}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{JK}^{-1}\\right) \\times(298 \\mathrm{~K})}=68.9 \\ldots\r\n\\end{aligned}\r\n$$\r\nand therefore\r\n$$\r\n\\begin{aligned}\r\n& \\frac{\\rho(450 \\mathrm{~nm}, 298 \\mathrm{~K})}{\\rho(700 \\mathrm{~nm}, 298 \\mathrm{~K})}=\\left(\\frac{700 \\times 10^{-9} \\mathrm{~m}}{450 \\times 10^{-9} \\mathrm{~m}}\\right)^5 \\times \\frac{\\left(\\mathrm{e}^{68.9 \\cdots}-1\\right)}{\\left(\\mathrm{e}^{107.2 \\cdots}-1\\right)} \\\\\r\n& =9.11 \\times\\left(2.30 \\times 10^{-17}\\right)=2.10 \\times 10^{-16}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Lead has $T_{\\mathrm{c}}=7.19 \\mathrm{~K}$ and $\\mathcal{H}_{\\mathrm{c}}(0)=63.9 \\mathrm{kA} \\mathrm{m}^{-1}$. At what temperature does lead become superconducting in a magnetic field of $20 \\mathrm{kA} \\mathrm{m}^{-1}$ ?", "answer_latex": " 6.0", "answer_number": "6.0", "unit": " $\\mathrm{~K}$", "source": "matter", "problemid": " 39.2", "comment": " with solution", "solution": "Rearrangement of eqn 39.7 gives\r\n$$\r\nT=T_{\\mathrm{c}}\\left(1-\\frac{\\mathcal{H}_{\\mathrm{c}}(T)}{\\mathcal{H}_{\\mathrm{c}}(0)}\\right)^{1 / 2}\r\n$$\r\nand substitution of the data gives\r\n$$\r\nT=(7.19 \\mathrm{~K}) \\times\\left(1-\\frac{20 \\mathrm{kAm}^{-1}}{63.9 \\mathrm{kAm}^{-1}}\\right)^{1 / 2}=6.0 \\mathrm{~K}\r\n$$\r\nThat is, lead becomes superconducting at temperatures below $6.0 \\mathrm{~K}$.\r\n"}, {"problem_text": "When an electric discharge is passed through gaseous hydrogen, the $\\mathrm{H}_2$ molecules are dissociated and energetically excited $\\mathrm{H}$ atoms are produced. If the electron in an excited $\\mathrm{H}$ atom makes a transition from $n=2$ to $n=1$, calculate the wavenumber of the corresponding line in the emission spectrum.", "answer_latex": " 82258", "answer_number": "82258", "unit": " $\\mathrm{~cm}^{-1}$", "source": "matter", "problemid": " 17.2", "comment": " with solution", "solution": "The wavenumber of the photon emitted when an electron makes a transition from $n_2=2$ to $n_1=1$ is given by\r\n$$\r\n\\begin{aligned}\r\n\\tilde{\\boldsymbol{v}} & =-\\tilde{R}_{\\mathrm{H}}\\left(\\frac{1}{n_2^2}-\\frac{1}{n_1^2}\\right) \\\\\r\n& =-\\left(109677 \\mathrm{~cm}^{-1}\\right) \\times\\left(\\frac{1}{2^2}-\\frac{1}{1^2}\\right) \\\\\r\n& =82258 \\mathrm{~cm}^{-1}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Calculate the shielding constant for the proton in a free $\\mathrm{H}$ atom.", "answer_latex": " 1.775", "answer_number": "1.775", "unit": " $10^{-5}$", "source": "matter", "problemid": " 48.2", "comment": " with solution", "solution": "The wavefunction for a hydrogen 1 s orbital is\r\n$$\r\n\\psi=\\left(\\frac{1}{\\pi a_0^3}\\right)^{1 / 2} \\mathrm{e}^{-r / a_0}\r\n$$\r\nso, because $\\mathrm{d} \\tau=r^2 \\mathrm{~d} r \\sin \\theta \\mathrm{d} \\theta \\mathrm{d} \\phi$, the expectation value of $1 / r$ is written as\r\n$$\r\n\\begin{aligned}\r\n\\left\\langle\\frac{1}{r}\\right\\rangle & =\\int \\frac{\\psi^* \\psi}{r} \\mathrm{~d} \\tau=\\frac{1}{\\pi a_0^3} \\int_0^{2 \\pi} \\mathrm{d} \\phi \\int_0^\\pi \\sin \\theta \\mathrm{d} \\theta \\int_0^{\\infty} r \\mathrm{e}^{-2 r / a_0} \\mathrm{~d} r \\\\\r\n& =\\frac{4}{a_0^3} \\overbrace{\\int_0^{\\infty} r \\mathrm{e}^{-2 r / a_0} \\mathrm{~d} r}^{a_0^2 / 4 \\text { (Integral E.1) }}=\\frac{1}{a_0}\r\n\\end{aligned}\r\n$$\r\nwhere we used the integral listed in the Resource section. Therefore,\r\n$$\r\n\\begin{aligned}\r\n& =\\frac{\\left(1.602 \\times 10^{-19} \\mathrm{C}\\right)^2 \\times(4 \\pi \\times 10^{-7} \\overbrace{\\mathrm{J}}^{\\mathrm{Jg} \\mathrm{m}^2 \\mathrm{~s}^{-2}} \\mathrm{~s}^2 \\mathrm{C}^{-2} \\mathrm{~m}^{-1})}{12 \\pi \\times\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right) \\times\\left(5.292 \\times 10^{-11} \\mathrm{~m}\\right)} \\\\\r\n& =1.775 \\times 10^{-5} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\n"}]
+[
+ {
+  "problem_text": "Calculating the maximum wavelength capable of photoejection\r\nA photon of radiation of wavelength $305 \\mathrm{~nm}$ ejects an electron from a metal with a kinetic energy of $1.77 \\mathrm{eV}$. Calculate the maximum wavelength of radiation capable of ejecting an electron from the metal.",
+  "answer_latex": " 540",
+  "answer_number": "540",
+  "unit": "$\\mathrm{~nm}$ ",
+  "source": "matter",
+  "problemid": " 4.1",
+  "comment": " with solution",
+  "solution": "From the expression for the work function $\\Phi=h \\nu-E_{\\mathrm{k}}$ the minimum frequency for photoejection is\r\n$$\r\n\\nu_{\\min }=\\frac{\\Phi}{h}=\\frac{h v-E_{\\mathrm{k}}}{h} \\stackrel{\\nu=c \\mid \\lambda}{=} \\frac{c}{\\lambda}-\\frac{E_{\\mathrm{k}}}{h}\r\n$$\r\nThe maximum wavelength is therefore\r\n$$\r\n\\lambda_{\\max }=\\frac{c}{v_{\\min }}=\\frac{c}{c / \\lambda-E_{\\mathrm{k}} / h}=\\frac{1}{1 / \\lambda-E_{\\mathrm{k}} / h c}\r\n$$\r\nNow we substitute the data. The kinetic energy of the electron is\r\n$$\r\n\\begin{aligned}\r\n& E_k=1.77 \\mathrm{eV} \\times\\left(1.602 \\times 10^{-19} \\mathrm{JeV}^{-1}\\right)=2.83 \\ldots \\times 10^{-19} \\mathrm{~J} \\\\\r\n& \\frac{E_k}{h c}=\\frac{2.83 \\ldots \\times 10^{-19} \\mathrm{~J}}{\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right) \\times\\left(2.998 \\times 10^8 \\mathrm{~m} \\mathrm{~s}^{-1}\\right)}=1.42 \\ldots \\times 10^6 \\mathrm{~m}^{-1}\r\n\\end{aligned}\r\n$$\r\nTherefore, with\r\n$$\r\n\\begin{aligned}\r\n& 1 / \\lambda=1 / 305 \\mathrm{~nm}=3.27 \\ldots \\times 10^6 \\mathrm{~m}^{-1}, \\\\\r\n& \\lambda_{\\max }=\\frac{1}{\\left(3.27 \\ldots \\times 10^6 \\mathrm{~m}^{-1}\\right)-\\left(1.42 \\ldots \\times 10^6 \\mathrm{~m}^{-1}\\right)}=5.40 \\times 10^{-7} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\nor $540 \\mathrm{~nm}$.\r\n"
+ },
+ {
+  "problem_text": "Estimate the molar volume of $\\mathrm{CO}_2$ at $500 \\mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.",
+  "answer_latex": " 0.366",
+  "answer_number": "0.366",
+  "unit": "$\\mathrm{dm}^3 \\mathrm{~mol}^{-1}$",
+  "source": "matter",
+  "problemid": " 36.1",
+  "comment": " with solution",
+  "solution": "\nFor van der Waals gas of $\\mathrm{CO}_2$ , $a=3.610 \\mathrm{dm}^6$ atm $\\mathrm{mol}^{-2}$ and $b=4.29 \\times 10^{-2} \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. Under the stated conditions, $R T / p=0.410 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. The coefficients in the equation for $V_{\\mathrm{m}}$ are therefore\r\n$$\r\n\\begin{aligned}\r\nb+R T / p & =0.453 \\mathrm{dm}^3 \\mathrm{~mol}^{-1} \\\\\r\na / p & =3.61 \\times 10^{-2}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^2 \\\\\r\na b / p & =1.55 \\times 10^{-3}\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)^3\r\n\\end{aligned}\r\n$$\r\nTherefore, on writing $x=V_{\\mathrm{m}} /\\left(\\mathrm{dm}^3 \\mathrm{~mol}^{-1}\\right)$, the equation to solve is\r\n$$\r\nx^3-0.453 x^2+\\left(3.61 \\times 10^{-2}\\right) x-\\left(1.55 \\times 10^{-3}\\right)=0\r\n$$\r\nThe acceptable root is $x=0.366$, which implies that $V_{\\mathrm{m}}=0.366$ $\\mathrm{dm}^3 \\mathrm{~mol}^{-1}$. The molar volume of a perfect gas under these conditions is $0.410 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$.\n"
+ },
+ {
+  "problem_text": "The single electron in a certain excited state of a hydrogenic $\\mathrm{He}^{+}$ion $(Z=2)$ is described by the wavefunction $R_{3,2}(r) \\times$ $Y_{2,-1}(\\theta, \\phi)$. What is the energy of its electron?",
+  "answer_latex": " -6.04697",
+  "answer_number": " -6.04697",
+  "unit": "$ \\mathrm{eV}$ ",
+  "source": "matter",
+  "problemid": " 17.1",
+  "comment": " with solution",
+  "solution": "\nReplacing $\\mu$ by $m_{\\mathrm{e}}$ and using $\\hbar=h / 2 \\pi$, we can write the expression for the energy as\r\n$$\r\nE_n=-\\frac{Z^2 m_e e^4}{8 \\varepsilon_0^2 h^2 n^2}=-\\frac{Z^2 h c \\tilde{R}_{\\infty}}{n^2}\r\n$$\r\nwith\r\n$$\n\\tilde{R}_{\\infty}=\\frac{9.10938 \\times 10^{-31} \\mathrm{~kg} \\times (1.602176 \\times 10^{-19} \\mathrm{C})^4}{8 \\times (8.85419 \\times 10^{-12} \\mathrm{~J}^{-1} \\mathrm{C}^2 \\mathrm{~m}^{-1} )^2 \\times(6.62608 \\times 10^{-34} \\mathrm{Js})^3} \\times \\underbrace{2.997926 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}}_c =109737 \\mathrm{~cm}^{-1} $$\r\nand\r\n$$\r\n\\begin{aligned}\r\nh c \\tilde{R}_{\\infty}= & \\left(6.62608 \\times 10^{-34} \\mathrm{Js}\\right) \\times\\left(2.997926 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right) \\\\\r\n& \\times\\left(109737 \\mathrm{~cm}^{-1}\\right) \\\\\r\n= & 2.17987 \\times 10^{-18} \\mathrm{~J}\r\n\\end{aligned}\r\n$$\r\nTherefore, for $n=3$, the energy is\r\n$$\r\n\\begin{aligned}\r\n& E_3=-\\frac{\\overbrace{4}^{Z^2} \\times \\overbrace{2.17987 \\times 10^{-18} \\mathrm{~J}}^{h c \\tilde{R}_{\\infty}}}{\\underset{\\tilde{n}^2}{9}} \\\\\r\n& =-9.68831 \\times 10^{-19} \\mathrm{~J} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nor $-0.968831 \\mathrm{aJ}$ (a, for atto, is the prefix that denotes $10^{-18}$ ). In some applications it is useful to express the energy in electronvolts $\\left(1 \\mathrm{eV}=1.602176 \\times 10^{-19} \\mathrm{~J}\\right)$; in this case, $E_3=-6.04697 \\mathrm{eV}$\n"
+ },
+ {
+  "problem_text": "Calculate the typical wavelength of neutrons after reaching thermal equilibrium with their surroundings at $373 \\mathrm{~K}$. For simplicity, assume that the particles are travelling in one dimension.",
+  "answer_latex": " 226",
+  "answer_number": "226",
+  "unit": "$\\mathrm{pm}$",
+  "source": "matter",
+  "problemid": " 37.4",
+  "comment": " with solution",
+  "solution": "From the equipartition principle, we know that the mean translational kinetic energy of a neutron at a temperature $T$ travelling in the $x$-direction is $E_{\\mathrm{k}}=\\frac{1}{2} k T$. The kinetic energy is also equal to $p^2 / 2 m$, where $p$ is the momentum of the neutron and $m$ is its mass. Hence, $p=(m k T)^{1 / 2}$. It follows from the de Broglie relation $\\lambda=h / p$ that the neutron's wavelength is\r\n$$\r\n\\lambda=\\frac{h}{(m k T)^{1 / 2}}\r\n$$\r\nTherefore, at $373 \\mathrm{~K}$,\r\n$$\r\n\\begin{aligned}\r\n\\lambda & =\\frac{6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left\\{\\left(1.675 \\times 10^{-27} \\mathrm{~kg}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\times(373 \\mathrm{~K})\\right\\}^{1 / 2}} \\\\\r\n& =\\frac{6.626 \\times 10^{-34}}{\\left(1.675 \\times 10^{-27} \\times 1.381 \\times 10^{-23} \\times 373\\right)^{1 / 2}} \\frac{\\mathrm{kg} \\mathrm{m}^2 \\mathrm{~s}^{-1}}{\\left(\\mathrm{~kg}^2 \\mathrm{~m}^2 \\mathrm{~s}^{-2}\\right)^{1 / 2}} \\\\\r\n& =2.26 \\times 10^{-10} \\mathrm{~m}=226 \\mathrm{pm}\r\n\\end{aligned}\r\n$$\r\nwhere we have used $1 \\mathrm{~J}=1 \\mathrm{~kg} \\mathrm{~m}^2 \\mathrm{~s}^{-2}$."
+ },
+ {
+  "problem_text": "Using the perfect gas equation\r\nCalculate the pressure in kilopascals exerted by $1.25 \\mathrm{~g}$ of nitrogen gas in a flask of volume $250 \\mathrm{~cm}^3$ at $20^{\\circ} \\mathrm{C}$.",
+  "answer_latex": " 435",
+  "answer_number": "435",
+  "unit": " $\\mathrm{kPa}$",
+  "source": "matter",
+  "problemid": " 1.1",
+  "comment": " with equation",
+  "solution": "The amount of $\\mathrm{N}_2$ molecules (of molar mass $28.02 \\mathrm{~g}$ $\\mathrm{mol}^{-1}$ ) present is\r\n$$\r\nn\\left(\\mathrm{~N}_2\\right)=\\frac{m}{M\\left(\\mathrm{~N}_2\\right)}=\\frac{1.25 \\mathrm{~g}}{28.02 \\mathrm{~g} \\mathrm{~mol}^{-1}}=\\frac{1.25}{28.02} \\mathrm{~mol}\r\n$$\r\nThe temperature of the sample is\r\n$$\r\nT / K=20+273.15, \\text { so } T=(20+273.15) \\mathrm{K}\r\n$$\r\nTherefore, after rewriting eqn 1.5 as $p=n R T / V$,\r\n$$\r\n\\begin{aligned}\r\np & =\\frac{\\overbrace{(1.25 / 28.02) \\mathrm{mol}}^n \\times \\overbrace{\\left(8.3145 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}\\right)}^R \\times \\overbrace{(20+273.15) \\mathrm{K}}^{\\left(2.50 \\times 10^{-4}\\right) \\mathrm{m}^3}}{\\underbrace{R}_V} \\\\\r\n& =\\frac{(1.25 / 28.02) \\times(8.3145) \\times(20+273.15)}{2.50 \\times 10^{-4}} \\frac{\\mathrm{J}}{\\mathrm{m}^3} \\\\\r\n1 \\mathrm{Jm}^{-3} & =1 \\mathrm{~Pa} \\\\\r\n& \\stackrel{\\infty}{=} 4.35 \\times 10^5 \\mathrm{~Pa}=435 \\mathrm{kPa}\r\n\\end{aligned}\r\n$$\r\n"
+ },
+ {
+  "problem_text": "Determine the energies and degeneracies of the lowest four energy levels of an ${ }^1 \\mathrm{H}^{35} \\mathrm{Cl}$ molecule freely rotating in three dimensions. What is the frequency of the transition between the lowest two rotational levels? The moment of inertia of an ${ }^1 \\mathrm{H}^{35} \\mathrm{Cl}$ molecule is $2.6422 \\times 10^{-47} \\mathrm{~kg} \\mathrm{~m}^2$.\r\n",
+  "answer_latex": " 635.7",
+  "answer_number": "635.7",
+  "unit": " $\\mathrm{GHz}$",
+  "source": "matter",
+  "problemid": " 14.1",
+  "comment": " with solution",
+  "solution": "First, note that\r\n$$\r\n\\frac{\\hbar^2}{2 I}=\\frac{\\left(1.055 \\times 10^{-34} \\mathrm{Js}^2\\right.}{2 \\times\\left(2.6422 \\times 10^{-47} \\mathrm{~kg} \\mathrm{~m}^2\\right)}=2.106 \\ldots \\times 10^{-22} \\mathrm{~J}\r\n$$\r\nor $0.2106 \\ldots$ zJ. We now draw up the following table, where the molar energies are obtained by multiplying the individual energies by Avogadro's constant:\r\n\\begin{tabular}{llll}\r\n\\hline$J$ & $E / z J$ & $E /\\left(\\mathrm{J} \\mathrm{mol}^{-1}\\right)$ & Degeneracy \\\\\r\n\\hline 0 & 0 & 0 & 1 \\\\\r\n1 & 0.4212 & 253.6 & 3 \\\\\r\n2 & 1.264 & 760.9 & 5 \\\\\r\n3 & 2.527 & 1522 & 7 \\\\\r\n\\hline\r\n\\end{tabular}\r\n\r\nThe energy separation between the two lowest rotational energy levels $\\left(J=0\\right.$ and 1 ) is $4.212 \\times 10^{-22} \\mathrm{~J}$, which corresponds to a photon frequency of\r\n$$\r\n\\nu=\\frac{\\Delta E}{h}=\\frac{4.212 \\times 10^{-22} \\mathrm{~J}}{6.626 \\times 10^{-34} \\mathrm{Js}}=6.357 \\times 10^{11} \\mathrm{~s}^{-1}=635.7 \\mathrm{GHz}\r\n$$"
+ },
+ {
+  "problem_text": "Using the Planck distribution\r\nCompare the energy output of a black-body radiator (such as an incandescent lamp) at two different wavelengths by calculating the ratio of the energy output at $450 \\mathrm{~nm}$ (blue light) to that at $700 \\mathrm{~nm}$ (red light) at $298 \\mathrm{~K}$.\r\n",
+  "answer_latex": " 2.10",
+  "answer_number": "2.10",
+  "unit": " $10^{-16}$",
+  "source": "matter",
+  "problemid": " 4.1",
+  "comment": " with solution",
+  "solution": "At a temperature $T$, the ratio of the spectral density of states at a wavelength $\\lambda_1$ to that at $\\lambda_2$ is given by\r\n$$\r\n\\frac{\\rho\\left(\\lambda_1, T\\right)}{\\rho\\left(\\lambda_2, T\\right)}=\\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^5 \\times \\frac{\\left(\\mathrm{e}^{h c / \\lambda_2 k T}-1\\right)}{\\left(\\mathrm{e}^{h c / \\lambda_1 k T}-1\\right)}\r\n$$\r\nInsert the data and evaluate this ratio.\r\nAnswer With $\\lambda_1=450 \\mathrm{~nm}$ and $\\lambda_2=700 \\mathrm{~nm}$,\r\n$$\r\n\\begin{aligned}\r\n\\frac{h c}{\\lambda_1 k T} & =\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right) \\times\\left(2.998 \\times 10^8 \\mathrm{~m} \\mathrm{~s}^{-1}\\right)}{\\left(450 \\times 10^{-9} \\mathrm{~m}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\times(298 \\mathrm{~K})}=107.2 \\ldots \\\\\r\n\\frac{h c}{\\lambda_2 k T} & =\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right) \\times\\left(2.998 \\times 10^8 \\mathrm{~m} \\mathrm{~s}^{-1}\\right)}{\\left(700 \\times 10^{-9} \\mathrm{~m}\\right) \\times\\left(1.381 \\times 10^{-23} \\mathrm{JK}^{-1}\\right) \\times(298 \\mathrm{~K})}=68.9 \\ldots\r\n\\end{aligned}\r\n$$\r\nand therefore\r\n$$\r\n\\begin{aligned}\r\n& \\frac{\\rho(450 \\mathrm{~nm}, 298 \\mathrm{~K})}{\\rho(700 \\mathrm{~nm}, 298 \\mathrm{~K})}=\\left(\\frac{700 \\times 10^{-9} \\mathrm{~m}}{450 \\times 10^{-9} \\mathrm{~m}}\\right)^5 \\times \\frac{\\left(\\mathrm{e}^{68.9 \\cdots}-1\\right)}{\\left(\\mathrm{e}^{107.2 \\cdots}-1\\right)} \\\\\r\n& =9.11 \\times\\left(2.30 \\times 10^{-17}\\right)=2.10 \\times 10^{-16}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Lead has $T_{\\mathrm{c}}=7.19 \\mathrm{~K}$ and $\\mathcal{H}_{\\mathrm{c}}(0)=63.9 \\mathrm{kA} \\mathrm{m}^{-1}$. At what temperature does lead become superconducting in a magnetic field of $20 \\mathrm{kA} \\mathrm{m}^{-1}$ ?",
+  "answer_latex": " 6.0",
+  "answer_number": "6.0",
+  "unit": " $\\mathrm{~K}$",
+  "source": "matter",
+  "problemid": " 39.2",
+  "comment": " with solution",
+  "solution": "\nRearrangement of eqn $$\n\\mathcal{H}_{\\mathrm{c}}(T)=\\mathcal{H}_{\\mathrm{c}}(0) (1-\\frac{T^2}{T_{\\mathrm{c}}^2})\n$$ gives\r\n$$\r\nT=T_{\\mathrm{c}}\\left(1-\\frac{\\mathcal{H}_{\\mathrm{c}}(T)}{\\mathcal{H}_{\\mathrm{c}}(0)}\\right)^{1 / 2}\r\n$$\r\nand substitution of the data gives\r\n$$\r\nT=(7.19 \\mathrm{~K}) \\times\\left(1-\\frac{20 \\mathrm{kAm}^{-1}}{63.9 \\mathrm{kAm}^{-1}}\\right)^{1 / 2}=6.0 \\mathrm{~K}\r\n$$\r\nThat is, lead becomes superconducting at temperatures below $6.0 \\mathrm{~K}$.\n"
+ },
+ {
+  "problem_text": "When an electric discharge is passed through gaseous hydrogen, the $\\mathrm{H}_2$ molecules are dissociated and energetically excited $\\mathrm{H}$ atoms are produced. If the electron in an excited $\\mathrm{H}$ atom makes a transition from $n=2$ to $n=1$, calculate the wavenumber of the corresponding line in the emission spectrum.",
+  "answer_latex": " 82258",
+  "answer_number": "82258",
+  "unit": " $\\mathrm{~cm}^{-1}$",
+  "source": "matter",
+  "problemid": " 17.2",
+  "comment": " with solution",
+  "solution": "The wavenumber of the photon emitted when an electron makes a transition from $n_2=2$ to $n_1=1$ is given by\r\n$$\r\n\\begin{aligned}\r\n\\tilde{\\boldsymbol{v}} & =-\\tilde{R}_{\\mathrm{H}}\\left(\\frac{1}{n_2^2}-\\frac{1}{n_1^2}\\right) \\\\\r\n& =-\\left(109677 \\mathrm{~cm}^{-1}\\right) \\times\\left(\\frac{1}{2^2}-\\frac{1}{1^2}\\right) \\\\\r\n& =82258 \\mathrm{~cm}^{-1}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Calculate the shielding constant for the proton in a free $\\mathrm{H}$ atom.",
+  "answer_latex": " 1.775",
+  "answer_number": "1.775",
+  "unit": " $10^{-5}$",
+  "source": "matter",
+  "problemid": " 48.2",
+  "comment": " with solution",
+  "solution": "The wavefunction for a hydrogen 1 s orbital is\r\n$$\r\n\\psi=\\left(\\frac{1}{\\pi a_0^3}\\right)^{1 / 2} \\mathrm{e}^{-r / a_0}\r\n$$\r\nso, because $\\mathrm{d} \\tau=r^2 \\mathrm{~d} r \\sin \\theta \\mathrm{d} \\theta \\mathrm{d} \\phi$, the expectation value of $1 / r$ is written as\r\n$$\r\n\\begin{aligned}\r\n\\left\\langle\\frac{1}{r}\\right\\rangle & =\\int \\frac{\\psi^* \\psi}{r} \\mathrm{~d} \\tau=\\frac{1}{\\pi a_0^3} \\int_0^{2 \\pi} \\mathrm{d} \\phi \\int_0^\\pi \\sin \\theta \\mathrm{d} \\theta \\int_0^{\\infty} r \\mathrm{e}^{-2 r / a_0} \\mathrm{~d} r \\\\\r\n& =\\frac{4}{a_0^3} \\overbrace{\\int_0^{\\infty} r \\mathrm{e}^{-2 r / a_0} \\mathrm{~d} r}^{a_0^2 / 4 \\text { (Integral E.1) }}=\\frac{1}{a_0}\r\n\\end{aligned}\r\n$$\r\nwhere we used the integral listed in the Resource section. Therefore,\r\n$$\r\n\\begin{aligned}\r\n& =\\frac{\\left(1.602 \\times 10^{-19} \\mathrm{C}\\right)^2 \\times(4 \\pi \\times 10^{-7} \\overbrace{\\mathrm{J}}^{\\mathrm{Jg} \\mathrm{m}^2 \\mathrm{~s}^{-2}} \\mathrm{~s}^2 \\mathrm{C}^{-2} \\mathrm{~m}^{-1})}{12 \\pi \\times\\left(9.109 \\times 10^{-31} \\mathrm{~kg}\\right) \\times\\left(5.292 \\times 10^{-11} \\mathrm{~m}\\right)} \\\\\r\n& =1.775 \\times 10^{-5} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\n"
+ }
+]

quan.json

@@ -1 +1,299 @@
-[{"problem_text": "Use the $D_0$ value of $\\mathrm{H}_2(4.478 \\mathrm{eV})$ and the $D_0$ value of $\\mathrm{H}_2^{+}(2.651 \\mathrm{eV})$ to calculate the first ionization energy of $\\mathrm{H}_2$ (that is, the energy needed to remove an electron from $\\mathrm{H}_2$ ).", "answer_latex": " 15.425", "answer_number": "15.425", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 13.3", "comment": " "}, {"problem_text": "Calculate the energy of one mole of UV photons of wavelength $300 \\mathrm{~nm}$ and compare it with a typical single-bond energy of $400 \\mathrm{~kJ} / \\mathrm{mol}$.", "answer_latex": " 399", "answer_number": "399", "unit": " $\\mathrm{~kJ} / \\mathrm{mol}$", "source": "quan", "problemid": " 1.3", "comment": " "}, {"problem_text": "Calculate the magnitude of the spin angular momentum of a proton. Give a numerical answer. ", "answer_latex": " 9.13", "answer_number": "9.13", "unit": " $10^{-35} \\mathrm{~J} \\mathrm{~s}$", "source": "quan", "problemid": " 10.1", "comment": " "}, {"problem_text": "The ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$ ground electronic state has $D_0=2.4287 \\mathrm{eV}, \\nu_e / c=1405.65 \\mathrm{~cm}^{-1}$, and $\\nu_e x_e / c=23.20 \\mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\\omega_e$ and $\\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$.", "answer_latex": " 2.5151", "answer_number": "2.5151", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 13.5", "comment": " "}, {"problem_text": "The positron has charge $+e$ and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an \"atom\" that consists of a positron and an electron.", "answer_latex": " -6.8", "answer_number": " -6.8", "unit": "$\\mathrm{eV}$", "source": "quan", "problemid": " 6.22", "comment": " "}, {"problem_text": "What is the value of the angular-momentum quantum number $l$ for a $t$ orbital?", "answer_latex": " 14", "answer_number": "14", "unit": " ", "source": "quan", "problemid": " 6.29", "comment": " "}, {"problem_text": "How many states belong to the carbon configurations $1 s^2 2 s^2 2 p^2$?", "answer_latex": " 15", "answer_number": "15", "unit": " ", "source": "quan", "problemid": " 11.22", "comment": " "}, {"problem_text": "Calculate the energy needed to compress three carbon-carbon single bonds and stretch three carbon-carbon double bonds to the benzene bond length $1.397 \u00c5$. Assume a harmonicoscillator potential-energy function for bond stretching and compression. Typical carboncarbon single- and double-bond lengths are 1.53 and $1.335 \u00c5$; typical stretching force constants for carbon-carbon single and double bonds are 500 and $950 \\mathrm{~N} / \\mathrm{m}$.", "answer_latex": " 27", "answer_number": "27", "unit": " $\\mathrm{kcal} / \\mathrm{mol}$", "source": "quan", "problemid": " 17.9", "comment": " Angstrom "}, {"problem_text": "When a particle of mass $9.1 \\times 10^{-28} \\mathrm{~g}$ in a certain one-dimensional box goes from the $n=5$ level to the $n=2$ level, it emits a photon of frequency $6.0 \\times 10^{14} \\mathrm{~s}^{-1}$. Find the length of the box.", "answer_latex": " 1.8", "answer_number": "1.8", "unit": "$\\mathrm{~nm}$", "source": "quan", "problemid": " 2.13", "comment": " "}, {"problem_text": "Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the $v=0$ state.", "answer_latex": " 0.16", "answer_number": "0.16", "unit": " ", "source": "quan", "problemid": " 4.42", "comment": " "}, {"problem_text": "Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.)", "answer_latex": " 0.332", "answer_number": "0.332", "unit": "$\\mathrm{~nm}$", "source": "quan", "problemid": " 1.6", "comment": " "}, {"problem_text": "Calculate the angle that the spin vector $S$ makes with the $z$ axis for an electron with spin function $\\alpha$.", "answer_latex": " 54.7", "answer_number": "54.7", "unit": " $^{\\circ}$", "source": "quan", "problemid": " 10.2", "comment": " "}, {"problem_text": "The AM1 valence electronic energies of the atoms $\\mathrm{H}$ and $\\mathrm{O}$ are $-11.396 \\mathrm{eV}$ and $-316.100 \\mathrm{eV}$, respectively. For $\\mathrm{H}_2 \\mathrm{O}$ at its AM1-calculated equilibrium geometry, the AM1 valence electronic energy (core-core repulsion omitted) is $-493.358 \\mathrm{eV}$ and the AM1 core-core repulsion energy is $144.796 \\mathrm{eV}$. For $\\mathrm{H}(g)$ and $\\mathrm{O}(g), \\Delta H_{f, 298}^{\\circ}$ values are 52.102 and $59.559 \\mathrm{kcal} / \\mathrm{mol}$, respectively. Find the AM1 prediction of $\\Delta H_{f, 298}^{\\circ}$ of $\\mathrm{H}_2 \\mathrm{O}(g)$.", "answer_latex": " -59.24", "answer_number": "-59.24", "unit": " $\\mathrm{kcal} / \\mathrm{mol}$", "source": "quan", "problemid": " 17.29", "comment": " "}, {"problem_text": "Given that $D_e=4.75 \\mathrm{eV}$ and $R_e=0.741 \u00c5$ for the ground electronic state of $\\mathrm{H}_2$, find $U\\left(R_e\\right)$ for this state.", "answer_latex": " -31.95", "answer_number": " -31.95", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 14.35", "comment": " Angstrom "}, {"problem_text": "For $\\mathrm{NaCl}, R_e=2.36 \u00c5$. The ionization energy of $\\mathrm{Na}$ is $5.14 \\mathrm{eV}$, and the electron affinity of $\\mathrm{Cl}$ is $3.61 \\mathrm{eV}$. Use the simple model of $\\mathrm{NaCl}$ as a pair of spherical ions in contact to estimate $D_e$. [One debye (D) is $3.33564 \\times 10^{-30} \\mathrm{C} \\mathrm{m}$.]", "answer_latex": " 4.56", "answer_number": " 4.56", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 14.5", "comment": " Angstrom "}, {"problem_text": "Find the number of CSFs in a full CI calculation of $\\mathrm{CH}_2 \\mathrm{SiHF}$ using a 6-31G** basis set.", "answer_latex": " 1.86", "answer_number": "1.86", "unit": "$10^{28} $", "source": "quan", "problemid": " 16.1", "comment": " "}, {"problem_text": "Calculate the ratio of the electrical and gravitational forces between a proton and an electron.", "answer_latex": " 2", "answer_number": "2", "unit": " $10^{39}$", "source": "quan", "problemid": " 6.15", "comment": " "}, {"problem_text": "A one-particle, one-dimensional system has the state function\r\n$$\r\n\\Psi=(\\sin a t)\\left(2 / \\pi c^2\\right)^{1 / 4} e^{-x^2 / c^2}+(\\cos a t)\\left(32 / \\pi c^6\\right)^{1 / 4} x e^{-x^2 / c^2}\r\n$$\r\nwhere $a$ is a constant and $c=2.000 \u00c5$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 \u00c5$ and $2.001 \u00c5$.", "answer_latex": " 0.000216", "answer_number": "0.000216", "unit": " ", "source": "quan", "problemid": " 1.13", "comment": " Angstrom"}, {"problem_text": "A one-particle, one-dimensional system has the state function\r\n$$\r\n\\Psi=(\\sin a t)\\left(2 / \\pi c^2\\right)^{1 / 4} e^{-x^2 / c^2}+(\\cos a t)\\left(32 / \\pi c^6\\right)^{1 / 4} x e^{-x^2 / c^2}\r\n$$\r\nwhere $a$ is a constant and $c=2.000 \u00c5$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 \u00c5$ and $2.001 \u00c5$.", "answer_latex": " 0.000216", "answer_number": "0.000216", "unit": " ", "source": "quan", "problemid": " 1.13", "comment": " Angstrom"}, {"problem_text": "The $J=2$ to 3 rotational transition in a certain diatomic molecule occurs at $126.4 \\mathrm{GHz}$, where $1 \\mathrm{GHz} \\equiv 10^9 \\mathrm{~Hz}$. Find the frequency of the $J=5$ to 6 absorption in this molecule.", "answer_latex": " 252.8", "answer_number": " 252.8", "unit": " $\\mathrm{GHz}$", "source": "quan", "problemid": " 6.10", "comment": " Approximated answer"}, {"problem_text": "Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \\mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance $r$ from the nuclear center is $-e Q / 4 \\pi \\varepsilon_0 r$, where $Q$ is the amount of proton charge within the sphere of radius $r$. The evaluation of the integral is simplified by noting that the exponential factor in $\\psi$ is essentially equal to 1 within the nucleus.\r\n", "answer_latex": " 1.2", "answer_number": "1.2", "unit": " $10^{-8} \\mathrm{eV}$", "source": "quan", "problemid": " 9.9", "comment": " "}, {"problem_text": "An electron in a three-dimensional rectangular box with dimensions of $5.00 \u00c5, 3.00 \u00c5$, and $6.00 \u00c5$ makes a radiative transition from the lowest-lying excited state to the ground state. Calculate the frequency of the photon emitted.", "answer_latex": "7.58", "answer_number": "7.58", "unit": " $10^{14} \\mathrm{~s}^{-1}$", "source": "quan", "problemid": " 3.35", "comment": " Angstrom "}, {"problem_text": "Do $\\mathrm{HF} / 6-31 \\mathrm{G}^*$ geometry optimizations on one conformers of $\\mathrm{HCOOH}$ with $\\mathrm{OCOH}$ dihedral angle of $0^{\\circ}$. Calculate the dipole moment.", "answer_latex": " 1.41", "answer_number": "1.41", "unit": " $\\mathrm{D}$", "source": "quan", "problemid": " 15.57", "comment": " "}, {"problem_text": "Frozen-core $\\mathrm{SCF} / \\mathrm{DZP}$ and CI-SD/DZP calculations on $\\mathrm{H}_2 \\mathrm{O}$ at its equilibrium geometry gave energies of -76.040542 and -76.243772 hartrees. Application of the Davidson correction brought the energy to -76.254549 hartrees. Find the coefficient of $\\Phi_0$ in the normalized CI-SD wave function.", "answer_latex": " 0.9731", "answer_number": "0.9731", "unit": " ", "source": "quan", "problemid": " 16.3", "comment": " "}, {"problem_text": "Let $w$ be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find $\\langle w\\rangle$.", "answer_latex": " 1", "answer_number": "1", "unit": " ", "source": "quan", "problemid": " 5.8", "comment": " "}, {"problem_text": "Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 \u00c5$.", "answer_latex": " 0.405", "answer_number": "0.405", "unit": " $\\mathrm{~N}$", "source": "quan", "problemid": " 1.31", "comment": " "}, {"problem_text": "When an electron in a certain excited energy level in a one-dimensional box of length $2.00  \u00c5$ makes a transition to the ground state, a photon of wavelength $8.79 \\mathrm{~nm}$ is emitted. Find the quantum number of the initial state.", "answer_latex": "4", "answer_number": "4", "unit": "", "source": "quan", "problemid": " 2.13", "comment": " Angstrom "}, {"problem_text": "For a macroscopic object of mass $1.0 \\mathrm{~g}$ moving with speed $1.0 \\mathrm{~cm} / \\mathrm{s}$ in a one-dimensional box of length $1.0 \\mathrm{~cm}$, find the quantum number $n$.", "answer_latex": " 3", "answer_number": "3", "unit": "$10^{26}$", "source": "quan", "problemid": " 2.11", "comment": " "}, {"problem_text": "For the $\\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \\mathrm{eV}$. Find $\\Delta H_0^{\\circ}$ for $\\mathrm{H}_2(g) \\rightarrow 2 \\mathrm{H}(g)$ in $\\mathrm{kJ} / \\mathrm{mol}$", "answer_latex": " 432.07", "answer_number": "432.07", "unit": " $\\mathrm{~kJ} / \\mathrm{mol}$", "source": "quan", "problemid": " 13.2", "comment": " "}, {"problem_text": "The contribution of molecular vibrations to the molar internal energy $U_{\\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\\mathrm{m}, \\mathrm{vib}}=R \\sum_{s=1}^{3 N-6} \\theta_s /\\left(e^{\\theta_s / T}-1\\right)$, where $\\theta_s \\equiv h \\nu_s / k$ and $\\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\\mathrm{m}, \\text { vib }}$ at $25^{\\circ} \\mathrm{C}$ of a normal mode with wavenumber $\\widetilde{v} \\equiv v_s / c$ of $900 \\mathrm{~cm}^{-1}$.", "answer_latex": " 0.14", "answer_number": "0.14", "unit": " $\\mathrm{kJ} / \\mathrm{mol}$", "source": "quan", "problemid": " 15.39", "comment": " "}, {"problem_text": "Calculate the magnitude of the spin magnetic moment of an electron.", "answer_latex": " 1.61", "answer_number": "1.61", "unit": " $10^{-23} \\mathrm{~J} / \\mathrm{T}$", "source": "quan", "problemid": " 10.17", "comment": " "}, {"problem_text": "A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \\mathrm{eV}$, calculate $\\langle V\\rangle$ for the ground state.", "answer_latex": " $3\frac{1}{3}$", "answer_number": "3.333333333", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 14.29", "comment": " screenshot answer is weird"}, {"problem_text": "For an electron in a certain rectangular well with a depth of $20.0 \\mathrm{eV}$, the lowest energy level lies $3.00 \\mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\\tan \\theta=\\sin \\theta / \\cos \\theta$", "answer_latex": " 0.264", "answer_number": "0.264", "unit": "$\\mathrm{~nm}$", "source": "quan", "problemid": " 2.27", "comment": " hint"}, {"problem_text": "Calculate the uncertainty $\\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$.", "answer_latex": " 0", "answer_number": "0", "unit": " ", "source": "quan", "problemid": " 7.56", "comment": " "}]
+[
+ {
+  "problem_text": "Use the $D_0$ value of $\\mathrm{H}_2(4.478 \\mathrm{eV})$ and the $D_0$ value of $\\mathrm{H}_2^{+}(2.651 \\mathrm{eV})$ to calculate the first ionization energy of $\\mathrm{H}_2$ (that is, the energy needed to remove an electron from $\\mathrm{H}_2$ ).",
+  "answer_latex": " 15.425",
+  "answer_number": "15.425",
+  "unit": " $\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 13.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the energy of one mole of UV photons of wavelength $300 \\mathrm{~nm}$ and compare it with a typical single-bond energy of $400 \\mathrm{~kJ} / \\mathrm{mol}$.",
+  "answer_latex": " 399",
+  "answer_number": "399",
+  "unit": " $\\mathrm{~kJ} / \\mathrm{mol}$",
+  "source": "quan",
+  "problemid": " 1.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the magnitude of the spin angular momentum of a proton. Give a numerical answer. ",
+  "answer_latex": " 9.13",
+  "answer_number": "9.13",
+  "unit": " $10^{-35} \\mathrm{~J} \\mathrm{~s}$",
+  "source": "quan",
+  "problemid": " 10.1",
+  "comment": " "
+ },
+ {
+  "problem_text": "The ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$ ground electronic state has $D_0=2.4287 \\mathrm{eV}, \\nu_e / c=1405.65 \\mathrm{~cm}^{-1}$, and $\\nu_e x_e / c=23.20 \\mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\\omega_e$ and $\\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$.",
+  "answer_latex": " 2.5151",
+  "answer_number": "2.5151",
+  "unit": " $\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 13.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "The positron has charge $+e$ and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an \"atom\" that consists of a positron and an electron.",
+  "answer_latex": " -6.8",
+  "answer_number": " -6.8",
+  "unit": "$\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 6.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "What is the value of the angular-momentum quantum number $l$ for a $t$ orbital?",
+  "answer_latex": " 14",
+  "answer_number": "14",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 6.29",
+  "comment": " "
+ },
+ {
+  "problem_text": "How many states belong to the carbon configurations $1 s^2 2 s^2 2 p^2$?",
+  "answer_latex": " 15",
+  "answer_number": "15",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 11.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the energy needed to compress three carbon-carbon single bonds and stretch three carbon-carbon double bonds to the benzene bond length $1.397 \u00c5$. Assume a harmonicoscillator potential-energy function for bond stretching and compression. Typical carboncarbon single- and double-bond lengths are 1.53 and $1.335 \u00c5$; typical stretching force constants for carbon-carbon single and double bonds are 500 and $950 \\mathrm{~N} / \\mathrm{m}$.",
+  "answer_latex": " 27",
+  "answer_number": "27",
+  "unit": " $\\mathrm{kcal} / \\mathrm{mol}$",
+  "source": "quan",
+  "problemid": " 17.9",
+  "comment": " Angstrom "
+ },
+ {
+  "problem_text": "When a particle of mass $9.1 \\times 10^{-28} \\mathrm{~g}$ in a certain one-dimensional box goes from the $n=5$ level to the $n=2$ level, it emits a photon of frequency $6.0 \\times 10^{14} \\mathrm{~s}^{-1}$. Find the length of the box.",
+  "answer_latex": " 1.8",
+  "answer_number": "1.8",
+  "unit": "$\\mathrm{~nm}$",
+  "source": "quan",
+  "problemid": " 2.13",
+  "comment": " "
+ },
+ {
+  "problem_text": "Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the $v=0$ state.",
+  "answer_latex": " 0.16",
+  "answer_number": "0.16",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 4.42",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.)",
+  "answer_latex": " 0.332",
+  "answer_number": "0.332",
+  "unit": "$\\mathrm{~nm}$",
+  "source": "quan",
+  "problemid": " 1.6",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the angle that the spin vector $S$ makes with the $z$ axis for an electron with spin function $\\alpha$.",
+  "answer_latex": " 54.7",
+  "answer_number": "54.7",
+  "unit": " $^{\\circ}$",
+  "source": "quan",
+  "problemid": " 10.2",
+  "comment": " "
+ },
+ {
+  "problem_text": "The AM1 valence electronic energies of the atoms $\\mathrm{H}$ and $\\mathrm{O}$ are $-11.396 \\mathrm{eV}$ and $-316.100 \\mathrm{eV}$, respectively. For $\\mathrm{H}_2 \\mathrm{O}$ at its AM1-calculated equilibrium geometry, the AM1 valence electronic energy (core-core repulsion omitted) is $-493.358 \\mathrm{eV}$ and the AM1 core-core repulsion energy is $144.796 \\mathrm{eV}$. For $\\mathrm{H}(g)$ and $\\mathrm{O}(g), \\Delta H_{f, 298}^{\\circ}$ values are 52.102 and $59.559 \\mathrm{kcal} / \\mathrm{mol}$, respectively. Find the AM1 prediction of $\\Delta H_{f, 298}^{\\circ}$ of $\\mathrm{H}_2 \\mathrm{O}(g)$.",
+  "answer_latex": " -59.24",
+  "answer_number": "-59.24",
+  "unit": " $\\mathrm{kcal} / \\mathrm{mol}$",
+  "source": "quan",
+  "problemid": " 17.29",
+  "comment": " "
+ },
+ {
+  "problem_text": "Given that $D_e=4.75 \\mathrm{eV}$ and $R_e=0.741 \u00c5$ for the ground electronic state of $\\mathrm{H}_2$, find $U\\left(R_e\\right)$ for this state.",
+  "answer_latex": " -31.95",
+  "answer_number": " -31.95",
+  "unit": " $\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 14.35",
+  "comment": " Angstrom "
+ },
+ {
+  "problem_text": "For $\\mathrm{NaCl}, R_e=2.36 \u00c5$. The ionization energy of $\\mathrm{Na}$ is $5.14 \\mathrm{eV}$, and the electron affinity of $\\mathrm{Cl}$ is $3.61 \\mathrm{eV}$. Use the simple model of $\\mathrm{NaCl}$ as a pair of spherical ions in contact to estimate $D_e$. [One debye (D) is $3.33564 \\times 10^{-30} \\mathrm{C} \\mathrm{m}$.]",
+  "answer_latex": " 4.56",
+  "answer_number": " 4.56",
+  "unit": " $\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 14.5",
+  "comment": " Angstrom "
+ },
+ {
+  "problem_text": "Find the number of CSFs in a full CI calculation of $\\mathrm{CH}_2 \\mathrm{SiHF}$ using a 6-31G** basis set.",
+  "answer_latex": " 1.86",
+  "answer_number": "1.86",
+  "unit": "$10^{28} $",
+  "source": "quan",
+  "problemid": " 16.1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the ratio of the electrical and gravitational forces between a proton and an electron.",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " $10^{39}$",
+  "source": "quan",
+  "problemid": " 6.15",
+  "comment": " "
+ },
+ {
+  "problem_text": "A one-particle, one-dimensional system has the state function\r\n$$\r\n\\Psi=(\\sin a t)\\left(2 / \\pi c^2\\right)^{1 / 4} e^{-x^2 / c^2}+(\\cos a t)\\left(32 / \\pi c^6\\right)^{1 / 4} x e^{-x^2 / c^2}\r\n$$\r\nwhere $a$ is a constant and $c=2.000 \u00c5$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 \u00c5$ and $2.001 \u00c5$.",
+  "answer_latex": " 0.000216",
+  "answer_number": "0.000216",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 1.13",
+  "comment": " Angstrom"
+ },
+ {
+  "problem_text": "The $J=2$ to 3 rotational transition in a certain diatomic molecule occurs at $126.4 \\mathrm{GHz}$, where $1 \\mathrm{GHz} \\equiv 10^9 \\mathrm{~Hz}$. Find the frequency of the $J=5$ to 6 absorption in this molecule.",
+  "answer_latex": " 252.8",
+  "answer_number": " 252.8",
+  "unit": " $\\mathrm{GHz}$",
+  "source": "quan",
+  "problemid": " 6.10",
+  "comment": " Approximated answer"
+ },
+ {
+  "problem_text": "Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \\mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance $r$ from the nuclear center is $-e Q / 4 \\pi \\varepsilon_0 r$, where $Q$ is the amount of proton charge within the sphere of radius $r$. The evaluation of the integral is simplified by noting that the exponential factor in $\\psi$ is essentially equal to 1 within the nucleus.\r\n",
+  "answer_latex": " 1.2",
+  "answer_number": "1.2",
+  "unit": " $10^{-8} \\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 9.9",
+  "comment": " "
+ },
+ {
+  "problem_text": "An electron in a three-dimensional rectangular box with dimensions of $5.00 \u00c5, 3.00 \u00c5$, and $6.00 \u00c5$ makes a radiative transition from the lowest-lying excited state to the ground state. Calculate the frequency of the photon emitted.",
+  "answer_latex": "7.58",
+  "answer_number": "7.58",
+  "unit": " $10^{14} \\mathrm{~s}^{-1}$",
+  "source": "quan",
+  "problemid": " 3.35",
+  "comment": " Angstrom "
+ },
+ {
+  "problem_text": "Do $\\mathrm{HF} / 6-31 \\mathrm{G}^*$ geometry optimizations on one conformers of $\\mathrm{HCOOH}$ with $\\mathrm{OCOH}$ dihedral angle of $0^{\\circ}$. Calculate the dipole moment.",
+  "answer_latex": " 1.41",
+  "answer_number": "1.41",
+  "unit": " $\\mathrm{D}$",
+  "source": "quan",
+  "problemid": " 15.57",
+  "comment": " "
+ },
+ {
+  "problem_text": "Frozen-core $\\mathrm{SCF} / \\mathrm{DZP}$ and CI-SD/DZP calculations on $\\mathrm{H}_2 \\mathrm{O}$ at its equilibrium geometry gave energies of -76.040542 and -76.243772 hartrees. Application of the Davidson correction brought the energy to -76.254549 hartrees. Find the coefficient of $\\Phi_0$ in the normalized CI-SD wave function.",
+  "answer_latex": " 0.9731",
+  "answer_number": "0.9731",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 16.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $w$ be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find $\\langle w\\rangle$.",
+  "answer_latex": " 1",
+  "answer_number": "1",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 5.8",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 \u00c5$.",
+  "answer_latex": " 0.405",
+  "answer_number": "0.405",
+  "unit": " $\\mathrm{~N}$",
+  "source": "quan",
+  "problemid": " 1.31",
+  "comment": " "
+ },
+ {
+  "problem_text": "When an electron in a certain excited energy level in a one-dimensional box of length $2.00  \u00c5$ makes a transition to the ground state, a photon of wavelength $8.79 \\mathrm{~nm}$ is emitted. Find the quantum number of the initial state.",
+  "answer_latex": "4",
+  "answer_number": "4",
+  "unit": "",
+  "source": "quan",
+  "problemid": " 2.13",
+  "comment": " Angstrom "
+ },
+ {
+  "problem_text": "For a macroscopic object of mass $1.0 \\mathrm{~g}$ moving with speed $1.0 \\mathrm{~cm} / \\mathrm{s}$ in a one-dimensional box of length $1.0 \\mathrm{~cm}$, find the quantum number $n$.",
+  "answer_latex": " 3",
+  "answer_number": "3",
+  "unit": "$10^{26}$",
+  "source": "quan",
+  "problemid": " 2.11",
+  "comment": " "
+ },
+ {
+  "problem_text": "For the $\\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \\mathrm{eV}$. Find $\\Delta H_0^{\\circ}$ for $\\mathrm{H}_2(g) \\rightarrow 2 \\mathrm{H}(g)$ in $\\mathrm{kJ} / \\mathrm{mol}$",
+  "answer_latex": " 432.07",
+  "answer_number": "432.07",
+  "unit": " $\\mathrm{~kJ} / \\mathrm{mol}$",
+  "source": "quan",
+  "problemid": " 13.2",
+  "comment": " "
+ },
+ {
+  "problem_text": "The contribution of molecular vibrations to the molar internal energy $U_{\\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\\mathrm{m}, \\mathrm{vib}}=R \\sum_{s=1}^{3 N-6} \\theta_s /\\left(e^{\\theta_s / T}-1\\right)$, where $\\theta_s \\equiv h \\nu_s / k$ and $\\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\\mathrm{m}, \\text { vib }}$ at $25^{\\circ} \\mathrm{C}$ of a normal mode with wavenumber $\\widetilde{v} \\equiv v_s / c$ of $900 \\mathrm{~cm}^{-1}$.",
+  "answer_latex": " 0.14",
+  "answer_number": "0.14",
+  "unit": " $\\mathrm{kJ} / \\mathrm{mol}$",
+  "source": "quan",
+  "problemid": " 15.39",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the magnitude of the spin magnetic moment of an electron.",
+  "answer_latex": " 1.61",
+  "answer_number": "1.61",
+  "unit": " $10^{-23} \\mathrm{~J} / \\mathrm{T}$",
+  "source": "quan",
+  "problemid": " 10.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \\mathrm{eV}$, calculate $\\langle V\\rangle$ for the ground state.",
+  "answer_latex": " $3\frac{1}{3}$",
+  "answer_number": "3.333333333",
+  "unit": " $\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 14.29",
+  "comment": " screenshot answer is weird"
+ },
+ {
+  "problem_text": "For an electron in a certain rectangular well with a depth of $20.0 \\mathrm{eV}$, the lowest energy level lies $3.00 \\mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\\tan \\theta=\\sin \\theta / \\cos \\theta$",
+  "answer_latex": " 0.264",
+  "answer_number": "0.264",
+  "unit": "$\\mathrm{~nm}$",
+  "source": "quan",
+  "problemid": " 2.27",
+  "comment": " hint"
+ },
+ {
+  "problem_text": "Calculate the uncertainty $\\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$.",
+  "answer_latex": " 0",
+  "answer_number": "0",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 7.56",
+  "comment": " "
+ }
+]

quan_sol.json

@@ -1 +1,82 @@
-[{"problem_text": "A one-particle, one-dimensional system has $\\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \\mathrm{~nm}$. At $t=0$, the particle's position is measured.  (b) Find the probability that the measured value is between $x=0$ and $x=2 \\mathrm{~nm}$.", "answer_latex": " 0.4908", "answer_number": "0.4908", "unit": " ", "source": "quan", "problemid": " 1.6.1_b", "comment": " ", "solution": "(b) Use of Eq. (1.23) and $|x|=x$ for $x \\geq 0$ gives\r\n$$\r\n\\begin{aligned}\r\n\\operatorname{Pr}(0 \\leq x \\leq 2 \\mathrm{~nm}) & =\\int_0^{2 \\mathrm{~nm}}|\\Psi|^2 d x=a^{-1} \\int_0^{2 \\mathrm{~nm}} e^{-2 x / a} d x \\\\\r\n& =-\\left.\\frac{1}{2} e^{-2 x / a}\\right|_0 ^{2 \\mathrm{~nm}}=-\\frac{1}{2}\\left(e^{-4}-1\\right)=0.4908\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Calculate the ground-state energy of the hydrogen atom using SI units and convert the result to electronvolts.", "answer_latex": "-13.598 ", "answer_number": "-13.598 ", "unit": " $\\mathrm{eV}$", "source": "quan", "problemid": " 6.6.1", "comment": " ", "solution": "The $\\mathrm{H}$ atom ground-state energy is given by (6.94) with $n=1$ and $Z=1$ as $E=-\\mu e^4 / 8 h^2 \\varepsilon_0^2$. Use of (6.105) for $\\mu$ gives\r\n$$\r\n\\begin{gathered}\r\nE=-\\frac{0.9994557\\left(9.109383 \\times 10^{-31} \\mathrm{~kg}\\right)\\left(1.6021766 \\times 10^{-19} \\mathrm{C}\\right)^4}{8\\left(6.626070 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)^2\\left(8.8541878 \\times 10^{-12} \\mathrm{C}^2 / \\mathrm{N}-\\mathrm{m}^2\\right)^2} \\frac{Z^2}{n^2} \\\\\r\nE=-\\left(2.178686 \\times 10^{-18} \\mathrm{~J}\\right)\\left(Z^2 / n^2\\right)\\left[(1 \\mathrm{eV}) /\\left(1.6021766 \\times 10^{-19} \\mathrm{~J}\\right)\\right]\r\n\\end{gathered}\r\n$$\r\n$$\r\nE=-(13.598 \\mathrm{eV})\\left(Z^2 / n^2\\right)=-13.598 \\mathrm{eV}\r\n$$\r\na number worth remembering. The minimum energy needed to ionize a ground-state hydrogen atom is $13.598 \\mathrm{eV}$.\r\n"}, {"problem_text": "Find the probability that the electron in the ground-state $\\mathrm{H}$ atom is less than a distance $a$ from the nucleus.", "answer_latex": " 0.323", "answer_number": "0.323", "unit": " ", "source": "quan", "problemid": "6.6.3 ", "comment": " ", "solution": "We want the probability that the radial coordinate lies between 0 and $a$. This is found by taking the infinitesimal probability (6.116) of being between $r$ and $r+d r$ and summing it over the range from 0 to $a$. This sum of infinitesimal quantities is the definite integral\r\n$$\r\n\\begin{aligned}\r\n\\int_0^a R_{n l}^2 r^2 d r & =\\frac{4}{a^3} \\int_0^a e^{-2 r / a} r^2 d r=\\left.\\frac{4}{a^3} e^{-2 r / a}\\left(-\\frac{r^2 a}{2}-\\frac{2 r a^2}{4}-\\frac{2 a^3}{8}\\right)\\right|_0 ^a \\\\\r\n& =4\\left[e^{-2}(-5 / 4)-(-1 / 4)\\right]=0.323\r\n\\end{aligned}\r\n$$"}, {"problem_text": "A one-particle, one-dimensional system has $\\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \\mathrm{~nm}$. At $t=0$, the particle's position is measured. (a) Find the probability that the measured value lies between $x=1.5000 \\mathrm{~nm}$ and $x=1.5001 \\mathrm{~nm}$.", "answer_latex": " 4.979", "answer_number": "4.979", "unit": " $10^{-6}$", "source": "quan", "problemid": " 1.6.1_a", "comment": " ", "solution": "(a) In this tiny interval, $x$ changes by only $0.0001 \\mathrm{~nm}$, and $\\Psi$ goes from $e^{-1.5000} \\mathrm{~nm}^{-1 / 2}=0.22313 \\mathrm{~nm}^{-1 / 2}$ to $e^{-1.5001} \\mathrm{~nm}^{-1 / 2}=0.22311 \\mathrm{~nm}^{-1 / 2}$, so $\\Psi$ is nearly constant in this interval, and it is a very good approximation to consider this interval as infinitesimal. The desired probability is given by (1.15) as\r\n$$\r\n\\begin{aligned}\r\n|\\Psi|^2 d x=a^{-1} e^{-2|x| / a} d x & =(1 \\mathrm{~nm})^{-1} e^{-2(1.5 \\mathrm{~nm}) /(1 \\mathrm{~nm})}(0.0001 \\mathrm{~nm}) \\\\\r\n& =4.979 \\times 10^{-6}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "In this example, $2.50 \\mathrm{~mol}$ of an ideal gas with $C_{V, m}=12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$ is expanded adiabatically against a constant external pressure of 1.00 bar. The initial temperature and pressure of the gas are $325 \\mathrm{~K}$ and $2.50 \\mathrm{bar}$, respectively. The final pressure is 1.25 bar. Calculate the final temperature, $q, w, \\Delta U$.", "answer_latex": " -1.78", "answer_number": "-1.78", "unit": " $\\mathrm{~kJ}$", "source": "quan", "problemid": " 2.6", "comment": " ", "solution": "Because the process is adiabatic, $q=0$, and $\\Delta U=w$. Therefore,\r\n$$\r\n\\Delta U=n C_{\\mathrm{v}, m}\\left(T_f-T_i\\right)=-P_{e x t e r n a l}\\left(V_f-V_i\\right)\r\n$$\r\nUsing the ideal gas law,\r\n$$\r\n\\begin{aligned}\r\n& n C_{\\mathrm{v}, m}\\left(T_f-T_i\\right)=-n R P_{\\text {external }}\\left(\\frac{T_f}{P_f}-\\frac{T_i}{P_i}\\right) \\\\\r\n& T_f\\left(n C_{\\mathrm{v}, m}+\\frac{n R P_{\\text {external }}}{P_f}\\right)=T_i\\left(n C_{\\mathrm{v}, m}+\\frac{n R P_{\\text {external }}}{P_i}\\right) \\\\\r\n& T_f=T_i\\left(\\frac{C_{\\mathrm{v}, m}+\\frac{R P_{\\text {external }}}{P_i}}{C_{\\mathrm{v}, m}+\\frac{R P_{\\text {external }}}{P_f}}\\right) \\\\\r\n& =325 \\mathrm{~K} \\times\\left(\\frac{12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}+\\frac{8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times 1.00 \\mathrm{bar}}{2.50 \\mathrm{bar}}}{12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}+\\frac{8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times 1.00 \\mathrm{bar}}{1.25 \\mathrm{bar}}}\\right)=268 \\mathrm{~K} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nWe calculate $\\Delta U=w$ from\r\n$$\r\n\\begin{aligned}\r\n\\Delta U & =n C_{V, m}\\left(T_f-T_i\\right)=2.5 \\mathrm{~mol} \\times 12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times(268 \\mathrm{~K}-325 \\mathrm{~K}) \\\\\r\n& =-1.78 \\mathrm{~kJ}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Find $Y_l^m(\\theta, \\phi)$  for  $l=0$.", "answer_latex": "$\\frac{1}{\\sqrt{4 \\pi}}$", "answer_number": "0.28209479", "unit": " ", "source": "quan", "problemid": " 5.1", "comment": " ", "solution": "For $l=0$, Eq. (5.96) gives $m=0$, and (5.97) becomes\r\n$$\r\nS_{0,0}(\\theta)=a_0\r\n$$\r\nThe normalization condition (5.80) gives\r\n$$\r\n\\begin{aligned}\r\n\\int_0^\\pi\\left|a_0\\right|^2 \\sin \\theta d \\theta=1 & =2\\left|a_0\\right|^2 \\\\\r\n\\left|a_0\\right| & =2^{-1 / 2}\r\n\\end{aligned}\r\n$$\r\nEquation (5.99) gives\r\n$$\r\nY_0^0(\\theta, \\phi)=\\frac{1}{\\sqrt{4 \\pi}}\r\n$$"}, {"problem_text": "The lowest-frequency pure-rotational absorption line of ${ }^{12} \\mathrm{C}^{32} \\mathrm{~S}$ occurs at $48991.0 \\mathrm{MHz}$. Find the bond distance in ${ }^{12} \\mathrm{C}^{32} \\mathrm{~S}$.", "answer_latex": " 1.5377", "answer_number": "1.5377", "unit": " $10^{-10} \\mathrm{~m}$", "source": "quan", "problemid": "6.4 ", "comment": " ", "solution": "The lowest-frequency rotational absorption is the $J=0 \\rightarrow 1$ line. Equations (1.4), $(6.52)$, and $(6.51)$ give\r\n$$\r\nh \\nu=E_{\\mathrm{upper}}-E_{\\mathrm{lower}}=\\frac{1(2) \\hbar^2}{2 \\mu d^2}-\\frac{0(1) \\hbar^2}{2 \\mu d^2}\r\n$$\r\nwhich gives $d=\\left(h / 4 \\pi^2 \\nu \\mu\\right)^{1 / 2}$. Table A.3 in the Appendix gives\r\n$$\r\n\\mu=\\frac{m_1 m_2}{m_1+m_2}=\\frac{12(31.97207)}{(12+31.97207)} \\frac{1}{6.02214 \\times 10^{23}} \\mathrm{~g}=1.44885 \\times 10^{-23} \\mathrm{~g}\r\n$$\r\nThe SI unit of mass is the kilogram, and\r\n$$\r\n\\begin{aligned}\r\nd=\\frac{1}{2 \\pi}\\left(\\frac{h}{\\nu_{0 \\rightarrow 1} \\mu}\\right)^{1 / 2} & =\\frac{1}{2 \\pi}\\left[\\frac{6.62607 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left(48991.0 \\times 10^6 \\mathrm{~s}^{-1}\\right)\\left(1.44885 \\times 10^{-26} \\mathrm{~kg}\\right)}\\right]^{1 / 2} \\\\\r\n& =1.5377 \\times 10^{-10} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\n"}, {"problem_text": "The strongest infrared band of ${ }^{12} \\mathrm{C}^{16} \\mathrm{O}$ occurs at $\\widetilde{\\nu}=2143 \\mathrm{~cm}^{-1}$. Find the force constant of ${ }^{12} \\mathrm{C}^{16} \\mathrm{O}$. ", "answer_latex": " 1855", "answer_number": "1855", "unit": " $\\mathrm{~N} / \\mathrm{m}$", "source": "quan", "problemid": " 4.3", "comment": " ", "solution": "The strongest infrared band corresponds to the $v=0 \\rightarrow 1$ transition. We approximate the molecular vibration as that of a harmonic oscillator. From (4.61), the equilibrium molecular vibrational frequency is approximately\r\n$$\r\n\\nu_e \\approx \\nu_{\\text {light }}=\\widetilde{\\nu} c=\\left(2143 \\mathrm{~cm}^{-1}\\right)\\left(2.9979 \\times 10^{10} \\mathrm{~cm} / \\mathrm{s}\\right)=6.424 \\times 10^{13} \\mathrm{~s}^{-1}\r\n$$\r\nTo relate $k$ to $\\nu_e$ in (4.59), we need the reduced mass $\\mu=m_1 m_2 /\\left(m_1+m_2\\right)$. One mole of ${ }^{12} \\mathrm{C}$ has a mass of $12 \\mathrm{~g}$ and contains Avogadro's number of atoms. Hence the mass of one atom of ${ }^{12} \\mathrm{C}$ is $(12 \\mathrm{~g}) /\\left(6.02214 \\times 10^{23}\\right)$. The reduced mass and force constant are\r\n$$\r\n\\begin{gathered}\r\n\\mu=\\frac{12(15.9949) \\mathrm{g}}{27.9949} \\frac{1}{6.02214 \\times 10^{23}}=1.1385 \\times 10^{-23} \\mathrm{~g} \\\\\r\nk=4 \\pi^2 \\nu_e^2 \\mu=4 \\pi^2\\left(6.424 \\times 10^{13} \\mathrm{~s}^{-1}\\right)^2\\left(1.1385 \\times 10^{-26} \\mathrm{~kg}\\right)=1855 \\mathrm{~N} / \\mathrm{m}\r\n\\end{gathered}\r\n$$"}]
+[
+ {
+  "problem_text": "A one-particle, one-dimensional system has $\\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \\mathrm{~nm}$. At $t=0$, the particle's position is measured. Find the probability that the measured value is between $x=0$ and $x=2 \\mathrm{~nm}$.",
+  "answer_latex": " 0.4908",
+  "answer_number": "0.4908",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 1.6.1_b",
+  "comment": " ",
+  "solution": "\n$$\r\n\\begin{aligned}\r\n\\operatorname{Pr}(0 \\leq x \\leq 2 \\mathrm{~nm}) & =\\int_0^{2 \\mathrm{~nm}}|\\Psi|^2 d x=a^{-1} \\int_0^{2 \\mathrm{~nm}} e^{-2 x / a} d x \\\\\r\n& =-\\left.\\frac{1}{2} e^{-2 x / a}\\right|_0 ^{2 \\mathrm{~nm}}=-\\frac{1}{2}\\left(e^{-4}-1\\right)=0.4908\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Calculate the ground-state energy of the hydrogen atom using SI units and convert the result to electronvolts.",
+  "answer_latex": "-13.598 ",
+  "answer_number": "-13.598 ",
+  "unit": " $\\mathrm{eV}$",
+  "source": "quan",
+  "problemid": " 6.6.1",
+  "comment": " ",
+  "solution": "\nThe $\\mathrm{H}$ atom ground-state energy with $n=1$ and $Z=1$ is $E=-\\mu e^4 / 8 h^2 \\varepsilon_0^2$. Use of equation $$\n\\mu_{\\mathrm{H}}=\\frac{m_e m_p}{m_e+m_p}=\\frac{m_e}{1+m_e / m_p}=\\frac{m_e}{1+0.000544617}=0.9994557 m_e\n$$ for $\\mu$ gives\r\n$$\r\n\\begin{gathered}\r\nE=-\\frac{0.9994557\\left(9.109383 \\times 10^{-31} \\mathrm{~kg}\\right)\\left(1.6021766 \\times 10^{-19} \\mathrm{C}\\right)^4}{8\\left(6.626070 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)^2\\left(8.8541878 \\times 10^{-12} \\mathrm{C}^2 / \\mathrm{N}-\\mathrm{m}^2\\right)^2} \\frac{Z^2}{n^2} \\\\\r\nE=-\\left(2.178686 \\times 10^{-18} \\mathrm{~J}\\right)\\left(Z^2 / n^2\\right)\\left[(1 \\mathrm{eV}) /\\left(1.6021766 \\times 10^{-19} \\mathrm{~J}\\right)\\right]\r\n\\end{gathered}\r\n$$\r\n$$\r\nE=-(13.598 \\mathrm{eV})\\left(Z^2 / n^2\\right)=-13.598 \\mathrm{eV}\r\n$$\r\na number worth remembering. The minimum energy needed to ionize a ground-state hydrogen atom is $13.598 \\mathrm{eV}$.\n"
+ },
+ {
+  "problem_text": "Find the probability that the electron in the ground-state $\\mathrm{H}$ atom is less than a distance $a$ from the nucleus.",
+  "answer_latex": " 0.323",
+  "answer_number": "0.323",
+  "unit": " ",
+  "source": "quan",
+  "problemid": "6.6.3 ",
+  "comment": " ",
+  "solution": "\nWe want the probability that the radial coordinate lies between 0 and $a$. This is found by taking the infinitesimal probability of being between $r$ and $r+d r$ and summing it over the range from 0 to $a$. This sum of infinitesimal quantities is the definite integral\r\n$$\r\n\\begin{aligned}\r\n\\int_0^a R_{n l}^2 r^2 d r & =\\frac{4}{a^3} \\int_0^a e^{-2 r / a} r^2 d r=\\left.\\frac{4}{a^3} e^{-2 r / a}\\left(-\\frac{r^2 a}{2}-\\frac{2 r a^2}{4}-\\frac{2 a^3}{8}\\right)\\right|_0 ^a \\\\\r\n& =4\\left[e^{-2}(-5 / 4)-(-1 / 4)\\right]=0.323\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "A one-particle, one-dimensional system has $\\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \\mathrm{~nm}$. At $t=0$, the particle's position is measured. Find the probability that the measured value lies between $x=1.5000 \\mathrm{~nm}$ and $x=1.5001 \\mathrm{~nm}$.",
+  "answer_latex": " 4.979",
+  "answer_number": "4.979",
+  "unit": " $10^{-6}$",
+  "source": "quan",
+  "problemid": " 1.6.1_a",
+  "comment": " ",
+  "solution": "\nIn this tiny interval, $x$ changes by only $0.0001 \\mathrm{~nm}$, and $\\Psi$ goes from $e^{-1.5000} \\mathrm{~nm}^{-1 / 2}=0.22313 \\mathrm{~nm}^{-1 / 2}$ to $e^{-1.5001} \\mathrm{~nm}^{-1 / 2}=0.22311 \\mathrm{~nm}^{-1 / 2}$, so $\\Psi$ is nearly constant in this interval, and it is a very good approximation to consider this interval as infinitesimal. The desired probability is given as\r\n$$\r\n\\begin{aligned}\r\n|\\Psi|^2 d x=a^{-1} e^{-2|x| / a} d x & =(1 \\mathrm{~nm})^{-1} e^{-2(1.5 \\mathrm{~nm}) /(1 \\mathrm{~nm})}(0.0001 \\mathrm{~nm}) \\\\\r\n& =4.979 \\times 10^{-6}\r\n\\end{aligned}\r\n$$\n\n"
+ },
+ {
+  "problem_text": "In this example, $2.50 \\mathrm{~mol}$ of an ideal gas with $C_{V, m}=12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$ is expanded adiabatically against a constant external pressure of 1.00 bar. The initial temperature and pressure of the gas are $325 \\mathrm{~K}$ and $2.50 \\mathrm{bar}$, respectively. The final pressure is 1.25 bar. Calculate the final $ \\Delta U$.",
+  "answer_latex": " -1.78",
+  "answer_number": "-1.78",
+  "unit": " $\\mathrm{~kJ}$",
+  "source": "quan",
+  "problemid": " 2.6",
+  "comment": " ",
+  "solution": "Because the process is adiabatic, $q=0$, and $\\Delta U=w$. Therefore,\r\n$$\r\n\\Delta U=n C_{\\mathrm{v}, m}\\left(T_f-T_i\\right)=-P_{e x t e r n a l}\\left(V_f-V_i\\right)\r\n$$\r\nUsing the ideal gas law,\r\n$$\r\n\\begin{aligned}\r\n& n C_{\\mathrm{v}, m}\\left(T_f-T_i\\right)=-n R P_{\\text {external }}\\left(\\frac{T_f}{P_f}-\\frac{T_i}{P_i}\\right) \\\\\r\n& T_f\\left(n C_{\\mathrm{v}, m}+\\frac{n R P_{\\text {external }}}{P_f}\\right)=T_i\\left(n C_{\\mathrm{v}, m}+\\frac{n R P_{\\text {external }}}{P_i}\\right) \\\\\r\n& T_f=T_i\\left(\\frac{C_{\\mathrm{v}, m}+\\frac{R P_{\\text {external }}}{P_i}}{C_{\\mathrm{v}, m}+\\frac{R P_{\\text {external }}}{P_f}}\\right) \\\\\r\n& =325 \\mathrm{~K} \\times\\left(\\frac{12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}+\\frac{8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times 1.00 \\mathrm{bar}}{2.50 \\mathrm{bar}}}{12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}+\\frac{8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times 1.00 \\mathrm{bar}}{1.25 \\mathrm{bar}}}\\right)=268 \\mathrm{~K} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nWe calculate $\\Delta U=w$ from\r\n$$\r\n\\begin{aligned}\r\n\\Delta U & =n C_{V, m}\\left(T_f-T_i\\right)=2.5 \\mathrm{~mol} \\times 12.47 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times(268 \\mathrm{~K}-325 \\mathrm{~K}) \\\\\r\n& =-1.78 \\mathrm{~kJ}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Find $Y_l^m(\\theta, \\phi)$  for  $l=0$.",
+  "answer_latex": "$\\frac{1}{\\sqrt{4 \\pi}}$",
+  "answer_number": "0.28209479",
+  "unit": " ",
+  "source": "quan",
+  "problemid": " 5.1",
+  "comment": " ",
+  "solution": "\nFor $l=0$, Eq. $$\nm=-l,-l+1,-l+2, \\ldots,-1,0,1, \\ldots, l-2, l-1, l\n$$ gives $m=0$, and $$\nS_{l, m}(\\theta)=\\sin ^{|m|} \\theta \\sum_{\\substack{j=1,3, \\ldots \\ \\text { or } j=0,2, \\ldots}}^{l-|m|} a_j \\cos ^j \\theta\n$$ becomes\r\n$$\r\nS_{0,0}(\\theta)=a_0\r\n$$\r\nThe normalization condition $$\n\\int_0^{\\infty}|R|^2 r^2 d r=1, \\quad \\int_0^\\pi|S|^2 \\sin \\theta d \\theta=1, \\quad \\int_0^{2 \\pi}|T|^2 d \\phi=1\n$$ gives$$\\begin{aligned}\r\n\\int_0^\\pi\\left|a_0\\right|^2 \\sin \\theta d \\theta=1 & =2\\left|a_0\\right|^2 \\\\\r\n\\left|a_0\\right| & =2^{-1 / 2}\r\n\\end{aligned}\r\n$$\r\nEquation $$\nY_l^m(\\theta, \\phi)=S_{l, m}(\\theta) T(\\phi)=\\frac{1}{\\sqrt{2 \\pi}} S_{l, m}(\\theta) e^{i m \\phi}\n$$ gives$$\r\nY_0^0(\\theta, \\phi)=\\frac{1}{\\sqrt{4 \\pi}}\r\n$$\n\n"
+ },
+ {
+  "problem_text": "The lowest-frequency pure-rotational absorption line of ${ }^{12} \\mathrm{C}^{32} \\mathrm{~S}$ occurs at $48991.0 \\mathrm{MHz}$. Find the bond distance in ${ }^{12} \\mathrm{C}^{32} \\mathrm{~S}$.",
+  "answer_latex": " 1.5377",
+  "answer_number": "1.5377",
+  "unit": " $10^{-10} \\mathrm{~m}$",
+  "source": "quan",
+  "problemid": "6.4 ",
+  "comment": " ",
+  "solution": "\nThe lowest-frequency rotational absorption is the $J=0 \\rightarrow 1$ line. $$h \\nu=E_{\\mathrm{upper}}-E_{\\mathrm{lower}}=\\frac{1(2) \\hbar^2}{2 \\mu d^2}-\\frac{0(1) \\hbar^2}{2 \\mu d^2}\r\n$$\r\nwhich gives $d=\\left(h / 4 \\pi^2 \\nu \\mu\\right)^{1 / 2}$. $$\r\n\\mu=\\frac{m_1 m_2}{m_1+m_2}=\\frac{12(31.97207)}{(12+31.97207)} \\frac{1}{6.02214 \\times 10^{23}} \\mathrm{~g}=1.44885 \\times 10^{-23} \\mathrm{~g}\r\n$$\r\nThe SI unit of mass is the kilogram, and\r\n$$\r\n\\begin{aligned}\r\nd=\\frac{1}{2 \\pi}\\left(\\frac{h}{\\nu_{0 \\rightarrow 1} \\mu}\\right)^{1 / 2} & =\\frac{1}{2 \\pi}\\left[\\frac{6.62607 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}}{\\left(48991.0 \\times 10^6 \\mathrm{~s}^{-1}\\right)\\left(1.44885 \\times 10^{-26} \\mathrm{~kg}\\right)}\\right]^{1 / 2} \\\\\r\n& =1.5377 \\times 10^{-10} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "The strongest infrared band of ${ }^{12} \\mathrm{C}^{16} \\mathrm{O}$ occurs at $\\widetilde{\\nu}=2143 \\mathrm{~cm}^{-1}$. Find the force constant of ${ }^{12} \\mathrm{C}^{16} \\mathrm{O}$. ",
+  "answer_latex": " 1855",
+  "answer_number": "1855",
+  "unit": " $\\mathrm{~N} / \\mathrm{m}$",
+  "source": "quan",
+  "problemid": " 4.3",
+  "comment": " ",
+  "solution": "\nThe strongest infrared band corresponds to the $v=0 \\rightarrow 1$ transition. We approximate the molecular vibration as that of a harmonic oscillator. The equilibrium molecular vibrational frequency is approximately\r\n$$\r\n\\nu_e \\approx \\nu_{\\text {light }}=\\widetilde{\\nu} c=\\left(2143 \\mathrm{~cm}^{-1}\\right)\\left(2.9979 \\times 10^{10} \\mathrm{~cm} / \\mathrm{s}\\right)=6.424 \\times 10^{13} \\mathrm{~s}^{-1}\r\n$$\r\nTo relate $k$ to $\\nu_e$, we need the reduced mass $\\mu=m_1 m_2 /\\left(m_1+m_2\\right)$. One mole of ${ }^{12} \\mathrm{C}$ has a mass of $12 \\mathrm{~g}$ and contains Avogadro's number of atoms. Hence the mass of one atom of ${ }^{12} \\mathrm{C}$ is $(12 \\mathrm{~g}) /\\left(6.02214 \\times 10^{23}\\right)$. The reduced mass and force constant are\r\n$$\r\n\\begin{gathered}\r\n\\mu=\\frac{12(15.9949) \\mathrm{g}}{27.9949} \\frac{1}{6.02214 \\times 10^{23}}=1.1385 \\times 10^{-23} \\mathrm{~g} \\\\\r\nk=4 \\pi^2 \\nu_e^2 \\mu=4 \\pi^2\\left(6.424 \\times 10^{13} \\mathrm{~s}^{-1}\\right)^2\\left(1.1385 \\times 10^{-26} \\mathrm{~kg}\\right)=1855 \\mathrm{~N} / \\mathrm{m}\r\n\\end{gathered}\r\n$$\n"
+ }
+]

stat.json

@@ -1 +1,650 @@
-[{"problem_text": "5.8-5. If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when\r\n(a) $n=100$.\r\n", "answer_latex": " $0.25$", "answer_number": "0.25", "unit": " ", "source": "stat", "problemid": " 5.8-5 (a)", "comment": " "}, {"problem_text": "5.3-13. A device contains three components, each of which has a lifetime in hours with the pdf\r\n$$\r\nf(x)=\\frac{2 x}{10^2} e^{-(x / 10)^2}, \\quad 0 < x < \\infty .\r\n$$\r\nThe device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \\leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).", "answer_latex": " $0.03$", "answer_number": "0.03", "unit": " ", "source": "stat", "problemid": " 5.3-13", "comment": " "}, {"problem_text": "5.6-13. The tensile strength $X$ of paper, in pounds per square inch, has $\\mu=30$ and $\\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\\bar{X}$ is greater than 29.5 pounds per square inch.", "answer_latex": " $0.9522$", "answer_number": "0.9522", "unit": " ", "source": "stat", "problemid": " 5.6-13", "comment": " "}, {"problem_text": "5.6-3. Let $\\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \\leq \\bar{X} \\leq 4)$\r\n", "answer_latex": " $0.8185$", "answer_number": "0.8185", "unit": " ", "source": "stat", "problemid": " 5.6-3", "comment": " "}, {"problem_text": "5.3-9. Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine\r\n(a) $P\\left(\\max X_i < 3 / 4\\right)=P\\left(X_1<3 / 4, X_2<3 / 4\\right)$", "answer_latex": " $\\frac{729}{4096}$", "answer_number": "0.178", "unit": " ", "source": "stat", "problemid": " 5.3-9", "comment": " "}, {"problem_text": "7.4-1. Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \\%$ confident that the maximum error of the estimate of $\\mu$ is 0.4 .\r\n", "answer_latex": " $117$", "answer_number": "117", "unit": " ", "source": "stat", "problemid": " 7.4-1", "comment": " "}, {"problem_text": "5.4-17. In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \\%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \\geq X+2$. HINT: What is the distribution of $Y-X+25$ ?\r\n", "answer_latex": " $0.3359$", "answer_number": "0.3359", "unit": " ", "source": "stat", "problemid": " 5.4-17", "comment": " "}, {"problem_text": "9.6-111. Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\\rho$. To test $H_0: \\rho=0$ against $H_1: \\rho \\neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.", "answer_latex": " $9$", "answer_number": "9", "unit": " ", "source": "stat", "problemid": " 9.6-11", "comment": " "}, {"problem_text": "7.3-5. In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval.\r\n(a) Give a point estimate of $p$.\r\n", "answer_latex": "$0.1800$", "answer_number": "0.1800", "unit": " ", "source": "stat", "problemid": " 7.3-5", "comment": " "}, {"problem_text": "7.4-15. If $\\bar{X}$ and $\\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\\bar{x}-\\bar{y} \\pm 4$ to be a $90 \\%$ confidence interval for $\\mu_X-\\mu_Y$. Assume that the standard deviations are known to be $\\sigma_X=15$ and $\\sigma_Y=25$.", "answer_latex": " $144$", "answer_number": "144", "unit": " ", "source": "stat", "problemid": " 7.4-15", "comment": " "}, {"problem_text": "7.4-7. For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to\r\n(a) 0.03 with $95 \\%$ confidence?", "answer_latex": " $1068$", "answer_number": "1068", "unit": " ", "source": "stat", "problemid": " 7.4-7", "comment": " "}, {"problem_text": "5.5-15. Let the distribution of $T$ be $t(17)$. Find\r\n(a) $t_{0.01}(17)$.\r\n", "answer_latex": "$2.567$", "answer_number": "2.567", "unit": " ", "source": "stat", "problemid": " 5.5-15 (a)", "comment": " "}, {"problem_text": "5.5-1. Let $X_1, X_2, \\ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute\r\n(a) $P(77<\\bar{X}<79.5)$.", "answer_latex": " $0.4772$", "answer_number": "0.4772", "unit": " ", "source": "stat", "problemid": " 5.5-1", "comment": " "}, {"problem_text": "5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?\r\n", "answer_latex": " $0.5768$", "answer_number": "0.5768", "unit": " ", "source": "stat", "problemid": " 5.4-19", "comment": " "}, {"problem_text": "7.3-9. Consider the following two groups of women: Group 1 consists of women who spend less than $\\$ 500$ annually on clothes; Group 2 comprises women who spend over $\\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive,\r\n(a) Give a point estimate of $p_1-p_2$.\r\n", "answer_latex": " $0.2115$", "answer_number": "0.2115", "unit": " ", "source": "stat", "problemid": " 7.3-9", "comment": " "}, {"problem_text": "5.6-9. In Example 5.6-4, compute $P(1.7 \\leq Y \\leq 3.2)$ with $n=4$ and compare your answer with the normal approximation of this probability.\r\n", "answer_latex": " $0.6749$", "answer_number": "0.6749", "unit": " ", "source": "stat", "problemid": "5.6-9", "comment": " "}, {"problem_text": "5.8-5. If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when\r\n(c) $n=1000$.", "answer_latex": " $0.925$", "answer_number": "0.925", "unit": " ", "source": "stat", "problemid": " 5.8-5", "comment": " "}, {"problem_text": "7.5-1. Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\\pi_p$. Compute\r\n(a) $P\\left(Y_2 < \\pi_{0.5} < Y_5\\right)$.", "answer_latex": " $0.7812$", "answer_number": "0.7812", "unit": " ", "source": "stat", "problemid": " 7.5-1", "comment": " "}, {"problem_text": "5.2-13. Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a\r\ndevice that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \\infty $\r\n(a) Find $G\\left(y_1, y_2\\right)=P\\left(Y_1 \\leq y_1, Y_2 \\leq y_2\\right)$.\r\n", "answer_latex": "0.5117 ", "answer_number": "0.5117", "unit": " ", "source": "stat", "problemid": " 5.2-13", "comment": " "}, {"problem_text": "5.4-5. Let $Z_1, Z_2, \\ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$\r\n", "answer_latex": " $0.925$", "answer_number": "0.925", "unit": " ", "source": "stat", "problemid": " 5.4-5", "comment": " "}, {"problem_text": "5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?", "answer_latex": " $5$", "answer_number": "5", "unit": " ", "source": "stat", "problemid": " 5.4-19", "comment": " "}, {"problem_text": "5.3-1. Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\\lambda_1=2$ and $\\lambda_2=3$. Find\r\n(a) $P\\left(X_1=3, X_2=5\\right)$.\r\nHINT. Note that this event can occur if and only if $\\left\\{X_1=1, X_2=0\\right\\}$ or $\\left\\{X_1=0, X_2=1\\right\\}$.", "answer_latex": " 0.0182", "answer_number": "0.0182", "unit": " ", "source": "stat", "problemid": " 5.3-1", "comment": " "}, {"problem_text": "5.9-1. Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3\r\n(a) By using the binomial distribution.\r\n", "answer_latex": " $0.9984$", "answer_number": "0.9984", "unit": " ", "source": "stat", "problemid": "5.9-1 (a) ", "comment": " "}, {"problem_text": "7.4-11. Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking.\r\n(a) How large a sample $n$ of rat embryos is needed for $y / n \\pm 0.10$ to be a $95 \\%$ confidence interval for $p$ ?", "answer_latex": " $38$", "answer_number": "38", "unit": " ", "source": "stat", "problemid": " 7.4-11", "comment": " "}, {"problem_text": "7.1-3. To determine the effect of $100 \\%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \\%$ nitrate every day. At the end of\r\ntwo weeks, the plants were measured. Here are data on seven of them:\r\n$$\r\n\\begin{array}{lllllll}\r\n17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8\r\n\\end{array}\r\n$$\r\nAssume that these data are a random sample from a normal distribution $N\\left(\\mu, \\sigma^2\\right)$.\r\n(a) Find the value of a point estimate of $\\mu$.\r\n", "answer_latex": " $15.757$", "answer_number": "15.757", "unit": " ", "source": "stat", "problemid": " 7.1-3", "comment": " "}, {"problem_text": "5.5-7. Suppose that the distribution of the weight of a prepackaged \"1-pound bag\" of carrots is $N\\left(1.18,0.07^2\\right)$ and the distribution of the weight of a prepackaged \"3-pound bag\" of carrots is $N\\left(3.22,0.09^2\\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag.\r\n", "answer_latex": "$0.9830$ ", "answer_number": "0.9830", "unit": " ", "source": "stat", "problemid": " 5.5-7", "comment": " "}, {"problem_text": "5.3-7. The distributions of incomes in two cities follow the two Pareto-type pdfs\r\n$$\r\nf(x)=\\frac{2}{x^3}, 1 < x < \\infty , \\text { and } g(y)= \\frac{3}{y^4} ,  \\quad 1 < y < \\infty,\r\n$$\r\nrespectively. Here one unit represents $\\$ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.", "answer_latex": " $\\frac{2}{5}$", "answer_number": "0.4", "unit": " ", "source": "stat", "problemid": " 5.3-7", "comment": " "}, {"problem_text": "7.3-3. Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year.\r\n(a) Use these data to give a point estimate of $p$.", "answer_latex": " $0.5061$", "answer_number": "0.5061", "unit": " ", "source": "stat", "problemid": " 7.3-3", "comment": " "}, {"problem_text": "6.I-I. One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:\r\n$\\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\\end{array}$\r\n(a) Find the sample mean, $\\bar{x}$.", "answer_latex": "$1.1$ ", "answer_number": "1.1", "unit": " ", "source": "stat", "problemid": " 6.1-1", "comment": " "}, {"problem_text": "5.5-1. Let $X_1, X_2, \\ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute\r\n(b) $P(74.2<\\bar{X}<78.4)$.", "answer_latex": " $0.8561$", "answer_number": "0.8561", "unit": " ", "source": "stat", "problemid": " 5.5-1 (b)", "comment": " "}, {"problem_text": "5.3-3. Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\\left(x_1\\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \\left(x_2\\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute\r\n(a) $P \\left(0.5 < X_1 < 1\\right.$ and $\\left.0.4 < X_2 < 0.8\\right)$.", "answer_latex": " $\\frac{36}{125}$\r\n", "answer_number": "1.44", "unit": " ", "source": "stat", "problemid": " 5.3-3", "comment": " "}, {"problem_text": "5.8-1. If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find\r\n(a) A lower bound for $P(23 < X < 43)$.", "answer_latex": " $0.84$", "answer_number": "0.84", "unit": " ", "source": "stat", "problemid": " 5.8-1 (a)", "comment": " "}, {"problem_text": "6.3-5. Let $Y_1 < Y_2 < \\cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\\pi_{0.7}=27.3$.\r\n(a) Determine $P\\left(Y_7<27.3\\right)$.\r\n", "answer_latex": " $0.2553$", "answer_number": "0.2553", "unit": " ", "source": "stat", "problemid": " 6.3-5", "comment": " "}, {"problem_text": "6.3-5. Let $Y_1 < Y_2 < \\cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\\pi_{0.7}=27.3$.\r\n(a) Determine $P\\left(Y_7<27.3\\right)$.\r\n", "answer_latex": " $0.2553$", "answer_number": "0.2553", "unit": " ", "source": "stat", "problemid": " 6.3-5", "comment": " "}, {"problem_text": "5.4-21. Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$.\r\n", "answer_latex": " $0.4207$", "answer_number": "0.4207", "unit": " ", "source": "stat", "problemid": " 5.4-21", "comment": " "}, {"problem_text": "7.4-5. A quality engineer wanted to be $98 \\%$ confident that the maximum error of the estimate of the mean strength, $\\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\\bar{x}=35.68$ and a standard deviation of $s=1.723$.\r\n(a) How large a sample is required?", "answer_latex": " $257$", "answer_number": "257", "unit": " ", "source": "stat", "problemid": " 7.4-5", "comment": " "}, {"problem_text": "5.2-5. Let the distribution of $W$ be $F(8,4)$. Find the following:\r\n(a) $F_{0.01}(8,4)$.\r\n", "answer_latex": " 14.80", "answer_number": "14.80", "unit": " ", "source": "stat", "problemid": " 5.2-5", "comment": " "}, {"problem_text": "5.8-5. If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when\r\n(b) $n=500$.", "answer_latex": " $0.85$", "answer_number": "0.85", "unit": " ", "source": "stat", "problemid": " 5.8-5", "comment": " "}, {"problem_text": "5.6-1. Let $\\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \\leq \\bar{X} \\leq 2 / 3)$.\r\n", "answer_latex": "$0.4772$", "answer_number": "0.4772", "unit": " ", "source": "stat", "problemid": " 5.6-1", "comment": " "}, {"problem_text": "5.2-9. Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.", "answer_latex": " 840", "answer_number": "840", "unit": " ", "source": "stat", "problemid": " 5.2-9", "comment": " "}, {"problem_text": "5.3-15. Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ \"triples\" and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$\r\n", "answer_latex": " $0.0384$", "answer_number": "0.0384", "unit": " ", "source": "stat", "problemid": " 5.3-15", "comment": " "}, {"problem_text": "5.2- II. Evaluate\r\n$$\r\n\\int_0^{0.4} \\frac{\\Gamma(7)}{\\Gamma(4) \\Gamma(3)} y^3(1-y)^2 d y\r\n$$\r\n(a) Using integration.", "answer_latex": " 0.1792", "answer_number": "0.1792", "unit": " ", "source": "stat", "problemid": " 5.2-11", "comment": " "}, {"problem_text": "5.6-7. Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\\mu=$ 54.030 and the standard deviation is $\\sigma=5.8$. Let $\\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \\leq \\bar{X} \\leq 54.453)$, approximately.\r\n", "answer_latex": " $0.6247$", "answer_number": "0.6247", "unit": " ", "source": "stat", "problemid": " 5.6-7", "comment": " "}, {"problem_text": "5.3-19. Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$. Compute $E(Z)$.\r\n", "answer_latex": " $5$", "answer_number": "5", "unit": " ", "source": "stat", "problemid": " 5.3-19", "comment": " "}, {"problem_text": "5.8-1. If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find\r\n(b) An upper bound for $P(|X-33| \\geq 14)$.\r\n", "answer_latex": " $0.082$", "answer_number": "0.082", "unit": " ", "source": "stat", "problemid": " 5.8-1", "comment": " "}, {"problem_text": "5.5-9. Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until \"death.\"\r\n(a) Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours.\r\n", "answer_latex": " $0.3085$", "answer_number": "0.3085", "unit": " ", "source": "stat", "problemid": " 5.5-9 (a)", "comment": " "}, {"problem_text": "An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.", "answer_latex": "$\\frac{1}{16}$", "answer_number": "0.0625", "unit": " ", "source": "stat", "problemid": " Problem 1.4.15", "comment": " "}, {"problem_text": " If $P(A)=0.8, P(B)=0.5$, and $P(A \\cup B)=0.9$. What is $P(A \\cap B)$?", "answer_latex": " 0.9", "answer_number": "0.9", "unit": " ", "source": "stat", "problemid": " Problem 1.4.5", "comment": " "}, {"problem_text": "Suppose that the alleles for eye color for a certain male fruit fly are $(R, W)$ and the alleles for eye color for the mating female fruit fly are $(R, W)$, where $R$ and $W$ represent red and white, respectively. Their offspring receive one allele for eye color from each parent. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offspring's eyes look white, what is the conditional probability that it has two white alleles for eye color?", "answer_latex": "$\\frac{1}{3}$", "answer_number": "0.33333333", "unit": " ", "source": "stat", "problemid": "Problem 1.3.5 ", "comment": " "}, {"problem_text": "Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let $A$ be the event that 3 is observed on the first trial. Let $B$ be the event that at least two trials are required to observe a 3 . Assuming that each side has probability $1 / 6$, find (a) $P(A)$.\r\n", "answer_latex": "$\\frac{1}{6}$", "answer_number": "0.166666666", "unit": " ", "source": "stat", "problemid": " Problem 1.1.5", "comment": " "}, {"problem_text": "An urn contains four balls numbered 1 through 4 . The balls are selected one at a time without replacement. A match occurs if the ball numbered $m$ is the $m$ th ball selected. Let the event $A_i$ denote a match on the $i$ th draw, $i=1,2,3,4$. Extend this exercise so that there are $n$ balls in the urn. What is the limit of this probability as $n$ increases without bound?", "answer_latex": " $1 - \\frac{1}{e}$", "answer_number": "0.6321205588", "unit": " ", "source": "stat", "problemid": " Problem 1.3.9", "comment": " "}, {"problem_text": " Of a group of patients having injuries, $28 \\%$ visit both a physical therapist and a chiropractor and $8 \\%$ visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by $16 \\%$. What is the probability of a randomly selected person from this group visiting a physical therapist?\r\n", "answer_latex": " 0.68", "answer_number": "0.68", "unit": " ", "source": "stat", "problemid": " Problem 1.1.1", "comment": " "}, {"problem_text": "A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or irregular and finds that 16\\% have high blood pressure; (b) 19\\% have low blood pressure; (c) $17 \\%$ have an irregular heartbeat; (d) of those with an irregular heartbeat, $35 \\%$ have high blood pressure; and (e) of those with normal blood pressure, $11 \\%$ have an irregular heartbeat. What percentage of her patients have a regular heartbeat and low blood pressure?", "answer_latex": " 15.1", "answer_number": "15.1", "unit": "% ", "source": "stat", "problemid": " 1.5.3", "comment": " "}, {"problem_text": "Roll a fair six-sided die three times. Let $A_1=$ $\\{1$ or 2 on the first roll $\\}, A_2=\\{3$ or 4 on the second roll $\\}$, and $A_3=\\{5$ or 6 on the third roll $\\}$. It is given that $P\\left(A_i\\right)=1 / 3, i=1,2,3 ; P\\left(A_i \\cap A_j\\right)=(1 / 3)^2, i \\neq j$; and $P\\left(A_1 \\cap A_2 \\cap A_3\\right)=(1 / 3)^3$. Use Theorem 1.1-6 to find $P\\left(A_1 \\cup A_2 \\cup A_3\\right)$.", "answer_latex": "$3(\\frac{1}{3})-3(\\frac{1}{3})^2+(\\frac{1}{3})^3$", "answer_number": "0.6296296296", "unit": " ", "source": "stat", "problemid": " Problem 1.1.9", "comment": " "}, {"problem_text": "Let $A$ and $B$ be independent events with $P(A)=$ $1 / 4$ and $P(B)=2 / 3$. Compute $P(A \\cap B)$", "answer_latex": " $\\frac{1}{6}$", "answer_number": "0.166666666", "unit": " ", "source": "stat", "problemid": " Problem 1.4.3", "comment": " "}, {"problem_text": "How many four-letter code words are possible using the letters in IOWA if The letters may not be repeated?", "answer_latex": " 24", "answer_number": "24", "unit": " ", "source": "stat", "problemid": " Problem 1.2.5", "comment": " "}, {"problem_text": "A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, $d_1 d_2 d_3 d_4$, where $d_i, i=1,2,3,4$, is selected from $1,2,3,4,5,6,7$, and 8 . How many different lock combinations are possible with such a lock?", "answer_latex": " 4096", "answer_number": "4096", "unit": " ", "source": "stat", "problemid": " Problem 1.2.1", "comment": " "}, {"problem_text": "An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is $151 / 300$. How many red balls are in the second urn?", "answer_latex": " 11", "answer_number": "11", "unit": " ", "source": "stat", "problemid": " Problem 1.3.15", "comment": " "}, {"problem_text": " A typical roulette wheel used in a casino has 38 slots that are numbered $1,2,3, \\ldots, 36,0,00$, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of $1 / 38$, and we are interested in the number of the slot into which the ball falls. Let $A=\\{0,00\\}$. Give the value of $P(A)$.", "answer_latex": "$\\frac{2}{38}$", "answer_number": "0.0526315789", "unit": "", "source": "stat", "problemid": "Problem 1.1.1 ", "comment": " "}, {"problem_text": "In the gambling game \"craps,\" a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2,3 , or 12 . If the sum is $4,5,6$, 8,9 , or 10 , that number is called the bettor's \"point.\" Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7 , the bettor wins. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11 , $P(7$ or 11$)$.", "answer_latex": " $\\frac{8}{36}$", "answer_number": "0.22222222", "unit": " ", "source": "stat", "problemid": "Problem 1.3.13 ", "comment": " "}, {"problem_text": "Given that $P(A \\cup B)=0.76$ and $P\\left(A \\cup B^{\\prime}\\right)=0.87$, find $P(A)$.", "answer_latex": " 0.63", "answer_number": "0.63", "unit": " ", "source": "stat", "problemid": " Problem 1.1.7", "comment": " "}, {"problem_text": "Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?", "answer_latex": " 0.00539", "answer_number": "0.00539", "unit": " ", "source": "stat", "problemid": " Problem 1.2.13", "comment": " "}, {"problem_text": "How many different license plates are possible if a state uses two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)?", "answer_latex": " 6760000", "answer_number": "6760000", "unit": " ", "source": "stat", "problemid": " Problem 1.2.3", "comment": " "}, {"problem_text": "Let $A$ and $B$ be independent events with $P(A)=$ 0.7 and $P(B)=0.2$. Compute $P(A \\cap B)$.\r\n", "answer_latex": " 0.14", "answer_number": "0.14", "unit": " ", "source": "stat", "problemid": "Problem 1.4.1 ", "comment": " "}, {"problem_text": "Suppose that $A, B$, and $C$ are mutually independent events and that $P(A)=0.5, P(B)=0.8$, and $P(C)=$ 0.9 . Find the probabilities that all three events occur?", "answer_latex": " 0.36", "answer_number": "0.36", "unit": " ", "source": "stat", "problemid": " Problem 1.4.9", "comment": " "}, {"problem_text": "A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of four of a kind (four cards of equal face value and one card of a different value).", "answer_latex": " 0.00024", "answer_number": "0.00024", "unit": " ", "source": "stat", "problemid": " Problem 1.2.17", "comment": " "}, {"problem_text": "Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?", "answer_latex": " 362880", "answer_number": "362880", "unit": " ", "source": "stat", "problemid": "Problem 1.2.11 ", "comment": " "}, {"problem_text": "Each of the 12 students in a class is given a fair 12 -sided die. In addition, each student is numbered from 1 to 12 . If the students roll their dice, what is the probability that there is at least one \"match\" (e.g., student 4 rolls a 4)?", "answer_latex": "$1-(11 / 12)^{12}$", "answer_number": "0.648004372", "unit": " ", "source": "stat", "problemid": " Problem 1.4.17", "comment": " "}, {"problem_text": "The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes four games?", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "stat", "problemid": " Problem 1.2.9", "comment": " "}, {"problem_text": "Draw one card at random from a standard deck of cards. The sample space $S$ is the collection of the 52 cards. Assume that the probability set function assigns $1 / 52$ to each of the 52 outcomes. Let\r\n$$\r\n\\begin{aligned}\r\nA & =\\{x: x \\text { is a jack, queen, or king }\\}, \\\\\r\nB & =\\{x: x \\text { is a } 9,10, \\text { or jack and } x \\text { is red }\\}, \\\\\r\nC & =\\{x: x \\text { is a club }\\}, \\\\\r\nD & =\\{x: x \\text { is a diamond, a heart, or a spade }\\} .\r\n\\end{aligned}\r\n$$\r\nFind $P(A)$", "answer_latex": "$\\frac{12}{52}$", "answer_number": "0.2307692308", "unit": " ", "source": "stat", "problemid": " Problem 1.1.3", "comment": " "}, {"problem_text": "An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?", "answer_latex": "$\\frac{1}{5}$", "answer_number": "0.2", "unit": " ", "source": "stat", "problemid": " Problem 1.3.7", "comment": " "}, {"problem_text": "Bowl $B_1$ contains two white chips, bowl $B_2$ contains two red chips, bowl $B_3$ contains two white and two red chips, and bowl $B_4$ contains three white chips and one red chip. The probabilities of selecting bowl $B_1, B_2, B_3$, or $B_4$ are $1 / 2,1 / 4,1 / 8$, and $1 / 8$, respectively. A bowl is selected using these probabilities and a chip is then drawn at random. Find $P(W)$, the probability of drawing a white chip.", "answer_latex": " $\\frac{21}{32}$", "answer_number": "0.65625", "unit": " ", "source": "stat", "problemid": "Problem 1.5.1 ", "comment": " "}, {"problem_text": "Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.", "answer_latex": " $\\frac{2}{3}$", "answer_number": "0.66666666666", "unit": " ", "source": "stat", "problemid": " Problem 1.1.13", "comment": " "}, {"problem_text": "In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select $6,7,8,9$.", "answer_latex": " 0.0024", "answer_number": " 0.0024", "unit": " ", "source": "stat", "problemid": " Problem 1.2.7", "comment": " "}, {"problem_text": "Extend Example 1.4-6 to an $n$-sided die. That is, suppose that a fair $n$-sided die is rolled $n$ independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1,2, \\ldots, n$. Find the limit of this probability as $n$ increases without bound.", "answer_latex": " $ 1-1 / e$", "answer_number": "0.6321205588", "unit": " ", "source": "stat", "problemid": " Problem 1.4.19", "comment": " "}]
+[
+ {
+  "problem_text": "If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=100$.",
+  "answer_latex": " $0.25$",
+  "answer_number": "0.25",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.8-5 (a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "A device contains three components, each of which has a lifetime in hours with the pdf\r\n$$\r\nf(x)=\\frac{2 x}{10^2} e^{-(x / 10)^2}, \\quad 0 < x < \\infty .\r\n$$\r\nThe device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \\leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).",
+  "answer_latex": " $0.03$",
+  "answer_number": "0.03",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-13",
+  "comment": " "
+ },
+ {
+  "problem_text": "The tensile strength $X$ of paper, in pounds per square inch, has $\\mu=30$ and $\\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\\bar{X}$ is greater than 29.5 pounds per square inch.",
+  "answer_latex": " $0.9522$",
+  "answer_number": "0.9522",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.6-13",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $\\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \\leq \\bar{X} \\leq 4)$",
+  "answer_latex": " $0.8185$",
+  "answer_number": "0.8185",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.6-3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine $P\\left(\\max X_i < 3 / 4\\right)=P\\left(X_1<3 / 4, X_2<3 / 4\\right)$",
+  "answer_latex": " $\\frac{729}{4096}$",
+  "answer_number": "0.178",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-9",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \\%$ confident that the maximum error of the estimate of $\\mu$ is 0.4 .",
+  "answer_latex": " $117$",
+  "answer_number": "117",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.4-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \\%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \\geq X+2$. HINT: What is the distribution of $Y-X+25$ ?",
+  "answer_latex": " $0.3359$",
+  "answer_number": "0.3359",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.4-17",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\\rho$. To test $H_0: \\rho=0$ against $H_1: \\rho \\neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.",
+  "answer_latex": " $9$",
+  "answer_number": "9",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 9.6-11",
+  "comment": " "
+ },
+ {
+  "problem_text": "In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval. Give a point estimate of $p$.",
+  "answer_latex": "$0.1800$",
+  "answer_number": "0.1800",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.3-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "If $\\bar{X}$ and $\\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\\bar{x}-\\bar{y} \\pm 4$ to be a $90 \\%$ confidence interval for $\\mu_X-\\mu_Y$. Assume that the standard deviations are known to be $\\sigma_X=15$ and $\\sigma_Y=25$.",
+  "answer_latex": " $144$",
+  "answer_number": "144",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.4-15",
+  "comment": " "
+ },
+ {
+  "problem_text": "For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to 0.03 with $95 \\%$ confidence?",
+  "answer_latex": " $1068$",
+  "answer_number": "1068",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.4-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let the distribution of $T$ be $t(17)$. Find $t_{0.01}(17)$.",
+  "answer_latex": "$2.567$",
+  "answer_number": "2.567",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.5-15 (a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X_1, X_2, \\ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(77<\\bar{X}<79.5)$.",
+  "answer_latex": " $0.4772$",
+  "answer_number": "0.4772",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.5-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?\r\n",
+  "answer_latex": " $0.5768$",
+  "answer_number": "0.5768",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.4-19",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the following two groups of women: Group 1 consists of women who spend less than $\\$ 500$ annually on clothes; Group 2 comprises women who spend over $\\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive, Give a point estimate of $p_1-p_2$.",
+  "answer_latex": " $0.2115$",
+  "answer_number": "0.2115",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.3-9",
+  "comment": " "
+ },
+ {
+  "problem_text": "Given below example: Approximate $P(39.75 \\leq \\bar{X} \\leq 41.25)$, where $\\bar{X}$ is the mean of a random sample of size 32 from a distribution with mean $\\mu=40$ and variance $\\sigma^2=8$. In the above example, compute $P(1.7 \\leq Y \\leq 3.2)$ with $n=4$",
+  "answer_latex": " $0.6749$",
+  "answer_number": "0.6749",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "5.6-9",
+  "comment": " "
+ },
+ {
+  "problem_text": "If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$.",
+  "answer_latex": " $0.925$",
+  "answer_number": "0.925",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.8-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\\pi_p$. Compute $P\\left(Y_2 < \\pi_{0.5} < Y_5\\right)$.",
+  "answer_latex": " $0.7812$",
+  "answer_number": "0.7812",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.5-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a\r\ndevice that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \\infty $ Find $G\\left(y_1, y_2\\right)=P\\left(Y_1 \\leq y_1, Y_2 \\leq y_2\\right)$.",
+  "answer_latex": "0.5117 ",
+  "answer_number": "0.5117",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.2-13",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $Z_1, Z_2, \\ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$",
+  "answer_latex": " $0.925$",
+  "answer_number": "0.925",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.4-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\\lambda_1=2$ and $\\lambda_2=3$. Find $P\\left(X_1=3, X_2=5\\right)$. HINT. Note that this event can occur if and only if $\\left\\{X_1=1, X_2=0\\right\\}$ or $\\left\\{X_1=0, X_2=1\\right\\}$.",
+  "answer_latex": " 0.0182",
+  "answer_number": "0.0182",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3 By using the binomial distribution.",
+  "answer_latex": " $0.9984$",
+  "answer_number": "0.9984",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "5.9-1 (a) ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking. How large a sample $n$ of rat embryos is needed for $y / n \\pm 0.10$ to be a $95 \\%$ confidence interval for $p$ ?",
+  "answer_latex": " $38$",
+  "answer_number": "38",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.4-11",
+  "comment": " "
+ },
+ {
+  "problem_text": "To determine the effect of $100 \\%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \\%$ nitrate every day. At the end of\r\ntwo weeks, the plants were measured. Here are data on seven of them:\r\n$$\r\n\\begin{array}{lllllll}\r\n17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8\r\n\\end{array}\r\n$$\r\nAssume that these data are a random sample from a normal distribution $N\\left(\\mu, \\sigma^2\\right)$. Find the value of a point estimate of $\\mu$.",
+  "answer_latex": " $15.757$",
+  "answer_number": "15.757",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.1-3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that the distribution of the weight of a prepackaged '1-pound bag' of carrots is $N\\left(1.18,0.07^2\\right)$ and the distribution of the weight of a prepackaged '3-pound bag' of carrots is $N\\left(3.22,0.09^2\\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag.",
+  "answer_latex": "$0.9830$ ",
+  "answer_number": "0.9830",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.5-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "The distributions of incomes in two cities follow the two Pareto-type pdfs $$ f(x)=\\frac{2}{x^3}, 1 < x < \\infty , \\text { and } g(y)= \\frac{3}{y^4} ,  \\quad 1 < y < \\infty,$$ respectively. Here one unit represents $ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.",
+  "answer_latex": " $\\frac{2}{5}$",
+  "answer_number": "0.4",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year. Use these data to give a point estimate of $p$.",
+  "answer_latex": " $0.5061$",
+  "answer_number": "0.5061",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.3-3",
+  "comment": " "
+ },
+ {
+  "problem_text": "One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:\r\n$\\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\\end{array}$ Find the sample mean, $\\bar{x}$.",
+  "answer_latex": "$1.1$ ",
+  "answer_number": "1.1",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 6.1-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X_1, X_2, \\ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(74.2<\\bar{X}<78.4)$.",
+  "answer_latex": " $0.8561$",
+  "answer_number": "0.8561",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.5-1 (b)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\\left(x_1\\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \\left(x_2\\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute $P \\left(0.5 < X_1 < 1\\right.$ and $\\left.0.4 < X_2 < 0.8\\right)$.",
+  "answer_latex": " $\\frac{36}{125}$\r\n",
+  "answer_number": "1.44",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-3",
+  "comment": " "
+ },
+ {
+  "problem_text": "If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find A lower bound for $P(23 < X < 43)$.",
+  "answer_latex": " $0.84$",
+  "answer_number": "0.84",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.8-1 (a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $Y_1 < Y_2 < \\cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\\pi_{0.7}=27.3$. Determine $P\\left(Y_7<27.3\\right)$.",
+  "answer_latex": " $0.2553$",
+  "answer_number": "0.2553",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 6.3-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$.",
+  "answer_latex": " $0.4207$",
+  "answer_number": "0.4207",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.4-21",
+  "comment": " "
+ },
+ {
+  "problem_text": "A quality engineer wanted to be $98 \\%$ confident that the maximum error of the estimate of the mean strength, $\\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\\bar{x}=35.68$ and a standard deviation of $s=1.723$. How large a sample is required?",
+  "answer_latex": " $257$",
+  "answer_number": "257",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 7.4-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let the distribution of $W$ be $F(8,4)$. Find the following: $F_{0.01}(8,4)$.",
+  "answer_latex": " 14.80",
+  "answer_number": "14.80",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.2-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=500$.",
+  "answer_latex": " $0.85$",
+  "answer_number": "0.85",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.8-5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $\\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \\leq \\bar{X} \\leq 2 / 3)$.",
+  "answer_latex": "$0.4772$",
+  "answer_number": "0.4772",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.6-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.",
+  "answer_latex": " 840",
+  "answer_number": "840",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.2-9",
+  "comment": " "
+ },
+ {
+  "problem_text": "Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ 'triples' and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$ ",
+  "answer_latex": " $0.0384$",
+  "answer_number": "0.0384",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-15",
+  "comment": " "
+ },
+ {
+  "problem_text": "Evaluate\r\n$$\r\n\\int_0^{0.4} \\frac{\\Gamma(7)}{\\Gamma(4) \\Gamma(3)} y^3(1-y)^2 d y\r\n$$ Using integration.",
+  "answer_latex": " 0.1792",
+  "answer_number": "0.1792",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.2-11",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\\mu=$ 54.030 and the standard deviation is $\\sigma=5.8$. Let $\\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \\leq \\bar{X} \\leq 54.453)$, approximately.",
+  "answer_latex": " $0.6247$",
+  "answer_number": "0.6247",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.6-7",
+  "comment": " "
+ },
+ {
+  "problem_text": "Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$. Compute $E(Z)$.",
+  "answer_latex": " $5$",
+  "answer_number": "5",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.3-19",
+  "comment": " "
+ },
+ {
+  "problem_text": "If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find An upper bound for $P(|X-33| \\geq 14)$.",
+  "answer_latex": " $0.082$",
+  "answer_number": "0.082",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.8-1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until 'death.' Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours.",
+  "answer_latex": " $0.3085$",
+  "answer_number": "0.3085",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " 5.5-9 (a)",
+  "comment": " "
+ },
+ {
+  "problem_text": "An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.",
+  "answer_latex": "$\\frac{1}{16}$",
+  "answer_number": "0.0625",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.4.15",
+  "comment": " "
+ },
+ {
+  "problem_text": " If $P(A)=0.8, P(B)=0.5$, and $P(A \\cup B)=0.9$. What is $P(A \\cap B)$?",
+  "answer_latex": " 0.9",
+  "answer_number": "0.9",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.4.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that the alleles for eye color for a certain male fruit fly are $(R, W)$ and the alleles for eye color for the mating female fruit fly are $(R, W)$, where $R$ and $W$ represent red and white, respectively. Their offspring receive one allele for eye color from each parent. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offspring's eyes look white, what is the conditional probability that it has two white alleles for eye color?",
+  "answer_latex": "$\\frac{1}{3}$",
+  "answer_number": "0.33333333",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Problem 1.3.5 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let $A$ be the event that 3 is observed on the first trial. Let $B$ be the event that at least two trials are required to observe a 3 . Assuming that each side has probability $1 / 6$, find $P(A)$.",
+  "answer_latex": "$\\frac{1}{6}$",
+  "answer_number": "0.166666666",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.1.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "An urn contains four balls numbered 1 through 4 . The balls are selected one at a time without replacement. A match occurs if the ball numbered $m$ is the $m$ th ball selected. Let the event $A_i$ denote a match on the $i$ th draw, $i=1,2,3,4$. Extend this exercise so that there are $n$ balls in the urn. What is the limit of this probability as $n$ increases without bound?",
+  "answer_latex": " $1 - \\frac{1}{e}$",
+  "answer_number": "0.6321205588",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.3.9",
+  "comment": " "
+ },
+ {
+  "problem_text": " Of a group of patients having injuries, $28 \\%$ visit both a physical therapist and a chiropractor and $8 \\%$ visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by $16 \\%$. What is the probability of a randomly selected person from this group visiting a physical therapist?\r\n",
+  "answer_latex": " 0.68",
+  "answer_number": "0.68",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.1.1",
+  "comment": " "
+ },
+ {
+  "problem_text": "A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or irregular and finds that 16\\% have high blood pressure; (b) 19\\% have low blood pressure; (c) $17 \\%$ have an irregular heartbeat; (d) of those with an irregular heartbeat, $35 \\%$ have high blood pressure; and (e) of those with normal blood pressure, $11 \\%$ have an irregular heartbeat. What percentage of her patients have a regular heartbeat and low blood pressure?",
+  "answer_latex": " 15.1",
+  "answer_number": "15.1",
+  "unit": "% ",
+  "source": "stat",
+  "problemid": " 1.5.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Roll a fair six-sided die three times. Let $A_1=$ $\\{1$ or 2 on the first roll $\\}, A_2=\\{3$ or 4 on the second roll $\\}$, and $A_3=\\{5$ or 6 on the third roll $\\}$. It is given that $P\\left(A_i\\right)=1 / 3, i=1,2,3 ; P\\left(A_i \\cap A_j\\right)=(1 / 3)^2, i \\neq j$; and $P\\left(A_1 \\cap A_2 \\cap A_3\\right)=(1 / 3)^3$. Use Theorem 1.1-6 to find $P\\left(A_1 \\cup A_2 \\cup A_3\\right)$.",
+  "answer_latex": "$3(\\frac{1}{3})-3(\\frac{1}{3})^2+(\\frac{1}{3})^3$",
+  "answer_number": "0.6296296296",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.1.9",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $A$ and $B$ be independent events with $P(A)=$ $1 / 4$ and $P(B)=2 / 3$. Compute $P(A \\cap B)$",
+  "answer_latex": " $\\frac{1}{6}$",
+  "answer_number": "0.166666666",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.4.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "How many four-letter code words are possible using the letters in IOWA if the letters may not be repeated?",
+  "answer_latex": " 24",
+  "answer_number": "24",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.2.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, $d_1 d_2 d_3 d_4$, where $d_i, i=1,2,3,4$, is selected from $1,2,3,4,5,6,7$, and 8 . How many different lock combinations are possible with such a lock?",
+  "answer_latex": " 4096",
+  "answer_number": "4096",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.2.1",
+  "comment": " "
+ },
+ {
+  "problem_text": "An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is $151 / 300$. How many red balls are in the second urn?",
+  "answer_latex": " 11",
+  "answer_number": "11",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.3.15",
+  "comment": " "
+ },
+ {
+  "problem_text": " A typical roulette wheel used in a casino has 38 slots that are numbered $1,2,3, \\ldots, 36,0,00$, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of $1 / 38$, and we are interested in the number of the slot into which the ball falls. Let $A=\\{0,00\\}$. Give the value of $P(A)$.",
+  "answer_latex": "$\\frac{2}{38}$",
+  "answer_number": "0.0526315789",
+  "unit": "",
+  "source": "stat",
+  "problemid": "Problem 1.1.1 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "In the gambling game \"craps,\" a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2,3 , or 12 . If the sum is $4,5,6$, 8,9 , or 10 , that number is called the bettor's \"point.\" Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7 , the bettor wins. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11 , $P(7$ or 11$)$.",
+  "answer_latex": " $\\frac{8}{36}$",
+  "answer_number": "0.22222222",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Problem 1.3.13 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Given that $P(A \\cup B)=0.76$ and $P\\left(A \\cup B^{\\prime}\\right)=0.87$, find $P(A)$.",
+  "answer_latex": " 0.63",
+  "answer_number": "0.63",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.1.7",
+  "comment": " "
+ },
+ {
+  "problem_text": "How many different license plates are possible if a state uses two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)?",
+  "answer_latex": " 6760000",
+  "answer_number": "6760000",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.2.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "Let $A$ and $B$ be independent events with $P(A)=$ 0.7 and $P(B)=0.2$. Compute $P(A \\cap B)$.\r\n",
+  "answer_latex": " 0.14",
+  "answer_number": "0.14",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Problem 1.4.1 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that $A, B$, and $C$ are mutually independent events and that $P(A)=0.5, P(B)=0.8$, and $P(C)=$ 0.9 . Find the probabilities that all three events occur?",
+  "answer_latex": " 0.36",
+  "answer_number": "0.36",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.4.9",
+  "comment": " "
+ },
+ {
+  "problem_text": "A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of four of a kind (four cards of equal face value and one card of a different value).",
+  "answer_latex": " 0.00024",
+  "answer_number": "0.00024",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.2.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?",
+  "answer_latex": " 362880",
+  "answer_number": "362880",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Problem 1.2.11 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Each of the 12 students in a class is given a fair 12 -sided die. In addition, each student is numbered from 1 to 12 . If the students roll their dice, what is the probability that there is at least one \"match\" (e.g., student 4 rolls a 4)?",
+  "answer_latex": "$1-(11 / 12)^{12}$",
+  "answer_number": "0.648004372",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.4.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes four games?",
+  "answer_latex": " 2",
+  "answer_number": "2",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.2.9",
+  "comment": " "
+ },
+ {
+  "problem_text": "Draw one card at random from a standard deck of cards. The sample space $S$ is the collection of the 52 cards. Assume that the probability set function assigns $1 / 52$ to each of the 52 outcomes. Let\r\n$$\r\n\\begin{aligned}\r\nA & =\\{x: x \\text { is a jack, queen, or king }\\}, \\\\\r\nB & =\\{x: x \\text { is a } 9,10, \\text { or jack and } x \\text { is red }\\}, \\\\\r\nC & =\\{x: x \\text { is a club }\\}, \\\\\r\nD & =\\{x: x \\text { is a diamond, a heart, or a spade }\\} .\r\n\\end{aligned}\r\n$$\r\nFind $P(A)$",
+  "answer_latex": "$\\frac{12}{52}$",
+  "answer_number": "0.2307692308",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.1.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?",
+  "answer_latex": "$\\frac{1}{5}$",
+  "answer_number": "0.2",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.3.7",
+  "comment": " "
+ },
+ {
+  "problem_text": "Bowl $B_1$ contains two white chips, bowl $B_2$ contains two red chips, bowl $B_3$ contains two white and two red chips, and bowl $B_4$ contains three white chips and one red chip. The probabilities of selecting bowl $B_1, B_2, B_3$, or $B_4$ are $1 / 2,1 / 4,1 / 8$, and $1 / 8$, respectively. A bowl is selected using these probabilities and a chip is then drawn at random. Find $P(W)$, the probability of drawing a white chip.",
+  "answer_latex": " $\\frac{21}{32}$",
+  "answer_number": "0.65625",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Problem 1.5.1 ",
+  "comment": " "
+ },
+ {
+  "problem_text": "Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.",
+  "answer_latex": " $\\frac{2}{3}$",
+  "answer_number": "0.66666666666",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.1.13",
+  "comment": " "
+ },
+ {
+  "problem_text": "In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select $6,7,8,9$.",
+  "answer_latex": " 0.0024",
+  "answer_number": " 0.0024",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.2.7",
+  "comment": " "
+ },
+ {
+  "problem_text": "Suppose that a fair $n$-sided die is rolled $n$ independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1,2, \\ldots, n$. Find the limit of this probability as $n$ increases without bound.",
+  "answer_latex": " $ 1-1 / e$",
+  "answer_number": "0.6321205588",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Problem 1.4.19",
+  "comment": " "
+ }
+]

stat_sol.json

@@ -1 +1,202 @@
-[{"problem_text": "An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let $A_1$ be those people with an auto policy only, $A_2$ those people with a homeowner policy only, and $A_3$ those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that $P\\left(A_1\\right)=0.3, P\\left(A_2\\right)=0.2$, and $P\\left(A_3\\right)=0.2$. Further, let $B$ be the event that the person will renew at least one of these policies. Say from past experience that we assign the conditional probabilities $P\\left(B \\mid A_1\\right)=0.6, P\\left(B \\mid A_2\\right)=0.7$, and $P\\left(B \\mid A_3\\right)=0.8$. Given that the person selected at random has an auto or homeowner policy, what is the conditional probability that the person will renew at least one of those policies?", "answer_latex": " 0.686", "answer_number": "0.686", "unit": " ", "source": "stat", "problemid": "Example 1.3.11 ", "comment": " ", "solution": "The desired probability is\r\n$$\r\n\\begin{aligned}\r\nP\\left(B \\mid A_1 \\cup A_2 \\cup A_3\\right) & =\\frac{P\\left(A_1 \\cap B\\right)+P\\left(A_2 \\cap B\\right)+P\\left(A_3 \\cap B\\right)}{P\\left(A_1\\right)+P\\left(A_2\\right)+P\\left(A_3\\right)} \\\\\r\n& =\\frac{(0.3)(0.6)+(0.2)(0.7)+(0.2)(0.8)}{0.3+0.2+0.2} \\\\\r\n& =\\frac{0.48}{0.70}=0.686 .\r\n\\end{aligned}\r\n$$"}, {"problem_text": "What is the number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards?", "answer_latex": " 635013559600", "answer_number": "635013559600", "unit": " ", "source": "stat", "problemid": " Example 1.2.10", "comment": " ", "solution": "The number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards is\r\n$$\r\n{ }_{52} C_{13}=\\left(\\begin{array}{l}\r\n52 \\\\\r\n13\r\n\\end{array}\\right)=\\frac{52 !}{13 ! 39 !}=635,013,559,600 .\r\n$$\r\n"}, {"problem_text": "What is the number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons?", "answer_latex": " 5040", "answer_number": "5040", "unit": " ", "source": "stat", "problemid": "Example 1.2.5 ", "comment": " ", "solution": "The number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons is\r\n$$\r\n{ }_{10} P_4=10 \\cdot 9 \\cdot 8 \\cdot 7=\\frac{10 !}{6 !}=5040 .\r\n$$"}, {"problem_text": "At a county fair carnival game there are 25 balloons on a board, of which 10 balloons 1.3-5 are yellow, 8 are red, and 7 are green. A player throws darts at the balloons to win a prize and randomly hits one of them. Given that the first balloon hit is yellow, what is the probability that the next balloon hit is also yellow?", "answer_latex": "$\\frac{9}{24}$", "answer_number": "0.375", "unit": " ", "source": "stat", "problemid": " Example 1.3.5", "comment": " ", "solution": "Of the 24 remaining balloons, 9 are yellow, so a natural value to assign to this conditional probability is $9 / 24$."}, {"problem_text": "What is the number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards?", "answer_latex": " 311875200", "answer_number": "311875200", "unit": " ", "source": "stat", "problemid": " Example 1.2.8", "comment": " ", "solution": "\r\nThe number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards is\r\n$$\r\n(52)(51)(50)(49)(48)=\\frac{52 !}{47 !}=311,875,200 .\r\n$$\r\n"}, {"problem_text": "A bowl contains seven blue chips and three red chips. Two chips are to be drawn successively at random and without replacement. We want to compute the probability that the first draw results in a red chip $(A)$ and the second draw results in a blue chip $(B)$. ", "answer_latex": "$\\frac{7}{30}$", "answer_number": "0.23333333333", "unit": " ", "source": "stat", "problemid": "Example 1.3.6 ", "comment": " ", "solution": "It is reasonable to assign the following probabilities:\r\n$$\r\nP(A)=\\frac{3}{10} \\text { and } P(B \\mid A)=\\frac{7}{9} \\text {. }\r\n$$\r\nThe probability of obtaining red on the first draw and blue on the second draw is\r\n$$\r\nP(A \\cap B)=\\frac{3}{10} \\cdot \\frac{7}{9}=\\frac{7}{30}\r\n$$"}, {"problem_text": "From an ordinary deck of playing cards, cards are to be drawn successively at random and without replacement. What is the probability that the third spade appears on the sixth draw?", "answer_latex": " 0.064", "answer_number": "0.064", "unit": " ", "source": "stat", "problemid": "Example 1.3.7 ", "comment": " ", "solution": "Let $A$ be the event of two spades in the first five cards drawn, and let $B$ be the event of a spade on the sixth draw. Thus, the probability that we wish to compute is $P(A \\cap B)$. It is reasonable to take\r\n$$\r\nP(A)=\\frac{\\left(\\begin{array}{c}\r\n13 \\\\\r\n2\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n39 \\\\\r\n3\r\n\\end{array}\\right)}{\\left(\\begin{array}{c}\r\n52 \\\\\r\n5\r\n\\end{array}\\right)}=0.274 \\quad \\text { and } \\quad P(B \\mid A)=\\frac{11}{47}=0.234\r\n$$\r\nThe desired probability, $P(A \\cap B)$, is the product of those numbers:\r\n$$\r\nP(A \\cap B)=(0.274)(0.234)=0.064\r\n$$\r\n"}, {"problem_text": "What is the probability of drawing three kings and two queens when drawing a five-card hand from a deck of 52 playing cards?", "answer_latex": " 0.0000092", "answer_number": "0.0000092", "unit": " ", "source": "stat", "problemid": " Example 1.2.11", "comment": " ", "solution": "Assume that each of the $\\left(\\begin{array}{c}52 \\\\ 5\\end{array}\\right)=2,598,960$ five-card hands drawn from a deck of 52 playing cards has the same probability of being selected. \r\nSuppose now that the event $B$ is the set of outcomes in which exactly three cards are kings and exactly two cards are queens. We can select the three kings in any one of $\\left(\\begin{array}{l}4 \\\\ 3\\end{array}\\right)$ ways and the two queens in any one of $\\left(\\begin{array}{l}4 \\\\ 2\\end{array}\\right)$ ways. By the multiplication principle, the number of outcomes in $B$ is\r\n$$\r\nN(B)=\\left(\\begin{array}{l}\r\n4 \\\\\r\n3\r\n\\end{array}\\right)\\left(\\begin{array}{l}\r\n4 \\\\\r\n2\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n44 \\\\\r\n0\r\n\\end{array}\\right)\r\n$$\r\nwhere $\\left(\\begin{array}{c}44 \\\\ 0\\end{array}\\right)$ gives the number of ways in which 0 cards are selected out of the nonkings and nonqueens and of course is equal to 1 . Thus,\r\n$$\r\nP(B)=\\frac{N(B)}{N(S)}=\\frac{\\left(\\begin{array}{l}\r\n4 \\\\\r\n3\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n4 \\\\\r\n2\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n44 \\\\\r\n0\r\n\\end{array}\\right)}{\\left(\\begin{array}{c}\r\n52 \\\\\r\n5\r\n\\end{array}\\right)}=\\frac{24}{2,598,960}=0.0000092 .\r\n$$"}, {"problem_text": "In an orchid show, seven orchids are to be placed along one side of the greenhouse. There are four lavender orchids and three white orchids. How many ways are there to lineup these orchids?", "answer_latex": " 35", "answer_number": "35", "unit": " ", "source": "stat", "problemid": "Example 1.2.13", "comment": " ", "solution": "Considering only the color of the orchids, we see that the number of lineups of the orchids is\r\n$$\r\n\\left(\\begin{array}{l}\r\n7 \\\\\r\n4\r\n\\end{array}\\right)=\\frac{7 !}{4 ! 3 !}=35 \\text {. }\r\n$$"}, {"problem_text": "If $P(A)=0.4, P(B)=0.5$, and $P(A \\cap B)=0.3$, find $P(B \\mid A)$.", "answer_latex": " 0.75", "answer_number": "0.75", "unit": " ", "source": "stat", "problemid": "Example 1.3.2 ", "comment": " ", "solution": "$P(B \\mid A)=P(A \\cap B) / P(A)=0.3 / 0.4=0.75$."}, {"problem_text": "What is the number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards?", "answer_latex": " 2598960", "answer_number": "2598960", "unit": " ", "source": "stat", "problemid": "Example 1.2.9 ", "comment": " ", "solution": "The number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards is\r\n$$\r\n{ }_{52} C_5=\\left(\\begin{array}{c}\r\n52 \\\\\r\n5\r\n\\end{array}\\right)=\\frac{52 !}{5 ! 47 !}=2,598,960\r\n$$\r\n"}, {"problem_text": "A certain food service gives the following choices for dinner: $E_1$, soup or tomato 1.2-2 juice; $E_2$, steak or shrimp; $E_3$, French fried potatoes, mashed potatoes, or a baked potato; $E_4$, corn or peas; $E_5$, jello, tossed salad, cottage cheese, or coleslaw; $E_6$, cake, cookies, pudding, brownie, vanilla ice cream, chocolate ice cream, or orange sherbet; $E_7$, coffee, tea, milk, or punch. How many different dinner selections are possible if one of the listed choices is made for each of $E_1, E_2, \\ldots$, and $E_7$ ?", "answer_latex": " 2688", "answer_number": "2688", "unit": " ", "source": "stat", "problemid": "Example 1.2.2", "comment": " ", "solution": " By the multiplication principle, there are\r\n$(2)(2)(3)(2)(4)(7)(4)=2688$\r\ndifferent combinations.\r\n"}, {"problem_text": "A rocket has a built-in redundant system. In this system, if component $K_1$ fails, it is bypassed and component $K_2$ is used. If component $K_2$ fails, it is bypassed and component $K_3$ is used. (An example of a system with these kinds of components is three computer systems.) Suppose that the probability of failure of any one component is 0.15 , and assume that the failures of these components are mutually independent events. Let $A_i$ denote the event that component $K_i$ fails for $i=1,2,3$. What is the probability that the system fails?", "answer_latex": " 0.9966", "answer_number": "0.9966", "unit": " ", "source": "stat", "problemid": "Example 1.4.5 ", "comment": " ", "solution": "\r\nBecause the system fails if $K_1$ fails and $K_2$ fails and $K_3$ fails, the probability that the system does not fail is given by\r\n$$\r\n\\begin{aligned}\r\nP\\left[\\left(A_1 \\cap A_2 \\cap A_3\\right)^{\\prime}\\right] & =1-P\\left(A_1 \\cap A_2 \\cap A_3\\right) \\\\\r\n& =1-P\\left(A_1\\right) P\\left(A_2\\right) P\\left(A_3\\right) \\\\\r\n& =1-(0.15)^3 \\\\\r\n& =0.9966 .\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Suppose that $P(A)=0.7, P(B)=0.3$, and $P(A \\cap B)=0.2$. These probabilities are 1.3-3 listed on the Venn diagram in Figure 1.3-1. Given that the outcome of the experiment belongs to $B$, what then is the probability of $A$ ? ", "answer_latex": " \\frac{2}{3}", "answer_number": "0.66666666666", "unit": " ", "source": "stat", "problemid": "Example 1.3.3 ", "comment": " ", "solution": "We are effectively restricting the sample space to $B$; of the probability $P(B)=0.3,0.2$ corresponds to $P(A \\cap B)$ and hence to $A$. That is, $0.2 / 0.3=2 / 3$ of the probability of $B$ corresponds to $A$. Of course, by the formal definition, we also obtain\r\n$$\r\nP(A \\mid B)=\\frac{P(A \\cap B)}{P(B)}=\\frac{0.2}{0.3}=\\frac{2}{3}\r\n$$"}, {"problem_text": "A coin is flipped 10 times and the sequence of heads and tails is observed. What is the number of possible 10-tuplets that result in four heads and six tails?", "answer_latex": " 210", "answer_number": "210", "unit": " ", "source": "stat", "problemid": "Example 1.2.12 ", "comment": " ", "solution": "A coin is flipped 10 times and the sequence of heads and tails is observed. The number of possible 10-tuplets that result in four heads and six tails is\r\n$$\r\n\\left(\\begin{array}{c}\r\n10 \\\\\r\n4\r\n\\end{array}\\right)=\\frac{10 !}{4 ! 6 !}=\\frac{10 !}{6 ! 4 !}=\\left(\\begin{array}{c}\r\n10 \\\\\r\n6\r\n\\end{array}\\right)=210 .\r\n$$\r\n"}, {"problem_text": "Among nine orchids for a line of orchids along one wall, three are white, four lavender, and two yellow. How many color displays are there?", "answer_latex": " 1260", "answer_number": "1260", "unit": " ", "source": "stat", "problemid": "Example 1.2.14 ", "comment": " ", "solution": "The number of different color displays is\r\n$$\r\n\\left(\\begin{array}{c}\r\n9 \\\\\r\n3,4,2\r\n\\end{array}\\right)=\\frac{9 !}{3 ! 4 ! 2 !}=1260\r\n$$\r\n"}, {"problem_text": "A survey was taken of a group's viewing habits of sporting events on TV during I.I-5 the last year. Let $A=\\{$ watched football $\\}, B=\\{$ watched basketball $\\}, C=\\{$ watched baseball $\\}$. The results indicate that if a person is selected at random from the surveyed group, then $P(A)=0.43, P(B)=0.40, P(C)=0.32, P(A \\cap B)=0.29$, $P(A \\cap C)=0.22, P(B \\cap C)=0.20$, and $P(A \\cap B \\cap C)=0.15$. Find $P(A \\cup B \\cup C)$.", "answer_latex": " 0.59", "answer_number": "0.59", "unit": " ", "source": "stat", "problemid": " Example 1.1.5", "comment": " ", "solution": "$$\r\n\\begin{aligned}\r\nP(A \\cup B \\cup C)= & P(A)+P(B)+P(C)-P(A \\cap B)-P(A \\cap C) \\\\\r\n& -P(B \\cap C)+P(A \\cap B \\cap C) \\\\\r\n= & 0.43+0.40+0.32-0.29-0.22-0.20+0.15 \\\\\r\n= & 0.59\r\n\\end{aligned}\r\n$$"}, {"problem_text": "A grade school boy has five blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of his then drawing a blue marble from his right pocket?", "answer_latex": " $\\frac{41}{90}$", "answer_number": "0.444444444444444 ", "unit": "", "source": "stat", "problemid": " Example 1.3.10", "comment": " ", "solution": "For notation, let $B L, B R$, and $W L$ denote drawing blue from left pocket, blue from right pocket, and white from left pocket, respectively. Then\r\n$$\r\n\\begin{aligned}\r\nP(B R) & =P(B L \\cap B R)+P(W L \\cap B R) \\\\\r\n& =P(B L) P(B R \\mid B L)+P(W L) P(B R \\mid W L) \\\\\r\n& =\\frac{5}{9} \\cdot \\frac{5}{10}+\\frac{4}{9} \\cdot \\frac{4}{10}=\\frac{41}{90}\r\n\\end{aligned}\r\n$$\r\nis the desired probability."}, {"problem_text": "A faculty leader was meeting two students in Paris, one arriving by train from Amsterdam and the other arriving by train from Brussels at approximately the same time. Let $A$ and $B$ be the events that the respective trains are on time. Suppose we know from past experience that $P(A)=0.93, P(B)=0.89$, and $P(A \\cap B)=0.87$. Find $P(A \\cup B)$.", "answer_latex": " 0.95", "answer_number": "0.95", "unit": " ", "source": "stat", "problemid": "Example 1.1.4 ", "comment": " ", "solution": "$$P(A \\cup B) =P(A)+P(B)-P(A \\cap B)=0.93+0.89-0.87=0.95$$"}, {"problem_text": "What is the number of possible four-letter code words, selecting from the 26 letters in the alphabet?", "answer_latex": " 358800", "answer_number": "358800", "unit": " ", "source": "stat", "problemid": "Example 1.2.4 ", "comment": " ", "solution": "The number of possible four-letter code words, selecting from the 26 letters in the  alphabet, in which all four letters are different is\r\n$$\r\n{ }_{26} P_4=(26)(25)(24)(23)=\\frac{26 !}{22 !}=358,800 .\r\n$$"}]
+[
+ {
+  "problem_text": "An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let $A_1$ be those people with an auto policy only, $A_2$ those people with a homeowner policy only, and $A_3$ those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that $P\\left(A_1\\right)=0.3, P\\left(A_2\\right)=0.2$, and $P\\left(A_3\\right)=0.2$. Further, let $B$ be the event that the person will renew at least one of these policies. Say from past experience that we assign the conditional probabilities $P\\left(B \\mid A_1\\right)=0.6, P\\left(B \\mid A_2\\right)=0.7$, and $P\\left(B \\mid A_3\\right)=0.8$. Given that the person selected at random has an auto or homeowner policy, what is the conditional probability that the person will renew at least one of those policies?",
+  "answer_latex": " 0.686",
+  "answer_number": "0.686",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.3.11 ",
+  "comment": " ",
+  "solution": "The desired probability is\r\n$$\r\n\\begin{aligned}\r\nP\\left(B \\mid A_1 \\cup A_2 \\cup A_3\\right) & =\\frac{P\\left(A_1 \\cap B\\right)+P\\left(A_2 \\cap B\\right)+P\\left(A_3 \\cap B\\right)}{P\\left(A_1\\right)+P\\left(A_2\\right)+P\\left(A_3\\right)} \\\\\r\n& =\\frac{(0.3)(0.6)+(0.2)(0.7)+(0.2)(0.8)}{0.3+0.2+0.2} \\\\\r\n& =\\frac{0.48}{0.70}=0.686 .\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "What is the number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards?",
+  "answer_latex": " 635013559600",
+  "answer_number": "635013559600",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Example 1.2.10",
+  "comment": " ",
+  "solution": "The number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards is\r\n$$\r\n{ }_{52} C_{13}=\\left(\\begin{array}{l}\r\n52 \\\\\r\n13\r\n\\end{array}\\right)=\\frac{52 !}{13 ! 39 !}=635,013,559,600 .\r\n$$\r\n"
+ },
+ {
+  "problem_text": "What is the number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons?",
+  "answer_latex": " 5040",
+  "answer_number": "5040",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.5 ",
+  "comment": " ",
+  "solution": "The number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons is\r\n$$\r\n{ }_{10} P_4=10 \\cdot 9 \\cdot 8 \\cdot 7=\\frac{10 !}{6 !}=5040 .\r\n$$"
+ },
+ {
+  "problem_text": "At a county fair carnival game there are 25 balloons on a board, of which 10 balloons 1.3-5 are yellow, 8 are red, and 7 are green. A player throws darts at the balloons to win a prize and randomly hits one of them. Given that the first balloon hit is yellow, what is the probability that the next balloon hit is also yellow?",
+  "answer_latex": "$\\frac{9}{24}$",
+  "answer_number": "0.375",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Example 1.3.5",
+  "comment": " ",
+  "solution": "Of the 24 remaining balloons, 9 are yellow, so a natural value to assign to this conditional probability is $9 / 24$."
+ },
+ {
+  "problem_text": "What is the number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards?",
+  "answer_latex": " 311875200",
+  "answer_number": "311875200",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Example 1.2.8",
+  "comment": " ",
+  "solution": "\r\nThe number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards is\r\n$$\r\n(52)(51)(50)(49)(48)=\\frac{52 !}{47 !}=311,875,200 .\r\n$$\r\n"
+ },
+ {
+  "problem_text": "A bowl contains seven blue chips and three red chips. Two chips are to be drawn successively at random and without replacement. We want to compute the probability that the first draw results in a red chip $(A)$ and the second draw results in a blue chip $(B)$. ",
+  "answer_latex": "$\\frac{7}{30}$",
+  "answer_number": "0.23333333333",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.3.6 ",
+  "comment": " ",
+  "solution": "It is reasonable to assign the following probabilities:\r\n$$\r\nP(A)=\\frac{3}{10} \\text { and } P(B \\mid A)=\\frac{7}{9} \\text {. }\r\n$$\r\nThe probability of obtaining red on the first draw and blue on the second draw is\r\n$$\r\nP(A \\cap B)=\\frac{3}{10} \\cdot \\frac{7}{9}=\\frac{7}{30}\r\n$$"
+ },
+ {
+  "problem_text": "From an ordinary deck of playing cards, cards are to be drawn successively at random and without replacement. What is the probability that the third spade appears on the sixth draw?",
+  "answer_latex": " 0.064",
+  "answer_number": "0.064",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.3.7 ",
+  "comment": " ",
+  "solution": "Let $A$ be the event of two spades in the first five cards drawn, and let $B$ be the event of a spade on the sixth draw. Thus, the probability that we wish to compute is $P(A \\cap B)$. It is reasonable to take\r\n$$\r\nP(A)=\\frac{\\left(\\begin{array}{c}\r\n13 \\\\\r\n2\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n39 \\\\\r\n3\r\n\\end{array}\\right)}{\\left(\\begin{array}{c}\r\n52 \\\\\r\n5\r\n\\end{array}\\right)}=0.274 \\quad \\text { and } \\quad P(B \\mid A)=\\frac{11}{47}=0.234\r\n$$\r\nThe desired probability, $P(A \\cap B)$, is the product of those numbers:\r\n$$\r\nP(A \\cap B)=(0.274)(0.234)=0.064\r\n$$\r\n"
+ },
+ {
+  "problem_text": "What is the probability of drawing three kings and two queens when drawing a five-card hand from a deck of 52 playing cards?",
+  "answer_latex": " 0.0000092",
+  "answer_number": "0.0000092",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Example 1.2.11",
+  "comment": " ",
+  "solution": "Assume that each of the $\\left(\\begin{array}{c}52 \\\\ 5\\end{array}\\right)=2,598,960$ five-card hands drawn from a deck of 52 playing cards has the same probability of being selected. \r\nSuppose now that the event $B$ is the set of outcomes in which exactly three cards are kings and exactly two cards are queens. We can select the three kings in any one of $\\left(\\begin{array}{l}4 \\\\ 3\\end{array}\\right)$ ways and the two queens in any one of $\\left(\\begin{array}{l}4 \\\\ 2\\end{array}\\right)$ ways. By the multiplication principle, the number of outcomes in $B$ is\r\n$$\r\nN(B)=\\left(\\begin{array}{l}\r\n4 \\\\\r\n3\r\n\\end{array}\\right)\\left(\\begin{array}{l}\r\n4 \\\\\r\n2\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n44 \\\\\r\n0\r\n\\end{array}\\right)\r\n$$\r\nwhere $\\left(\\begin{array}{c}44 \\\\ 0\\end{array}\\right)$ gives the number of ways in which 0 cards are selected out of the nonkings and nonqueens and of course is equal to 1 . Thus,\r\n$$\r\nP(B)=\\frac{N(B)}{N(S)}=\\frac{\\left(\\begin{array}{l}\r\n4 \\\\\r\n3\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n4 \\\\\r\n2\r\n\\end{array}\\right)\\left(\\begin{array}{c}\r\n44 \\\\\r\n0\r\n\\end{array}\\right)}{\\left(\\begin{array}{c}\r\n52 \\\\\r\n5\r\n\\end{array}\\right)}=\\frac{24}{2,598,960}=0.0000092 .\r\n$$"
+ },
+ {
+  "problem_text": "In an orchid show, seven orchids are to be placed along one side of the greenhouse. There are four lavender orchids and three white orchids. How many ways are there to lineup these orchids?",
+  "answer_latex": " 35",
+  "answer_number": "35",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.13",
+  "comment": " ",
+  "solution": "Considering only the color of the orchids, we see that the number of lineups of the orchids is\r\n$$\r\n\\left(\\begin{array}{l}\r\n7 \\\\\r\n4\r\n\\end{array}\\right)=\\frac{7 !}{4 ! 3 !}=35 \\text {. }\r\n$$"
+ },
+ {
+  "problem_text": "If $P(A)=0.4, P(B)=0.5$, and $P(A \\cap B)=0.3$, find $P(B \\mid A)$.",
+  "answer_latex": " 0.75",
+  "answer_number": "0.75",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.3.2 ",
+  "comment": " ",
+  "solution": "$P(B \\mid A)=P(A \\cap B) / P(A)=0.3 / 0.4=0.75$."
+ },
+ {
+  "problem_text": "What is the number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards?",
+  "answer_latex": " 2598960",
+  "answer_number": "2598960",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.9 ",
+  "comment": " ",
+  "solution": "The number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards is\r\n$$\r\n{ }_{52} C_5=\\left(\\begin{array}{c}\r\n52 \\\\\r\n5\r\n\\end{array}\\right)=\\frac{52 !}{5 ! 47 !}=2,598,960\r\n$$\r\n"
+ },
+ {
+  "problem_text": "A certain food service gives the following choices for dinner: $E_1$, soup or tomato 1.2-2 juice; $E_2$, steak or shrimp; $E_3$, French fried potatoes, mashed potatoes, or a baked potato; $E_4$, corn or peas; $E_5$, jello, tossed salad, cottage cheese, or coleslaw; $E_6$, cake, cookies, pudding, brownie, vanilla ice cream, chocolate ice cream, or orange sherbet; $E_7$, coffee, tea, milk, or punch. How many different dinner selections are possible if one of the listed choices is made for each of $E_1, E_2, \\ldots$, and $E_7$ ?",
+  "answer_latex": " 2688",
+  "answer_number": "2688",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.2",
+  "comment": " ",
+  "solution": " By the multiplication principle, there are\r\n$(2)(2)(3)(2)(4)(7)(4)=2688$\r\ndifferent combinations.\r\n"
+ },
+ {
+  "problem_text": "A rocket has a built-in redundant system. In this system, if component $K_1$ fails, it is bypassed and component $K_2$ is used. If component $K_2$ fails, it is bypassed and component $K_3$ is used. (An example of a system with these kinds of components is three computer systems.) Suppose that the probability of failure of any one component is 0.15 , and assume that the failures of these components are mutually independent events. Let $A_i$ denote the event that component $K_i$ fails for $i=1,2,3$. What is the probability that the system fails?",
+  "answer_latex": " 0.9966",
+  "answer_number": "0.9966",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.4.5 ",
+  "comment": " ",
+  "solution": "\r\nBecause the system fails if $K_1$ fails and $K_2$ fails and $K_3$ fails, the probability that the system does not fail is given by\r\n$$\r\n\\begin{aligned}\r\nP\\left[\\left(A_1 \\cap A_2 \\cap A_3\\right)^{\\prime}\\right] & =1-P\\left(A_1 \\cap A_2 \\cap A_3\\right) \\\\\r\n& =1-P\\left(A_1\\right) P\\left(A_2\\right) P\\left(A_3\\right) \\\\\r\n& =1-(0.15)^3 \\\\\r\n& =0.9966 .\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "Suppose that $P(A)=0.7, P(B)=0.3$, and $P(A \\cap B)=0.2$. Given that the outcome of the experiment belongs to $B$, what then is the probability of $A$ ?",
+  "answer_latex": " \\frac{2}{3}",
+  "answer_number": "0.66666666666",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.3.3 ",
+  "comment": " ",
+  "solution": "We are effectively restricting the sample space to $B$; of the probability $P(B)=0.3,0.2$ corresponds to $P(A \\cap B)$ and hence to $A$. That is, $0.2 / 0.3=2 / 3$ of the probability of $B$ corresponds to $A$. Of course, by the formal definition, we also obtain\r\n$$\r\nP(A \\mid B)=\\frac{P(A \\cap B)}{P(B)}=\\frac{0.2}{0.3}=\\frac{2}{3}\r\n$$"
+ },
+ {
+  "problem_text": "A coin is flipped 10 times and the sequence of heads and tails is observed. What is the number of possible 10-tuplets that result in four heads and six tails?",
+  "answer_latex": " 210",
+  "answer_number": "210",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.12 ",
+  "comment": " ",
+  "solution": "A coin is flipped 10 times and the sequence of heads and tails is observed. The number of possible 10-tuplets that result in four heads and six tails is\r\n$$\r\n\\left(\\begin{array}{c}\r\n10 \\\\\r\n4\r\n\\end{array}\\right)=\\frac{10 !}{4 ! 6 !}=\\frac{10 !}{6 ! 4 !}=\\left(\\begin{array}{c}\r\n10 \\\\\r\n6\r\n\\end{array}\\right)=210 .\r\n$$\r\n"
+ },
+ {
+  "problem_text": "Among nine orchids for a line of orchids along one wall, three are white, four lavender, and two yellow. How many color displays are there?",
+  "answer_latex": " 1260",
+  "answer_number": "1260",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.14 ",
+  "comment": " ",
+  "solution": "The number of different color displays is\r\n$$\r\n\\left(\\begin{array}{c}\r\n9 \\\\\r\n3,4,2\r\n\\end{array}\\right)=\\frac{9 !}{3 ! 4 ! 2 !}=1260\r\n$$\r\n"
+ },
+ {
+  "problem_text": "A survey was taken of a group's viewing habits of sporting events on TV during I.I-5 the last year. Let $A=\\{$ watched football $\\}, B=\\{$ watched basketball $\\}, C=\\{$ watched baseball $\\}$. The results indicate that if a person is selected at random from the surveyed group, then $P(A)=0.43, P(B)=0.40, P(C)=0.32, P(A \\cap B)=0.29$, $P(A \\cap C)=0.22, P(B \\cap C)=0.20$, and $P(A \\cap B \\cap C)=0.15$. Find $P(A \\cup B \\cup C)$.",
+  "answer_latex": " 0.59",
+  "answer_number": "0.59",
+  "unit": " ",
+  "source": "stat",
+  "problemid": " Example 1.1.5",
+  "comment": " ",
+  "solution": "$$\r\n\\begin{aligned}\r\nP(A \\cup B \\cup C)= & P(A)+P(B)+P(C)-P(A \\cap B)-P(A \\cap C) \\\\\r\n& -P(B \\cap C)+P(A \\cap B \\cap C) \\\\\r\n= & 0.43+0.40+0.32-0.29-0.22-0.20+0.15 \\\\\r\n= & 0.59\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "A grade school boy has five blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of his then drawing a blue marble from his right pocket?",
+  "answer_latex": " $\\frac{41}{90}$",
+  "answer_number": "0.444444444444444 ",
+  "unit": "",
+  "source": "stat",
+  "problemid": " Example 1.3.10",
+  "comment": " ",
+  "solution": "For notation, let $B L, B R$, and $W L$ denote drawing blue from left pocket, blue from right pocket, and white from left pocket, respectively. Then\r\n$$\r\n\\begin{aligned}\r\nP(B R) & =P(B L \\cap B R)+P(W L \\cap B R) \\\\\r\n& =P(B L) P(B R \\mid B L)+P(W L) P(B R \\mid W L) \\\\\r\n& =\\frac{5}{9} \\cdot \\frac{5}{10}+\\frac{4}{9} \\cdot \\frac{4}{10}=\\frac{41}{90}\r\n\\end{aligned}\r\n$$\r\nis the desired probability."
+ },
+ {
+  "problem_text": "A faculty leader was meeting two students in Paris, one arriving by train from Amsterdam and the other arriving by train from Brussels at approximately the same time. Let $A$ and $B$ be the events that the respective trains are on time. Suppose we know from past experience that $P(A)=0.93, P(B)=0.89$, and $P(A \\cap B)=0.87$. Find $P(A \\cup B)$.",
+  "answer_latex": " 0.95",
+  "answer_number": "0.95",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.1.4 ",
+  "comment": " ",
+  "solution": "$$P(A \\cup B) =P(A)+P(B)-P(A \\cap B)=0.93+0.89-0.87=0.95$$"
+ },
+ {
+  "problem_text": "What is the number of possible four-letter code words, selecting from the 26 letters in the alphabet?",
+  "answer_latex": " 358800",
+  "answer_number": "358800",
+  "unit": " ",
+  "source": "stat",
+  "problemid": "Example 1.2.4 ",
+  "comment": " ",
+  "solution": "The number of possible four-letter code words, selecting from the 26 letters in the  alphabet, in which all four letters are different is\r\n$$\r\n{ }_{26} P_4=(26)(25)(24)(23)=\\frac{26 !}{22 !}=358,800 .\r\n$$"
+ }
+]

thermo.json

@@ -1 +1,596 @@
-[{"problem_text": "Consider the half-cell reaction $\\operatorname{AgCl}(s)+\\mathrm{e}^{-} \\rightarrow$ $\\operatorname{Ag}(s)+\\mathrm{Cl}^{-}(a q)$. If $\\mu^{\\circ}(\\mathrm{AgCl}, s)=-109.71 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, and if $E^{\\circ}=+0.222 \\mathrm{~V}$ for this half-cell, calculate the standard Gibbs energy of formation of $\\mathrm{Cl}^{-}(a q)$.", "answer_latex": " -131.1", "answer_number": "-131.1", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 11.9", "comment": " "}, {"problem_text": "$\\mathrm{N}_2 \\mathrm{O}_3$ dissociates according to the equilibrium $\\mathrm{N}_2 \\mathrm{O}_3(\\mathrm{~g}) \\rightleftharpoons \\mathrm{NO}_2(\\mathrm{~g})+\\mathrm{NO}(\\mathrm{g})$. At $298 \\mathrm{~K}$ and one bar pressure, the degree of dissociation defined as the ratio of moles of $\\mathrm{NO}_2(g)$ or $\\mathrm{NO}(g)$ to the moles of the reactant assuming no dissociation occurs is $3.5 \\times 10^{-3}$. Calculate $\\Delta G_R^{\\circ}$ for this reaction.", "answer_latex": " 28", "answer_number": "28", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 6.37", "comment": " "}, {"problem_text": "Approximately how many oxygen molecules arrive each second at the mitochondrion of an active person with a mass of $84 \\mathrm{~kg}$ ? The following data are available: Oxygen consumption is about $40 . \\mathrm{mL}$ of $\\mathrm{O}_2$ per minute per kilogram of body weight, measured at $T=300 . \\mathrm{K}$ and $P=1.00 \\mathrm{~atm}$. In an adult there are about $1.6 \\times 10^{10}$ cells per kg body mass. Each cell contains about 800 . mitochondria.", "answer_latex": " 1.27", "answer_number": "1.27", "unit": "$10^6$ ", "source": "thermo", "problemid": " 1.1", "comment": " "}, {"problem_text": "In a FRET experiment designed to monitor conformational changes in T4 lysozyme, the fluorescence intensity fluctuates between 5000 and 10,000 counts per second.\r\nAssuming that 7500 counts represents a FRET efficiency of 0.5 , what is the change in FRET pair separation distance during the reaction? For the tetramethylrhodamine/texas red FRET pair employed $r_0=50 . \u00c5$.", "answer_latex": " 12", "answer_number": "12", "unit": " $\u00c5$", "source": "thermo", "problemid": " 19.46", "comment": " Angstrom "}, {"problem_text": "An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \\times 10^3 \\mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?", "answer_latex": " 4.85", "answer_number": "4.85", "unit": " $10^8 \\mathrm{~J}$", "source": "thermo", "problemid": " 5.4", "comment": " "}, {"problem_text": "An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \\times 10^3 \\mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?", "answer_latex": " 4.85", "answer_number": "4.85", "unit": " $10^8 \\mathrm{~J}$", "source": "thermo", "problemid": " 5.4", "comment": " "}, {"problem_text": "You have collected a tissue specimen that you would like to preserve by freeze drying. To ensure the integrity of the specimen, the temperature should not exceed $-5.00{ }^{\\circ} \\mathrm{C}$. The vapor pressure of ice at $273.16 \\mathrm{~K}$ is $624 \\mathrm{~Pa}$. What is the maximum pressure at which the freeze drying can be carried out?", "answer_latex": " 425", "answer_number": "425", "unit": " $\\mathrm{~Pa}$", "source": "thermo", "problemid": " 8.14", "comment": " "}, {"problem_text": "The molar constant volume heat capacity for $\\mathrm{I}_2(\\mathrm{~g})$ is $28.6 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$. What is the vibrational contribution to the heat capacity? You can assume that the contribution from the electronic degrees of freedom is negligible.", "answer_latex": " 7.82", "answer_number": "7.82", "unit": " $\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$", "source": "thermo", "problemid": " 15.22", "comment": " "}, {"problem_text": "The diffusion coefficient for $\\mathrm{CO}_2$ at $273 \\mathrm{~K}$ and $1 \\mathrm{~atm}$ is $1.00 \\times 10^{-5} \\mathrm{~m}^2 \\mathrm{~s}^{-1}$. Estimate the collisional cross section of $\\mathrm{CO}_2$ given this diffusion coefficient.", "answer_latex": " 0.318", "answer_number": "0.318", "unit": " $\\mathrm{~nm}^2$", "source": "thermo", "problemid": " 17.1", "comment": " "}, {"problem_text": "Benzoic acid, $1.35 \\mathrm{~g}$, is reacted with oxygen in a constant volume calorimeter to form $\\mathrm{H}_2 \\mathrm{O}(l)$ and $\\mathrm{CO}_2(g)$ at $298 \\mathrm{~K}$. The mass of the water in the inner bath is $1.55 \\times$ $10^3 \\mathrm{~g}$. The temperature of the calorimeter and its contents rises $2.76 \\mathrm{~K}$ as a result of this reaction. Calculate the calorimeter constant.", "answer_latex": " 6.64", "answer_number": "6.64", "unit": " $10^3 \\mathrm{~J}^{\\circ} \\mathrm{C}^{-1}$\r\n", "source": "thermo", "problemid": " 4.15", "comment": " "}, {"problem_text": "The activation energy for a reaction is $50 . \\mathrm{J} \\mathrm{mol}^{-1}$. Determine the effect on the rate constant for this reaction with a change in temperature from $273 \\mathrm{~K}$ to $298 \\mathrm{~K}$.", "answer_latex": " 0.15", "answer_number": "0.15", "unit": " ", "source": "thermo", "problemid": " 18.37", "comment": " "}, {"problem_text": "How long will it take to pass $200 . \\mathrm{mL}$ of $\\mathrm{H}_2$ at $273 \\mathrm{~K}$ through a $10 . \\mathrm{cm}$-long capillary tube of $0.25 \\mathrm{~mm}$ if the gas input and output pressures are 1.05 and $1.00 \\mathrm{~atm}$, respectively?", "answer_latex": " 22", "answer_number": "22", "unit": " $\\mathrm{~s}$", "source": "thermo", "problemid": " 17.21", "comment": " "}, {"problem_text": "Calculate the Debye-H\u00fcckel screening length $1 / \\kappa$ at $298 \\mathrm{~K}$ in a $0.0075 \\mathrm{~m}$ solution of $\\mathrm{K}_3 \\mathrm{PO}_4$.", "answer_latex": " 1.4", "answer_number": "1.4", "unit": " $\\mathrm{~nm}$", "source": "thermo", "problemid": " 10.23", "comment": " "}, {"problem_text": "A system consisting of $82.5 \\mathrm{~g}$ of liquid water at $300 . \\mathrm{K}$ is heated using an immersion heater at a constant pressure of 1.00 bar. If a current of $1.75 \\mathrm{~A}$ passes through the $25.0 \\mathrm{ohm}$ resistor for 100 .s, what is the final temperature of the water?", "answer_latex": " 322", "answer_number": "322", "unit": "$\\mathrm{~K}$", "source": "thermo", "problemid": " 2.13", "comment": " "}, {"problem_text": "For an ensemble consisting of a mole of particles having two energy levels separated by $1000 . \\mathrm{cm}^{-1}$, at what temperature will the internal energy equal $3.00 \\mathrm{~kJ}$ ?", "answer_latex": " 1310", "answer_number": "1310", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 15.4", "comment": " "}, {"problem_text": "A muscle fiber contracts by $3.5 \\mathrm{~cm}$ and in doing so lifts a weight. Calculate the work performed by the fiber. Assume the muscle fiber obeys Hooke's law $F=-k x$ with a force constant $k$ of $750 . \\mathrm{N} \\mathrm{m}^{-1}$.", "answer_latex": " 0.46", "answer_number": "0.46", "unit": "$\\mathrm{~J}$", "source": "thermo", "problemid": " 2.10", "comment": " "}, {"problem_text": "A gas sample is known to be a mixture of ethane and butane. A bulb having a $230.0 \\mathrm{~cm}^3$ capacity is filled with the gas to a pressure of $97.5 \\times 10^3 \\mathrm{~Pa}$ at $23.1^{\\circ} \\mathrm{C}$. If the mass of the gas in the bulb is $0.3554 \\mathrm{~g}$, what is the mole percent of butane in the mixture?\r\n", "answer_latex": " 32", "answer_number": "32", "unit": " %", "source": "thermo", "problemid": " 1.5", "comment": " "}, {"problem_text": "One liter of fully oxygenated blood can carry 0.18 liters of $\\mathrm{O}_2$ measured at $T=298 \\mathrm{~K}$ and $P=1.00 \\mathrm{~atm}$. Calculate the number of moles of $\\mathrm{O}_2$ carried per liter of blood. Hemoglobin, the oxygen transport protein in blood has four oxygen binding sites. How many hemoglobin molecules are required to transport the $\\mathrm{O}_2$ in $1.0 \\mathrm{~L}$ of fully oxygenated blood?", "answer_latex": " 1.11", "answer_number": "1.11", "unit": "$10^{21}$ ", "source": "thermo", "problemid": " 1.6", "comment": " "}, {"problem_text": "Consider a collection of molecules where each molecule has two nondegenerate energy levels that are separated by $6000 . \\mathrm{cm}^{-1}$. Measurement of the level populations demonstrates that there are exactly 8 times more molecules in the ground state than in the upper state. What is the temperature of the collection?", "answer_latex": " 4152", "answer_number": "4152", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 13.15", "comment": " "}, {"problem_text": "Calculate $\\Delta S^{\\circ}$ for the reaction $3 \\mathrm{H}_2(g)+\\mathrm{N}_2(g) \\rightarrow$ $2 \\mathrm{NH}_3(g)$ at $725 \\mathrm{~K}$. Omit terms in the temperature-dependent heat capacities higher than $T^2 / \\mathrm{K}^2$.", "answer_latex": " -191.2", "answer_number": "-191.2", "unit": " $\\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 5.14", "comment": " "}, {"problem_text": "The thermal conductivities of acetylene $\\left(\\mathrm{C}_2 \\mathrm{H}_2\\right)$ and $\\mathrm{N}_2$ at $273 \\mathrm{~K}$ and $1 \\mathrm{~atm}$ are 0.01866 and $0.0240 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~m}^{-1} \\mathrm{~s}^{-1}$, respectively. Based on these data, what is the ratio of the collisional cross section of acetylene relative to $\\mathrm{N}_2$ ?", "answer_latex": " 1.33", "answer_number": "1.33", "unit": " ", "source": "thermo", "problemid": " 17.15", "comment": " "}, {"problem_text": "Consider the gas phase thermal decomposition of 1.0 atm of $\\left(\\mathrm{CH}_3\\right)_3 \\mathrm{COOC}\\left(\\mathrm{CH}_3\\right)_3(\\mathrm{~g})$ to acetone $\\left(\\mathrm{CH}_3\\right)_2 \\mathrm{CO}(\\mathrm{g})$ and ethane $\\left(\\mathrm{C}_2 \\mathrm{H}_6\\right)(\\mathrm{g})$, which occurs with a rate constant of $0.0019 \\mathrm{~s}^{-1}$. After initiation of the reaction, at what time would you expect the pressure to be $1.8 \\mathrm{~atm}$ ?", "answer_latex": " 269", "answer_number": "269", "unit": " $\\mathrm{~s}$", "source": "thermo", "problemid": " 18.39", "comment": " "}, {"problem_text": "Autoclaves that are used to sterilize surgical tools require a temperature of $120 .{ }^{\\circ} \\mathrm{C}$ to kill some bacteria. If water is used for this purpose, at what pressure must the autoclave operate?", "answer_latex": "1.95 ", "answer_number": "1.95 ", "unit": " $\\mathrm{~atm}$", "source": "thermo", "problemid": " 8.13", "comment": " "}, {"problem_text": "Imagine gaseous $\\mathrm{Ar}$ at $298 \\mathrm{~K}$ confined to move in a two-dimensional plane of area $1.00 \\mathrm{~cm}^2$. What is the value of the translational partition function?", "answer_latex": " 3.9", "answer_number": "3.9", "unit": " $10^{17}$", "source": "thermo", "problemid": " 14.6", "comment": " "}, {"problem_text": "Determine the equilibrium constant for the dissociation of sodium at $298 \\mathrm{~K}: \\mathrm{Na}_2(g) \\rightleftharpoons 2 \\mathrm{Na}(g)$. For $\\mathrm{Na}_2$, $B=0.155 \\mathrm{~cm}^{-1}, \\widetilde{\\nu}=159 \\mathrm{~cm}^{-1}$, the dissociation energy is $70.4 \\mathrm{~kJ} / \\mathrm{mol}$, and the ground-state electronic degeneracy for $\\mathrm{Na}$ is 2 .", "answer_latex": " 2.25", "answer_number": "2.25", "unit": " $10^{-9}$", "source": "thermo", "problemid": " 15.47", "comment": " "}, {"problem_text": "At $298.15 \\mathrm{~K}, \\Delta G_f^{\\circ}(\\mathrm{HCOOH}, g)=-351.0 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ and $\\Delta G_f^{\\circ}(\\mathrm{HCOOH}, l)=-361.4 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Calculate the vapor pressure of formic acid at this temperature.", "answer_latex": "1.51", "answer_number": "1.51", "unit": " $10^3 \\mathrm{~Pa}$", "source": "thermo", "problemid": " 8.39", "comment": " "}, {"problem_text": "The collisional cross section of $\\mathrm{N}_2$ is $0.43 \\mathrm{~nm}^2$. What is the diffusion coefficient of $\\mathrm{N}_2$ at a pressure of $1 \\mathrm{~atm}$ and a temperature of $298 \\mathrm{~K}$ ?", "answer_latex": " 1.06", "answer_number": "1.06", "unit": " $10^{-5} \\mathrm{~m}^2 \\mathrm{~s}^{-1}$", "source": "thermo", "problemid": " 17.2", "comment": " "}, {"problem_text": "A vessel contains $1.15 \\mathrm{~g}$ liq $\\mathrm{H}_2 \\mathrm{O}$ in equilibrium with water vapor at $30 .{ }^{\\circ} \\mathrm{C}$. At this temperature, the vapor pressure of $\\mathrm{H}_2 \\mathrm{O}$ is 31.82 torr. What volume increase is necessary for all the water to evaporate?\r\n", "answer_latex": " 37.9", "answer_number": "37.9", "unit": "$\\mathrm{~L}$", "source": "thermo", "problemid": " 1.8", "comment": " "}, {"problem_text": "A cell is roughly spherical with a radius of $20.0 \\times 10^{-6} \\mathrm{~m}$. Calculate the work required to expand the cell surface against the surface tension of the surroundings if the radius increases by a factor of three. Assume the cell is surrounded by pure water and that $T=298.15 \\mathrm{~K}$.", "answer_latex": " 2.89", "answer_number": "2.89", "unit": " $10^{-9} \\mathrm{~J}$", "source": "thermo", "problemid": " 8.7", "comment": " "}, {"problem_text": "A vessel is filled completely with liquid water and sealed at $13.56^{\\circ} \\mathrm{C}$ and a pressure of 1.00 bar. What is the pressure if the temperature of the system is raised to $82.0^{\\circ} \\mathrm{C}$ ? Under these conditions, $\\beta_{\\text {water }}=2.04 \\times 10^{-4} \\mathrm{~K}^{-1}$, $\\beta_{\\text {vessel }}=1.42 \\times 10^{-4} \\mathrm{~K}^{-1}$, and $\\kappa_{\\text {water }}=4.59 \\times 10^{-5} \\mathrm{bar}^{-1}$.", "answer_latex": " 93.4", "answer_number": "93.4", "unit": "$\\mathrm{~bar}$", "source": "thermo", "problemid": " 3.6", "comment": " "}, {"problem_text": "A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height $h$ ) can be obtained by considering the potential energy due to gravity:\r\n$$\r\nP(h)=e^{-m g h / k T}\r\n$$\r\nIn this expression $m$ is the per-particle mass of the gas, $g$ is the acceleration due to gravity, $k$ is a constant equal to $1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}$, and $T$ is temperature. Determine $\\langle h\\rangle$ for methane $\\left(\\mathrm{CH}_4\\right)$ using this distribution function.", "answer_latex": " 1.6", "answer_number": "1.6", "unit": " $10^4 \\mathrm{~m}$", "source": "thermo", "problemid": " 12.32", "comment": " "}, {"problem_text": "A camper stranded in snowy weather loses heat by wind convection. The camper is packing emergency rations consisting of $58 \\%$ sucrose, $31 \\%$ fat, and $11 \\%$ protein by weight. Using the data provided in Problem P4.32 and assuming the fat content of the rations can be treated with palmitic acid data and the protein content similarly by the protein data in Problem P4.32, how much emergency rations must the camper consume in order to compensate for a reduction in body temperature of $3.5 \\mathrm{~K}$ ? Assume the heat capacity of the body equals that of water. Assume the camper weighs $67 \\mathrm{~kg}$.", "answer_latex": " 49", "answer_number": "49", "unit": " $\\mathrm{~g}$", "source": "thermo", "problemid": " 4.33", "comment": " "}, {"problem_text": "At 303 . K, the vapor pressure of benzene is 120 . Torr and that of hexane is 189 Torr. Calculate the vapor pressure of a solution for which $x_{\\text {benzene }}=0.28$ assuming ideal behavior.", "answer_latex": " 170", "answer_number": "170", "unit": " $\\mathrm{Torr}$", "source": "thermo", "problemid": " 9.8", "comment": " "}, {"problem_text": "Determine the molar standard Gibbs energy for ${ }^{35} \\mathrm{Cl}^{35} \\mathrm{Cl}$ where $\\widetilde{\\nu}=560 . \\mathrm{cm}^{-1}, B=0.244 \\mathrm{~cm}^{-1}$, and the ground electronic state is nondegenerate.", "answer_latex": " -57.2", "answer_number": "-57.2", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 15.45", "comment": " "}, {"problem_text": "For the reaction $\\mathrm{C}($ graphite $)+\\mathrm{H}_2 \\mathrm{O}(g) \\rightleftharpoons$ $\\mathrm{CO}(g)+\\mathrm{H}_2(g), \\Delta H_R^{\\circ}=131.28 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ at $298.15 \\mathrm{~K}$. Use the values of $C_{P, m}^{\\circ}$ at $298.15 \\mathrm{~K}$ in the data tables to calculate $\\Delta H_R^{\\circ}$ at $125.0^{\\circ} \\mathrm{C}$.", "answer_latex": " 132.9", "answer_number": "132.9", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 6.12", "comment": " "}, {"problem_text": "Calculate the mean ionic activity of a $0.0350 \\mathrm{~m} \\mathrm{Na}_3 \\mathrm{PO}_4$ solution for which the mean activity coefficient is 0.685 .", "answer_latex": " 0.0547", "answer_number": "0.0547", "unit": " ", "source": "thermo", "problemid": " 10.6", "comment": " "}, {"problem_text": " Consider the transition between two forms of solid tin, $\\mathrm{Sn}(s$, gray $) \\rightarrow \\mathrm{Sn}(s$, white $)$. The two phases are in equilibrium at 1 bar and $18^{\\circ} \\mathrm{C}$. The densities for gray and white tin are 5750 and $7280 \\mathrm{~kg} \\mathrm{~m}^{-3}$, respectively, and the molar entropies for gray and white tin are 44.14 and $51.18 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$, respectively. Calculate the temperature at which the two phases are in equilibrium at 350. bar.", "answer_latex": " -3.5", "answer_number": "-3.5", "unit": " $^{\\circ} \\mathrm{C}$", "source": "thermo", "problemid": " 8.43", "comment": " "}, {"problem_text": "The densities of pure water and ethanol are 997 and $789 \\mathrm{~kg} \\mathrm{~m}^{-3}$, respectively. For $x_{\\text {ethanol }}=0.35$, the partial molar volumes of ethanol and water are 55.2 and $17.8 \\times 10^{-3} \\mathrm{~L} \\mathrm{~mol}^{-1}$, respectively. Calculate the change in volume relative to the pure components when $2.50 \\mathrm{~L}$ of a solution with $x_{\\text {ethanol }}=0.35$ is prepared.", "answer_latex": " -0.10", "answer_number": "-0.10", "unit": " $\\mathrm{~L}$", "source": "thermo", "problemid": " 9.22", "comment": " "}, {"problem_text": "For $\\mathrm{N}_2$ at $298 \\mathrm{~K}$, what fraction of molecules has a speed between 200. and $300 . \\mathrm{m} / \\mathrm{s}$ ?", "answer_latex": " 0.132", "answer_number": "0.132", "unit": " ", "source": "thermo", "problemid": " 16.15", "comment": " "}, {"problem_text": "Calculate the pressure exerted by Ar for a molar volume of $1.31 \\mathrm{~L} \\mathrm{~mol}^{-1}$ at $426 \\mathrm{~K}$ using the van der Waals equation of state. The van der Waals parameters $a$ and $b$ for Ar are 1.355 bar dm${ }^6 \\mathrm{~mol}^{-2}$ and $0.0320 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$, respectively. Is the attractive or repulsive portion of the potential dominant under these conditions?\r\n", "answer_latex": " 26.9", "answer_number": "26.9", "unit": "$\\mathrm{~bar}$", "source": "thermo", "problemid": " 1.3", "comment": " "}, {"problem_text": "For water, $\\Delta H_{\\text {vaporization }}$ is $40.656 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, and the normal boiling point is $373.12 \\mathrm{~K}$. Calculate the boiling point for water on the top of Mt. Everest (elevation $8848 \\mathrm{~m}$ ), where the barometric pressure is 253 Torr.", "answer_latex": "344", "answer_number": "344", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 8.25", "comment": " "}, {"problem_text": "An ideal solution is formed by mixing liquids $\\mathrm{A}$ and $B$ at $298 \\mathrm{~K}$. The vapor pressure of pure A is 151 Torr and that of pure B is 84.3 Torr. If the mole fraction of $\\mathrm{A}$ in the vapor is 0.610 , what is the mole fraction of $\\mathrm{A}$ in the solution?", "answer_latex": " 0.466", "answer_number": "0.466", "unit": " ", "source": "thermo", "problemid": " 9.24", "comment": " "}, {"problem_text": "The mean solar flux at Earth's surface is $\\sim 2.00 \\mathrm{~J}$ $\\mathrm{cm}^{-2} \\mathrm{~min}^{-1}$. In a nonfocusing solar collector, the temperature reaches a value of $79.5^{\\circ} \\mathrm{C}$. A heat engine is operated using the collector as the hot reservoir and a cold reservoir at $298 \\mathrm{~K}$. Calculate the area of the collector needed to produce 1000. W. Assume that the engine operates at the maximum Carnot efficiency.", "answer_latex": " 19.4", "answer_number": "19.4", "unit": " $\\mathrm{~m}^2$", "source": "thermo", "problemid": " 5.42", "comment": " "}, {"problem_text": "A hiker caught in a thunderstorm loses heat when her clothing becomes wet. She is packing emergency rations that if completely metabolized will release $35 \\mathrm{~kJ}$ of heat per gram of rations consumed. How much rations must the hiker consume to avoid a reduction in body temperature of $2.5 \\mathrm{~K}$ as a result of heat loss? Assume the heat capacity of the body equals that of water and that the hiker weighs $51 \\mathrm{~kg}$.", "answer_latex": " 15", "answer_number": "15", "unit": "$\\mathrm{~g}$", "source": "thermo", "problemid": " 2.4", "comment": " "}, {"problem_text": "Calculate the degree of dissociation of $\\mathrm{N}_2 \\mathrm{O}_4$ in the reaction $\\mathrm{N}_2 \\mathrm{O}_4(g) \\rightleftharpoons 2 \\mathrm{NO}_2(g)$ at 300 . $\\mathrm{K}$ and a total pressure of 1.50 bar. Do you expect the degree of dissociation to increase or decrease as the temperature is increased to 550 . K? Assume that $\\Delta H_R^{\\circ}$ is independent of temperature.", "answer_latex": " 0.241", "answer_number": "0.241", "unit": " ", "source": "thermo", "problemid": " 6.30", "comment": " "}, {"problem_text": "Calculate $\\Delta G$ for the isothermal expansion of $2.25 \\mathrm{~mol}$ of an ideal gas at $325 \\mathrm{~K}$ from an initial pressure of 12.0 bar to a final pressure of 2.5 bar.", "answer_latex": " -9.54", "answer_number": "-9.54", "unit": " $10^3 \\mathrm{~J}$", "source": "thermo", "problemid": " 6.20", "comment": " "}, {"problem_text": "Determine the total collisional frequency for $\\mathrm{CO}_2$ at $1 \\mathrm{~atm}$ and $298 \\mathrm{~K}$.", "answer_latex": " 8.44", "answer_number": "8.44", "unit": " $10^{34} \\mathrm{~m}^{-3} \\mathrm{~s}^{-1}$", "source": "thermo", "problemid": " 16.30", "comment": " "}, {"problem_text": "The volatile liquids $A$ and $\\mathrm{B}$, for which $P_A^*=165$ Torr and $P_B^*=85.1$ Torr are confined to a piston and cylinder assembly. Initially, only the liquid phase is present. As the pressure is reduced, the first vapor is observed at a total pressure of 110 . Torr. Calculate $x_{\\mathrm{A}}$", "answer_latex": " 0.312", "answer_number": "0.312", "unit": " ", "source": "thermo", "problemid": " 9.9", "comment": " "}, {"problem_text": "The osmotic pressure of an unknown substance is measured at $298 \\mathrm{~K}$. Determine the molecular weight if the concentration of this substance is $31.2 \\mathrm{~kg} \\mathrm{~m}^{-3}$ and the osmotic pressure is $5.30 \\times 10^4 \\mathrm{~Pa}$. The density of the solution is $997 \\mathrm{~kg} \\mathrm{~m}^{-3}$.", "answer_latex": " 1.45", "answer_number": "1.45", "unit": " $10^3 \\mathrm{~g} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 9.7", "comment": " "}, {"problem_text": " One mole of Ar initially at 310 . K undergoes an adiabatic expansion against a pressure $P_{\\text {external }}=0$ from a volume of $8.5 \\mathrm{~L}$ to a volume of $82.0 \\mathrm{~L}$. Calculate the final temperature using the ideal gas", "answer_latex": " 310", "answer_number": "310", "unit": "$\\mathrm{~K}$", "source": "thermo", "problemid": " 7.4", "comment": " "}, {"problem_text": "A refrigerator is operated by a $0.25-\\mathrm{hp}(1 \\mathrm{hp}=$ 746 watts) motor. If the interior is to be maintained at $4.50^{\\circ} \\mathrm{C}$ and the room temperature on a hot day is $38^{\\circ} \\mathrm{C}$, what is the maximum heat leak (in watts) that can be tolerated? Assume that the coefficient of performance is $50 . \\%$ of the maximum theoretical value. What happens if the leak is greater than your calculated maximum value?", "answer_latex": " 773", "answer_number": "773", "unit": " $\\mathrm{~J} \\mathrm{~s}^{-1}$", "source": "thermo", "problemid": " 5.33", "comment": " "}, {"problem_text": "In order to get in shape for mountain climbing, an avid hiker with a mass of $60 . \\mathrm{kg}$ ascends the stairs in the world's tallest structure, the $828 \\mathrm{~m}$ tall Burj Khalifa in Dubai, United Arab Emirates. Assume that she eats energy bars on the way up and that her body is $25 \\%$ efficient in converting the energy content of the bars into the work of climbing. How many energy bars does she have to eat if a single bar produces $1.08 \\times 10^3 \\mathrm{~kJ}$ of energy upon metabolizing?", "answer_latex": " 1.8", "answer_number": "1.8", "unit": " ", "source": "thermo", "problemid": " 4.34", "comment": " "}, {"problem_text": "The half-life of ${ }^{238} \\mathrm{U}$ is $4.5 \\times 10^9$ years. How many disintegrations occur in $1 \\mathrm{~min}$ for a $10 \\mathrm{mg}$ sample of this element?", "answer_latex": " 1.43", "answer_number": "1.43", "unit": " $10^{24}$", "source": "thermo", "problemid": " 18.14", "comment": " "}, {"problem_text": "Calculate the ionic strength in a solution that is 0.0750 $m$ in $\\mathrm{K}_2 \\mathrm{SO}_4, 0.0085 \\mathrm{~m}$ in $\\mathrm{Na}_3 \\mathrm{PO}_4$, and $0.0150 \\mathrm{~m}$ in $\\mathrm{MgCl}_2$.", "answer_latex": " 0.321", "answer_number": "0.321", "unit": " $\\mathrm{~mol} \\mathrm{~kg}^{-1}$", "source": "thermo", "problemid": " 10.13", "comment": " "}, {"problem_text": "The interior of a refrigerator is typically held at $36^{\\circ} \\mathrm{F}$ and the interior of a freezer is typically held at $0.00^{\\circ} \\mathrm{F}$. If the room temperature is $65^{\\circ} \\mathrm{F}$, by what factor is it more expensive to extract the same amount of heat from the freezer than from the refrigerator? Assume that the theoretical limit for the performance of a reversible refrigerator is valid in this case.", "answer_latex": " 2.4", "answer_number": "2.4", "unit": " ", "source": "thermo", "problemid": " 5.17", "comment": " "}, {"problem_text": "Calculate the rotational partition function for $\\mathrm{SO}_2$ at $298 \\mathrm{~K}$ where $B_A=2.03 \\mathrm{~cm}^{-1}, B_B=0.344 \\mathrm{~cm}^{-1}$, and $B_C=0.293 \\mathrm{~cm}^{-1}$", "answer_latex": " 5840", "answer_number": "5840", "unit": " ", "source": "thermo", "problemid": " 14.16", "comment": " "}, {"problem_text": "For a two-level system where $v=1.50 \\times 10^{13} \\mathrm{~s}^{-1}$, determine the temperature at which the internal energy is equal to $0.25 \\mathrm{Nhv}$, or $1 / 2$ the limiting value of $0.50 \\mathrm{Nhv}$.", "answer_latex": " 655", "answer_number": "655", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 15.2", "comment": " "}, {"problem_text": "Calculate $K_P$ at $600 . \\mathrm{K}$ for the reaction $\\mathrm{N}_2 \\mathrm{O}_4(l) \\rightleftharpoons 2 \\mathrm{NO}_2(g)$ assuming that $\\Delta H_R^{\\circ}$ is constant over the interval 298-725 K.", "answer_latex": " 4.76", "answer_number": "4.76", "unit": " $10^6$", "source": "thermo", "problemid": " 6.10", "comment": " "}, {"problem_text": "Count Rumford observed that using cannon boring machinery a single horse could heat $11.6 \\mathrm{~kg}$ of ice water $(T=273 \\mathrm{~K})$ to $T=355 \\mathrm{~K}$ in 2.5 hours. Assuming the same rate of work, how high could a horse raise a $225 \\mathrm{~kg}$ weight in 2.5 minutes? Assume the heat capacity of water is $4.18 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~g}^{-1}$.", "answer_latex": " 30", "answer_number": "30", "unit": "$\\mathrm{~m}$", "source": "thermo", "problemid": " 2.5", "comment": " "}, {"problem_text": "The vibrational frequency of $I_2$ is $208 \\mathrm{~cm}^{-1}$. At what temperature will the population in the first excited state be half that of the ground state?", "answer_latex": " 432", "answer_number": "432", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 13.22", "comment": " "}, {"problem_text": "One mole of $\\mathrm{H}_2 \\mathrm{O}(l)$ is compressed from a state described by $P=1.00$ bar and $T=350$. K to a state described by $P=590$. bar and $T=750$. K. In addition, $\\beta=2.07 \\times 10^{-4} \\mathrm{~K}^{-1}$ and the density can be assumed to be constant at the value $997 \\mathrm{~kg} \\mathrm{~m}^{-3}$. Calculate $\\Delta S$ for this transformation, assuming that $\\kappa=0$.\r\n", "answer_latex": " 57.2", "answer_number": "57.2", "unit": " $\\mathrm{~K}^{-1}$", "source": "thermo", "problemid": " 5.5", "comment": " "}, {"problem_text": "A mass of $34.05 \\mathrm{~g}$ of $\\mathrm{H}_2 \\mathrm{O}(s)$ at $273 \\mathrm{~K}$ is dropped into $185 \\mathrm{~g}$ of $\\mathrm{H}_2 \\mathrm{O}(l)$ at $310 . \\mathrm{K}$ in an insulated container at 1 bar of pressure. Calculate the temperature of the system once equilibrium has been reached. Assume that $C_{P, m}$ for $\\mathrm{H}_2 \\mathrm{O}(l)$ is constant at its values for $298 \\mathrm{~K}$ throughout the temperature range of interest.", "answer_latex": " 292", "answer_number": "292", "unit": "$\\mathrm{~K}$", "source": "thermo", "problemid": " 3.5", "comment": " "}, {"problem_text": "Calculate $\\Delta H_f^{\\circ}$ for $N O(g)$ at $975 \\mathrm{~K}$, assuming that the heat capacities of reactants and products are constant over the temperature interval at their values at $298.15 \\mathrm{~K}$.", "answer_latex": " 91.7", "answer_number": "91.7", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 4.4", "comment": " "}, {"problem_text": "A two-level system is characterized by an energy separation of $1.30 \\times 10^{-18} \\mathrm{~J}$. At what temperature will the population of the ground state be 5 times greater than that of the excited state?", "answer_latex": " 5.85", "answer_number": "5.85", "unit": " $10^4$", "source": "thermo", "problemid": " 13.27", "comment": " "}, {"problem_text": "At what temperature are there Avogadro's number of translational states available for $\\mathrm{O}_2$ confined to a volume of 1000. $\\mathrm{cm}^3$ ?", "answer_latex": " 0.068", "answer_number": "0.068", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 14.5", "comment": " "}, {"problem_text": "The half-cell potential for the reaction $\\mathrm{O}_2(g)+4 \\mathrm{H}^{+}(a q)+4 \\mathrm{e}^{-} \\rightarrow 2 \\mathrm{H}_2 \\mathrm{O}(l)$ is $+1.03 \\mathrm{~V}$ at $298.15 \\mathrm{~K}$ when $a_{\\mathrm{O}_2}=1.00$. Determine $a_{\\mathrm{H}^{+}}$", "answer_latex": " 4.16", "answer_number": "4.16", "unit": " $10^{-4}$", "source": "thermo", "problemid": " 11.25", "comment": " "}, {"problem_text": "The partial molar volumes of water and ethanol in a solution with $x_{\\mathrm{H}_2 \\mathrm{O}}=0.45$ at $25^{\\circ} \\mathrm{C}$ are 17.0 and $57.5 \\mathrm{~cm}^3 \\mathrm{~mol}^{-1}$, respectively. Calculate the volume change upon mixing sufficient ethanol with $3.75 \\mathrm{~mol}$ of water to give this concentration. The densities of water and ethanol are 0.997 and $0.7893 \\mathrm{~g} \\mathrm{~cm}^{-3}$, respectively, at this temperature.", "answer_latex": " -8", "answer_number": "-8", "unit": " $\\mathrm{~cm}^3$", "source": "thermo", "problemid": " 9.5", "comment": " "}]
+[
+ {
+  "problem_text": "Consider the half-cell reaction $\\operatorname{AgCl}(s)+\\mathrm{e}^{-} \\rightarrow$ $\\operatorname{Ag}(s)+\\mathrm{Cl}^{-}(a q)$. If $\\mu^{\\circ}(\\mathrm{AgCl}, s)=-109.71 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, and if $E^{\\circ}=+0.222 \\mathrm{~V}$ for this half-cell, calculate the standard Gibbs energy of formation of $\\mathrm{Cl}^{-}(a q)$.",
+  "answer_latex": " -131.1",
+  "answer_number": "-131.1",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 11.9",
+  "comment": " "
+ },
+ {
+  "problem_text": "$\\mathrm{N}_2 \\mathrm{O}_3$ dissociates according to the equilibrium $\\mathrm{N}_2 \\mathrm{O}_3(\\mathrm{~g}) \\rightleftharpoons \\mathrm{NO}_2(\\mathrm{~g})+\\mathrm{NO}(\\mathrm{g})$. At $298 \\mathrm{~K}$ and one bar pressure, the degree of dissociation defined as the ratio of moles of $\\mathrm{NO}_2(g)$ or $\\mathrm{NO}(g)$ to the moles of the reactant assuming no dissociation occurs is $3.5 \\times 10^{-3}$. Calculate $\\Delta G_R^{\\circ}$ for this reaction.",
+  "answer_latex": " 28",
+  "answer_number": "28",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 6.37",
+  "comment": " "
+ },
+ {
+  "problem_text": "Approximately how many oxygen molecules arrive each second at the mitochondrion of an active person with a mass of $84 \\mathrm{~kg}$ ? The following data are available: Oxygen consumption is about $40 . \\mathrm{mL}$ of $\\mathrm{O}_2$ per minute per kilogram of body weight, measured at $T=300 . \\mathrm{K}$ and $P=1.00 \\mathrm{~atm}$. In an adult there are about $1.6 \\times 10^{10}$ cells per kg body mass. Each cell contains about 800 . mitochondria.",
+  "answer_latex": " 1.27",
+  "answer_number": "1.27",
+  "unit": "$10^6$ ",
+  "source": "thermo",
+  "problemid": " 1.1",
+  "comment": " "
+ },
+ {
+  "problem_text": "In a FRET experiment designed to monitor conformational changes in T4 lysozyme, the fluorescence intensity fluctuates between 5000 and 10,000 counts per second.\r\nAssuming that 7500 counts represents a FRET efficiency of 0.5 , what is the change in FRET pair separation distance during the reaction? For the tetramethylrhodamine/texas red FRET pair employed $r_0=50 . \u00c5$.",
+  "answer_latex": " 12",
+  "answer_number": "12",
+  "unit": " $\u00c5$",
+  "source": "thermo",
+  "problemid": " 19.46",
+  "comment": " Angstrom "
+ },
+ {
+  "problem_text": "An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \\times 10^3 \\mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?",
+  "answer_latex": " 4.85",
+  "answer_number": "4.85",
+  "unit": " $10^8 \\mathrm{~J}$",
+  "source": "thermo",
+  "problemid": " 5.4",
+  "comment": " "
+ },
+ {
+  "problem_text": "You have collected a tissue specimen that you would like to preserve by freeze drying. To ensure the integrity of the specimen, the temperature should not exceed $-5.00{ }^{\\circ} \\mathrm{C}$. The vapor pressure of ice at $273.16 \\mathrm{~K}$ is $624 \\mathrm{~Pa}$. What is the maximum pressure at which the freeze drying can be carried out?",
+  "answer_latex": " 425",
+  "answer_number": "425",
+  "unit": " $\\mathrm{~Pa}$",
+  "source": "thermo",
+  "problemid": " 8.14",
+  "comment": " "
+ },
+ {
+  "problem_text": "The molar constant volume heat capacity for $\\mathrm{I}_2(\\mathrm{~g})$ is $28.6 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$. What is the vibrational contribution to the heat capacity? You can assume that the contribution from the electronic degrees of freedom is negligible.",
+  "answer_latex": " 7.82",
+  "answer_number": "7.82",
+  "unit": " $\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$",
+  "source": "thermo",
+  "problemid": " 15.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "The diffusion coefficient for $\\mathrm{CO}_2$ at $273 \\mathrm{~K}$ and $1 \\mathrm{~atm}$ is $1.00 \\times 10^{-5} \\mathrm{~m}^2 \\mathrm{~s}^{-1}$. Estimate the collisional cross section of $\\mathrm{CO}_2$ given this diffusion coefficient.",
+  "answer_latex": " 0.318",
+  "answer_number": "0.318",
+  "unit": " $\\mathrm{~nm}^2$",
+  "source": "thermo",
+  "problemid": " 17.1",
+  "comment": " "
+ },
+ {
+  "problem_text": "Benzoic acid, $1.35 \\mathrm{~g}$, is reacted with oxygen in a constant volume calorimeter to form $\\mathrm{H}_2 \\mathrm{O}(l)$ and $\\mathrm{CO}_2(g)$ at $298 \\mathrm{~K}$. The mass of the water in the inner bath is $1.55 \\times$ $10^3 \\mathrm{~g}$. The temperature of the calorimeter and its contents rises $2.76 \\mathrm{~K}$ as a result of this reaction. Calculate the calorimeter constant.",
+  "answer_latex": " 6.64",
+  "answer_number": "6.64",
+  "unit": " $10^3 \\mathrm{~J}^{\\circ} \\mathrm{C}^{-1}$\r\n",
+  "source": "thermo",
+  "problemid": " 4.15",
+  "comment": " "
+ },
+ {
+  "problem_text": "The activation energy for a reaction is $50 . \\mathrm{J} \\mathrm{mol}^{-1}$. Determine the effect on the rate constant for this reaction with a change in temperature from $273 \\mathrm{~K}$ to $298 \\mathrm{~K}$.",
+  "answer_latex": " 0.15",
+  "answer_number": "0.15",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 18.37",
+  "comment": " "
+ },
+ {
+  "problem_text": "How long will it take to pass $200 . \\mathrm{mL}$ of $\\mathrm{H}_2$ at $273 \\mathrm{~K}$ through a $10 . \\mathrm{cm}$-long capillary tube of $0.25 \\mathrm{~mm}$ if the gas input and output pressures are 1.05 and $1.00 \\mathrm{~atm}$, respectively?",
+  "answer_latex": " 22",
+  "answer_number": "22",
+  "unit": " $\\mathrm{~s}$",
+  "source": "thermo",
+  "problemid": " 17.21",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the Debye-H\u00fcckel screening length $1 / \\kappa$ at $298 \\mathrm{~K}$ in a $0.0075 \\mathrm{~m}$ solution of $\\mathrm{K}_3 \\mathrm{PO}_4$.",
+  "answer_latex": " 1.4",
+  "answer_number": "1.4",
+  "unit": " $\\mathrm{~nm}$",
+  "source": "thermo",
+  "problemid": " 10.23",
+  "comment": " "
+ },
+ {
+  "problem_text": "A system consisting of $82.5 \\mathrm{~g}$ of liquid water at $300 . \\mathrm{K}$ is heated using an immersion heater at a constant pressure of 1.00 bar. If a current of $1.75 \\mathrm{~A}$ passes through the $25.0 \\mathrm{ohm}$ resistor for 100 .s, what is the final temperature of the water?",
+  "answer_latex": " 322",
+  "answer_number": "322",
+  "unit": "$\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 2.13",
+  "comment": " "
+ },
+ {
+  "problem_text": "For an ensemble consisting of a mole of particles having two energy levels separated by $1000 . \\mathrm{cm}^{-1}$, at what temperature will the internal energy equal $3.00 \\mathrm{~kJ}$ ?",
+  "answer_latex": " 1310",
+  "answer_number": "1310",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 15.4",
+  "comment": " "
+ },
+ {
+  "problem_text": "A muscle fiber contracts by $3.5 \\mathrm{~cm}$ and in doing so lifts a weight. Calculate the work performed by the fiber. Assume the muscle fiber obeys Hooke's law $F=-k x$ with a force constant $k$ of $750 . \\mathrm{N} \\mathrm{m}^{-1}$.",
+  "answer_latex": " 0.46",
+  "answer_number": "0.46",
+  "unit": "$\\mathrm{~J}$",
+  "source": "thermo",
+  "problemid": " 2.10",
+  "comment": " "
+ },
+ {
+  "problem_text": "A gas sample is known to be a mixture of ethane and butane. A bulb having a $230.0 \\mathrm{~cm}^3$ capacity is filled with the gas to a pressure of $97.5 \\times 10^3 \\mathrm{~Pa}$ at $23.1^{\\circ} \\mathrm{C}$. If the mass of the gas in the bulb is $0.3554 \\mathrm{~g}$, what is the mole percent of butane in the mixture?\r\n",
+  "answer_latex": " 32",
+  "answer_number": "32",
+  "unit": " %",
+  "source": "thermo",
+  "problemid": " 1.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "One liter of fully oxygenated blood can carry 0.18 liters of $\\mathrm{O}_2$ measured at $T=298 \\mathrm{~K}$ and $P=1.00 \\mathrm{~atm}$. Calculate the number of moles of $\\mathrm{O}_2$ carried per liter of blood. Hemoglobin, the oxygen transport protein in blood has four oxygen binding sites. How many hemoglobin molecules are required to transport the $\\mathrm{O}_2$ in $1.0 \\mathrm{~L}$ of fully oxygenated blood?",
+  "answer_latex": " 1.11",
+  "answer_number": "1.11",
+  "unit": "$10^{21}$ ",
+  "source": "thermo",
+  "problemid": " 1.6",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider a collection of molecules where each molecule has two nondegenerate energy levels that are separated by $6000 . \\mathrm{cm}^{-1}$. Measurement of the level populations demonstrates that there are exactly 8 times more molecules in the ground state than in the upper state. What is the temperature of the collection?",
+  "answer_latex": " 4152",
+  "answer_number": "4152",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 13.15",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate $\\Delta S^{\\circ}$ for the reaction $3 \\mathrm{H}_2(g)+\\mathrm{N}_2(g) \\rightarrow$ $2 \\mathrm{NH}_3(g)$ at $725 \\mathrm{~K}$. Omit terms in the temperature-dependent heat capacities higher than $T^2 / \\mathrm{K}^2$.",
+  "answer_latex": " -191.2",
+  "answer_number": "-191.2",
+  "unit": " $\\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 5.14",
+  "comment": " "
+ },
+ {
+  "problem_text": "The thermal conductivities of acetylene $\\left(\\mathrm{C}_2 \\mathrm{H}_2\\right)$ and $\\mathrm{N}_2$ at $273 \\mathrm{~K}$ and $1 \\mathrm{~atm}$ are 0.01866 and $0.0240 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~m}^{-1} \\mathrm{~s}^{-1}$, respectively. Based on these data, what is the ratio of the collisional cross section of acetylene relative to $\\mathrm{N}_2$ ?",
+  "answer_latex": " 1.33",
+  "answer_number": "1.33",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 17.15",
+  "comment": " "
+ },
+ {
+  "problem_text": "Consider the gas phase thermal decomposition of 1.0 atm of $\\left(\\mathrm{CH}_3\\right)_3 \\mathrm{COOC}\\left(\\mathrm{CH}_3\\right)_3(\\mathrm{~g})$ to acetone $\\left(\\mathrm{CH}_3\\right)_2 \\mathrm{CO}(\\mathrm{g})$ and ethane $\\left(\\mathrm{C}_2 \\mathrm{H}_6\\right)(\\mathrm{g})$, which occurs with a rate constant of $0.0019 \\mathrm{~s}^{-1}$. After initiation of the reaction, at what time would you expect the pressure to be $1.8 \\mathrm{~atm}$ ?",
+  "answer_latex": " 269",
+  "answer_number": "269",
+  "unit": " $\\mathrm{~s}$",
+  "source": "thermo",
+  "problemid": " 18.39",
+  "comment": " "
+ },
+ {
+  "problem_text": "Autoclaves that are used to sterilize surgical tools require a temperature of $120 .{ }^{\\circ} \\mathrm{C}$ to kill some bacteria. If water is used for this purpose, at what pressure must the autoclave operate?",
+  "answer_latex": "1.95 ",
+  "answer_number": "1.95 ",
+  "unit": " $\\mathrm{~atm}$",
+  "source": "thermo",
+  "problemid": " 8.13",
+  "comment": " "
+ },
+ {
+  "problem_text": "Imagine gaseous $\\mathrm{Ar}$ at $298 \\mathrm{~K}$ confined to move in a two-dimensional plane of area $1.00 \\mathrm{~cm}^2$. What is the value of the translational partition function?",
+  "answer_latex": " 3.9",
+  "answer_number": "3.9",
+  "unit": " $10^{17}$",
+  "source": "thermo",
+  "problemid": " 14.6",
+  "comment": " "
+ },
+ {
+  "problem_text": "Determine the equilibrium constant for the dissociation of sodium at $298 \\mathrm{~K}: \\mathrm{Na}_2(g) \\rightleftharpoons 2 \\mathrm{Na}(g)$. For $\\mathrm{Na}_2$, $B=0.155 \\mathrm{~cm}^{-1}, \\widetilde{\\nu}=159 \\mathrm{~cm}^{-1}$, the dissociation energy is $70.4 \\mathrm{~kJ} / \\mathrm{mol}$, and the ground-state electronic degeneracy for $\\mathrm{Na}$ is 2 .",
+  "answer_latex": " 2.25",
+  "answer_number": "2.25",
+  "unit": " $10^{-9}$",
+  "source": "thermo",
+  "problemid": " 15.47",
+  "comment": " "
+ },
+ {
+  "problem_text": "At $298.15 \\mathrm{~K}, \\Delta G_f^{\\circ}(\\mathrm{HCOOH}, g)=-351.0 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ and $\\Delta G_f^{\\circ}(\\mathrm{HCOOH}, l)=-361.4 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Calculate the vapor pressure of formic acid at this temperature.",
+  "answer_latex": "1.51",
+  "answer_number": "1.51",
+  "unit": " $10^3 \\mathrm{~Pa}$",
+  "source": "thermo",
+  "problemid": " 8.39",
+  "comment": " "
+ },
+ {
+  "problem_text": "The collisional cross section of $\\mathrm{N}_2$ is $0.43 \\mathrm{~nm}^2$. What is the diffusion coefficient of $\\mathrm{N}_2$ at a pressure of $1 \\mathrm{~atm}$ and a temperature of $298 \\mathrm{~K}$ ?",
+  "answer_latex": " 1.06",
+  "answer_number": "1.06",
+  "unit": " $10^{-5} \\mathrm{~m}^2 \\mathrm{~s}^{-1}$",
+  "source": "thermo",
+  "problemid": " 17.2",
+  "comment": " "
+ },
+ {
+  "problem_text": "A vessel contains $1.15 \\mathrm{~g}$ liq $\\mathrm{H}_2 \\mathrm{O}$ in equilibrium with water vapor at $30 .{ }^{\\circ} \\mathrm{C}$. At this temperature, the vapor pressure of $\\mathrm{H}_2 \\mathrm{O}$ is 31.82 torr. What volume increase is necessary for all the water to evaporate?\r\n",
+  "answer_latex": " 37.9",
+  "answer_number": "37.9",
+  "unit": "$\\mathrm{~L}$",
+  "source": "thermo",
+  "problemid": " 1.8",
+  "comment": " "
+ },
+ {
+  "problem_text": "A cell is roughly spherical with a radius of $20.0 \\times 10^{-6} \\mathrm{~m}$. Calculate the work required to expand the cell surface against the surface tension of the surroundings if the radius increases by a factor of three. Assume the cell is surrounded by pure water and that $T=298.15 \\mathrm{~K}$.",
+  "answer_latex": " 2.89",
+  "answer_number": "2.89",
+  "unit": " $10^{-9} \\mathrm{~J}$",
+  "source": "thermo",
+  "problemid": " 8.7",
+  "comment": " "
+ },
+ {
+  "problem_text": "A vessel is filled completely with liquid water and sealed at $13.56^{\\circ} \\mathrm{C}$ and a pressure of 1.00 bar. What is the pressure if the temperature of the system is raised to $82.0^{\\circ} \\mathrm{C}$ ? Under these conditions, $\\beta_{\\text {water }}=2.04 \\times 10^{-4} \\mathrm{~K}^{-1}$, $\\beta_{\\text {vessel }}=1.42 \\times 10^{-4} \\mathrm{~K}^{-1}$, and $\\kappa_{\\text {water }}=4.59 \\times 10^{-5} \\mathrm{bar}^{-1}$.",
+  "answer_latex": " 93.4",
+  "answer_number": "93.4",
+  "unit": "$\\mathrm{~bar}$",
+  "source": "thermo",
+  "problemid": " 3.6",
+  "comment": " "
+ },
+ {
+  "problem_text": "A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height $h$ ) can be obtained by considering the potential energy due to gravity:\r\n$$\r\nP(h)=e^{-m g h / k T}\r\n$$\r\nIn this expression $m$ is the per-particle mass of the gas, $g$ is the acceleration due to gravity, $k$ is a constant equal to $1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}$, and $T$ is temperature. Determine $\\langle h\\rangle$ for methane $\\left(\\mathrm{CH}_4\\right)$ using this distribution function.",
+  "answer_latex": " 1.6",
+  "answer_number": "1.6",
+  "unit": " $10^4 \\mathrm{~m}$",
+  "source": "thermo",
+  "problemid": " 12.32",
+  "comment": " "
+ },
+ {
+  "problem_text": "A camper stranded in snowy weather loses heat by wind convection. The camper is packing emergency rations consisting of $58 \\%$ sucrose, $31 \\%$ fat, and $11 \\%$ protein by weight. Using the data provided in Problem P4.32 and assuming the fat content of the rations can be treated with palmitic acid data and the protein content similarly by the protein data in Problem P4.32, how much emergency rations must the camper consume in order to compensate for a reduction in body temperature of $3.5 \\mathrm{~K}$ ? Assume the heat capacity of the body equals that of water. Assume the camper weighs $67 \\mathrm{~kg}$.",
+  "answer_latex": " 49",
+  "answer_number": "49",
+  "unit": " $\\mathrm{~g}$",
+  "source": "thermo",
+  "problemid": " 4.33",
+  "comment": " "
+ },
+ {
+  "problem_text": "At 303 . K, the vapor pressure of benzene is 120 . Torr and that of hexane is 189 Torr. Calculate the vapor pressure of a solution for which $x_{\\text {benzene }}=0.28$ assuming ideal behavior.",
+  "answer_latex": " 170",
+  "answer_number": "170",
+  "unit": " $\\mathrm{Torr}$",
+  "source": "thermo",
+  "problemid": " 9.8",
+  "comment": " "
+ },
+ {
+  "problem_text": "Determine the molar standard Gibbs energy for ${ }^{35} \\mathrm{Cl}^{35} \\mathrm{Cl}$ where $\\widetilde{\\nu}=560 . \\mathrm{cm}^{-1}, B=0.244 \\mathrm{~cm}^{-1}$, and the ground electronic state is nondegenerate.",
+  "answer_latex": " -57.2",
+  "answer_number": "-57.2",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 15.45",
+  "comment": " "
+ },
+ {
+  "problem_text": "For the reaction $\\mathrm{C}($ graphite $)+\\mathrm{H}_2 \\mathrm{O}(g) \\rightleftharpoons$ $\\mathrm{CO}(g)+\\mathrm{H}_2(g), \\Delta H_R^{\\circ}=131.28 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ at $298.15 \\mathrm{~K}$. Use the values of $C_{P, m}^{\\circ}$ at $298.15 \\mathrm{~K}$ in the data tables to calculate $\\Delta H_R^{\\circ}$ at $125.0^{\\circ} \\mathrm{C}$.",
+  "answer_latex": " 132.9",
+  "answer_number": "132.9",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 6.12",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the mean ionic activity of a $0.0350 \\mathrm{~m} \\mathrm{Na}_3 \\mathrm{PO}_4$ solution for which the mean activity coefficient is 0.685 .",
+  "answer_latex": " 0.0547",
+  "answer_number": "0.0547",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 10.6",
+  "comment": " "
+ },
+ {
+  "problem_text": " Consider the transition between two forms of solid tin, $\\mathrm{Sn}(s$, gray $) \\rightarrow \\mathrm{Sn}(s$, white $)$. The two phases are in equilibrium at 1 bar and $18^{\\circ} \\mathrm{C}$. The densities for gray and white tin are 5750 and $7280 \\mathrm{~kg} \\mathrm{~m}^{-3}$, respectively, and the molar entropies for gray and white tin are 44.14 and $51.18 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$, respectively. Calculate the temperature at which the two phases are in equilibrium at 350. bar.",
+  "answer_latex": " -3.5",
+  "answer_number": "-3.5",
+  "unit": " $^{\\circ} \\mathrm{C}$",
+  "source": "thermo",
+  "problemid": " 8.43",
+  "comment": " "
+ },
+ {
+  "problem_text": "The densities of pure water and ethanol are 997 and $789 \\mathrm{~kg} \\mathrm{~m}^{-3}$, respectively. For $x_{\\text {ethanol }}=0.35$, the partial molar volumes of ethanol and water are 55.2 and $17.8 \\times 10^{-3} \\mathrm{~L} \\mathrm{~mol}^{-1}$, respectively. Calculate the change in volume relative to the pure components when $2.50 \\mathrm{~L}$ of a solution with $x_{\\text {ethanol }}=0.35$ is prepared.",
+  "answer_latex": " -0.10",
+  "answer_number": "-0.10",
+  "unit": " $\\mathrm{~L}$",
+  "source": "thermo",
+  "problemid": " 9.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "For $\\mathrm{N}_2$ at $298 \\mathrm{~K}$, what fraction of molecules has a speed between 200. and $300 . \\mathrm{m} / \\mathrm{s}$ ?",
+  "answer_latex": " 0.132",
+  "answer_number": "0.132",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 16.15",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the pressure exerted by Ar for a molar volume of $1.31 \\mathrm{~L} \\mathrm{~mol}^{-1}$ at $426 \\mathrm{~K}$ using the van der Waals equation of state. The van der Waals parameters $a$ and $b$ for Ar are 1.355 bar dm ${ }^6 \\mathrm{~mol}^{-2}$ and $0.0320 \\mathrm{dm}^3 \\mathrm{~mol}^{-1}$, respectively. Is the attractive or repulsive portion of the potential dominant under these conditions?\r\n",
+  "answer_latex": " 26.9",
+  "answer_number": "26.9",
+  "unit": "$\\mathrm{~bar}$",
+  "source": "thermo",
+  "problemid": " 1.3",
+  "comment": " "
+ },
+ {
+  "problem_text": "For water, $\\Delta H_{\\text {vaporization }}$ is $40.656 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$, and the normal boiling point is $373.12 \\mathrm{~K}$. Calculate the boiling point for water on the top of Mt. Everest (elevation $8848 \\mathrm{~m}$ ), where the barometric pressure is 253 Torr.",
+  "answer_latex": "344",
+  "answer_number": "344",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 8.25",
+  "comment": " "
+ },
+ {
+  "problem_text": "An ideal solution is formed by mixing liquids $\\mathrm{A}$ and $B$ at $298 \\mathrm{~K}$. The vapor pressure of pure A is 151 Torr and that of pure B is 84.3 Torr. If the mole fraction of $\\mathrm{A}$ in the vapor is 0.610 , what is the mole fraction of $\\mathrm{A}$ in the solution?",
+  "answer_latex": " 0.466",
+  "answer_number": "0.466",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 9.24",
+  "comment": " "
+ },
+ {
+  "problem_text": "The mean solar flux at Earth's surface is $\\sim 2.00 \\mathrm{~J}$ $\\mathrm{cm}^{-2} \\mathrm{~min}^{-1}$. In a nonfocusing solar collector, the temperature reaches a value of $79.5^{\\circ} \\mathrm{C}$. A heat engine is operated using the collector as the hot reservoir and a cold reservoir at $298 \\mathrm{~K}$. Calculate the area of the collector needed to produce 1000. W. Assume that the engine operates at the maximum Carnot efficiency.",
+  "answer_latex": " 19.4",
+  "answer_number": "19.4",
+  "unit": " $\\mathrm{~m}^2$",
+  "source": "thermo",
+  "problemid": " 5.42",
+  "comment": " "
+ },
+ {
+  "problem_text": "A hiker caught in a thunderstorm loses heat when her clothing becomes wet. She is packing emergency rations that if completely metabolized will release $35 \\mathrm{~kJ}$ of heat per gram of rations consumed. How much rations must the hiker consume to avoid a reduction in body temperature of $2.5 \\mathrm{~K}$ as a result of heat loss? Assume the heat capacity of the body equals that of water and that the hiker weighs $51 \\mathrm{~kg}$.",
+  "answer_latex": " 15",
+  "answer_number": "15",
+  "unit": "$\\mathrm{~g}$",
+  "source": "thermo",
+  "problemid": " 2.4",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the degree of dissociation of $\\mathrm{N}_2 \\mathrm{O}_4$ in the reaction $\\mathrm{N}_2 \\mathrm{O}_4(g) \\rightleftharpoons 2 \\mathrm{NO}_2(g)$ at 300 . $\\mathrm{K}$ and a total pressure of 1.50 bar. Do you expect the degree of dissociation to increase or decrease as the temperature is increased to 550 . K? Assume that $\\Delta H_R^{\\circ}$ is independent of temperature.",
+  "answer_latex": " 0.241",
+  "answer_number": "0.241",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 6.30",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate $\\Delta G$ for the isothermal expansion of $2.25 \\mathrm{~mol}$ of an ideal gas at $325 \\mathrm{~K}$ from an initial pressure of 12.0 bar to a final pressure of 2.5 bar.",
+  "answer_latex": " -9.54",
+  "answer_number": "-9.54",
+  "unit": " $10^3 \\mathrm{~J}$",
+  "source": "thermo",
+  "problemid": " 6.20",
+  "comment": " "
+ },
+ {
+  "problem_text": "Determine the total collisional frequency for $\\mathrm{CO}_2$ at $1 \\mathrm{~atm}$ and $298 \\mathrm{~K}$.",
+  "answer_latex": " 8.44",
+  "answer_number": "8.44",
+  "unit": " $10^{34} \\mathrm{~m}^{-3} \\mathrm{~s}^{-1}$",
+  "source": "thermo",
+  "problemid": " 16.30",
+  "comment": " "
+ },
+ {
+  "problem_text": "The volatile liquids $A$ and $\\mathrm{B}$, for which $P_A^*=165$ Torr and $P_B^*=85.1$ Torr are confined to a piston and cylinder assembly. Initially, only the liquid phase is present. As the pressure is reduced, the first vapor is observed at a total pressure of 110 . Torr. Calculate $x_{\\mathrm{A}}$",
+  "answer_latex": " 0.312",
+  "answer_number": "0.312",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 9.9",
+  "comment": " "
+ },
+ {
+  "problem_text": "The osmotic pressure of an unknown substance is measured at $298 \\mathrm{~K}$. Determine the molecular weight if the concentration of this substance is $31.2 \\mathrm{~kg} \\mathrm{~m}^{-3}$ and the osmotic pressure is $5.30 \\times 10^4 \\mathrm{~Pa}$. The density of the solution is $997 \\mathrm{~kg} \\mathrm{~m}^{-3}$.",
+  "answer_latex": " 1.45",
+  "answer_number": "1.45",
+  "unit": " $10^3 \\mathrm{~g} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 9.7",
+  "comment": " "
+ },
+ {
+  "problem_text": " One mole of Ar initially at 310 . K undergoes an adiabatic expansion against a pressure $P_{\\text {external }}=0$ from a volume of $8.5 \\mathrm{~L}$ to a volume of $82.0 \\mathrm{~L}$. Calculate the final temperature using the ideal gas",
+  "answer_latex": " 310",
+  "answer_number": "310",
+  "unit": "$\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 7.4",
+  "comment": " "
+ },
+ {
+  "problem_text": "A refrigerator is operated by a $0.25-\\mathrm{hp}(1 \\mathrm{hp}=$ 746 watts) motor. If the interior is to be maintained at $4.50^{\\circ} \\mathrm{C}$ and the room temperature on a hot day is $38^{\\circ} \\mathrm{C}$, what is the maximum heat leak (in watts) that can be tolerated? Assume that the coefficient of performance is $50 . \\%$ of the maximum theoretical value.",
+  "answer_latex": " 773",
+  "answer_number": "773",
+  "unit": " $\\mathrm{~J} \\mathrm{~s}^{-1}$",
+  "source": "thermo",
+  "problemid": " 5.33",
+  "comment": " "
+ },
+ {
+  "problem_text": "In order to get in shape for mountain climbing, an avid hiker with a mass of $60 . \\mathrm{kg}$ ascends the stairs in the world's tallest structure, the $828 \\mathrm{~m}$ tall Burj Khalifa in Dubai, United Arab Emirates. Assume that she eats energy bars on the way up and that her body is $25 \\%$ efficient in converting the energy content of the bars into the work of climbing. How many energy bars does she have to eat if a single bar produces $1.08 \\times 10^3 \\mathrm{~kJ}$ of energy upon metabolizing?",
+  "answer_latex": " 1.8",
+  "answer_number": "1.8",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 4.34",
+  "comment": " "
+ },
+ {
+  "problem_text": "The half-life of ${ }^{238} \\mathrm{U}$ is $4.5 \\times 10^9$ years. How many disintegrations occur in $1 \\mathrm{~min}$ for a $10 \\mathrm{mg}$ sample of this element?",
+  "answer_latex": " 1.43",
+  "answer_number": "1.43",
+  "unit": " $10^{24}$",
+  "source": "thermo",
+  "problemid": " 18.14",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the ionic strength in a solution that is 0.0750 $m$ in $\\mathrm{K}_2 \\mathrm{SO}_4, 0.0085 \\mathrm{~m}$ in $\\mathrm{Na}_3 \\mathrm{PO}_4$, and $0.0150 \\mathrm{~m}$ in $\\mathrm{MgCl}_2$.",
+  "answer_latex": " 0.321",
+  "answer_number": "0.321",
+  "unit": " $\\mathrm{~mol} \\mathrm{~kg}^{-1}$",
+  "source": "thermo",
+  "problemid": " 10.13",
+  "comment": " "
+ },
+ {
+  "problem_text": "The interior of a refrigerator is typically held at $36^{\\circ} \\mathrm{F}$ and the interior of a freezer is typically held at $0.00^{\\circ} \\mathrm{F}$. If the room temperature is $65^{\\circ} \\mathrm{F}$, by what factor is it more expensive to extract the same amount of heat from the freezer than from the refrigerator? Assume that the theoretical limit for the performance of a reversible refrigerator is valid in this case.",
+  "answer_latex": " 2.4",
+  "answer_number": "2.4",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 5.17",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate the rotational partition function for $\\mathrm{SO}_2$ at $298 \\mathrm{~K}$ where $B_A=2.03 \\mathrm{~cm}^{-1}, B_B=0.344 \\mathrm{~cm}^{-1}$, and $B_C=0.293 \\mathrm{~cm}^{-1}$",
+  "answer_latex": " 5840",
+  "answer_number": "5840",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 14.16",
+  "comment": " "
+ },
+ {
+  "problem_text": "For a two-level system where $v=1.50 \\times 10^{13} \\mathrm{~s}^{-1}$, determine the temperature at which the internal energy is equal to $0.25 \\mathrm{Nhv}$, or $1 / 2$ the limiting value of $0.50 \\mathrm{Nhv}$.",
+  "answer_latex": " 655",
+  "answer_number": "655",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 15.2",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate $K_P$ at $600 . \\mathrm{K}$ for the reaction $\\mathrm{N}_2 \\mathrm{O}_4(l) \\rightleftharpoons 2 \\mathrm{NO}_2(g)$ assuming that $\\Delta H_R^{\\circ}$ is constant over the interval 298-725 K.",
+  "answer_latex": " 4.76",
+  "answer_number": "4.76",
+  "unit": " $10^6$",
+  "source": "thermo",
+  "problemid": " 6.10",
+  "comment": " "
+ },
+ {
+  "problem_text": "Count Rumford observed that using cannon boring machinery a single horse could heat $11.6 \\mathrm{~kg}$ of ice water $(T=273 \\mathrm{~K})$ to $T=355 \\mathrm{~K}$ in 2.5 hours. Assuming the same rate of work, how high could a horse raise a $225 \\mathrm{~kg}$ weight in 2.5 minutes? Assume the heat capacity of water is $4.18 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~g}^{-1}$.",
+  "answer_latex": " 30",
+  "answer_number": "30",
+  "unit": "$\\mathrm{~m}$",
+  "source": "thermo",
+  "problemid": " 2.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "The vibrational frequency of $I_2$ is $208 \\mathrm{~cm}^{-1}$. At what temperature will the population in the first excited state be half that of the ground state?",
+  "answer_latex": " 432",
+  "answer_number": "432",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 13.22",
+  "comment": " "
+ },
+ {
+  "problem_text": "One mole of $\\mathrm{H}_2 \\mathrm{O}(l)$ is compressed from a state described by $P=1.00$ bar and $T=350$. K to a state described by $P=590$. bar and $T=750$. K. In addition, $\\beta=2.07 \\times 10^{-4} \\mathrm{~K}^{-1}$ and the density can be assumed to be constant at the value $997 \\mathrm{~kg} \\mathrm{~m}^{-3}$. Calculate $\\Delta S$ for this transformation, assuming that $\\kappa=0$.\r\n",
+  "answer_latex": " 57.2",
+  "answer_number": "57.2",
+  "unit": " $\\mathrm{~K}^{-1}$",
+  "source": "thermo",
+  "problemid": " 5.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "A mass of $34.05 \\mathrm{~g}$ of $\\mathrm{H}_2 \\mathrm{O}(s)$ at $273 \\mathrm{~K}$ is dropped into $185 \\mathrm{~g}$ of $\\mathrm{H}_2 \\mathrm{O}(l)$ at $310 . \\mathrm{K}$ in an insulated container at 1 bar of pressure. Calculate the temperature of the system once equilibrium has been reached. Assume that $C_{P, m}$ for $\\mathrm{H}_2 \\mathrm{O}(l)$ is constant at its values for $298 \\mathrm{~K}$ throughout the temperature range of interest.",
+  "answer_latex": " 292",
+  "answer_number": "292",
+  "unit": "$\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 3.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "Calculate $\\Delta H_f^{\\circ}$ for $N O(g)$ at $975 \\mathrm{~K}$, assuming that the heat capacities of reactants and products are constant over the temperature interval at their values at $298.15 \\mathrm{~K}$.",
+  "answer_latex": " 91.7",
+  "answer_number": "91.7",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 4.4",
+  "comment": " "
+ },
+ {
+  "problem_text": "A two-level system is characterized by an energy separation of $1.30 \\times 10^{-18} \\mathrm{~J}$. At what temperature will the population of the ground state be 5 times greater than that of the excited state?",
+  "answer_latex": " 5.85",
+  "answer_number": "5.85",
+  "unit": " $10^4$",
+  "source": "thermo",
+  "problemid": " 13.27",
+  "comment": " "
+ },
+ {
+  "problem_text": "At what temperature are there Avogadro's number of translational states available for $\\mathrm{O}_2$ confined to a volume of 1000. $\\mathrm{cm}^3$ ?",
+  "answer_latex": " 0.068",
+  "answer_number": "0.068",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 14.5",
+  "comment": " "
+ },
+ {
+  "problem_text": "The half-cell potential for the reaction $\\mathrm{O}_2(g)+4 \\mathrm{H}^{+}(a q)+4 \\mathrm{e}^{-} \\rightarrow 2 \\mathrm{H}_2 \\mathrm{O}(l)$ is $+1.03 \\mathrm{~V}$ at $298.15 \\mathrm{~K}$ when $a_{\\mathrm{O}_2}=1.00$. Determine $a_{\\mathrm{H}^{+}}$",
+  "answer_latex": " 4.16",
+  "answer_number": "4.16",
+  "unit": " $10^{-4}$",
+  "source": "thermo",
+  "problemid": " 11.25",
+  "comment": " "
+ },
+ {
+  "problem_text": "The partial molar volumes of water and ethanol in a solution with $x_{\\mathrm{H}_2 \\mathrm{O}}=0.45$ at $25^{\\circ} \\mathrm{C}$ are 17.0 and $57.5 \\mathrm{~cm}^3 \\mathrm{~mol}^{-1}$, respectively. Calculate the volume change upon mixing sufficient ethanol with $3.75 \\mathrm{~mol}$ of water to give this concentration. The densities of water and ethanol are 0.997 and $0.7893 \\mathrm{~g} \\mathrm{~cm}^{-3}$, respectively, at this temperature.",
+  "answer_latex": " -8",
+  "answer_number": "-8",
+  "unit": " $\\mathrm{~cm}^3$",
+  "source": "thermo",
+  "problemid": " 9.5",
+  "comment": " "
+ }
+]

thermo_sol.json

@@ -1 +1,172 @@
-[{"problem_text": "In this problem, 3.00 mol of liquid mercury is transformed from an initial state characterized by $T_i=300 . \\mathrm{K}$ and $P_i=1.00$ bar to a final state characterized by $T_f=600 . \\mathrm{K}$ and $P_f=3.00$ bar.\r\na. Calculate $\\Delta S$ for this process; $\\beta=1.81 \\times 10^{-4} \\mathrm{~K}^{-1}, \\rho=13.54 \\mathrm{~g} \\mathrm{~cm}^{-3}$, and $C_{P, m}$ for $\\mathrm{Hg}(l)=27.98 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$.", "answer_latex": " 58.2", "answer_number": "58.2", "unit": "$\\mathrm{~J}\\mathrm{~K}^{-1}$", "source": "thermo", "problemid": " 5.6", "comment": " ", "solution": "a. Because the volume changes only slightly with temperature and pressure over the range indicated,\r\n$$\r\n\\begin{aligned}\r\n& \\Delta S=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-\\int_{P_i}^{P_f} V \\beta d P \\approx n C_{P, m} \\ln \\frac{T_f}{T_i}-n V_{m, i} \\beta\\left(P_f-P_i\\right) \\\\\r\n& =3.00 \\mathrm{~mol} \\times 27.98 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times \\ln \\frac{600 . \\mathrm{K}}{300 . \\mathrm{K}} \\\\\r\n& -3.00 \\mathrm{~mol} \\times \\frac{200.59 \\mathrm{~g} \\mathrm{~mol}^{-1}}{13.54 \\mathrm{~g} \\mathrm{~cm}^{-3} \\times \\frac{10^6 \\mathrm{~cm}^3}{\\mathrm{~m}^3}} \\times 1.81 \\times 10^{-4} \\mathrm{~K}^{-1} \\times 2.00 \\mathrm{bar} \\\\\r\n& \\times 10^5 \\mathrm{Pabar}^{-1} \\\\\r\n& =58.2 \\mathrm{~J} \\mathrm{~K}^{-1}-1.61 \\times 10^{-3} \\mathrm{~J} \\mathrm{~K}^{-1}=58.2 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$"}, {"problem_text": "For an ensemble consisting of 1.00 moles of particles having two energy levels separated by $h v=1.00 \\times 10^{-20} \\mathrm{~J}$, at what temperature will the internal energy of this system equal $1.00 \\mathrm{~kJ}$ ?", "answer_latex": " 449", "answer_number": "449", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 15.2", "comment": " ", "solution": "Using the expression for total energy and recognizing that $N=n N_A$,\r\n$$\r\nU=-\\left(\\frac{\\partial \\ln Q}{\\partial \\beta}\\right)_V=-n N_A\\left(\\frac{\\partial \\ln q}{\\partial \\beta}\\right)_V\r\n$$\r\nEvaluating the preceding expression and paying particular attention to units, we get\r\n$$\r\n\\begin{aligned}\r\n& U=-n N_A\\left(\\frac{\\partial}{\\partial \\beta} \\ln q\\right)_V=-\\frac{n N_A}{q}\\left(\\frac{\\partial q}{\\partial \\beta}\\right)_V \\\\\r\n& \\frac{U}{n N_A}=\\frac{-1}{\\left(1+e^{-\\beta h \\nu}\\right)}\\left(\\frac{\\partial}{\\partial \\beta}\\left(1+e^{-\\beta h \\nu}\\right)\\right)_V \\\\\r\n&=\\frac{h \\nu e^{-\\beta h \\nu}}{1+e^{-\\beta h \\nu}}=\\frac{h \\nu}{e^{\\beta h \\nu}+1} \\\\\r\n& \\frac{n N_A h \\nu}{U}-1=e^{\\beta h \\nu} \\\\\r\n& \\ln \\left(\\frac{n N_A h \\nu}{U}-1\\right)=\\beta h \\nu=\\frac{h \\nu}{k T}\r\n\\end{aligned}\r\n$$\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{h \\nu}{k \\ln \\left(\\frac{n N_A h \\nu}{U}-1\\right)} \\\\\r\n= & \\frac{1.00 \\times 10^{-20} \\mathrm{~J}}{\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\ln \\left(\\frac{(1.00 \\mathrm{~mol})\\left(6.022 \\times 10^{23} \\mathrm{~mol}^{-1}\\right)\\left(1.00 \\times 10^{-20} \\mathrm{~J}\\right)}{\\left(1.00 \\times 10^3 \\mathrm{~J}\\right)}-1\\right)} \\\\\r\n& =449 \\mathrm{~K}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "The sedimentation coefficient of lysozyme $\\left(\\mathrm{M}=14,100 \\mathrm{~g} \\mathrm{~mol}^{-1}\\right)$ in water at $20^{\\circ} \\mathrm{C}$ is $1.91 \\times 10^{-13} \\mathrm{~s}$ and the specific volume is $0.703 \\mathrm{~cm}^3 \\mathrm{~g}^{-1}$. The density of water at this temperature is $0.998 \\mathrm{~g} \\mathrm{~cm}^{-3}$ and $\\eta=1.002 \\mathrm{cP}$. Assuming lysozyme is spherical, what is the radius of this protein?", "answer_latex": "1.94", "answer_number": "1.94", "unit": " $\\mathrm{~nm}$", "source": "thermo", "problemid": " 17.9", "comment": " ", "solution": "The frictional coefficient is dependent on molecular radius. Solving Equation (17.56) for $f$, we obtain\r\n$$\r\n\\begin{aligned}\r\n& f=\\frac{m(1-\\overline{\\mathrm{V}} \\rho)}{\\overline{\\mathrm{s}}}=\\frac{\\frac{\\left(14,100 \\mathrm{~g} \\mathrm{~mol}^{-1}\\right)}{\\left(6.022 \\times 10^{23} \\mathrm{~mol}^{-1}\\right)}\\left(1-(0.703 \\mathrm{~mL} \\mathrm{~g})\\left(0.998 \\mathrm{~g} \\mathrm{~mL}^{-1}\\right)\\right)}{1.91 \\times 10^{-13} \\mathrm{~s}} \\\\\r\n& =3.66 \\times 10^{-8} \\mathrm{~g} \\mathrm{~s}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nThe frictional coefficient is related to the radius for a spherical particle by Equation (17.51) such that\r\n$$\r\nr=\\frac{f}{6 \\pi \\eta}=\\frac{3.66 \\times 10^{-8} \\mathrm{~g} \\mathrm{~s}^{-1}}{6 \\pi\\left(1.002 \\mathrm{~g} \\mathrm{~m}^{-1} \\mathrm{~s}^{-1}\\right)}=1.94 \\times 10^{-9} \\mathrm{~m}=1.94 \\mathrm{~nm}\r\n$$"}, {"problem_text": "Determine the standard molar entropy of $\\mathrm{Ne}$ and $\\mathrm{Kr}$ under standard thermodynamic conditions.", "answer_latex": "164 ", "answer_number": "164", "unit": " $ \\mathrm{Jmol}^{-1} \\mathrm{~K}^{-1}$", "source": "thermo", "problemid": " 15.5", "comment": " ", "solution": "Beginning with the expression for entropy [see Equation (15.63)]:\r\n$$\r\n\\begin{aligned}\r\nS & =\\frac{5}{2} R+R \\ln \\left(\\frac{V}{\\Lambda^3}\\right)-R \\ln N_A \\\\\r\n& =\\frac{5}{2} R+R \\ln \\left(\\frac{V}{\\Lambda^3}\\right)-54.75 R \\\\\r\n& =R \\ln \\left(\\frac{V}{\\Lambda^3}\\right)-52.25 R\r\n\\end{aligned}\r\n$$\r\nThe conventional standard state is defined by $T=298 \\mathrm{~K}$ and $V_m=24.4 \\mathrm{~L}\\left(0.0244 \\mathrm{~m}^3\\right)$. The thermal wavelength for $\\mathrm{Ne}$ is\r\n$$\r\n\\begin{aligned}\r\n\\Lambda & =\\left(\\frac{h^2}{2 \\pi m k T}\\right)^{1 / 2} \\\\\r\n& =\\left(\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)^2}{2 \\pi\\left(\\frac{0.02018 \\mathrm{~kg} \\mathrm{~mol}^{-1}}{N_A}\\right)\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)(298 \\mathrm{~K})}\\right)^{1 / 2} \\\\\r\n& =2.25 \\times 10^{-11} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\nUsing this value for the thermal wavelength, the entropy becomes\r\n$$\r\n\\begin{aligned}\r\nS & =R \\ln \\left(\\frac{0.0244 \\mathrm{~m}^3}{\\left(2.25 \\times 10^{-11} \\mathrm{~m}\\right)^3}\\right)-52.25 R \\\\\r\n& =69.84 R-52.25 R=17.59 R=146 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}\r\n\\end{aligned}\r\n$$\r\nThe experimental value is $146.48 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$ ! Rather than determining the entropy of $\\mathrm{Kr}$ directly, it is easier to determine the difference in entropy relative to Ne:\r\n$$\r\n\\begin{aligned}\r\n\\Delta S & =S_{K r}-S_{N e}=S=R \\ln \\left(\\frac{V}{\\Lambda_{K r}^3}\\right)-R \\ln \\left(\\frac{V}{\\Lambda_{N e}^3}\\right) \\\\\r\n& =R \\ln \\left(\\frac{\\Lambda_{N e}}{\\Lambda_{K r}}\\right)^3 \\\\\r\n& =3 R \\ln \\left(\\frac{\\Lambda_{N e}}{\\Lambda_{K r}}\\right) \\\\\r\n& =3 R \\ln \\left(\\frac{m_{K r}}{m_{N e}}\\right)^{1 / 2} \\\\\r\n& =\\frac{3}{2} R \\ln \\left(\\frac{m_{K r}}{m_{N e}}\\right)=\\frac{3}{2} R \\ln (4.15) \\\\\r\n& =17.7 \\mathrm{Jmol} ^{-1}\\mathrm{K}^{-1}\r\n\\end{aligned}\r\n$$\r\nUsing this difference, the standard molar entropy of $\\mathrm{Kr}$ becomes\r\n$$\r\nS_{K r}=\\Delta S+S_{N e}=164 \\mathrm{Jmol}^{-1} \\mathrm{~K}^{-1}\r\n$$\r\nThe experimental value is $163.89 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$, and the calculated value is again in excellent agreement!"}, {"problem_text": "Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \\mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \\mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \\mathrm{C}$ decay events per minute. How old is the wood?", "answer_latex": "4.86 ", "answer_number": "4.86", "unit": " $10^{11} \\mathrm{~s}$", "source": "thermo", "problemid": " 18.4", "comment": " ", "solution": "The ratio of decay events yields the amount of ${ }^{14} \\mathrm{C}$ present currently versus the amount that was present when the tree died:\r\n$$\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}=\\frac{2.40 \\mathrm{~min}^{-1}}{15.3 \\mathrm{~min}^{-1}}=0.157\r\n$$\r\nThe rate constant for isotope decay is related to the half-life as follows:\r\n$$\r\nk=\\frac{\\ln 2}{t_{1 / 2}}=\\frac{\\ln 2}{5760 \\text { years }}=\\frac{\\ln 2}{1.82 \\times 10^{11} \\mathrm{~s}}=3.81 \\times 10^{-12} \\mathrm{~s}^{-1}\r\n$$\r\nWith the rate constant and ratio of isotope concentrations, the age of the fossilized wood is readily determined:\r\n$$\r\n\\begin{aligned}\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0} & =e^{-k t} \\\\\r\n\\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-k t \\\\\r\n-\\frac{1}{k} \\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-\\frac{1}{3.81 \\times 10^{-12} \\mathrm{~s}} \\ln (0.157)=t \\\\\r\n4.86 \\times 10^{11} \\mathrm{~s} & =t\r\n\\end{aligned}\r\n$$\r\nThis time corresponds to an age of roughly 15,400 years."}, {"problem_text": "Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \\mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \\mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \\mathrm{C}$ decay events per minute. How old is the wood?", "answer_latex": "4.86 ", "answer_number": "4.86", "unit": " $10^{11} \\mathrm{~s}$", "source": "thermo", "problemid": " 18.4", "comment": " ", "solution": "The ratio of decay events yields the amount of ${ }^{14} \\mathrm{C}$ present currently versus the amount that was present when the tree died:\r\n$$\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}=\\frac{2.40 \\mathrm{~min}^{-1}}{15.3 \\mathrm{~min}^{-1}}=0.157\r\n$$\r\nThe rate constant for isotope decay is related to the half-life as follows:\r\n$$\r\nk=\\frac{\\ln 2}{t_{1 / 2}}=\\frac{\\ln 2}{5760 \\text { years }}=\\frac{\\ln 2}{1.82 \\times 10^{11} \\mathrm{~s}}=3.81 \\times 10^{-12} \\mathrm{~s}^{-1}\r\n$$\r\nWith the rate constant and ratio of isotope concentrations, the age of the fossilized wood is readily determined:\r\n$$\r\n\\begin{aligned}\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0} & =e^{-k t} \\\\\r\n\\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-k t \\\\\r\n-\\frac{1}{k} \\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-\\frac{1}{3.81 \\times 10^{-12} \\mathrm{~s}} \\ln (0.157)=t \\\\\r\n4.86 \\times 10^{11} \\mathrm{~s} & =t\r\n\\end{aligned}\r\n$$\r\nThis time corresponds to an age of roughly 15,400 years."}, {"problem_text": "Determine the diffusion coefficient for Ar at $298 \\mathrm{~K}$ and a pressure of $1.00 \\mathrm{~atm}$.", "answer_latex": "1.1 ", "answer_number": "1.1", "unit": " $10^{-5}~\\mathrm{m^2~s^{-1}}$", "source": "thermo", "problemid": " 17.1", "comment": " ", "solution": "\ufffc\ufffcUsing Equation (17.10) and the collisional cross section for Ar provided in Table 17.1,\r\n $$\r\n\\begin{aligned}\r\nD_{Ar} &= \\frac{1}{3} \\nu_{ave, Ar} \\lambda_{Ar} \\\\\r\n&= \\frac{1}{3} \\left(\\frac{8RT}{\\pi M_{Ar}}\\right)^{\\frac{1}{2}} \\left(\\frac{RT}{PN_A\\sqrt{2}\\sigma_{Ar}}\\right) \\\\\r\n&= \\frac{1}{3} \\left(\\frac{8(8.314~\\mathrm{J~mol^{-1}~K^{-1}}) \\times 298~\\mathrm{K}}{\\pi(0.040~\\mathrm{kg~mol^{-1}})}\\right)^{\\frac{1}{2}} \\\\\r\n&\\quad \\times \\left(\\frac{(8.314~\\mathrm{J~mol^{-1}~K^{-1}}) \\times 298~\\mathrm{K}}{(101,325~\\mathrm{Pa}) \\times (6.022 \\times 10^{23}~\\mathrm{mol^{-1}})} \\times \\frac{1}{\\sqrt{2}(3.6 \\times 10^{-19}~\\mathrm{m^2})}\\right) \\\\\r\n&= \\frac{1}{3} \\times (397~\\mathrm{m~s^{-1}}) \\times (7.98 \\times 10^{-8}~\\mathrm{m}) \\\\\r\n&= 1.1 \\times 10^{-5}~\\mathrm{m^2~s^{-1}}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "Gas cylinders of $\\mathrm{CO}_2$ are sold in terms of weight of $\\mathrm{CO}_2$. A cylinder contains $50 \\mathrm{lb}$ (22.7 $\\mathrm{kg}$ ) of $\\mathrm{CO}_2$. How long can this cylinder be used in an experiment that requires flowing $\\mathrm{CO}_2$ at $293 \\mathrm{~K}(\\eta=146 \\mu \\mathrm{P})$ through a 1.00-m-long tube (diameter $\\left.=0.75 \\mathrm{~mm}\\right)$ with an input pressure of $1.05 \\mathrm{~atm}$ and output pressure of $1.00 \\mathrm{~atm}$ ? The flow is measured at the tube output.", "answer_latex": "4.49 ", "answer_number": "4.49", "unit": " $10^6 \\mathrm{~s}$", "source": "thermo", "problemid": "17.7 ", "comment": " ", "solution": "Using Equation (17.45), the gas flow rate $\\Delta V / \\Delta t$ is\r\n$$\r\n\\begin{aligned}\r\n\\frac{\\Delta V}{\\Delta t}= & \\frac{\\pi r^4}{16 \\eta L P_0}\\left(P_2^2-P_1^2\\right) \\\\\r\n= & \\frac{\\pi\\left(0.375 \\times 10^{-3} \\mathrm{~m}\\right)^4}{16\\left(1.46 \\times 10^{-5} \\mathrm{~kg} \\mathrm{~m}^{-1} \\mathrm{~s}^{-1}\\right)(1.00 \\mathrm{~m})(101,325 \\mathrm{~Pa})} \\\\\r\n& \\times\\left((106,391 \\mathrm{~Pa})^2-(101,325 \\mathrm{~Pa})^2\\right) \\\\\r\n= & 2.76 \\times 10^{-6} \\mathrm{~m}^3 \\mathrm{~s}^{-1}\r\n\\end{aligned}\r\n$$\r\nConverting the $\\mathrm{CO}_2$ contained in the cylinder to the volume occupied at $298 \\mathrm{~K}$ and 1 atm pressure, we get\r\n$$\r\n\\begin{aligned}\r\nn_{\\mathrm{CO}_2} & =22.7 \\mathrm{~kg}\\left(\\frac{1}{0.044 \\mathrm{~kg} \\mathrm{~mol}^{-1}}\\right)=516 \\mathrm{~mol} \\\\\r\nV & =\\frac{n R T}{P}=1.24 \\times 10^4 \\mathrm{~L}\\left(\\frac{10^{-3} \\mathrm{~m}^3}{\\mathrm{~L}}\\right)=12.4 \\mathrm{~m}^3\r\n\\end{aligned}\r\n$$\r\nGiven the effective volume of $\\mathrm{CO}_2$ contained in the cylinder, the duration over which the cylinder can be used is\r\n$$\r\n\\frac{12.4 \\mathrm{~m}^3}{2.76 \\times 10^{-6} \\mathrm{~m}^3 \\mathrm{~s}^{-1}}=4.49 \\times 10^6 \\mathrm{~s}\r\n$$\r\nThis time corresponds to roughly 52 days."}, {"problem_text": "The vibrational frequency of $I_2$ is $208 \\mathrm{~cm}^{-1}$. What is the probability of $I_2$ populating the $n=2$ vibrational level if the molecular temperature is $298 \\mathrm{~K}$ ?", "answer_latex": " 0.086", "answer_number": "0.086", "unit": " ", "source": "thermo", "problemid": " 13.5", "comment": " ", "solution": "Molecular vibrational energy levels can be modeled as harmonic oscillators; therefore, this problem can be solved by employing a strategy identical to the one just presented. To evaluate the partition function $q$, the \"trick\" used earlier was to write the partition function as a series and use the equivalent series expression:\r\n$$\r\n\\begin{aligned}\r\nq & =\\sum_n e^{-\\beta \\varepsilon_n}=1+e^{-\\beta h c \\widetilde{\\nu}}+e^{-2 \\beta h c \\tilde{\\nu}}+e^{-3 \\beta h c \\widetilde{\\nu}}+\\ldots \\\\\r\n& =\\frac{1}{1-e^{-\\beta h c \\widetilde{\\nu}}}\r\n\\end{aligned}\r\n$$\r\nSince $\\tilde{\\nu}=208 \\mathrm{~cm}^{-1}$ and $T=298 \\mathrm{~K}$, the partition function is\r\n$$\r\n\\begin{aligned}\r\nq & =\\frac{1}{1-e^{-\\beta h c \\widetilde{\\nu}}} \\\\\r\n& =\\frac{1}{1-e^{-h c \\widetilde{\\nu} / k T}} \\\\\r\n& =\\frac{1}{1-\\exp \\left[-\\left(\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right)\\left(3.00 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right)\\left(208 \\mathrm{~cm}^{-1}\\right)}{\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)(298 \\mathrm{~K})}\\right)\\right]} \\\\\r\n& =\\frac{1}{1-e^{-1}}=1.58\r\n\\end{aligned}\r\n$$\r\nThis result is then used to evaluate the probability of occupying the second vibrational state $(n=2)$ as follows:\r\n$$\r\n\\begin{aligned}\r\np_2 & =\\frac{e^{-2 \\beta h c \\tilde{\\nu}}}{q} \\\\\r\n& =\\frac{\\exp \\left[-2\\left(\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}^{-1}\\right)\\left(3.00 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right)\\left(208 \\mathrm{~cm}^{-1}\\right)}{\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)(298 \\mathrm{~K})}\\right)\\right]}{1.58} \\\\\r\n& =0.086\r\n\\end{aligned}\r\n$$"}, {"problem_text": "The value of $\\Delta G_f^{\\circ}$ for $\\mathrm{Fe}(g)$ is $370.7 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ at $298.15 \\mathrm{~K}$, and $\\Delta H_f^{\\circ}$ for $\\mathrm{Fe}(g)$ is $416.3 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ at the same temperature. Assuming that $\\Delta H_f^{\\circ}$ is constant in the interval $250-400 \\mathrm{~K}$, calculate $\\Delta G_f^{\\circ}$ for $\\mathrm{Fe}(g)$ at 400. K.", "answer_latex": "355.1 ", "answer_number": "355.1", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "source": "thermo", "problemid": " 6.4", "comment": "\n ", "solution": "$$\r\n\\begin{aligned}\r\n\\Delta G_f^{\\circ}\\left(T_2\\right) & =T_2\\left[\\frac{\\Delta G_f^{\\circ}\\left(T_1\\right)}{T_1}+\\Delta H_f^{\\circ}\\left(T_1\\right) \\times\\left(\\frac{1}{T_2}-\\frac{1}{T_1}\\right)\\right] \\\\\r\n& =400 . \\mathrm{K} \\times\\left[\\begin{array}{l}\r\n\\left.\\frac{370.7 \\times 10^3 \\mathrm{~J} \\mathrm{~mol}^{-1}}{298.15 \\mathrm{~K}}+416.3 \\times 10^3 \\mathrm{~J} \\mathrm{~mol}^{-1}\\right] \\\\\r\n\\times\\left(\\frac{1}{400 . \\mathrm{K}}-\\frac{1}{298.15 \\mathrm{~K}}\\right)\r\n\\end{array}\\right] \\\\\r\n\\Delta G_f^{\\circ}(400 . \\mathrm{K}) & =355.1 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "The reactant 1,3-cyclohexadiene can be photochemically converted to cis-hexatriene. In an experiment, $2.5 \\mathrm{mmol}$ of cyclohexadiene are converted to cis-hexatriene when irradiated with 100. W of 280. nm light for $27.0 \\mathrm{~s}$. All of the light is absorbed by the sample. What is the overall quantum yield for this photochemical process?", "answer_latex": " 0.396", "answer_number": "0.396", "unit": " ", "source": "thermo", "problemid": " 19.6", "comment": " ", "solution": "First, the total photon energy absorbed by the sample, $E_{a b s}$, is\r\n$$\r\nE_{a b s}=(\\text { power }) \\Delta t=\\left(100 . \\mathrm{Js}^{-1}\\right)(27.0 \\mathrm{~s})=2.70 \\times 10^3 \\mathrm{~J}\r\n$$\r\nNext, the photon energy at $280 . \\mathrm{nm}$ is\r\n$$\r\n\\mathrm{E}_{p h}=\\frac{h c}{\\lambda}=\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right)\\left(2.998 \\times 10^8 \\mathrm{~ms}^{-1}\\right)}{2.80 \\times 10^{-7} \\mathrm{~m}}=7.10 \\times 10^{-19} \\mathrm{~J}\r\n$$\r\nThe total number of photons absorbed by the sample is therefore\r\n$$\r\n\\frac{E_{a b s}}{E_{p h}}=\\frac{2.70 \\times 10^3 \\mathrm{~J}}{7.10 \\times 10^{-19} \\mathrm{~J} \\text { photon }^{-1}}=3.80 \\times 10^{21} \\text { photons }\r\n$$\r\nDividing this result by Avogadro's number results in $6.31 \\times 10^{-3}$ Einsteins or moles of photons. Therefore, the overall quantum yield is\r\n$$\r\n\\phi=\\frac{\\text { moles }_{\\text {react }}}{\\text { moles }_{\\text {photon }}}=\\frac{2.50 \\times 10^{-3} \\mathrm{~mol}}{6.31 \\times 10^{-3} \\mathrm{~mol}}=0.396 \\approx 0.40\r\n$$"}, {"problem_text": "In this problem, $2.50 \\mathrm{~mol}$ of $\\mathrm{CO}_2$ gas is transformed from an initial state characterized by $T_i=450 . \\mathrm{K}$ and $P_i=1.35$ bar to a final state characterized by $T_f=800 . \\mathrm{K}$ and $P_f=$ 3.45 bar. Using Equation (5.23), calculate $\\Delta S$ for this process. Assume ideal gas behavior and use the ideal gas value for $\\beta$. For $\\mathrm{CO}_2$,\r\n$$\r\n\\frac{C_{P, m}}{\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}}=18.86+7.937 \\times 10^{-2} \\frac{T}{\\mathrm{~K}}-6.7834 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}+2.4426 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\r\n$$", "answer_latex": " 48.6", "answer_number": "48.6", "unit": " $\\mathrm{~J} \\mathrm{~K}^{-1}$", "source": "thermo", "problemid": " 5.5", "comment": " ", "solution": "Consider the following reversible process in order to calculate $\\Delta S$. The gas is first heated reversibly from 450 . to 800 . $\\mathrm{K}$ at a constant pressure of 1.35 bar. Subsequently, the gas is reversibly compressed at a constant temperature of 800 . $\\mathrm{K}$ from a pressure of 1.35 bar to a pressure of 3.45 bar. The entropy change for this process is obtained using Equation (5.23) with the value of $\\beta=1 / T$ from Example Problem 5.4.\r\n$$\r\n\\begin{aligned}\r\n& \\Delta S=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-\\int_{P_i}^{P_f} V \\beta d P=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-n R \\int_{P_i}^{P_f} \\frac{d P}{P}=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-n R \\ln \\frac{P_f}{P_i} \\\\\r\n& =2.50 \\times \\int_{450 .}^{800 .} \\frac{\\left(18.86+7.937 \\times 10^{-2} \\frac{T}{\\mathrm{~K}}-6.7834 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}+2.4426 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\\right)}{\\frac{T}{\\mathrm{~K}}} d \\frac{T}{\\mathrm{~K}} \\\\\r\n& -2.50 \\mathrm{~mol} \\times 8.314 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} \\times \\ln \\frac{3.45 \\mathrm{bar}}{1.35 \\mathrm{bar}} \\\\\r\n& =27.13 \\mathrm{~J} \\mathrm{~K}^{-1}+69.45 \\mathrm{~J} \\mathrm{~K}^{-1}-37.10 \\mathrm{~J} \\mathrm{~K}^{-1}+8.57 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n& -19.50 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n& =48.6 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$"}, {"problem_text": "One mole of $\\mathrm{CO}$ gas is transformed from an initial state characterized by $T_i=320 . \\mathrm{K}$ and $V_i=80.0 \\mathrm{~L}$ to a final state characterized by $T_f=650 . \\mathrm{K}$ and $V_f=120.0 \\mathrm{~L}$. Using Equation (5.22), calculate $\\Delta S$ for this process. Use the ideal gas values for $\\beta$ and $\\kappa$. For CO,\r\n$$\r\n\\frac{C_{V, m}}{\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}}=31.08-0.01452 \\frac{T}{\\mathrm{~K}}+3.1415 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}-1.4973 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\r\n$$", "answer_latex": " 24.4", "answer_number": "24.4", "unit": " $\\mathrm{~J} \\mathrm{~K}^{-1}$", "source": "thermo", "problemid": " 5.4", "comment": " ", "solution": "For an ideal gas,\r\n$$\r\n\\begin{gathered}\r\n\\beta=\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial T}\\right)_P=\\frac{1}{V}\\left(\\frac{\\partial[n R T / P]}{\\partial T}\\right)_P=\\frac{1}{T} \\text { and } \\\\\r\n\\kappa=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_T=-\\frac{1}{V}\\left(\\frac{\\partial[n R T / P]}{\\partial P}\\right)_T=\\frac{1}{P}\r\n\\end{gathered}\r\n$$\r\nConsider the following reversible process in order to calculate $\\Delta S$. The gas is first heated reversibly from 320 . to $650 . \\mathrm{K}$ at a constant volume of $80.0 \\mathrm{~L}$. Subsequently, the gas is reversibly expanded at a constant temperature of 650 . $\\mathrm{K}$ from a volume of $80.0 \\mathrm{~L}$ to a volume of $120.0 \\mathrm{~L}$. The entropy change for this process is obtained using the integral form of Equation (5.22) with the values of $\\beta$ and $\\kappa$ cited earlier. The result is\r\n$$\r\n\\Delta S=\\int_{T_i}^{T_f} \\frac{C_V}{T} d T+n R \\ln \\frac{V_f}{V_i}\r\n$$\r\n$\\Delta S=$\r\n$$\r\n\\begin{aligned}\r\n1 \\mathrm{~mol} \\times & \\int_{320 .}^{650 .} \\frac{\\left(31.08-0.01452 \\frac{T}{\\mathrm{~K}}+3.1415 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}-1.4973 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\\right)}{\\mathrm{K}} d \\frac{T}{\\mathrm{~K}} \\\\\r\n& +1 \\mathrm{~mol} \\times 8.314 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} \\times \\ln \\frac{120.0 \\mathrm{~L}}{80.0 \\mathrm{~L}} \\\\\r\n= & 22.025 \\mathrm{~J} \\mathrm{~K}^{-1}-4.792 \\mathrm{~J} \\mathrm{~K}^{-1}+5.028 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n& -1.207 \\mathrm{~J} \\mathrm{~K}^{-1}+3.371 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n= & 24.4 \\mathrm{~J} \\mathrm{~K}^{-1}\r\n\\end{aligned}\r\n$$"}, {"problem_text": "You are given the following reduction reactions and $E^{\\circ}$ values:\r\n$$\r\n\\begin{array}{ll}\r\n\\mathrm{Fe}^{3+}(a q)+\\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}^{2+}(a q) & E^{\\circ}=+0.771 \\mathrm{~V} \\\\\r\n\\mathrm{Fe}^{2+}(a q)+2 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s) & E^{\\circ}=-0.447 \\mathrm{~V}\r\n\\end{array}\r\n$$\r\nCalculate $E^{\\circ}$ for the half-cell reaction $\\mathrm{Fe}^{3+}(a q)+3 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s)$.", "answer_latex": " -0.041", "answer_number": "-0.041", "unit": " $\\mathrm{~V}$", "source": "thermo", "problemid": " 11.3", "comment": " ", "solution": "We calculate the desired value of $E^{\\circ}$ by converting the given $E^{\\circ}$ values to $\\Delta G^{\\circ}$ values, and combining these reduction reactions to obtain the desired equation.\r\n$$\r\n\\begin{gathered}\r\n\\mathrm{Fe}^{3+}(a q)+\\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}^{2+}(a q) \\\\\r\n\\Delta G^{\\circ}=-n F E^{\\circ}=-1 \\times 96485 \\mathrm{C} \\mathrm{mol}^{-1} \\times 0.771 \\mathrm{~V}=-74.39 \\mathrm{~kJ} \\mathrm{~mol}^{-1} \\\\\r\n\\mathrm{Fe}^{2+}(a q)+2 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s) \\\\\r\n\\Delta G^{\\circ}=-n F E^{\\circ}=-2 \\times 96485 \\mathrm{C} \\mathrm{mol}^{-1} \\times(-0.447 \\mathrm{~V})=86.26 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{gathered}\r\n$$\r\nWe next add the two equations as well as their $\\Delta G^{\\circ}$ to obtain\r\n$$\r\n\\begin{gathered}\r\n\\mathrm{Fe}^{3+}(a q)+3 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s) \\\\\r\n\\Delta G^{\\circ}=-74.39 \\mathrm{~kJ} \\mathrm{~mol}^{-1}+86.26 \\mathrm{~kJ} \\mathrm{~mol}^{-1}=11.87 \\mathrm{~kJ} \\mathrm{~mol}^{-1} \\\\\r\nE_{F e^{3+} / F e}^{\\circ}=-\\frac{\\Delta G^{\\circ}}{n F}=\\frac{-11.87 \\times 10^3 \\mathrm{~J} \\mathrm{~mol}^{-1}}{3 \\times 96485 \\mathrm{C} \\mathrm{mol}^{-1}}=-0.041 \\mathrm{~V}\r\n\\end{gathered}\r\n$$\r\nThe $E^{\\circ}$ values cannot be combined directly, because they are intensive rather than extensive quantities.\r\nThe preceding calculation can be generalized as follows. Assume that $n_1$ electrons are transferred in the reaction with the potential $E_{A / B}^{\\circ}$, and $n_2$ electrons are transferred in the reaction with the potential $E_{B / C}^{\\circ}$. If $n_3$ electrons are transferred in the reaction with the potential $E_{A / C}^{\\circ}$, then $n_3 E_{A / C}^{\\circ}=n_1 E_{A / B}^{\\circ}+n_2 E_{B / C}^{\\circ}$."}, {"problem_text": "At $298.15 \\mathrm{~K}, \\Delta G_f^{\\circ}(\\mathrm{C}$, graphite $)=0$, and $\\Delta G_f^{\\circ}(\\mathrm{C}$, diamond $)=2.90 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Therefore, graphite is the more stable solid phase at this temperature at $P=P^{\\circ}=1$ bar. Given that the densities of graphite and diamond are 2.25 and $3.52 \\mathrm{~kg} / \\mathrm{L}$, respectively, at what pressure will graphite and diamond be in equilibrium at $298.15 \\mathrm{~K}$ ?", "answer_latex": "1.51 ", "answer_number": "1.51", "unit": " $10^4 \\mathrm{bar}$", "source": "thermo", "problemid": " 6.13", "comment": " ", "solution": "At equilibrium $\\Delta G=G(\\mathrm{C}$, graphite $)-G(\\mathrm{C}$, diamond $)=0$. Using the pressure dependence of $G,\\left(\\partial G_m / \\partial P\\right)_T=V_m$, we establish the condition for equilibrium:\r\n$$\r\n\\begin{gathered}\r\n\\Delta G=\\Delta G_f^{\\circ}(\\mathrm{C}, \\text { graphite })-\\Delta G_f^{\\circ}(\\mathrm{C}, \\text { diamond }) \\\\\r\n+\\left(V_m^{\\text {graphite }}-V_m^{\\text {diamond }}\\right)(\\Delta P)=0 \\\\\r\n0=0-2.90 \\times 10^3+\\left(V_m^{\\text {graphite }}-V_m^{\\text {diamond }}\\right)(P-1 \\mathrm{bar}) \\\\\r\nP=1 \\mathrm{bar}+\\frac{2.90 \\times 10^3}{M_C\\left(\\frac{1}{\\rho_{\\text {graphite }}}-\\frac{1}{\\rho_{\\text {diamond }}}\\right)} \\\\\r\n=1 \\mathrm{bar}+\\frac{2.90 \\times 10^3}{12.00 \\times 10^{-3} \\mathrm{~kg} \\mathrm{~mol}^{-1} \\times\\left(\\frac{1}{2.25 \\times 10^3 \\mathrm{~kg} \\mathrm{~m}^{-3}}-\\frac{1}{3.52 \\times 10^3 \\mathrm{~kg} \\mathrm{~m}^{-3}}\\right)}\\\\\r\n=10^5 \\mathrm{~Pa}+1.51 \\times 10^9 \\mathrm{~Pa}=1.51 \\times 10^4 \\mathrm{bar}\r\n\\end{gathered}\r\n$$\r\nFortunately for all those with diamond rings, although the conversion of diamond to graphite at $1 \\mathrm{bar}$ and $298 \\mathrm{~K}$ is spontaneous, the rate of conversion is vanishingly small.\r\n"}, {"problem_text": "Imagine tossing a coin 50 times. What are the probabilities of observing heads 25 times (i.e., 25 successful experiments)?", "answer_latex": "0.11 ", "answer_number": "0.11", "unit": " ", "source": "thermo", "problemid": " 12.8", "comment": " ", "solution": "The trial of interest consists of 50 separate experiments; therefore, $n=50$. Considering the case of 25 successful experiments where $j=25$. The probability $\\left(P_{25}\\right)$ is\r\n$$\r\n\\begin{aligned}\r\nP_{25} & =C(n, j)\\left(P_E\\right)^j\\left(1-P_E\\right)^{n-j} \\\\\r\n& =C(50,25)\\left(P_E\\right)^{25}\\left(1-P_E\\right)^{25} \\\\\r\n& =\\left(\\frac{50 !}{(25 !)(25 !)}\\right)\\left(\\frac{1}{2}\\right)^{25}\\left(\\frac{1}{2}\\right)^{25}=\\left(1.26 \\times 10^{14}\\right)\\left(8.88 \\times 10^{-16}\\right)=0.11\r\n\\end{aligned}\r\n$$"}, {"problem_text": "In a rotational spectrum of $\\operatorname{HBr}\\left(B=8.46 \\mathrm{~cm}^{-1}\\right)$, the maximum intensity transition in the R-branch corresponds to the $J=4$ to 5 transition. At what temperature was the spectrum obtained?", "answer_latex": "4943 ", "answer_number": "4943", "unit": " $\\mathrm{~K}$", "source": "thermo", "problemid": " 14.4", "comment": " ", "solution": "The information provided for this problem dictates that the $J=4$ rotational energy level was the most populated at the temperature at which the spectrum was taken. To determine the temperature, we first determine the change in occupation number for the rotational energy level, $a_J$, versus $J$ as follows:\r\n$$\r\n\\begin{aligned}\r\na_J & =\\frac{N(2 J+1) e^{-\\beta h c B J(J+1)}}{q_R}=\\frac{N(2 J+1) e^{-\\beta h c B J(J+1)}}{\\left(\\frac{1}{\\beta h c B}\\right)} \\\\\r\n& =N \\beta h c B(2 J+1) e^{-\\beta h c B J(J+1)}\r\n\\end{aligned}\r\n$$\r\nNext, we take the derivative of $a_J$ with respect to $J$ and set the derivative equal to zero to find the maximum of the function:\r\n$$\r\n\\begin{aligned}\r\n\\frac{d a_J}{d J} & =0=\\frac{d}{d J} N \\beta h c B(2 J+1) e^{-\\beta h c B J(J+1)} \\\\\r\n0 & =\\frac{d}{d J}(2 J+1) e^{-\\beta h c B J(J+1)} \\\\\r\n0 & =2 e^{-\\beta h c B J(J+1)}-\\beta h c B(2 J+1)^2 e^{-\\beta h c B J(J+1)} \\\\\r\n0 & =2-\\beta h c B(2 J+1)^2 \\\\\r\n2 & =\\beta h c B(2 J+1)^2=\\frac{h c B}{k T}(2 J+1)^2 \\\\\r\nT & =\\frac{(2 J+1)^2 h c B}{2 k}\r\n\\end{aligned}\r\n$$\r\nSubstitution of $J=4$ into the preceding expression results in the following temperature at which the spectrum was obtained:\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{(2 J+1)^2 h c B}{2 k} \\\\\r\n& =\\frac{(2(4)+1)^2\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)\\left(3.00 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right)\\left(8.46 \\mathrm{~cm}^{-1}\\right)}{2\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)} \\\\\r\n& =4943 \\mathrm{~K}\r\n\\end{aligned}\r\n$$"}]
+[
+ {
+  "problem_text": "In this problem, 3.00 mol of liquid mercury is transformed from an initial state characterized by $T_i=300 . \\mathrm{K}$ and $P_i=1.00$ bar to a final state characterized by $T_f=600 . \\mathrm{K}$ and $P_f=3.00$ bar.\r\nCalculate $\\Delta S$ for this process; $\\beta=1.81 \\times 10^{-4} \\mathrm{~K}^{-1}, \\rho=13.54 \\mathrm{~g} \\mathrm{~cm}^{-3}$, and $C_{P, m}$ for $\\mathrm{Hg}(l)=27.98 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$.",
+  "answer_latex": " 58.2",
+  "answer_number": "58.2",
+  "unit": "$\\mathrm{~J}\\mathrm{~K}^{-1}$",
+  "source": "thermo",
+  "problemid": " 5.6",
+  "comment": " ",
+  "solution": "\nBecause the volume changes only slightly with temperature and pressure over the range indicated,\r\n$$\r\n\\begin{aligned}\r\n& \\Delta S=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-\\int_{P_i}^{P_f} V \\beta d P \\approx n C_{P, m} \\ln \\frac{T_f}{T_i}-n V_{m, i} \\beta\\left(P_f-P_i\\right) \\\\\r\n& =3.00 \\mathrm{~mol} \\times 27.98 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1} \\times \\ln \\frac{600 . \\mathrm{K}}{300 . \\mathrm{K}} \\\\\r\n& -3.00 \\mathrm{~mol} \\times \\frac{200.59 \\mathrm{~g} \\mathrm{~mol}^{-1}}{13.54 \\mathrm{~g} \\mathrm{~cm}^{-3} \\times \\frac{10^6 \\mathrm{~cm}^3}{\\mathrm{~m}^3}} \\times 1.81 \\times 10^{-4} \\mathrm{~K}^{-1} \\times 2.00 \\mathrm{bar} \\\\\r\n& \\times 10^5 \\mathrm{Pabar}^{-1} \\\\\r\n& =58.2 \\mathrm{~J} \\mathrm{~K}^{-1}-1.61 \\times 10^{-3} \\mathrm{~J} \\mathrm{~K}^{-1}=58.2 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "For an ensemble consisting of 1.00 moles of particles having two energy levels separated by $h v=1.00 \\times 10^{-20} \\mathrm{~J}$, at what temperature will the internal energy of this system equal $1.00 \\mathrm{~kJ}$ ?",
+  "answer_latex": " 449",
+  "answer_number": "449",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 15.2",
+  "comment": " ",
+  "solution": "Using the expression for total energy and recognizing that $N=n N_A$,\r\n$$\r\nU=-\\left(\\frac{\\partial \\ln Q}{\\partial \\beta}\\right)_V=-n N_A\\left(\\frac{\\partial \\ln q}{\\partial \\beta}\\right)_V\r\n$$\r\nEvaluating the preceding expression and paying particular attention to units, we get\r\n$$\r\n\\begin{aligned}\r\n& U=-n N_A\\left(\\frac{\\partial}{\\partial \\beta} \\ln q\\right)_V=-\\frac{n N_A}{q}\\left(\\frac{\\partial q}{\\partial \\beta}\\right)_V \\\\\r\n& \\frac{U}{n N_A}=\\frac{-1}{\\left(1+e^{-\\beta h \\nu}\\right)}\\left(\\frac{\\partial}{\\partial \\beta}\\left(1+e^{-\\beta h \\nu}\\right)\\right)_V \\\\\r\n&=\\frac{h \\nu e^{-\\beta h \\nu}}{1+e^{-\\beta h \\nu}}=\\frac{h \\nu}{e^{\\beta h \\nu}+1} \\\\\r\n& \\frac{n N_A h \\nu}{U}-1=e^{\\beta h \\nu} \\\\\r\n& \\ln \\left(\\frac{n N_A h \\nu}{U}-1\\right)=\\beta h \\nu=\\frac{h \\nu}{k T}\r\n\\end{aligned}\r\n$$\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{h \\nu}{k \\ln \\left(\\frac{n N_A h \\nu}{U}-1\\right)} \\\\\r\n= & \\frac{1.00 \\times 10^{-20} \\mathrm{~J}}{\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right) \\ln \\left(\\frac{(1.00 \\mathrm{~mol})\\left(6.022 \\times 10^{23} \\mathrm{~mol}^{-1}\\right)\\left(1.00 \\times 10^{-20} \\mathrm{~J}\\right)}{\\left(1.00 \\times 10^3 \\mathrm{~J}\\right)}-1\\right)} \\\\\r\n& =449 \\mathrm{~K}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "The sedimentation coefficient of lysozyme $\\left(\\mathrm{M}=14,100 \\mathrm{~g} \\mathrm{~mol}^{-1}\\right)$ in water at $20^{\\circ} \\mathrm{C}$ is $1.91 \\times 10^{-13} \\mathrm{~s}$ and the specific volume is $0.703 \\mathrm{~cm}^3 \\mathrm{~g}^{-1}$. The density of water at this temperature is $0.998 \\mathrm{~g} \\mathrm{~cm}^{-3}$ and $\\eta=1.002 \\mathrm{cP}$. Assuming lysozyme is spherical, what is the radius of this protein?",
+  "answer_latex": "1.94",
+  "answer_number": "1.94",
+  "unit": " $\\mathrm{~nm}$",
+  "source": "thermo",
+  "problemid": " 17.9",
+  "comment": " ",
+  "solution": "\nThe frictional coefficient is dependent on molecular radius. Solving Equation for $f$, we obtain\r\n$$\r\n\\begin{aligned}\r\n& f=\\frac{m(1-\\overline{\\mathrm{V}} \\rho)}{\\overline{\\mathrm{s}}}=\\frac{\\frac{\\left(14,100 \\mathrm{~g} \\mathrm{~mol}^{-1}\\right)}{\\left(6.022 \\times 10^{23} \\mathrm{~mol}^{-1}\\right)}\\left(1-(0.703 \\mathrm{~mL} \\mathrm{~g})\\left(0.998 \\mathrm{~g} \\mathrm{~mL}^{-1}\\right)\\right)}{1.91 \\times 10^{-13} \\mathrm{~s}} \\\\\r\n& =3.66 \\times 10^{-8} \\mathrm{~g} \\mathrm{~s}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$\r\nThe frictional coefficient is related to the radius for a spherical particle by Equation such that\r\n$$\r\nr=\\frac{f}{6 \\pi \\eta}=\\frac{3.66 \\times 10^{-8} \\mathrm{~g} \\mathrm{~s}^{-1}}{6 \\pi\\left(1.002 \\mathrm{~g} \\mathrm{~m}^{-1} \\mathrm{~s}^{-1}\\right)}=1.94 \\times 10^{-9} \\mathrm{~m}=1.94 \\mathrm{~nm}\r\n$$\n"
+ },
+ {
+  "problem_text": "Determine the standard molar entropy of $\\mathrm{Ne}$ under standard thermodynamic conditions.",
+  "answer_latex": "164 ",
+  "answer_number": "146",
+  "unit": " $ \\mathrm{Jmol}^{-1} \\mathrm{~K}^{-1}$",
+  "source": "thermo",
+  "problemid": " 15.5",
+  "comment": " ",
+  "solution": "\nBeginning with the expression for entropy:\r\n$$\r\n\\begin{aligned}\r\nS & =\\frac{5}{2} R+R \\ln \\left(\\frac{V}{\\Lambda^3}\\right)-R \\ln N_A \\\\\r\n& =\\frac{5}{2} R+R \\ln \\left(\\frac{V}{\\Lambda^3}\\right)-54.75 R \\\\\r\n& =R \\ln \\left(\\frac{V}{\\Lambda^3}\\right)-52.25 R\r\n\\end{aligned}\r\n$$\r\nThe conventional standard state is defined by $T=298 \\mathrm{~K}$ and $V_m=24.4 \\mathrm{~L}\\left(0.0244 \\mathrm{~m}^3\\right)$. The thermal wavelength for $\\mathrm{Ne}$ is\r\n$$\r\n\\begin{aligned}\r\n\\Lambda & =\\left(\\frac{h^2}{2 \\pi m k T}\\right)^{1 / 2} \\\\\r\n& =\\left(\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)^2}{2 \\pi\\left(\\frac{0.02018 \\mathrm{~kg} \\mathrm{~mol}^{-1}}{N_A}\\right)\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)(298 \\mathrm{~K})}\\right)^{1 / 2} \\\\\r\n& =2.25 \\times 10^{-11} \\mathrm{~m}\r\n\\end{aligned}\r\n$$\r\nUsing this value for the thermal wavelength, the entropy becomes\r\n$$\r\n\\begin{aligned}\r\nS & =R \\ln \\left(\\frac{0.0244 \\mathrm{~m}^3}{\\left(2.25 \\times 10^{-11} \\mathrm{~m}\\right)^3}\\right)-52.25 R \\\\\r\n& =69.84 R-52.25 R=17.59 R=146 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \\mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \\mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \\mathrm{C}$ decay events per minute. How old is the wood?",
+  "answer_latex": "4.86 ",
+  "answer_number": "4.86",
+  "unit": " $10^{11} \\mathrm{~s}$",
+  "source": "thermo",
+  "problemid": " 18.4",
+  "comment": " ",
+  "solution": "The ratio of decay events yields the amount of ${ }^{14} \\mathrm{C}$ present currently versus the amount that was present when the tree died:\r\n$$\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}=\\frac{2.40 \\mathrm{~min}^{-1}}{15.3 \\mathrm{~min}^{-1}}=0.157\r\n$$\r\nThe rate constant for isotope decay is related to the half-life as follows:\r\n$$\r\nk=\\frac{\\ln 2}{t_{1 / 2}}=\\frac{\\ln 2}{5760 \\text { years }}=\\frac{\\ln 2}{1.82 \\times 10^{11} \\mathrm{~s}}=3.81 \\times 10^{-12} \\mathrm{~s}^{-1}\r\n$$\r\nWith the rate constant and ratio of isotope concentrations, the age of the fossilized wood is readily determined:\r\n$$\r\n\\begin{aligned}\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0} & =e^{-k t} \\\\\r\n\\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-k t \\\\\r\n-\\frac{1}{k} \\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-\\frac{1}{3.81 \\times 10^{-12} \\mathrm{~s}} \\ln (0.157)=t \\\\\r\n4.86 \\times 10^{11} \\mathrm{~s} & =t\r\n\\end{aligned}\r\n$$\r\nThis time corresponds to an age of roughly 15,400 years."
+ },
+ {
+  "problem_text": "Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \\mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \\mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \\mathrm{C}$ decay events per minute. How old is the wood?",
+  "answer_latex": "4.86 ",
+  "answer_number": "4.86",
+  "unit": " $10^{11} \\mathrm{~s}$",
+  "source": "thermo",
+  "problemid": " 18.4",
+  "comment": " ",
+  "solution": "The ratio of decay events yields the amount of ${ }^{14} \\mathrm{C}$ present currently versus the amount that was present when the tree died:\r\n$$\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}=\\frac{2.40 \\mathrm{~min}^{-1}}{15.3 \\mathrm{~min}^{-1}}=0.157\r\n$$\r\nThe rate constant for isotope decay is related to the half-life as follows:\r\n$$\r\nk=\\frac{\\ln 2}{t_{1 / 2}}=\\frac{\\ln 2}{5760 \\text { years }}=\\frac{\\ln 2}{1.82 \\times 10^{11} \\mathrm{~s}}=3.81 \\times 10^{-12} \\mathrm{~s}^{-1}\r\n$$\r\nWith the rate constant and ratio of isotope concentrations, the age of the fossilized wood is readily determined:\r\n$$\r\n\\begin{aligned}\r\n\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0} & =e^{-k t} \\\\\r\n\\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-k t \\\\\r\n-\\frac{1}{k} \\ln \\left(\\frac{\\left[{ }^{14} \\mathrm{C}\\right]}{\\left[{ }^{14} \\mathrm{C}\\right]_0}\\right) & =-\\frac{1}{3.81 \\times 10^{-12} \\mathrm{~s}} \\ln (0.157)=t \\\\\r\n4.86 \\times 10^{11} \\mathrm{~s} & =t\r\n\\end{aligned}\r\n$$\r\nThis time corresponds to an age of roughly 15,400 years."
+ },
+ {
+  "problem_text": "Determine the diffusion coefficient for Ar at $298 \\mathrm{~K}$ and a pressure of $1.00 \\mathrm{~atm}$.",
+  "answer_latex": "1.1 ",
+  "answer_number": "1.1",
+  "unit": " $10^{-5}~\\mathrm{m^2~s^{-1}}$",
+  "source": "thermo",
+  "problemid": " 17.1",
+  "comment": " ",
+  "solution": "\n$$\r\n\\begin{aligned}\r\nD_{Ar} &= \\frac{1}{3} \\nu_{ave, Ar} \\lambda_{Ar} \\\\\r\n&= \\frac{1}{3} \\left(\\frac{8RT}{\\pi M_{Ar}}\\right)^{\\frac{1}{2}} \\left(\\frac{RT}{PN_A\\sqrt{2}\\sigma_{Ar}}\\right) \\\\\r\n&= \\frac{1}{3} \\left(\\frac{8(8.314~\\mathrm{J~mol^{-1}~K^{-1}}) \\times 298~\\mathrm{K}}{\\pi(0.040~\\mathrm{kg~mol^{-1}})}\\right)^{\\frac{1}{2}} \\\\\r\n&\\quad \\times \\left(\\frac{(8.314~\\mathrm{J~mol^{-1}~K^{-1}}) \\times 298~\\mathrm{K}}{(101,325~\\mathrm{Pa}) \\times (6.022 \\times 10^{23}~\\mathrm{mol^{-1}})} \\times \\frac{1}{\\sqrt{2}(3.6 \\times 10^{-19}~\\mathrm{m^2})}\\right) \\\\\r\n&= \\frac{1}{3} \\times (397~\\mathrm{m~s^{-1}}) \\times (7.98 \\times 10^{-8}~\\mathrm{m}) \\\\\r\n&= 1.1 \\times 10^{-5}~\\mathrm{m^2~s^{-1}}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "Gas cylinders of $\\mathrm{CO}_2$ are sold in terms of weight of $\\mathrm{CO}_2$. A cylinder contains $50 \\mathrm{lb}$ (22.7 $\\mathrm{kg}$ ) of $\\mathrm{CO}_2$. How long can this cylinder be used in an experiment that requires flowing $\\mathrm{CO}_2$ at $293 \\mathrm{~K}(\\eta=146 \\mu \\mathrm{P})$ through a 1.00-m-long tube (diameter $\\left.=0.75 \\mathrm{~mm}\\right)$ with an input pressure of $1.05 \\mathrm{~atm}$ and output pressure of $1.00 \\mathrm{~atm}$ ? The flow is measured at the tube output.",
+  "answer_latex": "4.49 ",
+  "answer_number": "4.49",
+  "unit": " $10^6 \\mathrm{~s}$",
+  "source": "thermo",
+  "problemid": "17.7 ",
+  "comment": " ",
+  "solution": "\nThe gas flow rate $\\Delta V / \\Delta t$ is\r\n$$\r\n\\begin{aligned}\r\n\\frac{\\Delta V}{\\Delta t}= & \\frac{\\pi r^4}{16 \\eta L P_0}\\left(P_2^2-P_1^2\\right) \\\\\r\n= & \\frac{\\pi\\left(0.375 \\times 10^{-3} \\mathrm{~m}\\right)^4}{16\\left(1.46 \\times 10^{-5} \\mathrm{~kg} \\mathrm{~m}^{-1} \\mathrm{~s}^{-1}\\right)(1.00 \\mathrm{~m})(101,325 \\mathrm{~Pa})} \\\\\r\n& \\times\\left((106,391 \\mathrm{~Pa})^2-(101,325 \\mathrm{~Pa})^2\\right) \\\\\r\n= & 2.76 \\times 10^{-6} \\mathrm{~m}^3 \\mathrm{~s}^{-1}\r\n\\end{aligned}\r\n$$\r\nConverting the $\\mathrm{CO}_2$ contained in the cylinder to the volume occupied at $298 \\mathrm{~K}$ and 1 atm pressure, we get\r\n$$\r\n\\begin{aligned}\r\nn_{\\mathrm{CO}_2} & =22.7 \\mathrm{~kg}\\left(\\frac{1}{0.044 \\mathrm{~kg} \\mathrm{~mol}^{-1}}\\right)=516 \\mathrm{~mol} \\\\\r\nV & =\\frac{n R T}{P}=1.24 \\times 10^4 \\mathrm{~L}\\left(\\frac{10^{-3} \\mathrm{~m}^3}{\\mathrm{~L}}\\right)=12.4 \\mathrm{~m}^3\r\n\\end{aligned}\r\n$$\r\nGiven the effective volume of $\\mathrm{CO}_2$ contained in the cylinder, the duration over which the cylinder can be used is\r\n$$\r\n\\frac{12.4 \\mathrm{~m}^3}{2.76 \\times 10^{-6} \\mathrm{~m}^3 \\mathrm{~s}^{-1}}=4.49 \\times 10^6 \\mathrm{~s}\r\n$$\r\nThis time corresponds to roughly 52 days.\n"
+ },
+ {
+  "problem_text": "The vibrational frequency of $I_2$ is $208 \\mathrm{~cm}^{-1}$. What is the probability of $I_2$ populating the $n=2$ vibrational level if the molecular temperature is $298 \\mathrm{~K}$ ?",
+  "answer_latex": " 0.086",
+  "answer_number": "0.086",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 13.5",
+  "comment": " ",
+  "solution": "Molecular vibrational energy levels can be modeled as harmonic oscillators; therefore, this problem can be solved by employing a strategy identical to the one just presented. To evaluate the partition function $q$, the \"trick\" used earlier was to write the partition function as a series and use the equivalent series expression:\r\n$$\r\n\\begin{aligned}\r\nq & =\\sum_n e^{-\\beta \\varepsilon_n}=1+e^{-\\beta h c \\widetilde{\\nu}}+e^{-2 \\beta h c \\tilde{\\nu}}+e^{-3 \\beta h c \\widetilde{\\nu}}+\\ldots \\\\\r\n& =\\frac{1}{1-e^{-\\beta h c \\widetilde{\\nu}}}\r\n\\end{aligned}\r\n$$\r\nSince $\\tilde{\\nu}=208 \\mathrm{~cm}^{-1}$ and $T=298 \\mathrm{~K}$, the partition function is\r\n$$\r\n\\begin{aligned}\r\nq & =\\frac{1}{1-e^{-\\beta h c \\widetilde{\\nu}}} \\\\\r\n& =\\frac{1}{1-e^{-h c \\widetilde{\\nu} / k T}} \\\\\r\n& =\\frac{1}{1-\\exp \\left[-\\left(\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right)\\left(3.00 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right)\\left(208 \\mathrm{~cm}^{-1}\\right)}{\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)(298 \\mathrm{~K})}\\right)\\right]} \\\\\r\n& =\\frac{1}{1-e^{-1}}=1.58\r\n\\end{aligned}\r\n$$\r\nThis result is then used to evaluate the probability of occupying the second vibrational state $(n=2)$ as follows:\r\n$$\r\n\\begin{aligned}\r\np_2 & =\\frac{e^{-2 \\beta h c \\tilde{\\nu}}}{q} \\\\\r\n& =\\frac{\\exp \\left[-2\\left(\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}^{-1}\\right)\\left(3.00 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right)\\left(208 \\mathrm{~cm}^{-1}\\right)}{\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)(298 \\mathrm{~K})}\\right)\\right]}{1.58} \\\\\r\n& =0.086\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "The value of $\\Delta G_f^{\\circ}$ for $\\mathrm{Fe}(g)$ is $370.7 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ at $298.15 \\mathrm{~K}$, and $\\Delta H_f^{\\circ}$ for $\\mathrm{Fe}(g)$ is $416.3 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$ at the same temperature. Assuming that $\\Delta H_f^{\\circ}$ is constant in the interval $250-400 \\mathrm{~K}$, calculate $\\Delta G_f^{\\circ}$ for $\\mathrm{Fe}(g)$ at 400. K.",
+  "answer_latex": "355.1 ",
+  "answer_number": "355.1",
+  "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$",
+  "source": "thermo",
+  "problemid": " 6.4",
+  "comment": "\n ",
+  "solution": "$$\r\n\\begin{aligned}\r\n\\Delta G_f^{\\circ}\\left(T_2\\right) & =T_2\\left[\\frac{\\Delta G_f^{\\circ}\\left(T_1\\right)}{T_1}+\\Delta H_f^{\\circ}\\left(T_1\\right) \\times\\left(\\frac{1}{T_2}-\\frac{1}{T_1}\\right)\\right] \\\\\r\n& =400 . \\mathrm{K} \\times\\left[\\begin{array}{l}\r\n\\left.\\frac{370.7 \\times 10^3 \\mathrm{~J} \\mathrm{~mol}^{-1}}{298.15 \\mathrm{~K}}+416.3 \\times 10^3 \\mathrm{~J} \\mathrm{~mol}^{-1}\\right] \\\\\r\n\\times\\left(\\frac{1}{400 . \\mathrm{K}}-\\frac{1}{298.15 \\mathrm{~K}}\\right)\r\n\\end{array}\\right] \\\\\r\n\\Delta G_f^{\\circ}(400 . \\mathrm{K}) & =355.1 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "The reactant 1,3-cyclohexadiene can be photochemically converted to cis-hexatriene. In an experiment, $2.5 \\mathrm{mmol}$ of cyclohexadiene are converted to cis-hexatriene when irradiated with 100. W of 280. nm light for $27.0 \\mathrm{~s}$. All of the light is absorbed by the sample. What is the overall quantum yield for this photochemical process?",
+  "answer_latex": " 0.396",
+  "answer_number": "0.396",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 19.6",
+  "comment": " ",
+  "solution": "First, the total photon energy absorbed by the sample, $E_{a b s}$, is\r\n$$\r\nE_{a b s}=(\\text { power }) \\Delta t=\\left(100 . \\mathrm{Js}^{-1}\\right)(27.0 \\mathrm{~s})=2.70 \\times 10^3 \\mathrm{~J}\r\n$$\r\nNext, the photon energy at $280 . \\mathrm{nm}$ is\r\n$$\r\n\\mathrm{E}_{p h}=\\frac{h c}{\\lambda}=\\frac{\\left(6.626 \\times 10^{-34} \\mathrm{Js}\\right)\\left(2.998 \\times 10^8 \\mathrm{~ms}^{-1}\\right)}{2.80 \\times 10^{-7} \\mathrm{~m}}=7.10 \\times 10^{-19} \\mathrm{~J}\r\n$$\r\nThe total number of photons absorbed by the sample is therefore\r\n$$\r\n\\frac{E_{a b s}}{E_{p h}}=\\frac{2.70 \\times 10^3 \\mathrm{~J}}{7.10 \\times 10^{-19} \\mathrm{~J} \\text { photon }^{-1}}=3.80 \\times 10^{21} \\text { photons }\r\n$$\r\nDividing this result by Avogadro's number results in $6.31 \\times 10^{-3}$ Einsteins or moles of photons. Therefore, the overall quantum yield is\r\n$$\r\n\\phi=\\frac{\\text { moles }_{\\text {react }}}{\\text { moles }_{\\text {photon }}}=\\frac{2.50 \\times 10^{-3} \\mathrm{~mol}}{6.31 \\times 10^{-3} \\mathrm{~mol}}=0.396 \\approx 0.40\r\n$$"
+ },
+ {
+  "problem_text": "In this problem, $2.50 \\mathrm{~mol}$ of $\\mathrm{CO}_2$ gas is transformed from an initial state characterized by $T_i=450 . \\mathrm{K}$ and $P_i=1.35$ bar to a final state characterized by $T_f=800 . \\mathrm{K}$ and $P_f=$ 3.45 bar. Using Equation (5.23), calculate $\\Delta S$ for this process. Assume ideal gas behavior and use the ideal gas value for $\\beta$. For $\\mathrm{CO}_2$,\r\n$$\r\n\\frac{C_{P, m}}{\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}}=18.86+7.937 \\times 10^{-2} \\frac{T}{\\mathrm{~K}}-6.7834 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}+2.4426 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\r\n$$",
+  "answer_latex": " 48.6",
+  "answer_number": "48.6",
+  "unit": " $\\mathrm{~J} \\mathrm{~K}^{-1}$",
+  "source": "thermo",
+  "problemid": " 5.5",
+  "comment": " ",
+  "solution": "\nConsider the following reversible process in order to calculate $\\Delta S$. The gas is first heated reversibly from 450 . to 800 . $\\mathrm{K}$ at a constant pressure of 1.35 bar. Subsequently, the gas is reversibly compressed at a constant temperature of 800 . $\\mathrm{K}$ from a pressure of 1.35 bar to a pressure of 3.45 bar.\r\n$$\r\n\\begin{aligned}\r\n& \\Delta S=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-\\int_{P_i}^{P_f} V \\beta d P=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-n R \\int_{P_i}^{P_f} \\frac{d P}{P}=\\int_{T_i}^{T_f} \\frac{C_P}{T} d T-n R \\ln \\frac{P_f}{P_i} \\\\\r\n& =2.50 \\times \\int_{450 .}^{800 .} \\frac{\\left(18.86+7.937 \\times 10^{-2} \\frac{T}{\\mathrm{~K}}-6.7834 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}+2.4426 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\\right)}{\\frac{T}{\\mathrm{~K}}} d \\frac{T}{\\mathrm{~K}} \\\\\r\n& -2.50 \\mathrm{~mol} \\times 8.314 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} \\times \\ln \\frac{3.45 \\mathrm{bar}}{1.35 \\mathrm{bar}} \\\\\r\n& =27.13 \\mathrm{~J} \\mathrm{~K}^{-1}+69.45 \\mathrm{~J} \\mathrm{~K}^{-1}-37.10 \\mathrm{~J} \\mathrm{~K}^{-1}+8.57 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n& -19.50 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n& =48.6 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n&\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "One mole of $\\mathrm{CO}$ gas is transformed from an initial state characterized by $T_i=320 . \\mathrm{K}$ and $V_i=80.0 \\mathrm{~L}$ to a final state characterized by $T_f=650 . \\mathrm{K}$ and $V_f=120.0 \\mathrm{~L}$. Using Equation (5.22), calculate $\\Delta S$ for this process. Use the ideal gas values for $\\beta$ and $\\kappa$. For CO,\r\n$$\r\n\\frac{C_{V, m}}{\\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}}=31.08-0.01452 \\frac{T}{\\mathrm{~K}}+3.1415 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}-1.4973 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\r\n$$",
+  "answer_latex": " 24.4",
+  "answer_number": "24.4",
+  "unit": " $\\mathrm{~J} \\mathrm{~K}^{-1}$",
+  "source": "thermo",
+  "problemid": " 5.4",
+  "comment": " ",
+  "solution": "\nFor an ideal gas,\r\n$$\r\n\\begin{gathered}\r\n\\beta=\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial T}\\right)_P=\\frac{1}{V}\\left(\\frac{\\partial[n R T / P]}{\\partial T}\\right)_P=\\frac{1}{T} \\text { and } \\\\\r\n\\kappa=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_T=-\\frac{1}{V}\\left(\\frac{\\partial[n R T / P]}{\\partial P}\\right)_T=\\frac{1}{P}\r\n\\end{gathered}\r\n$$\r\nConsider the following reversible process in order to calculate $\\Delta S$. The gas is first heated reversibly from 320 . to $650 . \\mathrm{K}$ at a constant volume of $80.0 \\mathrm{~L}$. Subsequently, the gas is reversibly expanded at a constant temperature of 650 . $\\mathrm{K}$ from a volume of $80.0 \\mathrm{~L}$ to a volume of $120.0 \\mathrm{~L}$. The entropy change for this process is obtained using the integral form with the values of $\\beta$ and $\\kappa$ cited earlier. The result is\r\n$$\r\n\\Delta S=\\int_{T_i}^{T_f} \\frac{C_V}{T} d T+n R \\ln \\frac{V_f}{V_i}\r\n$$\r\n$\\Delta S=$\r\n$$\r\n\\begin{aligned}\r\n1 \\mathrm{~mol} \\times & \\int_{320 .}^{650 .} \\frac{\\left(31.08-0.01452 \\frac{T}{\\mathrm{~K}}+3.1415 \\times 10^{-5} \\frac{T^2}{\\mathrm{~K}^2}-1.4973 \\times 10^{-8} \\frac{T^3}{\\mathrm{~K}^3}\\right)}{\\mathrm{K}} d \\frac{T}{\\mathrm{~K}} \\\\\r\n& +1 \\mathrm{~mol} \\times 8.314 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} \\times \\ln \\frac{120.0 \\mathrm{~L}}{80.0 \\mathrm{~L}} \\\\\r\n= & 22.025 \\mathrm{~J} \\mathrm{~K}^{-1}-4.792 \\mathrm{~J} \\mathrm{~K}^{-1}+5.028 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n& -1.207 \\mathrm{~J} \\mathrm{~K}^{-1}+3.371 \\mathrm{~J} \\mathrm{~K}^{-1} \\\\\r\n= & 24.4 \\mathrm{~J} \\mathrm{~K}^{-1}\r\n\\end{aligned}\r\n$$\n"
+ },
+ {
+  "problem_text": "You are given the following reduction reactions and $E^{\\circ}$ values:\r\n$$\r\n\\begin{array}{ll}\r\n\\mathrm{Fe}^{3+}(a q)+\\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}^{2+}(a q) & E^{\\circ}=+0.771 \\mathrm{~V} \\\\\r\n\\mathrm{Fe}^{2+}(a q)+2 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s) & E^{\\circ}=-0.447 \\mathrm{~V}\r\n\\end{array}\r\n$$\r\nCalculate $E^{\\circ}$ for the half-cell reaction $\\mathrm{Fe}^{3+}(a q)+3 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s)$.",
+  "answer_latex": " -0.041",
+  "answer_number": "-0.041",
+  "unit": " $\\mathrm{~V}$",
+  "source": "thermo",
+  "problemid": " 11.3",
+  "comment": " ",
+  "solution": "We calculate the desired value of $E^{\\circ}$ by converting the given $E^{\\circ}$ values to $\\Delta G^{\\circ}$ values, and combining these reduction reactions to obtain the desired equation.\r\n$$\r\n\\begin{gathered}\r\n\\mathrm{Fe}^{3+}(a q)+\\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}^{2+}(a q) \\\\\r\n\\Delta G^{\\circ}=-n F E^{\\circ}=-1 \\times 96485 \\mathrm{C} \\mathrm{mol}^{-1} \\times 0.771 \\mathrm{~V}=-74.39 \\mathrm{~kJ} \\mathrm{~mol}^{-1} \\\\\r\n\\mathrm{Fe}^{2+}(a q)+2 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s) \\\\\r\n\\Delta G^{\\circ}=-n F E^{\\circ}=-2 \\times 96485 \\mathrm{C} \\mathrm{mol}^{-1} \\times(-0.447 \\mathrm{~V})=86.26 \\mathrm{~kJ} \\mathrm{~mol}^{-1}\r\n\\end{gathered}\r\n$$\r\nWe next add the two equations as well as their $\\Delta G^{\\circ}$ to obtain\r\n$$\r\n\\begin{gathered}\r\n\\mathrm{Fe}^{3+}(a q)+3 \\mathrm{e}^{-} \\rightarrow \\mathrm{Fe}(s) \\\\\r\n\\Delta G^{\\circ}=-74.39 \\mathrm{~kJ} \\mathrm{~mol}^{-1}+86.26 \\mathrm{~kJ} \\mathrm{~mol}^{-1}=11.87 \\mathrm{~kJ} \\mathrm{~mol}^{-1} \\\\\r\nE_{F e^{3+} / F e}^{\\circ}=-\\frac{\\Delta G^{\\circ}}{n F}=\\frac{-11.87 \\times 10^3 \\mathrm{~J} \\mathrm{~mol}^{-1}}{3 \\times 96485 \\mathrm{C} \\mathrm{mol}^{-1}}=-0.041 \\mathrm{~V}\r\n\\end{gathered}\r\n$$\r\nThe $E^{\\circ}$ values cannot be combined directly, because they are intensive rather than extensive quantities.\r\nThe preceding calculation can be generalized as follows. Assume that $n_1$ electrons are transferred in the reaction with the potential $E_{A / B}^{\\circ}$, and $n_2$ electrons are transferred in the reaction with the potential $E_{B / C}^{\\circ}$. If $n_3$ electrons are transferred in the reaction with the potential $E_{A / C}^{\\circ}$, then $n_3 E_{A / C}^{\\circ}=n_1 E_{A / B}^{\\circ}+n_2 E_{B / C}^{\\circ}$."
+ },
+ {
+  "problem_text": "At $298.15 \\mathrm{~K}, \\Delta G_f^{\\circ}(\\mathrm{C}$, graphite $)=0$, and $\\Delta G_f^{\\circ}(\\mathrm{C}$, diamond $)=2.90 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Therefore, graphite is the more stable solid phase at this temperature at $P=P^{\\circ}=1$ bar. Given that the densities of graphite and diamond are 2.25 and $3.52 \\mathrm{~kg} / \\mathrm{L}$, respectively, at what pressure will graphite and diamond be in equilibrium at $298.15 \\mathrm{~K}$ ?",
+  "answer_latex": "1.51 ",
+  "answer_number": "1.51",
+  "unit": " $10^4 \\mathrm{bar}$",
+  "source": "thermo",
+  "problemid": " 6.13",
+  "comment": " ",
+  "solution": "At equilibrium $\\Delta G=G(\\mathrm{C}$, graphite $)-G(\\mathrm{C}$, diamond $)=0$. Using the pressure dependence of $G,\\left(\\partial G_m / \\partial P\\right)_T=V_m$, we establish the condition for equilibrium:\r\n$$\r\n\\begin{gathered}\r\n\\Delta G=\\Delta G_f^{\\circ}(\\mathrm{C}, \\text { graphite })-\\Delta G_f^{\\circ}(\\mathrm{C}, \\text { diamond }) \\\\\r\n+\\left(V_m^{\\text {graphite }}-V_m^{\\text {diamond }}\\right)(\\Delta P)=0 \\\\\r\n0=0-2.90 \\times 10^3+\\left(V_m^{\\text {graphite }}-V_m^{\\text {diamond }}\\right)(P-1 \\mathrm{bar}) \\\\\r\nP=1 \\mathrm{bar}+\\frac{2.90 \\times 10^3}{M_C\\left(\\frac{1}{\\rho_{\\text {graphite }}}-\\frac{1}{\\rho_{\\text {diamond }}}\\right)} \\\\\r\n=1 \\mathrm{bar}+\\frac{2.90 \\times 10^3}{12.00 \\times 10^{-3} \\mathrm{~kg} \\mathrm{~mol}^{-1} \\times\\left(\\frac{1}{2.25 \\times 10^3 \\mathrm{~kg} \\mathrm{~m}^{-3}}-\\frac{1}{3.52 \\times 10^3 \\mathrm{~kg} \\mathrm{~m}^{-3}}\\right)}\\\\\r\n=10^5 \\mathrm{~Pa}+1.51 \\times 10^9 \\mathrm{~Pa}=1.51 \\times 10^4 \\mathrm{bar}\r\n\\end{gathered}\r\n$$\r\nFortunately for all those with diamond rings, although the conversion of diamond to graphite at $1 \\mathrm{bar}$ and $298 \\mathrm{~K}$ is spontaneous, the rate of conversion is vanishingly small.\r\n"
+ },
+ {
+  "problem_text": "Imagine tossing a coin 50 times. What are the probabilities of observing heads 25 times (i.e., 25 successful experiments)?",
+  "answer_latex": "0.11 ",
+  "answer_number": "0.11",
+  "unit": " ",
+  "source": "thermo",
+  "problemid": " 12.8",
+  "comment": " ",
+  "solution": "The trial of interest consists of 50 separate experiments; therefore, $n=50$. Considering the case of 25 successful experiments where $j=25$. The probability $\\left(P_{25}\\right)$ is\r\n$$\r\n\\begin{aligned}\r\nP_{25} & =C(n, j)\\left(P_E\\right)^j\\left(1-P_E\\right)^{n-j} \\\\\r\n& =C(50,25)\\left(P_E\\right)^{25}\\left(1-P_E\\right)^{25} \\\\\r\n& =\\left(\\frac{50 !}{(25 !)(25 !)}\\right)\\left(\\frac{1}{2}\\right)^{25}\\left(\\frac{1}{2}\\right)^{25}=\\left(1.26 \\times 10^{14}\\right)\\left(8.88 \\times 10^{-16}\\right)=0.11\r\n\\end{aligned}\r\n$$"
+ },
+ {
+  "problem_text": "In a rotational spectrum of $\\operatorname{HBr}\\left(B=8.46 \\mathrm{~cm}^{-1}\\right)$, the maximum intensity transition in the R-branch corresponds to the $J=4$ to 5 transition. At what temperature was the spectrum obtained?",
+  "answer_latex": "4943 ",
+  "answer_number": "4943",
+  "unit": " $\\mathrm{~K}$",
+  "source": "thermo",
+  "problemid": " 14.4",
+  "comment": " ",
+  "solution": "The information provided for this problem dictates that the $J=4$ rotational energy level was the most populated at the temperature at which the spectrum was taken. To determine the temperature, we first determine the change in occupation number for the rotational energy level, $a_J$, versus $J$ as follows:\r\n$$\r\n\\begin{aligned}\r\na_J & =\\frac{N(2 J+1) e^{-\\beta h c B J(J+1)}}{q_R}=\\frac{N(2 J+1) e^{-\\beta h c B J(J+1)}}{\\left(\\frac{1}{\\beta h c B}\\right)} \\\\\r\n& =N \\beta h c B(2 J+1) e^{-\\beta h c B J(J+1)}\r\n\\end{aligned}\r\n$$\r\nNext, we take the derivative of $a_J$ with respect to $J$ and set the derivative equal to zero to find the maximum of the function:\r\n$$\r\n\\begin{aligned}\r\n\\frac{d a_J}{d J} & =0=\\frac{d}{d J} N \\beta h c B(2 J+1) e^{-\\beta h c B J(J+1)} \\\\\r\n0 & =\\frac{d}{d J}(2 J+1) e^{-\\beta h c B J(J+1)} \\\\\r\n0 & =2 e^{-\\beta h c B J(J+1)}-\\beta h c B(2 J+1)^2 e^{-\\beta h c B J(J+1)} \\\\\r\n0 & =2-\\beta h c B(2 J+1)^2 \\\\\r\n2 & =\\beta h c B(2 J+1)^2=\\frac{h c B}{k T}(2 J+1)^2 \\\\\r\nT & =\\frac{(2 J+1)^2 h c B}{2 k}\r\n\\end{aligned}\r\n$$\r\nSubstitution of $J=4$ into the preceding expression results in the following temperature at which the spectrum was obtained:\r\n$$\r\n\\begin{aligned}\r\nT & =\\frac{(2 J+1)^2 h c B}{2 k} \\\\\r\n& =\\frac{(2(4)+1)^2\\left(6.626 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}\\right)\\left(3.00 \\times 10^{10} \\mathrm{~cm} \\mathrm{~s}^{-1}\\right)\\left(8.46 \\mathrm{~cm}^{-1}\\right)}{2\\left(1.38 \\times 10^{-23} \\mathrm{~J} \\mathrm{~K}^{-1}\\right)} \\\\\r\n& =4943 \\mathrm{~K}\r\n\\end{aligned}\r\n$$"
+ }
+]