Virial equation of state

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The virial equation of state is used to describe the behavior of diluted gases. It is usually written as an expansion of the compressibility factor, Z, in terms of either the density or the pressure. Such an expansion was first introduced by Heike Kamerlingh Onnes in 1901 (Ref. 1 and 2). In the first case:

\frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}.

where

[edit] Virial coefficients

The second virial coefficient represents the initial departure from ideal-gas behavior

B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2

where NA is Avogadros number and dτ1 and dτ2 are volume elements of two different molecules in configuration space.

One can write the third virial coefficient as

B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23} dr_1 dr_2 dr_3

where f is the Mayer f-function (see also: Cluster integrals). See also:

[edit] Convergence

For a commentary on the convergence of the virial equation of state see Ref 4 and section 3 of Ref. 5.

[edit] References

  1. H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden 71 pp. 3-25 (1901)
  2. H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 4 pp. 125-147 (1902)
  3. James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics 7 pp. 195-229 (1940)
  4. J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics 5 pp. 841-847 (1964)
  5. A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter 20 283102 (2008)
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