One-Dimensional Mixture of Fluids

Note that the sum over all orderings of species is restricted to fixed values of and Nb Once we open the system, we carry the sum over Na and Nb in two steps. First we %xN= Na + Nb. Now the sum over all Na and Nb with a fixed N is the same as the sum over all possible vectors s. This is followed by the sum over all A, which is the result (4.5.76). 

If we choose P as the thermodynamic pressure P, then oSf(E) is referred to as the generalized PF. It is clear from (4.5.81) that the integral in this case diverges. The reason is that E(T, L, X) is a function of the single extensive variable L. Transforming L into the thermodynamic intensive variable P gives a partition function which is a function of the intensive variables T, P, X only. However, the Gibbs-Duhem relation states that 

Hence the intensive variables T, P, /j, or T, P, X are not independent. For this reason we have denoted by P the new variable in the Laplace transform taken in (4.5.75). Suppose that we choose P P then the integral in (4.5.81) converges, and we have 

It is clear that, since in the limit P P the Ihs of (4.5.83) diverges, there must be at least one of yj P ) equal to 1. The secular equation of the matrix M is 

Since the elements of M are functions of 7, P, and A we can use the implicit Eq. (4.5.84) to derive all thermodynamic quantities of interest. We treat a two-component system of A and B for which the secular equation is 

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