 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/36c/b/iii/solution

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

Since

⟨ z ⟩ = (β m g)^{- 1}

Z_{N} = \frac{1}{N !} (\frac{A ⟨ z ⟩}{λ ^{3}})^{N} .

Using

lo g N! ≃ N lo g N - N

and

S = k_{B} (lo g Z_{N} + β ⟨ E ⟩)

gives

S = N k_{B} [lo g (\frac{A ⟨ z ⟩}{N λ ^{3}}) + \frac{7}{2}] .

This has the Sackur-Tetrode form with the effective volume

A ⟨ z ⟩

7/2

rather than

5/2

. The extra unit arises because the atmosphere's vertical extent grows with temperature and its mean gravitational energy per atom is

k_{B} T

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