Cluster integrals: Difference between revisions

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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]]
[[Image:ijk.png]]
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The first three cluster integrals are (Ref. 1 Eq. 13.6)
The first three cluster integrals are (Ref. 1 Eq. 13.6)
:<math>b_1 = \frac{1}{V}\int d\tau_1 =1</math>
:<math>b_1 = \frac{1}{1!V}\int d\tau_1 =1</math>


Ref. 1 Eq. 13.7:
Ref. 1 Eq. 13.7:


:<math>b_2 = \frac{1}{2V} \iint f(r_{12}) d\tau_2 d\tau_1 = \frac{1}{2}\int_0^\infty 4\pi r^2 f(r) dr</math>
:<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:
and Ref. 1 Eq. 13.8:


:<math>b_3 = \frac{1}{6V} \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>
:<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.
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==
==See also==
*[[Cluster diagrams]]
*[[Cluster diagrams]]

Revision as of 15:23, 31 July 2007

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)

Ref. 1 Eq. 13.7:

and Ref. 1 Eq. 13.8:

using the Mayer f-function notation.

Irreducible clusters

Irreducible clusters are denoted by

note .


note

note

See also

References

  1. Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.