# 7.10: Chapter 7 Problems

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
7.1 Derive the following relations from the definitions of $$\alpha, \kappa_{T}$$, and $$\rho$$ :
$\alpha=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_{p} \quad \kappa_{T}=\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right)_{T}$

7.2 Use equations in this chapter to derive the following expressions for an ideal gas:
$\alpha=1 / T \quad \kappa_{T}=1 / p$

7.3 For a gas with the simple equation of state
$V_{\mathrm{m}}=\frac{R T}{p}+B$
(Eq. 2.2.8), where $$B$$ is the second virial coefficient (a function of $$T$$ ), find expressions for $$\alpha$$, $$\kappa_{T}$$, and $$\left(\partial U_{\mathrm{m}} / \partial V\right)_{T}$$ in terms of $$\mathrm{d} B / \mathrm{d} T$$ and other state functions.

7.4 Show that when the virial equation $$p V_{\mathrm{m}}=R T\left(1+B_{p} p+C_{p} p^{2}+\cdots\right)$$ (Eq. 2.2.3) adequately represents the equation of state of a real gas, the Joule-Thomson coefficient is given by
$\mu_{\mathrm{JT}}=\frac{R T^{2}\left[\mathrm{~d} B_{p} / \mathrm{d} T+\left(\mathrm{d} C_{p} / \mathrm{d} T\right) p+\cdots\right]}{C_{p, \mathrm{~m}}}$
Note that the limiting value at low pressure, $$R T^{2}\left(\mathrm{~d} B_{p} / \mathrm{d} T\right) / C_{p, \mathrm{~m}}$$, is not necessarily equal to zero even though the equation of state approaches that of an ideal gas in this limit.

7.5 The quantity $$(\partial T / \partial V)_{U}$$ is called the Joule coefficient. James Joule attempted to evaluate this quantity by measuring the temperature change accompanying the expansion of air into a vacuum - the "Joule experiment." Write an expression for the total differential of $$U$$ with $$T$$ and $$V$$ as independent variables, and by a procedure similar to that used in Sec. 7.5.2 show that the Joule coefficient is equal to
$\frac{p-\alpha T / \kappa_{T}}{C_{V}}$

7.6 $$p-V-T$$ data for several organic liquids were measured by Gibson and Loeffler. $${ }^{11}$$ The following formulas describe the results for aniline.
Molar volume as a function of temperature at $$p=1$$ bar $$(298-358 \mathrm{~K})$$ :
$V_{\mathrm{m}}=a+b T+c T^{2}+d T^{3}$
where the parameters have the values
$\begin{array}{ll} a=69.287 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} & c=-1.0443 \times 10^{-4} \mathrm{~cm}^{3} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \\ b=0.08852 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} & d=1.940 \times 10^{-7} \mathrm{~cm}^{3} \mathrm{~K}^{-3} \mathrm{~mol}^{-1} \end{array}$
Molar volume as a function of pressure at $$T=298.15 \mathrm{~K}$$ (1-1000 bar):
$V_{\mathrm{m}}=e-f \ln (g+p / \text { bar })$
where the parameter values are
$e=156.812 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \quad f=8.5834 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \quad g=2006.6$

(a) Use these formulas to evaluate $$\alpha, \kappa_{T},(\partial p / \partial T)_{V}$$, and $$(\partial U / \partial V)_{T}$$ (the internal pressure) for aniline at $$T=298.15 \mathrm{~K}$$ and $$p=1.000$$ bar.

(b) Estimate the pressure increase if the temperature of a fixed amount of aniline is increased by $$0.10 \mathrm{~K}$$ at constant volume.

7.7 (a) From the total differential of $$H$$ with $$T$$ and $$p$$ as independent variables, derive the relation $$\left(\partial C_{p, \mathrm{~m}} / \partial p\right)_{T}=-T\left(\partial^{2} V_{\mathrm{m}} / \partial T^{2}\right)_{p}$$
(b) Evaluate $$\left(\partial C_{p, \mathrm{~m}} / \partial p\right)_{T}$$ for liquid aniline at $$300.0 \mathrm{~K}$$ and 1 bar using data in Prob. $$7.6 .$$

7.8 (a) From the total differential of $$V$$ with $$T$$ and $$p$$ as independent variables, derive the relation $$(\partial \alpha / \partial p)_{T}=-\left(\partial \kappa_{T} / \partial T\right)_{p}$$.
(b) Use this relation to estimate the value of $$\alpha$$ for benzene at $$25^{\circ} \mathrm{C}$$ and 500 bar, given that the value of $$\alpha$$ is $$1.2 \times 10^{-3} \mathrm{~K}^{-1}$$ at $$25^{\circ} \mathrm{C}$$ and 1 bar. (Use information from Fig. $$7.2$$ on page 168.)

7.9 Certain equations of state supposed to be applicable to nonpolar liquids and gases are of the form $$p=T f\left(V_{\mathrm{m}}\right)-a / V_{\mathrm{m}}^{2}$$, where $$f\left(V_{\mathrm{m}}\right)$$ is a function of the molar volume only and $$a$$ is a constant.
(a) Show that the van der Waals equation of state $$\left(p+a / V_{\mathrm{m}}^{2}\right)\left(V_{\mathrm{m}}-b\right)=R T$$ (where $$a$$ and $$b$$ are constants) is of this form.
(b) Show that any fluid with an equation of state of this form has an internal pressure equal to $$a / V_{\mathrm{m}}^{2}$$.

7.10 Suppose that the molar heat capacity at constant pressure of a substance has a temperature dependence given by $$C_{p, \mathrm{~m}}=a+b T+c T^{2}$$, where $$a, b$$, and $$c$$ are constants. Consider the heating of an amount $$n$$ of the substance from $$T_{1}$$ to $$T_{2}$$ at constant pressure. Find expressions for $$\Delta H$$ and $$\Delta S$$ for this process in terms of $$a, b, c, n, T_{1}$$, and $$T_{2}$$.

7.11 At $$p=1 \mathrm{~atm}$$, the molar heat capacity at constant pressure of aluminum is given by
$C_{p, \mathrm{~m}}=a+b T$
where the constants have the values
$a=20.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \quad b=0.01238 \mathrm{JK}^{-2} \mathrm{~mol}^{-1}$
Calculate the quantity of electrical work needed to heat $$2.000 \mathrm{~mol}$$ of aluminum from $$300.00 \mathrm{~K}$$ to $$400.00 \mathrm{~K}$$ at $$1 \mathrm{~atm}$$ in an adiabatic enclosure.

7.12 The temperature dependence of the standard molar heat capacity of gaseous carbon dioxide in the temperature range $$298 \mathrm{~K}-2000 \mathrm{~K}$$ is given by
$C_{p, \mathrm{~m}}^{\circ}=a+b T+\frac{c}{T^{2}}$
where the constants have the values
$a=44.2 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \quad b=8.8 \times 10^{-3} \mathrm{~J} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \quad c=-8.6 \times 10^{5} \mathrm{~J} \mathrm{~K} \mathrm{~mol}^{-1}$
Calculate the enthalpy and entropy changes when one mole of $$\mathrm{CO}_{2}$$ is heated at 1 bar from $$300.00 \mathrm{~K}$$ to $$800.00 \mathrm{~K}$$. You can assume that at this pressure $$C_{p, \mathrm{~m}}$$ is practically equal to $$C_{p, \mathrm{~m}}^{\circ}$$.

7.13 This problem concerns gaseous carbon dioxide. At $$400 \mathrm{~K}$$, the relation between $$p$$ and $$V_{\mathrm{m}}$$ at pressures up to at least 100 bar is given to good accuracy by a virial equation of state truncated

at the second virial coefficient, $$B$$. In the temperature range $$300 \mathrm{~K}-800 \mathrm{~K}$$ the dependence of $$B$$ on temperature is given by
$B=a^{\prime}+b^{\prime} T+c^{\prime} T^{2}+d^{\prime} T^{3}$
where the constants have the values
\begin{aligned} &a^{\prime}=-521 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \\ &b^{\prime}=2.08 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \\ &c^{\prime}=-2.89 \times 10^{-3} \mathrm{~cm}^{3} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \\ &d^{\prime}=1.397 \times 10^{-6} \mathrm{~cm}^{3} \mathrm{~K}^{-3} \mathrm{~mol}^{-1} \end{aligned}

(a) From information in Prob. 7.12, calculate the standard molar heat capacity at constant pressure, $$C_{p, \mathrm{~m}}^{\circ}$$, at $$T=400.0 \mathrm{~K}$$.

(b) Estimate the value of $$C_{p, \mathrm{~m}}$$ under the conditions $$T=400.0 \mathrm{~K}$$ and $$p=100.0$$ bar.

7.14 A chemist, needing to determine the specific heat capacity of a certain liquid but not having an electrically heated calorimeter at her disposal, used the following simple procedure known as drop calorimetry. She placed $$500.0 \mathrm{~g}$$ of the liquid in a thermally insulated container equipped with a lid and a thermometer. After recording the initial temperature of the liquid, $$24.80^{\circ} \mathrm{C}$$, she removed a $$60.17$$-g block of aluminum metal from a boiling water bath at $$100.00^{\circ} \mathrm{C}$$ and quickly immersed it in the liquid in the container. After the contents of the container had become thermally equilibrated, she recorded a final temperature of $$27.92^{\circ} \mathrm{C}$$. She calculated the specific heat capacity $$C_{p} / m$$ of the liquid from these data, making use of the molar mass of aluminum $$\left(M=26.9815 \mathrm{~g} \mathrm{~mol}^{-1}\right)$$ and the formula for the molar heat capacity of aluminum given in Prob. $$7.11 .$$

(a) From these data, find the specific heat capacity of the liquid under the assumption that its value does not vary with temperature. Hint: Treat the temperature equilibration process as adiabatic and isobaric $$(\Delta H=0)$$, and equate $$\Delta H$$ to the sum of the enthalpy changes in the two phases.

(b) Show that the value obtained in part (a) is actually an average value of $$C_{p} / m$$ over the temperature range between the initial and final temperatures of the liquid given by
$\frac{\int_{T_{1}}^{T_{2}}\left(C_{p} / m\right) \mathrm{d} T}{T_{2}-T_{1}}$

7.15 Suppose a gas has the virial equation of state $$p V_{\mathrm{m}}=R T\left(1+B_{p} p+C_{p} p^{2}\right)$$, where $$B_{p}$$ and $$C_{p}$$ depend only on $$T$$, and higher powers of $$p$$ can be ignored.
(a) Derive an expression for the fugacity coefficient, $$\phi$$, of this gas as a function of $$p$$.
(b) For $$\mathrm{CO}_{2}(\mathrm{~g})$$ at $$0.00^{\circ} \mathrm{C}$$, the virial coefficients have the values $$B_{p}=-6.67 \times 10^{-3}$$ bar $$^{-1}$$ and $$C_{p}=-3.4 \times 10^{-5} \mathrm{bar}^{-2}$$. Evaluate the fugacity $$f$$ at $$0.00^{\circ} \mathrm{C}$$ and $$p=20.0$$ bar.

7.16 Table $$7.6$$ on the next page lists values of the molar volume of gaseous $$\mathrm{H}_{2} \mathrm{O}$$ at $$400.00^{\circ} \mathrm{C}$$ and 12 pressures.
(a) Evaluate the fugacity coefficient and fugacity of $$\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$ at $$400.00^{\circ} \mathrm{C}$$ and 200 bar.
(b) Show that the second virial coefficient $$B$$ in the virial equation of state, $$p V_{\mathrm{m}}=R T(1+$$ $$\left.B / V_{\mathrm{m}}+C / V_{\mathrm{m}}^{2}+\cdots\right)$$, is given by
$B=R T \lim _{p \rightarrow 0}\left(\frac{V_{\mathrm{m}}}{R T}-\frac{1}{p}\right)$
where the limit is taken at constant $$T$$. Then evaluate $$B$$ for $$\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$ at $$400.00^{\circ} \mathrm{C}$$.

Table 7.6 Molar volume of $$\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$ at $$400.00^{\circ} \mathrm{C}^{a}$$
$\begin{tabular}{cccc} \hline$$p / 10^{5} \mathrm{~Pa}$$ & $$V_{\mathrm{m}} / 10^{-3} \mathrm{~m}^{3} \mathrm{~mol}^{-1}$$ & $$p / 10^{5} \mathrm{~Pa}$$ & $$V_{\mathrm{m}} / 10^{-3} \mathrm{~m}^{3} \mathrm{~mol}^{-1}$$ \\ \hline 1 & $$55.896$$ & 100 & $$0.47575$$ \\ 10 & $$5.5231$$ & 120 & $$0.37976$$ \\ 20 & $$2.7237$$ & 140 & $$0.31020$$ \\ 40 & $$1.3224$$ & 160 & $$0.25699$$ \\ 60 & $$0.85374$$ & 180 & $$0.21447$$ \\ 80 & $$0.61817$$ & 200 & $$0.17918$$ \\ \hline \end{tabular}$
$${ }^{a}$$ based on data in Ref. [75]

This page titled 7.10: Chapter 7 Problems is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Howard DeVoe via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.