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

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.