Combinatorial factor

Combinatory factors occurring in size distribution functions for nonlinear condensation polymers (see Eqs. IX-18 and -34). 

Within the present approximation the partition function of the mixture has the same structure (8) as a pure component, except for the ideal combinatorial factor ... 

Of course, the natural order need not be maintained and thus we have to sum over all the particle labellings. Because of the integrals over the impulsions, this gives rise simply to combinatorial factors. 

MSE.ll. I. Prigogine, L. Sarolea, and L. Van Hove, On the combinatory factor in regular assemblies, Tram. Faraday Soc. 48, 485M-92 (1952). 

Inserting the electron line factor [47, 48], the proton slope contribution (6.1), and the combinatorial factor 2 in the skeleton integral in (3.33), one obtains an integral for the electron-line contribution which does not depend on any parameters, and can be easily calculated numerically with an arbitrary precision [49]. Like a more complicated integral for the corrections of order a Za) ui [48] this integral also admits an analytic evaluation, and the analytic result was obtained in [50, 51]... 

The second factor on the right-hand side of Eq. (5.1) is called the combinatorial factor. It is the partition sum of the equivalent ensemble of molecules that interact only through steric restraints. The combinatorial factor takes into account all size and shape effects of the molecules. There is no exact expression for Zc but, by fitting the simplified theoretical models to thermodynamic data of... 

The derivation of the equation for AB)) is a bit tricky, because of some combinatorial factors that cire involved. It is easier not to work with the number of AB pairs, but with the number of AB pairs where B has a specific position with respect to A. Let s assume that we have a square grid, and lets define ABp the number of AB pairs in /3 with B to the right of A. It is reasonable to assume that each of the four positions that B can have with respect to A is equally likely, so that AB)) — 4(AH), and... 

The above reasoning is instructive from two points of view. In the first place, to obtain (20.19) we have to assume that molecules 1 and 2 are of approximately the same size, otherwise they would not each occupy a single lattice point in the liquid. If, for example, the molecules of kind 1 occupy three places, while molecules 2 occupy one place, then the combinatory factor (20.19) must be replaced by an expression of a different form which leads to a different entropy of mixing (c/. chap. XXV). 

The combinatorial factor Q is related to the fraction of the three-dimensional space taken by polymeric chains and the other part taken by the solvent and is independent of the compactness (or otherwise) of the chain. The free volume factor Qfv depends on the experimental values of molar volume V (or segmental volume v), of isobaric expansivity a and of isothermal compressibility top. The configurational (or interactional, or potential... 

Again, we have two combinatorial factors, one each for the associated cation vacancies. Here, the total number of lattices sites, N Ng, = =... 

The behavior of one-dimensional mixtures of hard spheres of different diameters a- and follows directly from the exact validity of the quasi-chemical approximation for the one-dimensional combinatorial factor for particles interacting only with their nearest neighbois. Unfortunately, this result is valid only for one-dimensional systems. It would be of considerable interest to extend the scaled particle theory to deal with mixtures of hard spheres. 

The configurational partition function per chain molecule, q, is expressed in terms of combinatorial factor P, free volume, Vf, and the mean potential energy of a cell, eo ... 

Table 2. Combinatorial factors [10] and integrals [13-16] for Mayer expansions of the virial coefficients B3, B4, and Bs at low D.

As an example, consider B4 for hard spheres. Using the combinatorial factors and numerical integrals in Table 2, the virial coefficient and the two integral equation approximations to it are... 

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