Isothermal-isobaric ensemble ($NpT$)

An equilibrium statistical ensemble with fixed particle number, pressure and temperature, allowing volume and energy to fluctuate. With a temperature-independent volume measure, its joint weight is proportional to $e^{-\beta (E+PV)}$. Its normalization and any temperature-dependent prefactor must be specified when taking thermodynamic derivatives.

With temperature-independent microstate enthalpies and measure, differentiating the normalized weight $e^{-\beta H}/\Delta$ gives ${\partial }_{\beta }⟨H⟩=-\mathrm{Var}(H)$. Since $d\beta /dT=-1/(k_B{T}^2)$, the displayed response follows. A stable finite-system equilibrium distribution therefore has nonnegative heat capacity at constant pressure in this setting.

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.