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\sect{Heat capacity of an ideal Bose gas}

Consider a volume $\mathsf{V}$ that contains $N$ mass $\mass$ bosons. 
The gas is in a thermal equilibrium at temperature~$T$. 

\begin{enumerate}

\item
Write an explicit expression for the condenstation temperature $T_c$.

\item
Calculate the chemical potential, the energy and the pressure in the Boltzmann approximation ${T \gg T_c}$.

\item
Calculate the chemical potential, the energy and the pressure in the regime ${T < T_c}$.

\item
Calculate  $C_v$  for  ${T<T_c}$

\item
Calculate  $C_v$  for  ${T=T_c}$

\item
Calculate  $C_v$  for ${T\gg T_c}$

\item
Express the ratio $C_p/C_v$ using the polylogaritmic functions.  
Explain why $C_p \rightarrow  \infty$  in the condensed phase?

% \frac{5g_\frac{5}{2}(z) g_\frac{1}{2}(z)}{3g^2_\frac{3}{2}(z) 
        
\item
Find the $\gamma$ in the adiabatic equation of state.  
Note that in general it does not equal $C_p/C_v$.

\end{enumerate}


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