Associated Solution Model for the Liquid Phase

Relation between Species Mole Fractions and Composition 

If Nf is the number of gram-atoms of component element i and n is the number of moles of species j, then the conservation of atom types requires that 

Defining atom fractions of the components x, and mole fractions of species yi by a generalization of Eqs. (28) and (29), the mole fractions of the uncombined species are obtained from Eqs. (54)—(56) as 

This equation is used in an iterative scheme to obtain the mole fraction of the species for a fixed T and composition. First k4 and ks are approximated by setting the activity coefficients in Eqs. (63) and (64) equal to one. Equation (67) becomes a cubic in y3, which is solved by Cardan s method. Equations (65) and (66) are then used to obtain y4 and y5 and Eq. (57) to obtain yj and y2 In the second step the approximate values for the mole fractions are used to calculate approximate values for the activity coefficients and then for jc4 and k5. The new values for k4 and k5 are inserted into Eq. (67), which is solved to give a second approximation for y3. These steps are repeated until k4 and k5 change by less than 10 9 in successive steps. A generally more reliable criterion for termination would involve the fractional changes in k4 and k5 upon successive iterations. 

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