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

Assume the material gas is monatomic. Its specific enthalpy together with the radiation enthalpy isAt constant total pressure, logarithmic differentiation of
$P=\mathcal{R}\rho T+a_{\mathrm{rad}}{T}^4/3$ givesConsequently the specific heat capacity at constant pressure is

The three stellar adiabatic exponents are defined byFor the gas-radiation mixture,andIt follows thatTheir defining derivatives imply the identity

For any equation of state,Eliminating $P/(\rho T{c}_V)$ gives the requested ratio of specific heats:

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.