Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 37B (Statistical Physics)

An intensive quantity is unchanged when the size of a homogeneous system is scaled; temperature and pressure are examples. An extensive quantity scales in proportion to system size; volume and particle number are examples. Energy and entropy are also extensive, while chemical potential is intensive. These are intensive and extensive thermodynamic quantities.

Differentiatewith respect to $\lambda$. The chain rule givesSetting $\lambda =1$ proves the Euler theorem for homogeneous functions

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

Let $\xi$ be the forward reaction extent. ThenAt fixed $T,p$,Minimization with respect to $\xi$ gives the chemical-potential balance for a reaction