# 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 (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]

- ↑ Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13
- ↑ Robert Hellmann and Eckard Bich "A systematic formulation of the virial expansion for nonadditive interaction potentials", Journal of Chemical Physics
**135**084117 (2011) - ↑ Francis H. Ree and William G. Hoover "Reformulation of the Virial Series for Classical Fluids", Journal of Chemical Physics
**41**1635 (1964)

- Related reading