Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 4 / ii / Solution

Write $\mathcal{R}=k_B/(\mu m_u)$. At fixed density,The specific internal energy isApplying the first law of thermodynamics adiabatically givesCombining these logarithmic derivatives yieldsAs gas pressure dominates,the result for a monatomic perfect gas. In the radiation-pressure limit it tends to $1/4$.