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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 [[compressibility factor]], <math> Z </math>, in terms of either the | | It is usually written as an expansion of the [[compresiblity 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> | | density or the pressure. Such an expansion was first introduced by Kammerlingh Onnes. In the first case: |
| <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
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| :<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 |
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| * <math> p </math> is the [[pressure]] | | * <math> p </math> is the pressure |
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| *<math> V </math> is the volume | | *<math> V </math> is the volume |
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| *<math> N </math> is the number of molecules | | *<math> N </math> is the number of molecules |
| *<math> T </math> is the [[temperature]]
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| *<math>k_B</math> is the [[Boltzmann constant]]
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| *<math> \rho \equiv \frac{N}{V} </math> is the (number) density | | *<math> \rho \equiv \frac{N}{V} </math> is the (number) density |
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| *<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 |ideal-gas]] behaviour | | The second virial coefficient represents the initial departure from ideal-gas behavior |
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| | <math>B_{2}(T)= \frac{N_0}{2V} \int .... \int (1-e^{-u/kT}) ~d\tau_1 d\tau_2 |
| | </math> |
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| :<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_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 |
| | 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 |
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| 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
| | :<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> |
| in configuration space.
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| One can write the third virial coefficient as
| | Using the [[Mayer f-function]] |
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| :<math>B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23} dr_1 dr_2 dr_3</math> | | :<math>f_{ij}=f(r_{ij})= \exp\left(-\frac{u(r)}{k_BT}\right) -1 </math> |
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| where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]).
| | one can write the third virial coefficient more compactly as |
| 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>
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| ==Convergence==
| | :<math>B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23} dr_1 dr_2 dr_3 |
| 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>.
| | </math> |
| ==Quantum virial coefficients==
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| 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>.
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| ==References== | | ==References== |
| <references/>
| | # H. Kammerlingh Onnes "", Communications from the Physical Laboratory Leiden '''71''' (1901) |
| '''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)] |
| *[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)]
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| *Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928
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| *[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)]
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| *[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)]
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| [[category:equations of state]] | | [[category:equations of state]] |