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. 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.

In the ensemble, the probability of a microstate ${\displaystyle i}$ is ${\displaystyle Z^{-1}e^{-\beta (E(i)+pV(i))}}$, where ${\displaystyle Z}$ is the partition function, ${\displaystyle E(i)}$ is the internal energy of the system in microstate ${\displaystyle i}$, and ${\displaystyle V(i)}$ is the volume of the system in microstate ${\displaystyle i}$.

The probability of a macrostate is ${\displaystyle Z^{-1}e^{-\beta (E+pV-TS)}=Z^{-1}e^{-\beta G}}$, where ${\displaystyle G}$ is the Gibbs free energy.