Virial coefficient

Virial coefficients $B\_i$ appear as coefficients in the virial expansion of the pressure of a many-particle 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 interactionbetween the particles, the third ($B\_3$) 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 [T. L. Hill, "Introduction to Statistical Thermodynamics", Addison-Wesley (1960).] of the grand canonical partition function:$Xi=sum\_n\; lambda^n\; Q\_n\; =\; exp\; [p\; V\; /\; (k\_B\; T\; )]\; .$ Here $p$ is the pressure, $V$ is the volume of the vessel containing the particles, $k\_B$ is Boltzmann's constant, $T$ is the absolutetemperature, $lambda\; =exp\; [mu/(k\_BT)]$, with $mu$ the chemical potential. The quantity $Q\_n$ is the canonical partition function of a subsystem of $n$ particles::$Q\_n\; =\; operatorname\{tr\}\; [\; e^\{-\; H(1,2,ldots,n)/(k\_B\; T)\}\; ]\; .$Here $H(1,2,ldots,n)$ is the Hamiltonian (energy operator) of a subsystem of$n$ particles. The Hamiltonian is a sum of the kinetic energies of the particlesand the total $n$-particle potential energy (interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions.The grand partition function $Xi$ 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 $ln\; Xi$ equals $p\; V\; /\; (k\_B\; T\; )$. In this manner one derives:$B\_2\; =\; V\; left(frac\{1\}\{2\}-frac\{Q\_2\}\{Q\_1^2\}\; ight)$:$B\_3\; =\; V^2\; left\; [\; frac\{2Q\_2\}\{Q\_1^2\}Big(\; frac\{2Q\_2\}\{Q\_1^2\}-1Big)\; -frac\{1\}\{3\}Big(frac\{6Q\_3\}\{Q\_1^3\}-1Big)\; ight]$.These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function $Q\_1$ contains only a kinetic energy term. In the classical limit$hbar\; =\; 0$ the kinetic energy operators commute with the potential operators andthe 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 complexcombinatorial problem. Making the classical approximation andneglecting non-additive interactions (if present), the combinatoricscan be handled graphically as first shown by Joseph E. Mayer and Maria Goeppert-Mayer [J. E. Mayer and M. G. Mayer, "Statistical Mechanics," Wiley, New York, (1940). ] .They introduced the function :$f(1,2)\; =\; expleft\; [-\; frac\{u(|vec\{r\}\_1-\; vec\{r\}\_2|)\}\{k\_B\; T\}\; ight]\; -\; 1$ and wrote the cluster expansion in terms of these functions. Here $u(|vec\{r\}\_1-\; vec\{r\}\_2|)$ is the interaction between particle 1 and 2 (which are assumed to be identical particles).

Definition in terms of graphs

The virial coeffcients $B\_i$ 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:
# Take a graph and label its white vertex by $k=0$ and the remaining black vertices with $k=1,..,i$.
# 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
# With each bond linking two vertices associate the Mayer f-function corresponding to the interparticle potential
# Integrate over all coordinates assigned to the black vertices
# 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

:In particular we get:$B\_2\; =\; -frac\{1\}\{2\}\; int\; Big(\; e^\{-u(|vec\{r\}\_1|)/(k\_BT)\}\; -\; 1\; Big)\; dvec\{r\}\_1\; ,$where particle 2 was assumed to define the origin ($vec\{r\}\_2\; =\; vec\{0\}$).This classical expression for the second virial coefficient was first derived by L. S. Ornstein in his 1908 Leiden University Ph.D. thesis.

ee also

Boyle temperature - temperature at which the second virial coefficient $B\_\{2\}$ vanishes

Literature

See further:

*J. P. Hansen and I. R. McDonald, "The theory of Simple Liquids," Academic Press, London (1986).
*J. H. Dymond and E. B. Smith, "The Virial Coefficients of Pure Gases and Mixtures: a Critical Compilation," Clarendon, Oxford (1980).