= Finite-size normalization of the isothermal-isobaric ensemble
{title2=$-\partial_\beta\ln(\beta P\Delta)=\langle H\rangle-k_BT$}
For a classical monatomic <ideal gas>, direct volume integration gives $\Delta=\Lambda^{-3N}(\beta P)^{-N-1}$. Consequently the continuous-volume ensemble has $\langle H\rangle=(5N/2+1)k_BT$. Defining a dimensionless thermodynamic function using $\widetilde\Delta=\beta P\Delta$ instead gives $-\partial_\beta\ln\widetilde\Delta=(5N/2)k_BT$. The normalized volume distribution is identical, because the prefactor cancels in probabilities. Its temperature derivative nevertheless contributes $-k_BT$ to the derived enthalpy and $-k_B$ to its heat capacity. The two conventions agree at leading extensive order, but cannot be mixed in an exact finite-$N$ fluctuation identity.
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