Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 35E / b / Solution

For van der Waals, the identity gives $(\partial E/\partial V{)}_T=a{N}^2/V^2$, so $E(T,V)=f(T)-a{N}^2/V$. The dilute monatomic limit fixes $f(T)=3NkT/2$, hence $E=3NkT/2-a{N}^2/V$ and $C_V=3Nk/2$ at every volume.

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