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.
Derivation The first step in obtaining a closed expression for virial coefficients is a cluster expansion of the grand canonical partition function
Ξ = ∑ n λ n Q n = e ( p V ) / ( k B T ) {\displaystyle \Xi =\sum _{n}{\lambda ^{n}Q_{n}}=e^{\left(pV\right)/\left(k_{\text{B}}T\right)}}
Here p {\displaystyle p} is the pressure, V {\displaystyle V} is the volume of the vessel containing the particles, k B {\displaystyle k_{\text{B}}} is the Boltzmann constant, T {\displaystyle T} is the absolute temperature, λ = exp [ μ / ( k B T ) ] {\displaystyle \lambda =\exp[\mu /(k_{\text{B}}T)]} 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:
Q n = tr [ e − H ( 1 , 2 , … , n ) / ( k B T ) ] . {\displaystyle Q_{n}=\operatorname {tr} [e^{-H(1,2,\ldots ,n)/(k_{\text{B}}T)}].}
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
B 2 = V ( 1 2 − Q 2 Q 1 2 ) {\displaystyle B_{2}=V\left({\frac {1}{2}}-{\frac {Q_{2}}{Q_{1}^{2}}}\right)}
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