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 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 (Ref. 1 Eq. 13.6)

$b_{1}={\frac {1}{1!V}}\int d\tau _{1}=1$

Ref. 1 Eq. 13.7:

$b_{2}={\frac {1}{2!V}}\iint f(r_{12})d\tau _{2}d\tau _{1}={\frac {1}{2}}\int _{0}^{\infty }4\pi r^{2}f(r)dr$

and Ref. 1 Eq. 13.8:

$b_{3}={\frac {1}{3!V}}\iiint (f_{31}f_{21}+f_{32}f_{31}+f_{32}f_{21}+f_{32}f_{31}f_{21})d\tau _{3}d\tau _{2}d\tau _{1}$

using the Mayer f-function notation.

Irreducible clusters

Irreducible clusters are denoted by $\beta _{k}$

$\beta _{1}=\int f_{31}d\tau _{3}={\frac {1}{V}}\iint f_{12}d\tau _{1}d\tau _{2}=\int _{0}^{\infty }4\pi r^{2}f(r)dr$

note $b_{2}={\frac {1}{2}}\beta _{1}$.

$\beta _{2}={\frac {1}{2V}}\iiint f_{32}f_{31}f_{21}d\tau _{1}d\tau _{2}d\tau _{3}$

note $b_{3}={\frac {1}{2}}\beta _{1}^{2}+{\frac {1}{3}}\beta _{2}$

$\beta _{3}={\frac {1}{6V}}\iiiint (3f_{43}f_{32}f_{21}f_{41}+6f_{43}f_{32}f_{21}f_{41}f_{31}+f_{43}f_{32}f_{21}f_{41}f_{31}f_{42})d\tau _{1}d\tau _{2}d\tau _{3}d\tau _{4}$

note $b_{4}={\frac {2}{3}}\beta _{1}^{3}+\beta _{1}\beta _{2}+{\frac {1}{4}}\beta _{3}$

See also

• Cluster diagrams

References

1. Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.
2. Edwin E. Salpeter "On Mayer's theory of cluster expansions", Annals of Physics 5 pp. 183-223 (1958)