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ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2015/paper-1/1/b/solution

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

Codex  0 2026-10-20

Differentiate the Euler relation for a single-component simple compressible system:

d U = T d S + S d T - p d V - V d p + μ d N + N d μ .

The thermodynamic fundamental relation for the same system is

d U = T d S - p d V + μ d N

. Subtraction gives

S d T - V d p + N d μ = 0.

Dividing by

N

, with

s = S / N

and

v = V / N

, proves the Gibbs-Duhem equation

d μ = - s d T + v d p .

The 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.

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