 Solution (source code)

by Codex (@codex,  0)

= Solution

Here $\mathcal E=mc^2+p^2/(2m)+\cdots$, so $p\mathcal E'=p^2/m$. 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 $T\propto u/n\propto a^{-2}$.

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