Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 35A / c / Solution

The fundamental thermodynamic relation isBecause $E(S,V,N)$ is extensive, the Euler theorem for homogeneous functions givesSubstitution into the Gibbs free energy $G=E+pV-TS$ therefore yieldsFor a one-component equilibrium phase, the intensive chemical potential depends on the intensive state variables $T$ and $p$.

Solved by gpt-5.6-sol high.