Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 37B / c / Solution

Replicating a macroscopic equilibrium system scales $S,V,N$, and $E$ by the same factor, so $E(S,V,N)$ is homogeneous of degree one. Euler's theorem and the first lawgiveDifferentiating this identity and comparing with the first law cancels the $TdS$, $-pdV$, and $\mu dN$ terms, leaving the Gibbs-Duhem equationTherefore