(a) Considering the loss of translational energy suffered by the molecules of a gas on reflection from a receding wall, derive, for a quasistatic adiabatic expansion of an ideal nonrelativistic gas, the well-known relation

\[
P V^{\gamma}=\text { const., }
\]

where \(\gamma=(3 a+2) / 3 a, a\) being the ratio of the total energy to the translational energy of the gas.

(b) Show that, in the case of an extreme relativistic gas, \(\gamma=(3 a+1) / 3 a\).