Isothermal-isobaric partition function ($\Delta ={\int }_0^{\infty }Z(N,V,T)e^{-\beta PV}dV$)

Integrating a canonical partition function over volume gives the displayed continuous-volume normalization. For a temperature-independent measure and Hamiltonian, $-{\partial }_{\beta }\ln \Delta =⟨E+PV⟩$. Multiplying by a volume-independent prefactor leaves normalized volume probabilities unchanged, but a temperature-dependent prefactor changes this derivative.

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.