Ternary and higher order chemical mixtures

The interpretation of equation (k) for a ternary or higher-order solution is no more difficult in principle than the interpretation already given for a binary solution. However, every additional component requires the addition of one compositional variable to the free energy of mixing function, and therefore one additional dimension in free energy - composition space. The graphical depiction of a binary system thus requires two dimensions (Gip, X2) that of a ternary system, three dimensions (G, X2, X3), and so on. The graphical representation of multicomponent solutions with c 3 is virtually impossible, except in projection, and it becomes necessary to rely on strictly fimctional treatments of the properties of state. 

In a ternary system (i = 1,2,3), the ideal molar free energy of mixing is 

---