Virial coefficients 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) 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

is the Boltzmann constant is the fugacity the chemical potential is the canonical partition particles:

particles. The Hamiltonian is a sum of the kinetic energies-particle potential energy (interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions. The grand partition function. In this manner one derives

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.

virial coefficients becomes quickly a complex combinatorial problem. Making the classical approximation and neglecting 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:

is the interaction potential between particle 1 and 2 (which are assumed to be identical particles).

Definition in terms of graphs

are related to the irreducible Mayer cluster integrals through

The latter are concisely defined in terms of graphs.

The rule for turning these graphs into integrals is as follows:

1. Take a graph and label.

2. Associate a labelled coordinate k to each of the vertices, representing the continuous degrees of freedom associated with that particle. The coordinate 0 is reserved for the white vertex

3. With each bond linking two vertices associate the Mayer f-function corresponding to the interparticle potential

4. Integrate over all coordinates assigned to the black vertices

5. Multiply the end result with the symmetry number of the graph, defined as the inverse of the number of permutations of the black labelled vertices that leave the graph topologically invariant.

The first two cluster integrals are

1. {|

|| || |} The expression of the second virial coefficient is thus:

). This classical expression for the second virial coefficient was first derived by Leonard Ornstein in his 1908 Leiden University Ph.D. thesis.

See also

• Boyle temperature vanishes

• Excess property

• Compressibility factor

References

Further reading

• http://scitation.aip.org/content/aip/journal/jcp/50/10/10.1063/1.1670902

• http://scitation.aip.org/content/aip/journal/jcp/50/11/10.1063/1.1670994

• Reid, C. R., Prausnitz, J. M., Poling B. E., Properties of gases and liquids, IV edition, Mc Graw-Hill, 1987