= Solution
The <volume> <integral> is a <gamma function> <integral>,
$$
\int_0^\infty V^N e^{-\beta PV}\,dV=\frac{N!}{(\beta P)^{N+1}},
\qquad
\boxed{Q_P=\frac{1}{\Lambda^{3N}(\beta P)^{N+1}}}.
$$
For the prescribed bridge relation, $\beta PQ_P=(\beta P\Lambda^3)^{-N}$, so $G=Nk_BT\ln(\beta P\Lambda^3)$. Since $\Lambda\propto T^{-1/2}$, this gives the macroscopic ideal-gas result
$$
\boxed{H=G-T(\partial_TG)_P=\frac52Nk_BT,\qquad C_P=\frac52Nk_B},
\qquad C_P-C_V=Nk_B.
$$
Heating 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 $\langle H\rangle_P=(5N/2+1)k_BT$ and $C_P=(5N/2+1)k_B$. The two answers differ because $-\partial_\beta\ln(\beta PQ_P)=-\partial_\beta\ln Q_P-k_BT$. The normalized continuous-volume distribution has $\langle PV\rangle=(N+1)k_BT$; 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.
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