# Solid-Solid-Liquid Equilibrium

This Demonstration illustrates the behavior of a system of two pure substances ( and ) and a solid compound of the two (). Solid-solid equilibrium, solid-liquid equilibrium, and solid-solid-liquid equilibrium are represented in the phase diagram. Mixtures of the pure solids ( + {A}2{B}3 and + {A}2{B}3) are immiscible. At the selected temperature and mole fraction of (represented by the center black point on the -- diagram), the relative amounts of the four possible phases are shown in the bar graph. The mole fraction of in the liquid phase (mixture of and ) is given above the liquid bar in the bar graph. Move the black point by adjusting the sliders for mole fraction and heat added. When heat is added, the temperature increases, except when the point is on one of the two horizontal lines (at about 200 °C and 300 °C) or when pure {A}2{B}3 is in equilibrium with the liquid phase. On the lines at 200 °C and 300 °C, three phases can be in equilibrium: solid , liquid, and either solid or . A mole balance is used to find the relative amounts of each phase on these lines. The amount of heat added represents what phases are present in this system; it is not meant to represent a real system. In the two-phase regions, the relative amounts of each phase are obtained using the lever rule, and the mole fraction of in the liquid phase is shown by a vertical dashed line.

### DETAILS

The lever rule is used to find the relative molar contents of each phase. An example in the solid + liquid region is given by:
,
,
where is the overall mole fraction of the mixture (the mole fraction of the point on the -- diagram), is the mole fraction of in the liquid phase, the mole fraction of in the solid phase is zero, is the relative amount of liquid, and is the relative amount of solid .
When the system is in solid-solid-liquid equilibrium, the relative amounts of each phase are found from mass balances. For example, using a value of 10% melted and an initial mole fraction of of 0.7 in the diagram:
1. Determine the initial mole fraction of :
.
2. Determine the mole fraction of component in each phase:
,
,
.
3. Set a basis for the amount of total moles in the system:
.
4. Determine the percentage melted:
.
5. Determine the amount of liquid in the system:
.
6. Perform a mole balance on the whole system:
,
.
7. Perform a mole balance on component :
,
,
.
8. Simultaneously solve the equations 6 and 7 with the unknown variables (the number of equations should equal the number of unknown variables):
.
.
9. Relative amounts of each phase:
(relative amount of solid ),
(relative amount of solid ),
(relative amount of liquid).

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