Editing Cluster integrals

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.
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 (Ref. 1 Eq. 13.6)
:<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>
==See also==
*[[Cluster diagrams]]
==References==
# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.
#[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]]