Cluster integrals

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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 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])

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[edit]

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

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 B_6 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]


Related reading