# 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, $Z(N, V, T) \,$ .

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

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