Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 3 / iv / Solution

In the gas-pressure limit $\beta \to 1$, the mixture becomes a monatomic ideal gas, soIn the radiation-pressure limit $\beta \to 0$,whereas ${\chi }_{\rho }=\beta \to 0$ and thereforeThe divergence reflects the singular constant-pressure heat capacity of pure radiation: its pressure fixes its temperature independently of density.