Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 35C / i / Solution

The one-particle partition function isThe classical indistinguishable noninteracting gas has $Z_N=Z_1^N/N!$; for labeled particles omit $N!$, which changes none of the requested derivatives. With $\beta =(k_{B}T{)}^{-1}$,The allowed equation of state gives $PV=N{k}_{B}T$, so the enthalpy is $E+PV=5N{k}_{B}T/2$ and $\boxed{C_P=5N{k}_B/2}$, at fixed particle number.