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The number of configurations in an assembly and cooperative phenomena

  • T. S. Chang (a1)

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We consider an ideal problem of adsorption of single and double particles upon a solid surface which has its sites of accommodation regularly arranged, and by comparing the equilibrium properties obtained by Bethe's method with the ordinary statistical formulae, we obtain approximate expressions for:

(1) g(N, n, X), the number of ways of arranging n particles upon N sites of a lattice so that the number of neighbouring sites occupied by the particles is X.

(2) g2(N, n, X), the number of ways of arranging n double particles upon N sites so that each of the double particles takes up two adjacent sites and the number of neighbouring sites occupied by two different particles is X.

Both these expressions are found to agree with the exact values when the N sites lie on a straight line. When we use the first expression to construct the configurational partition functions of certain physical assemblies and expand them in powers of 1/kT, they are found to agree with the corresponding rigorous expressions as far as (1/kT)3, which is the highest power which we can find rigorously at present. With the help of the first expression, formal equations for superlattice formation in an alloy with the composition 1: 1 and equations for the separation into phases of regular liquids are given. Lastly we show that atoms and molecules in a regular liquid may dissociate or recombine suddenly accompanied by a latent heat. This is a new cooperative phenomenon, which may bear some resemblance to the melting process between the solid and liquid states.

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Peierls, R., Proc. Cambridge Phil. Soc. 32 (1936), 471.

Bethe, H., Proc. Roy. Soc. A, 150 (1935), 552.

Fowler, R. H., Proc. Cambridge Phil. Soc. 34 (1938), 382.

Peierls, R., Proc. Cambridge Phil. Soc. 32 (1936), 471;Wang, J. S., Proc. Roy. Soc. A, 161 (1937), 127.

Rushbrooke, G. S., Proc. Roy. Soc. A, 166 (1938), 296.

Chang, T. S., Proc. Roy. Soc. A, 169 (1939), 512. Equation (26) is not given explicitly, but it can easily be found from the g.p.f. of the typical aggregate given there.

Fowler, R. H. and Rushbrooke, G. S., Trans. Faraday Soc. 33 (1937), 1272.

Fowler, R. H., Statistical mechanics, 2nd ed. (Cambridge, 1937), p. 36.

A further check of (36) is afforded by writing it in the form

and summing both sides for all X. The left-hand side must give N!/{n!(N-n)!}. The right-hand side can be summed by Gauss's lemma (Whittaker and Watson, Modern analysis, 4th ed. (Cambridge, 1927), p. 281), and is found to give N!/{n!(N-n)!}. as it should.

Kirkwood, J., J. Chem. Phys. 6 (1938), 70.

Fowler, R. H., Proc. Cambridge Phil. Soc. 32 (1936), 144.

Chang, T. S., Proc. Cambridge Phil. Soc. 34 (1938), 224.

Chang, T. S., Proc. Cambridge Phil. Soc. 35 (1939), 70.

Bethe, H., Proc. Roy. Soc. 150 (1935), 552.

A generalization allowing z to be different for different phases can be easily made. The only change necessary is to define ø as the right-hand side of (58), where V and z are both to be chosen so as to make ø a minimum. The curve ø, c described below may be discontinuous, and trivial changes in determining c 1 and c 2 by drawing tangents have to be made. It is also easy to make such generalizations as allowing z to change in a regular liquid which does not tend to separate into phases, or in an alloy with a superlattice of the type AB provided that the lattice structure of the alloy does not change into one which can no longer be divided into two sublattices.

We do not write out (65).

Rushbrooke, G. S., Proc. Roy. Soc. A, 166 (1938), 296.

A solid-liquid fusion theory has recently been proposed by Lennard-Jones and Devonshire (Proc. Roy. Soc. 169 (1939), 317) in which the lattice of a solid is taken to be one of the two sublattices of a lattice of the type AB. The solid and the liquid states are distinguished from one another by assuming that in the liquid state there are equal numbers of atoms on the two sublattices, while in the solid state nearly all the atoms are on one sublattice. Thus the fusion process is merely superlattice formation. According to our discussion in § 4, there cannot be any latent heat phenomena unless drastic assumptions are made about Y. The mere fact of the expansion of the lattice constant during the ordering will not introduce a latent heat. An evident defect of the theory is that there is no a priori reason for taking the lattice of the solid to be a sublattice of a lattice of the type AB, and it may just as well be taken as one of the four sublattices of a lattice of the type AB 3. The wide difference between these two possibilities shows that the quantitative results calculated from such theories are extremely doubtful.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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