The quasi-chemical model

The quasi-chemical model is denoted by the symbol QC . We will now use a method similar to that which we used in section 2.3.1. 

We will conduct the calculation in the simple case of the alloy AB, in which the atomic fiactions and the fiactions of sites are both equal to /2. More complex calculations, which are nevertheless similar on all points to those carried out above, can be used to solve more general cases. 

In the circumstances of our scenario, we again refer to the last row of Table 2.3 for the expressions of the numbers of t)q)es of atoms as a function of the degree of order. 

On the basis of Table 2.5, we will now calculate the number of pairs of neighboring sites occupied in the four possible cases Aa-Bb, Aa-Ab, Aa-Ba and Ba-Bb for any value of the degree of order 5. The total number of pairs, obviously, is Nz 2. If we let zX represent the number of pairs Aa-Ab, it is easy to fdl in the cells of Table 2.5. 

In view of relation [2.112], we know that the configurational partition function for all the states characterized by a pair of values 

---

Another important feature of the quasi-chemical model is that the model allows for mixing of A and B with different coordination numbers, za and zb- The number of pairs generated by the total number of A and B atoms is then... 

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

The quasi-chemical model was derived by Guggenheim for application to organic fluid mixtures. Applying it to crystalline solids is not immediate, because it necessitates conceptual modifications of operative parameters, such as the above-mentioned contact factor. Empirical methods of derivation of the above parameters, based on structural data, are available in the literature (Green, 1970 Sax-ena, 1972). We will not treat this model, because it is of scanty application in geochemistry. More exhaustive treatment can be found in Guggenheim (1952) and Ganguly and Saxena (1987). 

Figure 6.2 Gibbs free energy of mixing in the quasi-chemical model of Toop and Samis (1962a,b), compared with values experimentally observed in Pb0-Si02 and Ca0-Si02 melts. Reprinted from Toop and Samis (1962b), with kind permission of ASM International, Materials Park, Ohio.

All these quantities define the number of the 1 1, 2 2, and 1 2 bonds in the quasi chemical model, nn, 022, and ni2- The total number of bonds to each component 1 cation is 2a and to component 2, 2b so that if n, is the number of moles of component i, then... 

This is referred to as the quasi-chemical model. More detailed solid solution models can be found in textbooks on metallurgical thermodynamics, for example, Chemical Thennodynamics of Materials, by C. H. P. Lupis (Elsevier Science Publishers, Amsterdam, 1983). 

This notion of short-distance order comes into play in certain solution models, and is encountered in the quasi-chemical model (see section 2.3.4). 

The quasi-chemical model of a solution postulates the existence of a true thermod5mamic equilibrium between one pair A-A, one pair B-B and two pairs A-B, as follows ... 

The solution thus modeled is called the quasi-chemical model. 

Note.- We have seen that relation [2.70] of the strictly-iegular solution model was called the Bragg and Williams zero-order approximation. Similarly, in view of relation [2.89], the quasichemical solution is called an approximation of order 1. The order at hand is, in fact, the power to which the exponential appearing in relation [2.89] is raised power zero for the Bragg and Williams model, and power 1 for the quasi-chemical model. 

Figure 2.12 shows the variations in Warren and Cowley s order parameter for the quasi-chemical model, with the following values of the parameters T = 800 K, NsWab = 30 kJ and z = 12. The result obtained respects the symmetry of the model, with a minimum for the composition Xa = Xb = 0.5. 

Figure 2.12. Variation of the degree of order as a function of the composition of a binary solution in the quasi-chemical model (data taken om [DES 10])...

In order to evaluate the functions g(5) and E s), we need to know the distribution of the atoms on the lattice for the given value of s. Two models have been developed the Gorsky, Bragg and Williams model and the quasi-chemical model. The hypotheses upon which these models are based are similar, respectively, to those used for the model of a strictly-regular solution (see section 2.3.3) and those used for Fowler and Guggenheim s quasi-chemical solution model (see section 2.3.5). 

The quasi-chemical model is based on the hypothesis of distribution of the pairs represented by the equilibrium [2R.2]. This hypothesis has been proven by Bethe, using the grand partition function. 

Figure 3.2. Comparison of the excess Gibbs energies of a strictly-regular solution and the quasi-chemical model (reproducedfrom [DES10], p.62 - see Bibliography)...

---