3.27 Maxwell relations, example for vdW gas

We now apply Maxwell relations to the equation of a vdW gas to find some astonishing parameter dependencies:

• Dependency of the entropy on the volume for a vdW gas:

  \begin{equation*} \begin{split} &\left( \frac{\partial S}{\partial V}\right)_T = \left( \frac{\partial p}{\partial T}\right)_V \quad \Rightarrow \quad \Delta S = \int_{V_1}^{V_2} \left( \frac{\partial p}{\partial T}\right)_V dV \\ \Delta S & = \int_{V_1}^{V_2} \left[\left( \frac{\partial }{\partial T}\right)_V \left(\frac{n\,R\,T}{V - n\,b}-\frac{a\,n^2}{V^2} \right)\right] dV = n\,R \ln \frac{V_2-n\,b}{V_1-n\,b} \end{split} \label{eq:S_V_vdW} \end{equation*}

  (3.65)

  So (astonishingly) \(\Delta S\) does not depend on \(a\).

• Internal pressure \(\pi_T\) of a vdW gas.
  We start with the total differential of \(U(S,V)\):

  \begin{equation*} \begin{split} \pi_T = \left( \frac{\partial U}{\partial V}\right)_T = &\;\; \left( \frac{\partial U}{\partial S}\right)_V \left( \frac{\partial S}{\partial V}\right)_T + \left( \frac{\partial U}{\partial V}\right)_S = T \left( \frac{\partial S}{\partial V}\right)_T - p = T \left( \frac{\partial p}{\partial T}\right)_V - p \\ p = &\;\; \frac{n\,R\,T}{V - n\,b}-\frac{a\,n^2}{V^2} \quad \Rightarrow \quad \left( \frac{\partial p}{\partial T}\right)_V = \frac{n\,R}{V - n\,b} \\ \pi_T = \left( \frac{\partial U}{\partial V}\right)_T = &\;\; T \left( \frac{\partial p}{\partial T}\right)_V - p = \frac{n\,R\,T}{V - n\,b} - \left(\frac{n\,R\,T}{V - n\,b} - \frac{a\,n^2}{V^2} \right) = \frac{a\,n^2}{V^2}\\ \end{split} \label{eq:dU_dV_vdW} \end{equation*}

  (3.66)

  So (astonishingly) the internal pressure \(\pi_T\) does not depend on \(b\).

© J. Carstensen (TD Kin I)