 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-4/34c/iii/solution

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

Z (T, μ) = \sum_{i} e^{- (E_{i} - μ N_{i}) / (k T)} .

The probabilities in part (i) and their logarithms give

k lo g Z = S - \frac{E}{T} + \frac{μ N}{T} .

At fixed volume and fixed

μ

, differentiate the partition sum:

\partial_{T} lo g Z = \frac{E - μ N}{k T ^{2}} .

Combining with the preceding identity yields

\frac{\partial}{\partial T} (k T lo g Z)_{μ} = S .

The variable held fixed is essential; this is the grand canonical derivative.

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