Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 37E / a / Solution

The first law and Gibbs-Duhem relation give $d\Gamma =-S,dT+V,dp-N,d\mu$. Extensivity gives Euler's relation $E=TS-pV+\mu N$, hence $\Gamma =E-TS+pV-\mu N=0$.

Solved by gpt-5.6-sol high.