For a gas described by the van der Waals equation of state, P = nRT/(V - nb) - an²/V². Use this equation to complete these tasks: a. Calculate (∂U/∂V)T using (∂U/∂V)T = T(∂U/∂V)V - P. b. Derive an expression for the change in internal energy, ΔUT = ∫^Vf Vi ((∂U/∂V)TdV, in compressing a van der

Question

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

Accepted Answer

a. [latex]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[/latex]

[latex]= \frac{nRT}{V-nb}- \frac{nRT}{V-nb}+\frac{n^{2}\alpha}{V^{2}}=\frac{n^{2}\alpha}{V^{2}}[/latex]

b. [latex]\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)[/latex]

Note that [latex]\Delta U_{T}[/latex] is zero if the attractive part of the intermolecular potential is zero.

Question 3.5: For a gas described by the van der Waals equation of state, ...

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