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\sect{Classical gas with general dispersion relation}

Consider a gas of $N$ non-interacting particles in a $d$ dimensional box. 
The kinetic energy of a particle is $\epsilon_p = c|p|^s$. 

\begin{itemize}

\item[(a)] Find the partition function of the gas for a given temperature is $T$.

\item[(b)] Define $\gamma = 1+(s/d)$ and using (a) show that the energy is $E = \frac{N T}{\gamma-1}$.

\item[(c)] Show that the entropy is $S = \frac{N}{\gamma -1}\,\text{ln} \,(P V^\gamma)+ f(N)$.

\item[(d)] Deduce that in an adiabatic process ${PV^{\gamma }=\const}$.
 
\item[(e)] Show that the heat capacity ratio is ${C_P/C_V = \gamma}$.

\end{itemize}