Solutions with more than one sub-lattice

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

A binary ionic solution must contain at least three kinds of species. One example is a solution of AC and BC. Here we have two cation species A+ and B+ and one common anion species C . The sum of the charge of the cations and the anions must be equal to satisfy electro-neutrality. Hence NA+ + NB+ = N(. = N where NA+, AB+ and Nc are the total number of each of the ions and N is the total number of sites in each sub-lattice. The total number of distinguishable arrangements of A+ and B+ cations on the cation sub-lattice is M/N A, JVg+ . The expression for the molar Gibbs energy of mixing of the ideal ionic solution AC-BC is thus analogous to that derived in Section 9.1 and can be expressed as 

The entropy of mixing is obtained by differentiation of eq. (9.65) with regard to temperature. 

An analogous expression can be derived for an ionic solution with a common cation, and the ideal entropy for a system AC-BD is twice as large as that for the AC-BC system. This approach can also be used for an alloy Aj B C, where the atoms A and B are randomly distributed on one sub-lattice and C fills completely the second separate sub-lattice. 

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In simple solutions such as binary alloys, the components are distributed on a single lattice. More complex solutions may consist of two or more sub-lattices, and in a solution of simple ionic salts like NaCl and NaBr there is one sub-lattice for cations and one for anions. In these cases the interactions considered in the models are between next neighbouring pairs of atoms rather than nearest neighbour atoms, as is the case with a single lattice. Two sub-lattice models can also be applied to... 

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