The Entropy of Mixing according to Liquid Lattice Theory

These assumptions having been accepted as required, we consider the total number of ways of arranging the ui identical molecules of the solvent and identical molecules of the solute on the lattice comprising no = ni+ri2 cells. This is just the number of combinations of no things taken Ui at a time, or 

This treatment, resting essentially on the assumed approximate interchangeability of molecules of solvent and solute in the solution, cannot possibly hold for polymer solutions in which the solute molecule may be a thousand or more times the size of the solvent. The long chain polymer may be considered to consist of x chain segTneTits each of which is equal in size to a solvent molecule x is, of course, the ratio of the molar volumes of the solute and solvent. A segment and a solvent molecule may replace one another in the liquid lattice. In other respects the assumptions required are equivalent to those used above. The polymer solution differs from that containing an equal proportion of monomeric solute in the one important respect that sets of x contiguous cells in the lattice are required for accommodation of polymer molecules, whereas no such restriction applies to the solution of the monomeric solute. The situation is illustrated in Fig. 110. 

We wish to calculate first the total coTifiguratioTial eTitropy of the poly- 

If each of the n2 polymer molecules to be added to the lattice were 

Arrangements in which the sets of x contiguous lattice cells chosen for occupation by polymer molecules are identical but which differ only in the permutation of the polymer molecules over these sets would be counted as different in this enumeration scheme. Since the polymer molecules actually are identical, it is appropriate to eliminate this redundancy. We require merely the number Q. of ways in which ri2 sets of X consecutively adjacent cells may be chosen from the lattice the order in which the sets are filled by polymer molecules is immaterial. Since U2 differentiable polymer molecules (labeled, for example, in the order of their insertion into the lattice) could be assigned to a given arrangement of U2 sets of sites in ri2 different ways, it follows that 

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