 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-2/9d/c/solution

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

Here

E = m c^{2} + p^{2} / (2 m) + \dots

, so

p E^{'} = p^{2} / m

. Consequently

P = (3 V)^{- 1} \int p^{2} / m, \overset{n}{ˉ}, d p = 2 U / (3 V)

. For an adiabatic monatomic gas,

u \propto a^{- 5}

and

n \propto a^{- 3}

, hence

T \propto u / n \propto a^{- 2}

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