1.11 Ternary system representation
Equilibrium data of ternary systems are usually graphically shown using either right triangular or equilateral triangular diagrams.
The here below diagram shows the data for a ternary liquid system containing acetone, toluene and water. Any point located within the triangle or on one of its edge represents a different composition of the system.

In particular:

The points at the three apex, A, B, C , represent the three pure compounds. For example, in A, it is x_{A} = 1, x_{B} = 0, x_{C} = 0;
Any point on a side (A-B; B-C; A-C ), of the triangle represents a binary mixture of the two components which are at its extremity.
Any point, like S or R , located inside the triangle represents a mixture of the three components A, B and C.
Such diagram has the property that the sum of the lengths of the perpendicular lines drawn from any interior point, like R, to the sides is equal to the height of the triangle. Thus if the height is scaled from 0 to 100, the percentage of component A in a system represented by R is the distance of R from the side opposite to the A apex. In a similar way, the content of B and C can be determined.

The degrees of freedom can be determined with Gibbs phase rule. In any point like S , it is:

F = C - P + 2 = 3 - 1 + 2 = 4;

where
C = 3 components;
P = 1 (outside the immiscibility region);

and since this diagram is drawn at constant pressure and temperature, we have only 2 degrees of freedom; thus meaning that once having fixed the composition of two of the three components, the third one is consequently determined.
While in any point like R :

F = C - P + 2 = 3 - 2 + 2 = 3;

where
C = 3 components;
P = 2 (within the immiscibility region);

and since again pressure and temperature are fixed, the system has only 1 degree of freedom and the compositions of the two phases are univocally determined by the end-points (M and P) of the tie-line passing through the point R.

The solubility envelope
The curve LMNOPQ represents the saturation curve for the ternary system ABC. This line is also called solubility envelope since it delimits a region where the three components are not fully miscible. Any system, represented by point like R within the immiscibility region , will split into two phases, represented by M and P in equilibrium with each other. The immiscibility region is a biphasic region . The two points M and P lie on the same equilibrium line, called tie-line .

Any point of the solubility envelope behaves like R. Point O is called the plait point and it is characterised by the fact that the two phases have identical compositions. In this point the tie-line converge in a point and the two phases become only one phase.
All the pairs of points lying into solubility envelope, on one side and on the other of the plait point , are related by an equilibrium correlation represented by the tie-line .
The solubility envelope for the same ternary system is going to change at different temperatures and pressures , since different equilibria are present among the components.
The three following pictures show the effect of the temperature on the solubility envelope for the same ternary system. This is going to be smaller and smaller for increasing temperature since the solubility in most of the time increases with the temperature.

When the process design of a ternary system requires the use of graphical methods, the use of the right triangular diagrams could be more convenient since in this case the drawing of perpendicular lines and heights could result a lot easier.

Typicall a diagram does not contain all the tie-lines needed for the graphical solution of problems with ternary system. Additional tie-lines can be draw following the here below illustrated graphical procedure.