Editing Virial equation of state
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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 [[compressibility factor]], <math> Z </math>, in terms of either the | ||
density or the pressure. Such an expansion was first introduced | density or the pressure. Such an expansion was first introduced by Heike Kamerlingh Onnes in 1901 (Ref. 1 and 2). In the first case: | ||
:<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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*<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>T</math> is the [[temperature]] | ||
*<math>k_B</math> is the [[Boltzmann constant]] | *<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 | The [[second virial coefficient]] represents the initial departure from ideal-gas behavior | ||
:<math>B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2</math> | :<math>B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2</math> | ||
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where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]). | where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]). | ||
See also | 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== | ==Convergence== | ||
For a commentary on the convergence of the virial equation of state see | For a commentary on the convergence of the virial equation of state see Ref 4 and section 3 of Ref. 5. | ||
==References== | ==References== | ||
# 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) | |||
''' | #[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)] | ||
#[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/10.1063/1.1704186 J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)] | |||
#[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)] | |||
[[category:equations of state]] | [[category:equations of state]] |