Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 34D / i / Solution

The Gibbs free energy is $G=U-TS+pV$. For an extensive thermodynamic system, scaling the amount of matter at fixed pressure, temperature and composition scales $G$ by the same factor, hence $G(T,p,N)=N\mu (T,p)$. Euler's relation $U=TS-pV+\mu N$ gives the same conclusion. This uses the ordinary thermodynamic-limit extensivity assumption, rather than a universal claim about finite-size or long-range-interacting systems.

The fundamental thermodynamic identity gives $dG=-SdT+Vdp+\mu dN$. Comparing with $d(N\mu )$ at fixed $N$ yieldsThus the chemical potential changes with pressure at a rate equal to the volume per particle.