Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 9 / b / Solution

Differentiate $H=U+pV$ at fixed pressure and subtract the heat capacity at constant volume:Now use $U=U(T,V)$ and the multivariable chain rule along a constant-pressure path:Substitution cancels the fixed-volume derivative and gives the volume derivative form of the heat-capacity difference,For the ideal gas, the first term in brackets vanishes and $V=NkT/p$, so $(\partial V/\partial T{)}_p=Nk/p$. Consequently $\boxed{C_p-C_V=Nk}$ for the entire gas, with fixed particle number $N$. The extra heat capacity accounts for expansion work at constant pressure.