An Analogue of the Isomerization Equilibrium

We now discuss a different system, the treatment of which is equivalent to the treatment of the isomerization reaction discussed above. Consider again a one-component system at 7, K, N. The PF of this system is, as before, 

Suppose we now place a partition that divides the volume V into two compartments of volumes Va and Vb, with Na and Nb in Va and Vb, respectively (Fig. 2.5). Note that the molecules in the two compartments are the same molecules. However, we may refer to the molecule in the volume Va as the A molecules and in Vb as the B molecules. Although the molecules are of the same species, they are distinguishable by the very fact that they occupy different compartments. 

The partition function for the system with the inserted partition is 

In general, we can choose arbitrary densities, and hence Pa Pb- However, there is one density for which we obtain the condition of chemical equilibrium 

at equilibrium, the number of particles in each compartment is proportional to the volume of that compartment. The last relation is the analogue of the equilibrium constant which in the previous case is given by (2.4.23). In both cases we have shown that the classification into two (or more) components is a matter of choice. There are infinitely many ways of achieving such a classification. 

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