Editing 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]], <math> Z </math>, in terms of either the
density or the pressure. Such an expansion was first introduced in 1885 by Thiesen <ref>[http://dx.doi.org/10.1002/andp.18852600308 M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik '''24''' pp. 467-492 (1885)]</ref> and extensively studied by Heike Kamerlingh Onnes <ref> 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)</ref>
<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 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)]</ref>, and mathematically by Ursell <ref>[http://dx.doi.org/10.1017/S0305004100011191 H. D. Ursell "The evaluation of Gibbs' phase-integral for imperfect gases", Mathematical Proceedings of the Cambridge Philosophical Society '''23''' pp. 685-697 (1927)]</ref>. One has
 
:<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}</math>.

where 

* <math> p </math> is the [[pressure]]
*<math> V </math>  is the volume
*<math> N </math> is the number of molecules
*<math> T </math> is the [[temperature]]
*<math>k_B</math> is the [[Boltzmann constant]]
*<math> \rho \equiv \frac{N}{V} </math> is the (number) density
*<math> B_k\left( T \right) </math> is called the k-th virial coefficient
==Virial coefficients==
The [[second virial coefficient]] represents the initial departure from [[Ideal gas |ideal-gas]] behaviour 

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

where <math>N_A</math> is [[Avogadro constant | Avogadros number]] and <math>d\tau_1</math> and <math>d\tau_2</math> are volume elements of two different molecules
in configuration space. 

One can write the third virial coefficient as

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

where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]).
See also:
*[http://dx.doi.org/10.1080/002689796173453 M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics '''89''' pp. 1005-1017 (1996)]

==Convergence==
For a commentary on the convergence of the virial equation of state see <ref>[http://dx.doi.org/10.1063/1.1704186     J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)]</ref> and section 3 of <ref>[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]</ref>
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 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
[[category:equations of state]]