# Isothermal–isobaric ensemble

The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature ${\displaystyle T\,}$ and constant pressure ${\displaystyle P\,}$ applied. It is also called the ${\displaystyle NpT}$-ensemble, where the number of particles ${\displaystyle N\,}$ 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, ${\displaystyle Z(N,V,T)\,}$ .

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

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

There are several candidates for the normalization factor ${\displaystyle C\,}$, e.g., ${\displaystyle C=N/V\,}$, or ${\displaystyle C=\beta P\,}$. 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,

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

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

${\displaystyle G=F+PV.\;\,}$

## References

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