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Obtain an expression for the isothermal compressibility \(\kappa=-1 / V(\partial V / \partial P)_{T}\) for a van der Waals gas.

Use the relation \((\partial U / \partial V)_{T}=T(\partial P / \partial T)_{V}-P\) and the cyclic rule to obtain an expression for the internal pressure, \((\partial U / \partial V)_{T},\) in terms of \(P, \beta, T,\) and \(\kappa\).

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 \(\kappa_{\text {water}}=4.59 \times 10^{-5} \mathrm{bar}^{-1}\).

Integrate the expression \(\beta=1 / V(\partial V / \partial T)_{P}\) assuming that \(\beta\) is independent of temperature. By doing \(s o,\) obtain an expression for \(V\) as a function of \(T\) and \(\beta\) at constant \(P\).

Because \((\partial H / \partial P)_{T}=-C_{P} \mu_{J-T}\), the change in enthalpy of a gas expanded at constant temperature can be calculated. To do so, the functional dependence of \(\mu_{J-T}\) on \(P\) must be known. Treating Ar as a van der Waals gas, calculate \(\Delta H\) when 1 mole of \(\mathrm{Ar}\) is expanded from 325 bar to 1.75 bar at 375 K. Assume that \(\mu_{J-T}\) is independent of pressure and is given by \(\mu_{J-T}=[(2 a / R T)-b] / C_{P, m},\) and \(C_{P, m}=5 R / 2\) for Ar. What value would \(\Delta H\) have if the gas exhibited ideal gas behavior?

Calculate \(w, q, \Delta H,\) and \(\Delta U\) for the process in which 1.75 moles of water undergoes the transition \(\mathrm{H}_{2} \mathrm{O}(l, 373 \mathrm{K}) \rightarrow\) \(\mathrm{H}_{2} \mathrm{O}(g, 610 . \mathrm{K})\) at 1 bar of pressure. The volume of liquid water at \(373 \mathrm{K}\) is \(1.89 \times 10^{-5} \mathrm{m}^{3} \mathrm{mol}^{-1}\) and the molar volume of steam at \(373 \mathrm{K}\) and \(610 . \mathrm{K}\) is 3.03 and \(5.06 \times 10^{-2} \mathrm{m}^{3} \mathrm{mol}^{-1}\) respectively. For steam, \(C_{P, m}\) can be considered constant over the temperature interval of interest at \(33.58 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\).

Starting with the van der Waals equation of state, find an expression for the total differential \(d P\) in terms of \(d V\) and \(d T .\) By calculating the mixed partial derivatives \(\left(\partial(\partial P / \partial V)_{T} / \partial T\right)_{V}\) and \(\left(\partial(\partial P / \partial T)_{V} / \partial V\right)_{T},\) determine if \(d P\) is an exact differential.

Use \((\partial U / \partial V)_{T}=(\beta T-\kappa P) / \kappa\) to calculate \((\partial U / \partial V)_{T}\) for an ideal gas P3.23 Derive the following relation, \\[ \left(\frac{\partial U}{\partial V_{m}}\right)_{T}=\frac{3 a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} \\] for the internal pressure of a gas that obeys the Redlich-Kwong equation of state, \\[ P=\frac{R T}{V_{m}-b}-\frac{a}{\sqrt{T}} \frac{1}{V_{m}\left(V_{m}+b\right)} \\]

Because \(V\) is a state function, \(\left(\partial(\partial V / \partial T)_{P} / \partial P\right)_{T}=\) \(\left(\partial(\partial V / \partial P)_{T} / \partial T\right)_{P} .\) Using this relationship, show that the isothermal compressibility and isobaric expansion coefficient are related by \((\partial \beta / \partial P)_{T}=-(\partial \kappa / \partial T)_{P}\).

This problem will give you practice in using the cyclic rule. Use the ideal gas law to obtain the three functions \(P=f(V, T), V=g(P, T),\) and \(T=h(P, V) .\) Show that the cyclic rule \((\partial P / \partial V)_{T}(\partial V / \partial T)_{P}(\partial T / \partial P)_{V}=-1\) is obeyed.