 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-4/35e/b/solution

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

For van der Waals, the identity gives

(\partial E / \partial V)_{T} = a N^{2} / V^{2}

, so

E (T, V) = f (T) - a N^{2} / V

. The dilute monatomic limit fixes

f (T) = 3 N k T /2

, hence

E = 3 N k T /2 - a N^{2} / V

and

C_{V} = 3 N k /2

at every volume.

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