7.10: Chapter 7 Problems
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]