## The Lever Rule

At temperature-composition points within a single-phase region, the abscissa of the phase diagram is the actual composition. In two-phase regions, on the other hand, the abscissa gives the overall composition of the two co-existing phases. Figure 8.16 shows a two-phase region (shaded area) bounded on the left by phase I and on the right by phase II. This diagram represents any of the two-phase regions with single-phase neighbors in the phase diagrams depicted in this chapter. At point P in the two-phase region of Fig. 8.16, the overall composition is xB. The compositions of the two phases present are located at the intersections of the horizontal line through P with the upper and lower

Fig. 8.16 Illustration of the lever rule

phase boundaries of the adjacent single phases - that is, at points a and b. The lengths of the line segments a-P and P-b give the relative quantities (in moles) of phases I and II in the mixture. This can be shown by material balances. If the mixture contains nI moles of phase I of composition xBI and nII moles of phase II of composition xBII, the total number of moles is N = nI + nII and the total moles of component B in the mixture is:

xbN = xBInI + xBIInn

These two equations are solved for the fraction of phase I in the mixture:

fraction phase I = ^ = —^ = ^ N x„„ - x„ ab

This formula is known as the lever rule. It applies to any two-phase region in a phase diagram but has no meaning if applied to a single-phase zone.

Reference

1. D. R. Gaskell, Introduction to Metallurgical Thermodynamics, 2nd Ed., McGraw-Hill (1981)

2. K. Valsaraj, Elements of Environmental Engineering, 2nd Ed. Lewis (2000) p. 648