# Virial equation of state

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, , in terms of either the
density or the pressure. Such an expansion was first introduced in 1885 by Thiesen ^{[1]} and extensively studied by Heike Kamerlingh Onnes ^{[2]}
^{[3]}, and mathematically by Ursell ^{[4]}. One has

- .

where

- is the pressure
- is the volume
- is the number of molecules
- is the temperature
- is the Boltzmann constant
- is the (number) density
- is called the k-th virial coefficient

## Virial coefficients[edit]

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

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

One can write the third virial coefficient as

where *f* is the Mayer f-function (see also: Cluster integrals).
See also ^{[5]}

## Convergence[edit]

For a commentary on the convergence of the virial equation of state see ^{[6]} and section 3 of ^{[7]}.

## Quantum virial coefficients[edit]

Using the path integral formulation one can also calculate the virial coefficients of quantum systems ^{[8]}.

## References[edit]

- ↑ M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik
**24**pp. 467-492 (1885) - ↑ 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) - ↑ 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) - ↑ H. D. Ursell "The evaluation of Gibbs' phase-integral for imperfect gases", Mathematical Proceedings of the Cambridge Philosophical Society
**23**pp. 685-697 (1927) - ↑ M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics
**89**pp. 1005-1017 (1996) - ↑ J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics
**5**pp. 841-847 (1964) - ↑ A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter
**20**283102 (2008) - ↑ Giovanni Garberoglio and Allan H. Harvey "Path-integral calculation of the third virial coefficient of quantum gases at low temperatures", Journal of Chemical Physics 134, 134106 (2011)

**Related reading**

- James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics
**7**pp. 195-229 (1940) - Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928
- Nathaniel S. Barlow, Andrew J. Schultz, Steven J. Weinstein and David A. Kofke "Analytic continuation of the virial series through the critical point using parametric approximants", Journal of Chemical Physics
**143**071103 (2015) - Harold W. Hatch, Sally Jiao, Nathan A. Mahynski, Marco A. Blanco, and Vincent K. Shen "Communication: Predicting virial coefficients and alchemical transformations by extrapolating Mayer-sampling Monte Carlo simulations", Journal of Chemical Physics
**147**231102 (2017)