1. The Lever Rule

2. We do not in general, measure free energies in order to determine phase composition.
It is much easier to allow samples of known composition to come to equilibrium at various T
And then use of optical microscopy, electrical or X-ray techniques for the presence of the various phases.

3. These data can be recorded directly on the diagram.
Consider the alloy system of Fig. 2.13,
If we start with pure A at temperature T1 and slowly add B our entire specimen will be alpha phase until the composition becomes x alpha.

4. As the B fraction in the alloy exceeds X alpha, beta phase will appear.
Once in the two phase region, the composition of alpha and beta at temperature T1 remain fixed.
Further increase in the B contents lead to the increase amount of beta phase present and decreases in the alpha phase.
This goes on until the total composition reaches x beta at which point the alloy is totally beta.

5. The relative amount of each phase may be computed for any total compositon b/t x alpha and x beta.
Material conservation is used.
If N-alpha moles of alpha and N-beta moles of B are present.
N-alpha + N-beta = N
N= total number of moles of alloy

6. Likewise total moles of B in the alloy must equal the sum of B in the both phases.
X-alpha N-alpha + X-beta N-beta = Xo N
Solution of the above pair of equation for N-alpha and N-beta gives
N-alpha/N = (X-beta – Xo / X-beta – X-alpha)
N-beta/N = (X-o – X-alpha / X-beta – X-alpha)
Now the total composition is fulcrum and X-beta – Xo and Xo – X-alpha are regarded as lever arms. The fraction of the alloy which comprises each phase is simply the opposite lever arm divided by the total length.

7. Now the total composition is fulcrum and X-beta – Xo and Xo – X-alpha are regarded as lever arms.
The fraction of the alloy which comprises each phase is simply the opposite lever arm divided by the total length.