Figure 1: Two-phase region on a schematic phase diagram of a binary system. Let

Question

Figure 1: Two-phase region on a schematic phase diagram of a binary system.

Let the point (C0, T0) on the phase diagram belong to a region where two phases, 1 and 2, are present. Let W1 and W2 denote the weight fractions of the phases 1 and 2, respectively:

Wk =mk /m, where the index k enumerates the phases (k = 1, 2), mk   is the mass of the phase   k , and   m = m1   + m2   is the total mass of the mixture of the phases 1 and 2.

(a) Derive the lever rule: W1 ?C1 =W2 ?C2 ,   (*)

where ?Ck = |Ck - C0| , Ck being the borderline composition of phase k (k = 1, 2) at the temperature T0 , as shown in Figure 1 above.

1(b) Using the lever rule (*), express the weight fraction of each phase Wk (k = 1, 2) through their compositions Ck .

Hint for Problem 1(a):

Start with the fact that the mass of the component B in the system equals m(B) = m1(B) + m2(B) ,

where mk(B) is the mass of the component B in the phase k (k = 1, 2), and demonstrate that

W1 C1 +W2 C2 =C0 .   (**) Remember that, by definition, the composition of phase k is

Ck =mk(B)/mk , and the weight fraction of the component B in the system is

C0   = m(B) / m . Then use the obvious equality

W1 +W2 =1 together with Eq. (**) to prove the lever rule (*).

Explanation / Answer

Mass balance of component B:

Mass of a component that is present in both phases equal to the mass of the component in one phase + mass of the component in the second phase => W1 C1 +W2 C2 =C0

Total Mass balance

Sum of mass fractions =1 => W1 +W2 =1

Solving the above two equations we get W1 = (C2-C0)/(C2-C1)

and W2 = (C1-C0)/(C1-C2)

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