For a real gas obeying the van der Waals equation, the inversion temperature $T_i$ is defined as the temperature at which the Joule-Thomson coefficient is zero. What is the expression for $T_i$ in terms of $a$ and $b$?

Solution

1. The Joule-Thomson coefficient $$\begin{array}{rl}{\mu }_{JT}& =(\frac{\partial T}{\partial P}{)}_H\\ & =0\end{array}$$ implies $(\frac{\partial H}{\partial P}{)}_T=0$.
2. For a van der Waals gas, $H=U+PV$.
3. Using the thermodynamic relation $(\frac{\partial H}{\partial P}{)}_T=V-T(\frac{\partial V}{\partial T}{)}_P$.
4. Substituting the van der Waals equation leads to $T_i=\frac{2a}{Rb}$.
5. Hence the answer is (B).