## LEVER RULE

To determine compositions of phases and the relative proportions of phases to each other in Binary diagrams the LEVER RULE is used.

##### Compositions by the Lever Rule
Once again back to our hypothetical binary system
1. Point "I" lies above the liquidus within the liquid field.

What is the composition, in terms of the two end member components, A and B, of the liquid represented by this point?

To determine the composition of "I" you must complete the following steps:
1. Draw a line through "I" perpendicular to the AB join, i.e., the base of the diagram. This line represents a line of constant composition and is referred to as an isopleth.
2. The liquid at "I" consists of a mixture of A and B, the proportions of which can be determined simply by measuring the length of three lines, AI', BI' and AB and then ratio these lengths.
%A = I'B/AB *100
%B = I'A/AB *100

This gives us the bulk composition of the liquid at this point. If the composition point for the moves then we get a new bulk composition for that point represented by the new liquid.

With reference to the hypothetical binary system:

2. Point Z
At the point represented by Z there are several questions to be considered:
1. What is the bulk composition of Point Z?
Point Z has the same bulk composition as Point I, used above, as it lies on the same isopleth, but at a lower temperature.
2. What phases are present?
Point Z lies in the field where two phases, B + L, are in equilibrium, therfore the two phases present have to be soild B and Liquid.
3. What are the proportions of the phases present?
To determine the proportions of B + L at Z, carry out the following steps:
1. Draw a line through Z, parallel to the base of the diagram (This line is at a constant temperature and is an isotherm.) This line should extend only to the boundaries of the B + L field - Points X and Y.
2. Measure the three line segments - ZX, Zy and XY and ratio these lengths using the lever rule.
% B = ZX/XY * 100 = 38% B
%L = ZY/XY * 100 = 62% L
4. What are the composition of the phases present?
At Point Z if we were to examine the system we would see crystals of B in a glassy matrix (the Liquid) in equilibrium. The points X and Y assist us in determining the compostions of the two phases. As this is the simplest binary system possible, one which does not exhibit solid solution, all of the solid phases are of fixed composition.
• Point Y lies on the right hand side of the binary system where we have 100% B and 0% A, therfore the solid represented by Y must have a composition of Pure B.
The composition of the liquid in equilibrium with the Pure B is represented by X, on the liquidus surface.
• To determine the composition of the Liquid at X, draw an isopleth down to the base of the diagram.
• Now measure three line segments AX', BX' and AB, ratio these using the Lever Rule to get the composition of the liquid X, in terms of A and B, the two components which define the system.
%A = BX'/AB *100 = 40% A
%B = AX'/AB*100 = 60% B

Liquid compositions are always expressed in terms of the two end member components which define the system. Likewise the composition of any solid phase (100% B, 0% A) and the Bulk composition (40% A, 60% B) of any liquid are also expressed in terms of the end member components.