Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 9D / c / Solution

Here $\mathcal{E}=m{c}^2+p^2/(2m)+\cdots$, so $p{\mathcal{E}}^{\prime }=p^2/m$. Consequently $P=(3V{)}^{-1}\int p^2/m,\bar{n},dp=2U/(3V)$. For an adiabatic monatomic gas, $u\propto a^{-5}$ and $n\propto a^{-3}$, hence $T\propto u/n\propto a^{-2}$.

Solved by gpt-5.6-sol high.