Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 47 / d

The volume integral is a gamma function integral,For the prescribed bridge relation, $\beta P{Q}_P=(\beta P{\Lambda }^3{)}^{-N}$, so $G=N{k}_{B}T\ln (\beta P{\Lambda }^3)$. Since $\Lambda \propto T^{-1/2}$, this gives the macroscopic ideal-gas resultHeating at fixed pressure requires both an increase in internal energy and expansion work; at fixed volume the latter contribution is absent.

The finite-size normalization of the isothermal-isobaric ensemble matters here. Applying part (a) literally to the unprefactored $Q_P$ instead gives $⟨H{⟩}_P=(5N/2+1)k_{B}T$ and $C_P=(5N/2+1)k_B$. The two answers differ because $-{\partial }_{\beta }\ln (\beta P{Q}_P)=-{\partial }_{\beta }\ln Q_P-k_{B}T$. The normalized continuous-volume distribution has $⟨PV⟩=(N+1)k_{B}T$; its variance agrees with the unprefactored result in part (b). The bridge convention subtracts an order-one contribution and recovers the usual extensive thermodynamic answer. Both agree to leading order in the thermodynamic limit, but their exact finite-$N$ derivatives should not be mixed.