Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 35E (Statistical Physics)

From $dF=-S,dT-p,dV$, Maxwell gives $(\partial S/\partial V{)}_T=(\partial p/\partial T{)}_V$. With $dE=T,dS-p,dV$, $(\partial E/\partial V{)}_T=T(\partial p/\partial T{)}_V-p=T^2{\partial }_T(p/T){|}_V$.

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.

On an isotherm, $\Delta Q=T\Delta S=NkT\log [(V_2-bN)/(V_1-bN)]$.

Setting $dS=0$ gives $C_{V}dT+NkT,dV/(V-bN)=0$, hence $T(V-bN{)}^{2/3}=\text{constant}$.

For any reversible Carnot cycle, $\eta =1-T_C/T_H$. The parameters $a,b$ alter the volumes and heat amounts but cancel from the ratio, as required by the second law's universality.