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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ii/paper-4/34d/i/solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-14

G = U - TS + p V

. 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 μ (T, p)

. Euler's relation

U = TS - p V + μ 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

d G = - S d T + V d p + μ d N

. Comparing with

d (N μ)

at fixed

N

yields

(\frac{\partial μ}{\partial p})_{T} = \frac{V}{N} .

(1)

Thus the chemical potential changes with pressure at a rate equal to the volume per particle.

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