Solid-Solid-Liquid Equilibrium

This Demonstration illustrates the behavior of a system of two pure substances (A and B) and a solid compound of the two ((Subscript[A, 2])(Subscript[B, 3])). Solid-solid equilibrium, solid-liquid equilibrium, and solid-solid-liquid equilibrium are represented in the phase diagram. Mixtures of the pure solids (A + (Subscript[A, 2])(Subscript[B, 3]) and B + (Subscript[A, 2])(Subscript[B, 3])) are immiscible. At the selected  temperature and mole fraction of B (represented by the center black point on the T-x-y diagram), the relative amounts of the four possible phases are shown in the bar graph. The mole fraction of B in the liquid phase (mixture of A and B) is given above the liquid bar in the bar graph. Move the black point by adjusting the sliders for mole fraction B 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 (Subscript[A, 2])(Subscript[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 (Subscript[A, 2])(Subscript[B, 3]), liquid, and either solid A or B. 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 B in the liquid phase is shown by a vertical dashed line.

THINGS TO TRY

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DETAILS

The lever rule is used to find the relative molar contents of each phase. An example in the solid A + liquid region is given by:

(Subscript[r, liquid])= (((Subscript[z, B])-0)/((Subscript[x, B])-0))=((Subscript[z, B])/(Subscript[x, B])),

(Subscript[r, solid of A])= (((Subscript[z, B])-(Subscript[x, B]))/(0-(Subscript[x, B])))=1-(((Subscript[z, B])-0)/((Subscript[x, B])-0))=1-(Subscript[r, liquid]),

where (Subscript[z, B]) is the overall mole fraction of the mixture (the mole fraction of the point on the T-x-y diagram), (Subscript[x, B]) is the mole fraction of B in the liquid phase, the mole fraction of B in the solid A phase is zero, (Subscript[r, liquid]) is the relative amount of liquid, and (Subscript[r, solid]) is the relative amount of solid A.

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 B of 0.7 in the diagram:

1. Determine the initial mole fraction of B:

(Subscript[z, B])=0.7.

2. Determine the mole fraction of component B in each phase:

(Subscript[S, solid, (Subscript[A, 2])(Subscript[B, 3])])=0.6,

(Subscript[S, solid, B])=1,

(Subscript[x, B])=0.8.

3. Set a basis for the amount of total moles in the system:

(Subscript[S, total])=1.

4. Determine the percentage melted:

μ=0.1.

5. Determine the amount of liquid in the system:

L = μ (Subscript[S, total])=0.1 (1)=0.1.

6. Perform a mole balance on the whole system:

(Subscript[S, total])=(Subscript[S, solid, (Subscript[A, 2])(Subscript[B, 3])])+(Subscript[S, solid, B])+L,

1=(Subscript[S, solid, (Subscript[A, 2])(Subscript[B, 3])])+(Subscript[S, solid, B])+0.1.

7. Perform a mole balance on component B:

(Subscript[z, B])(Subscript[S, total])=0.6(Subscript[S, (Subscript[A, 2])(Subscript[B, 3])])+1(Subscript[S, B])+(Subscript[x, B])L ,

0.7 (1)=0.6(Subscript[S, (Subscript[A, 2])(Subscript[B, 3])])+1(Subscript[S, B])+0.8 (0.1),

0.7=0.6(Subscript[S, (Subscript[A, 2])(Subscript[B, 3])])+(Subscript[S, B])+0.08.

8. Simultaneously solve the equations 6 and 7 with the unknown variables (the number of equations should equal the number of unknown variables):

1=(Subscript[S, solid, (Subscript[A, 2])(Subscript[B, 3])])+(Subscript[S, solid, B])+0.1.

0.7=0.6(Subscript[S, (Subscript[A, 2])(Subscript[B, 3])])+(Subscript[S, B])+0.08.

9. Relative amounts of each phase:

(Subscript[S, solid, (Subscript[A, 2])(Subscript[B, 3])])=0.7 (relative amount of solid (Subscript[A, 2])(Subscript[B, 3])),

(Subscript[S, B])=0.2 (relative amount of solid B),

V=0.1 (relative amount of liquid).

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Megan E. Maguire and Neil Hendren

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