Isothermal–isobaric ensemble

The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature $T\,$ and constant pressure $P\,$ applied. It is also called the $NpT$-ensemble, where the number of particles $N\,$ is also kept as a constant. This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition.^[1] The partition function can be written as the weighted sum of the partition function of canonical ensemble, $Z(N,V,T)\,$ .

$\Delta (N,P,T)=\int Z(N,V,T)\exp(-\beta PV)CdV.\,\;$

where $\beta =1/k_{B}T\,$ ($k_{B}\,$ is the Boltzmann constant), and $V\,$ is volume of the system.

There are several candidates for the normalization factor $C\,$, e.g., $C=N/V\,$, or $C=\beta P\,$. These choices make the partition function a nondimensional quantity. The differences vanish in the thermodynamic limit, i.e., in the limit of infinite number of particles.

The characteristic state function of this ensemble is the Gibbs free energy,

$G(N,P,T)=-k_{B}T\ln \Delta (N,P,T)\;\,$

This thermodynamic potential is related to the Helmholtz free energy (logarithm of the canonical partition function), $F\,$, in the following way:^[1]

$G=F+PV.\;\,$

References

1. Dill, Ken A.; Bromberg, Sarina; Stigter, Dirk (2003). Molecular Driving Forces. New York: Garland Science.