Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 1 / a / Solution

The approximation requires a nonrelativistic monatomic gas whose particles collide often enough for local thermodynamic equilibrium, with a mean free path much shorter than every macroscopic scale. Translational motion must dominate its heat capacity, giving the adiabatic exponent $\gamma =5/3$. Heat conduction, viscosity, shocks, radiative heating and cooling, ionization, chemical reactions, and excitation of internal degrees of freedom must be negligible over the flow time. Under those conditions entropy is advected and the gas behaves as an adiabatic perfect gas.