Virial coefficient

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Virial coefficients Bi 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 B2 depends only on the pair interaction between the particles, the third (B3) 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[1] of the grand canonical partition function

Ξ = ∑ λnQn = exp[pV / (kBT)].
n

Here p is the pressure, V is the volume of the vessel containing the particles, kB is Boltzmann's constant, T is the absolute temperature, λ = exp[μ / (kBT)], with μ the chemical potential. The quantity Qn 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 particles and 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 Ξ 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 lnΞ = pV / (kBT). In this manner one derives

B_2 = V \left(\frac{1}{2}-\frac{Q_2}{Q_1^2}\right)
B_3 = V^2 \left[ \frac{2Q_2}{Q_1^2}\Big( \frac{2Q_2}{Q_1^2}-1\Big) -\frac{1}{3}\Big(\frac{6Q_3}{Q_1^3}-1\Big) \right].

These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function Q1 contains only a kinetic energy term. In the classical limit \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 B3 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 [2]. They introduced the function

f(1,2) = \exp\left[- \frac{u(|\vec{r}_1- \vec{r}_2|)}{k_B T}\right] - 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).

[edit] Definition in terms of graphs

The virial coeffcients Bi are related to the irreducible Mayer cluster integrals βi through

B_{i+1}=-\frac{i}{i+1}\beta_i


The latter are concisely defined in terms of graphs.

\beta_i=\mbox{The sum of all connected, irreducible graphs with one white and}\ i\ \mbox{black vertices}

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

  1. Take a graph and label its white vertex by k = 0 and the remaining black vertices with k = 1,..,i.
  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

β1 = =\int d\mathbf{1} f(\mathbf{0},\mathbf{1})
β2 = =\frac{1}{2}\int d\mathbf{1} \int d\mathbf{2} f(\mathbf{0},\mathbf{1})f(\mathbf{0},\mathbf{2})f(\mathbf{1},\mathbf{2})

In particular we get

B_2 = -\frac{1}{2} \int \Big( e^{-u(|\vec{r}_1|)/(k_BT)} - 1 \Big) d\vec{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 Leyden University Ph.D. thesis.


[edit] Literature

  1. ^ T. L. Hill, Introduction to Statistical Thermodynamics, Addison-Wesley (1960).
  2. ^ J. E. Mayer and M. G. Mayer, Statistical Mechanics, Wiley, New York, (1940).

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).
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