Gas Model of a Binary Mixture

Consider a linear system of M sites, Na of which are occupied by molecules of type A and Nb sites are occupied by molecules of type B. If we let N = Na Nb, then we have a system the PF of which is isomorphous to the simple Ising model studied in section 4.1 i.e., each site can be in either one of two states, A or B, depending on the type of molecule occupying the site. In this section we treat a slightly more general case. Each site can be in one of three states empty, occupied by A, and occupied by B. Thus we have M sites, Na sites occupied by A and Nb occupied by B. But Na + Nb M. 

The energy levels are determined by the number of nearest-neighbor pairs of each kind. 

The sum (4.4.2) is over all possible states consistent with the fixed values of N, and Nb-We now write the grand PF for a system opened with respect to both A and B, i.e., 

Thus for each configuration s of the system, we have a factor Xa for each site occupied by A, a, factor Xb for a site occupied by B, and a factor Ao = 1 for an empty site. For each pair of neighbor sites we have a factor exp( —where the interaction energies are given in (4.4.1). 

The sum over all configurations gives the usual trace of the matrix P. 

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