Isothermal–isobaric ensemble

The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature

This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition.

[1] The NPT ensemble is also useful for measuring the equation of state of model systems whose virial expansion for pressure cannot be evaluated, or systems near first-order phase transitions.

is the internal energy of the system in microstate

-ensemble can be derived from statistical mechanics by beginning with a system of

identical atoms described by a Hamiltonian of the form

This system is described by the partition function of the canonical ensemble in 3 dimensions: where

, the thermal de Broglie wavelength (

(which accounts for indistinguishability of particles) both ensure normalization of entropy in the quasi-classical limit.

[2] It is convenient to adopt a new set of coordinates defined by

such that the partition function becomes If this system is then brought into contact with a bath of volume

at constant temperature and pressure containing an ideal gas with total particle number

, the partition function of the whole system is simply the product of the partition functions of the subsystems: The integral over the

Substituting this into the above expression for the partition function, multiplying by a factor

(see below for justification for this step), and integrating over the volume V then gives The partition function for the bath is simply

Separating this term out of the overall expression gives the partition function for the

, the partition function can be rewritten as which can be written more generally as a weighted sum over the partition function for the canonical ensemble The quantity

is simply some constant with units of inverse volume, which is necessary to make the integral dimensionless.

, but in general it can take on multiple values.

The ambiguity in its choice stems from the fact that volume is not a quantity that can be counted (unlike e.g. the number of particles), and so there is no “natural metric” for the final volume integration performed in the above derivation.

[2] This problem has been addressed in multiple ways by various authors,[3][4] leading to values for C with the same units of inverse volume.

becomes arbitrary) in the thermodynamic limit, where the number of particles goes to infinity.

-ensemble can also be viewed as a special case of the Gibbs canonical ensemble, in which the macrostates of the system are defined according to external temperature

and external forces acting on the system

is the system's Hamiltonian in the absence of external forces and

of the system then occur with probability defined by [6] where the normalization factor

Then the normalization factor becomes where the Hamiltonian has been written in terms of the particle momenta

term is a Gaussian integral, and can be evaluated explicitly as Inserting this result into

gives a familiar expression: This is almost the partition function for the

From the preceding analysis it is clear that the characteristic state function of this ensemble is the Gibbs free energy, This thermodynamic potential is related to the Helmholtz free energy (logarithm of the canonical partition function),