Regular solution model for a two sub-lattice system

The regular model for an ionic solution is similarly analogous to the regular solution derived in Section 9.1. Recall that the energy of the regular solution model was calculated as a sum of pairwise interactions. With two sub-lattices, pair interactions between species in one sub-lattice with species in the other sub-lattice (nearest neighbour interactions) and pair interactions within each sub-lattice (next nearest neighbour interactions), must be accounted for. 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of 

Introducing once more a pairwise interaction parameter co 

NA i [ l is easily derived when the cations are assumed to be randomly distributed on the cation sub-lattice. The probability of finding an AB (or BA) pair is 2Xa+Xb+ in analogy with the derivation of the regular solution in Section 9.1. iVA+B+ is then the product of the total number of cation-cation pairs multiplied by this probability 

The partition function for the regular ionic solution model is now 

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