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The virial equation of state is used to describe the behavior of diluted gases.  
The '''virial equation of state''' is used to describe the behavior of diluted gases.  
It is usually written as an expansion of the [[compresiblity factor]], <math> Z </math>, in terms of either the
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 by Kammerlingh Onnes. In the first case:
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>.
:<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}</math>.
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where  
where  


* <math> p </math> is the pressure
* <math> p </math> is the [[pressure]]
 
*<math> V </math>  is the volume
*<math> V </math>  is the volume
*<math> N </math> is the number of molecules
*<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> \rho \equiv \frac{N}{V} </math> is the (number) density
*<math> B_k\left( T \right) </math> is called the k-th virial coefficient
*<math> B_k\left( T \right) </math> is called the k-th virial coefficient
==Virial coefficients==
==Virial coefficients==
The second virial coefficient represents the initial departure from ideal-gas behavior
The [[second virial coefficient]] represents the initial departure from [[Ideal gas |ideal-gas]] behaviour
 
<math>B_{2}(T)= \frac{N_0}{2V} \int .... \int (1-e^{-u/kT}) ~d\tau_1 d\tau_2
</math>


where <math>N_0</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
:<math>B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2</math>
in configuration space. The integration is to be performed over all available phase-space; that is,
over the volume of the containing vessel.
For the special case where the molecules posses spherical symmetry, so that <math>u</math> depends not on
orientation, but only on the separation <math>r</math> of a pair of molecules, the equation can be simplified to


:<math>B_{2}(T)= - \frac{1}{2} \int_0^\infty \left(\langle \exp\left(-\frac{u(r)}{k_BT}\right)\rangle -1 \right) 4 \pi r^2 dr</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.


Using the [[Mayer f-function]]
One can write the third virial coefficient as


:<math>f_{ij}=f(r_{ij})= \exp\left(-\frac{u(r)}{k_BT}\right) -1 </math>  
:<math>B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23}  dr_1 dr_2 dr_3</math>


one can write the third virial coefficient more compactly as
where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]).
See also <ref>[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)]</ref>


:<math>B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23}  dr_1 dr_2 dr_3
==Convergence==
</math>
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>.
==Quantum virial coefficients==
Using the [[path integral formulation]] one can also calculate the virial coefficients of quantum systems  <ref>[http://dx.doi.org/10.1063/1.3573564 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)]</ref>.
==References==
==References==
# H. Kammerlingh Onnes "", Communications from the Physical Laboratory Leiden '''71''' (1901)
<references/>
#[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)]
'''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
*[http://dx.doi.org/10.1063/1.4929392  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)]
*[https://doi.org/10.1063/1.5016165 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)]


[[category:equations of state]]
[[category:equations of state]]

Latest revision as of 15:51, 17 November 2020

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]

  1. ↑ M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik 24 pp. 467-492 (1885)
  2. ↑ 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)
  3. ↑ 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)
  4. ↑ H. D. Ursell "The evaluation of Gibbs' phase-integral for imperfect gases", Mathematical Proceedings of the Cambridge Philosophical Society 23 pp. 685-697 (1927)
  5. ↑ M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics 89 pp. 1005-1017 (1996)
  6. ↑ J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics 5 pp. 841-847 (1964)
  7. ↑ A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter 20 283102 (2008)
  8. ↑ 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

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