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| In an [[ideal gas]] there are no intermolecular interactions. However, in an imperfect or real gas, this is not so, and the
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| [[second virial coefficient]] is other than zero. Mayer and Mayer developed a theoretical treatment of the
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| [[Virial equation of state |virial coefficients]] in terms of '''cluster integrals'''.
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| The simplest cluster is that consisting of a single molecule, not bound to any other. | | The simplest cluster is that consisting of a single molecule, not bound to any other. |
| A cluster of three specified identical molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways: | | A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways: |
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| [[Image:ijk.png]]
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| The first three cluster integrals are (Eq. 13.6 in <ref>Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13</ref>)
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| :<math>b_1 = \frac{1}{1!V}\int d\tau_1 =1</math>
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| Ref. 1 Eq. 13.7:
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| :<math>b_2 = \frac{1}{2!V} \iint f(r_{12}) d\tau_2 d\tau_1 = \frac{1}{2}\int_0^\infty 4\pi r^2 f(r) dr</math>
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| and Ref. 1 Eq. 13.8:
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| :<math>b_3 = \frac{1}{3!V} \iiint (f_{31} f_{21} + f_{32}f_{31} + f_{32}f_{21} + f_{32}f_{31}f_{21}) d\tau_3 d\tau_2 d\tau_1</math>
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| using the [[Mayer f-function]] notation.
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| ==Irreducible clusters==
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| Irreducible clusters are denoted by <math>\beta_k</math>
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| :<math>\beta_1 = \int f_{31} d\tau_3 = \frac{1}{V}\iint f_{12}d\tau_1 d\tau_2 =\int_0^\infty 4 \pi r^2 f(r) dr</math>
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| note <math>b_2 = \frac{1}{2}\beta_1</math>.
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| :<math>\beta_2 = \frac{1}{2V}\iiint f_{32}f_{31}f_{21} d\tau_1 d\tau_2 d\tau_3 </math>
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| note <math>b_3 = \frac{1}{2} \beta_1^2 + \frac{1}{3}\beta_2</math>
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| :<math>\beta_3 = \frac{1}{6V}\iiiint (3f_{43}f_{32}f_{21}f_{41}+6f_{43}f_{32}f_{21}f_{41}f_{31} + f_{43}f_{32}f_{21}f_{41}f_{31}f_{42})d\tau_1 d\tau_2 d\tau_3 d\tau_4 </math>
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| note <math>b_4 = \frac{2}{3}\beta_1^3 + \beta_1 \beta_2 + \frac{1}{4}\beta_3</math>
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| ==Hellmann and Bich diagrams==
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| Hellmann and Bich have rederived the [[virial equation of state]] from the [[Grand canonical ensemble#Grand canonical partition function | grand canonical partition function]] without restricting themselves to pairwise [[intermolecular pair potential]]s <ref>[http://dx.doi.org/10.1063/1.3626524 Robert Hellmann and Eckard Bich "A systematic formulation of the virial expansion for nonadditive interaction potentials", Journal of Chemical Physics '''135''' 084117 (2011)]</ref>. This leads to expressions for the virial coefficients that, for <math>B_6</math> and beyond, require the evaluation of far fewer diagrams when compared to the original diagrams of Mayer or to the reformulation of Ree and Hoover <ref>[http://dx.doi.org/10.1063/1.1726136 Francis H. Ree and William G. Hoover "Reformulation of the Virial Series for Classical Fluids", Journal of Chemical Physics '''41''' 1635 (1964)]</ref>.
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| ==See also== | | ==See also== |
| | *[[Mayer f-function]] |
| *[[Cluster diagrams]] | | *[[Cluster diagrams]] |
| ==References== | | ==References== |
| <references/>
| | # Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) |
| ;Related reading
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| *[http://dx.doi.org/10.1016/0003-4916(58)90058-7 Edwin E. Salpeter "On Mayer's theory of cluster expansions", Annals of Physics '''5''' pp. 183-223 (1958)]
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| [[Category: Statistical mechanics]] | | [[Category: Statistical mechanics]] |