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\sect{Entropy and heat capacity of quantum ideal gases}

Consider an $N$ particle ideal gas confined in volume $V$. 
Find (a) the entropy~$S$ and (b) the heat capacity~$C$, 
highlighting its dependence on the temperature~$T$.

\begin{enumerate}
	
\item[(1)] Consider classical gas.
	
\item[(2)] Consider Fermi gas at low temperatures, using leading order
	Sommerfeld expansion.
	
\item[(3)] Consider Bose gas below the condensation temperature.
	
\item[(4)] Consider Bose gas above the condensation temperature.
	
\item[(5)] What is $C_{Bose}/C_{classical}$ at the condensation temperature?
	
\item[(6)] For temperatures that are above but very close to the condensation temperature, find an approximation for $C_V$
in terms of elementary functions.	

\end{enumerate}

Hints: In (4) use the Grand-Canonical formalism to express $N$ and
$E$ as a function of the temperature $T$ and the fugacity $z$. Use the
equation for $N$ in order to deduce an expression for $(\partial z/\partial T)_N$. 
Note that the derivative of the polylogarithmic function $L_{\alpha}(z)$ is
$(1/z)L_{\alpha-1}(z)$.
Final results should be expressed in terms of $(N,V,T)$,
but it is allowed to define and use the notations
$\lambda_T$ and $\epsilon_F$ and $T_c$.
In item (4) the final result can include
ratios of polylogarithmic functions,
with the fugacity~$z$ as an implicit variable.
Note that such ratios are all of order unity
throughout the whole temperature range
provided ${\alpha>1}$, while functions
with ${\alpha<1}$ are singular at~$z=1$.



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