Evaluate (left(partial^{2} P / partial T^{2}ight)_{u},left(partial^{2} mu / partial T^{2}ight)_{u}), and (left(partial^{2} mu / partial T^{2}ight)_{P}) of

Question:

Evaluate $\left(\partial^{2} P / \partial T^{2}ight)_{u},\left(\partial^{2} \mu / \partial T^{2}ight)_{u}$, and $\left(\partial^{2} \mu / \partial T^{2}ight)_{P}$ of an ideal Fermi gas and check that your results satisfy the thermodynamic relations

\[
C_{V}=V T\left(\frac{\partial^{2} P}{\partial T^{2}}ight)_{v}-N T\left(\frac{\partial^{2} \mu}{\partial T^{2}}ight)_{v}
\]

and

\[
C_{P}=-N T\left(\frac{\partial^{2} \mu}{\partial T^{2}}ight)_{P}
\]

Examine the low-temperature behavior of these quantities.