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Virial coefficientVirial coefficients B_{i} appear as coefficients in the virial expansion of the pressure of a manyparticle system in powers of the density. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient B_{2} depends only on the pair interaction between the particles, the third (B_{3}) depends on 2 and nonadditive 3body interactions, and so on. Additional recommended knowledgeThe first step in obtaining a closed expression for virial coefficients is a cluster expansion^{[1]} of the grand canonical partition function
Here p is the pressure, V is the volume of the vessel containing the particles, k_{B} is Boltzmann's constant, T is the absolute temperature, λ = exp[μ / (k_{B}T)], with μ the chemical potential. The quantity Q_{n} is the canonical partition function of a subsystem of n particles: Here is the Hamiltonian (energy operator) of a subsystem of n particles. The Hamiltonian is a sum of the kinetic energies of the particles and the total nparticle potential energy (interaction energy). The latter includes pair interactions and possibly 3body and higherbody interactions. The grand partition function Ξ can be expanded in a sum of contributions from onebody, twobody, etc. clusters. The virial expansion is obtained from this expansion by observing that lnΞ equals pV / (k_{B}T). In this manner one derives
These are quantumstatistical expressions containing kinetic energies. Note that the oneparticle partition function Q_{1} contains only a kinetic energy term. In the classical limit 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} virial coefficients becomes quickly a complex combinatorial problem. Making the classical approximation and neglecting nonadditive interactions (if present), the combinatorics can be handled graphically as first shown by Joseph E. Mayer and Maria GoeppertMayer ^{[2]}. They introduced the function and wrote the cluster expansion in terms of these functions. Here is the interaction between particle 1 and 2 (which are assumed to be identical particles). Definition in terms of graphsThe virial coeffcients B_{i} are related to the irreducible Mayer cluster integrals β_{i} through
The rule for turning these graphs into integrals is as follows:
The first two cluster integrals are
In particular we get where particle 2 was assumed to define the origin (). This classical expression for the second virial coefficient was first derived by L. S. Ornstein in his 1908 Leiden University Ph.D. thesis. See alsoBoyle temperature  temperature at which the second virial coefficient B_{2} vanishes Literature
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Virial_coefficient". A list of authors is available in Wikipedia. 