Question 3.5: For a gas described by the van der Waals equation of state, ...
For a gas described by the van der Waals equation of state, P =nRT/\left( V-nb \right)-an^{2}/V^{2}. Use this equation to complete these tasks: a. Calculate \left(\partial U/ \partial V\right)_{T} using \left(\partial U/ \partial V\right)_{T}=T\left( \partial P/\partial T \right)_{V}-P. b. Derive an expression for the change in internal energy, \Delta U_{T}=\int_{V_{i}}^{V_{f}}\left( \partial U/\partial V \right)_{T}dV, in compressing a van der Waals gas from an initial molar volume V_{i} to a final molar volume V_{f} at constant temperature.
a. T\left( \frac{\partial P}{\partial T} \right)_{V}-P=T\left( \frac{\partial \left[ \frac{nRT}{V-nb}-\frac{n^{2}\alpha}{V^{2}} \right]}{\partial T} \right)_{V}- P= \frac{nRT}{V-nb}- P
= \frac{nRT}{V-nb}- \frac{nRT}{V-nb}+\frac{n^{2}\alpha}{V^{2}}=\frac{n^{2}\alpha}{V^{2}}
b. \Delta U_{T}=\int_{V_{i}}^{V_{f}}\left( \frac{\partial U}{\partial V} \right)_{T}dV =\int_{V_{i}}^{V_{f}}\frac{n^{2}\alpha}{V^{2}}dV=n^{2}\alpha\left( \frac{1}{V_{i}}-\frac{1}{V_{f}} \right)
Note that \Delta U_{T} is zero if the attractive part of the intermolecular potential is zero.