Past exam of the thermal and statistical physics course of the University of Cambridge / 2015 / Paper 1 / 1 / b / Solution

Differentiate the Euler relation for a single-component simple compressible system:The thermodynamic fundamental relation for the same system is $dU=TdS-pdV+\mu dN$. Subtraction givesDividing by $N$, with $s=S/N$ and $v=V/N$, proves the Gibbs-Duhem equationThe underlying Euler relation assumes extensive bulk thermodynamics and no additional work coordinates. The result says that the intensive quantities are constrained; the chemical potential cannot be varied independently of temperature and pressure within one homogeneous phase.