Isothermal–isobaric ensemble

Statistical mechanics

Thermodynamics Kinetic theory
Particle Statistics Spin-Statistics Theorem Identical Particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic Statistics Braid Statistics
Thermodynamic Ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric µVT Open statistical
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi Bose

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

Statistical mechanics

Statistical ensembles	Microcanonical Canonical Grand canonical Isothermal–isobaric Isoenthalpic–isobaric ensemble Open

Statistical thermodynamics	Characteristic state functions

Partition functions	Translational Vibrational Rotational

Equations of state	Dieterici Van der Waals Ideal gas law Birch–Murnaghan

Entropy	Sackur–Tetrode equation Tsallis entropy Von Neumann entropy

Particle statistics	Maxwell–Boltzmann statistics Fermi–Dirac statistics Bose–Einstein statistics

Statistical field theory	Conformal field theory Osterwalder–Schrader axioms

Quantum statistical mechanics	Density matrix Gibbs measure Partition function Phase space formulation of quantum mechanics Slater determinant

See also	Probability distribution Elementary particles

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