Finite-size normalization of the isothermal-isobaric ensemble ($-{\partial }_{\beta }\ln (\beta P\Delta )=⟨H⟩-k_{B}T$)

For a classical monatomic ideal gas, direct volume integration gives $\Delta ={\Lambda }^{-3N}(\beta P{)}^{-N-1}$. Consequently the continuous-volume ensemble has $⟨H⟩=(5N/2+1)k_{B}T$. Defining a dimensionless thermodynamic function using $\tilde{\Delta }=\beta P\Delta$ instead gives $-{\partial }_{\beta }\ln \tilde{\Delta }=(5N/2)k_{B}T$. The normalized volume distribution is identical, because the prefactor cancels in probabilities. Its temperature derivative nevertheless contributes $-k_{B}T$ 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.