 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/37b/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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 law

d E = T d S - p d V + μ d N

give

E = TS - p V + μ N .

Differentiating this identity and comparing with the first law cancels the

T d S

- p d V

, and

μ d N

S d T - V d p + N d μ = 0.

Therefore

d μ = \frac{V d p - S d T}{N} .

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