Cluster integrals: Difference between revisions

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(New page: The simplest cluster is that consisting of a single molecule, not bound to any other. A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways: ==Se...)
 
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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
[[second virial coefficient]] is other than zero. Mayer and Mayer developed a theoretical treatment of the
[[Virial equation of state |virial coefficients]] in terms of '''cluster integrals'''.
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 molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:
A cluster of three specified identical molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:
 
[[Image:ijk.png]]
 
 
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>)
:<math>b_1 = \frac{1}{1!V}\int d\tau_1 =1</math>
 
Ref. 1 Eq. 13.7:
 
:<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>
 
and Ref. 1 Eq. 13.8:
 
:<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>
 
using the [[Mayer f-function]] notation.
==Irreducible clusters==
Irreducible clusters are denoted by <math>\beta_k</math>
 
:<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>
 
note <math>b_2 = \frac{1}{2}\beta_1</math>.
 
 
 
:<math>\beta_2 = \frac{1}{2V}\iiint f_{32}f_{31}f_{21} d\tau_1 d\tau_2 d\tau_3 </math>
 
note <math>b_3 = \frac{1}{2} \beta_1^2 + \frac{1}{3}\beta_2</math>
 
:<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>
 
note <math>b_4 = \frac{2}{3}\beta_1^3 + \beta_1 \beta_2 + \frac{1}{4}\beta_3</math>
==Hellmann and Bich diagrams==
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>.
==See also==
==See also==
*[[Mayer f-function]]
*[[Cluster diagrams]]
*[[Cluster diagrams]]
==References==
==References==
# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940)
<references/>
;Related reading
*[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)]
[[Category: Statistical mechanics]]
[[Category: Statistical mechanics]]

Latest revision as of 14:21, 1 September 2011

In an ideal gas there are no intermolecular interactions. However, in an imperfect or real gas, this is not so, and the second virial coefficient is other than zero. Mayer and Mayer developed a theoretical treatment of the virial coefficients in terms of cluster integrals.

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:


The first three cluster integrals are (Eq. 13.6 in [1])

Ref. 1 Eq. 13.7:

and Ref. 1 Eq. 13.8:

using the Mayer f-function notation.

Irreducible clusters[edit]

Irreducible clusters are denoted by

note .


note

note

Hellmann and Bich diagrams[edit]

Hellmann and Bich have rederived the virial equation of state from the grand canonical partition function without restricting themselves to pairwise intermolecular pair potentials [2]. This leads to expressions for the virial coefficients that, for 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 [3].

See also[edit]

References[edit]

Related reading