Virial coefficient

Virial coefficients B i {displaystyle B_{i}} appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient B 2 {displaystyle B_{2}} depends only on the pair interaction between the particles, the third ( B 3 {displaystyle B_{3}} ) depends on 2- and non-additive 3-body interactions, and so on. Virial coefficients B i {displaystyle B_{i}} appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient B 2 {displaystyle B_{2}} depends only on the pair interaction between the particles, the third ( B 3 {displaystyle B_{3}} ) depends on 2- and non-additive 3-body interactions, and so on. The first step in obtaining a closed expression for virial coefficients is a cluster expansion of the grand canonical partition function Here p {displaystyle p} is the pressure, V {displaystyle V} is the volume of the vessel containing the particles, k B {displaystyle k_{B}} is Boltzmann's constant, T {displaystyle T} is the absolute temperature, λ = exp ⁡ [ μ / ( k B T ) ] {displaystyle lambda =exp} is the fugacity, with μ {displaystyle mu } the chemical potential. The quantity Q n {displaystyle Q_{n}} is the canonical partition function of a subsystem of n {displaystyle n} particles: Here H ( 1 , 2 , … , n ) {displaystyle H(1,2,ldots ,n)} is the Hamiltonian (energy operator) of a subsystem of n {displaystyle n} particles. The Hamiltonian is a sum of the kinetic energies of the particles and the total n {displaystyle n} -particle potential energy (interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions. The grand partition function Ξ {displaystyle Xi } can be expanded in a sum of contributions from one-body, two-body, etc. clusters. The virial expansion is obtained from this expansion by observing that ln ⁡ Ξ {displaystyle ln Xi } equals p V / ( k B T ) {displaystyle pV/(k_{B}T)} . In this manner one derives These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function Q 1 {displaystyle Q_{1}} contains only a kinetic energy term. In the classical limit ℏ = 0 {displaystyle hbar =0} the kinetic energy operators commute with the potential operators and the kinetic energies in numerator and denominator cancel mutually. The trace (tr) becomes an integral over the configuration space. It follows that classical virial coefficients depend on the interactions between the particles only and are given as integrals over the particle coordinates. The derivation of higher than B 3 {displaystyle B_{3}} virial coefficients becomes quickly a complex combinatorial problem. Making the classical approximation andneglecting non-additive interactions (if present), the combinatorics can be handled graphically as first shown by Joseph E. Mayer and Maria Goeppert-Mayer. They introduced what is now known as the Mayer function: and wrote the cluster expansion in terms of these functions. Here u ( | r → 1 − r → 2 | ) {displaystyle u(|{vec {r}}_{1}-{vec {r}}_{2}|)} is the interaction potential between particle 1 and 2 (which are assumed to be identical particles). The virial coefficients B i {displaystyle B_{i}} are related to the irreducible Mayer cluster integrals β i {displaystyle eta _{i}} through