Now, what we need do in order for us to determine how much of one phase we have,

we recognize because we're dealing with a fixed amount of material.

What we know is once we get into a two phase field,

what we have are the two phases.

A solid phase and a liquid phase.

In this case what we're going to say is that the fraction

of those two phases has got to be equal to 1.

Or in other words whatever percentage of solid we have, whatever percentage liquid

we have, we have to recognize that we have 100% all the time of those two phases.

Now what we can do is, we know that the composition, that is XB0,

the composition of our alloy, is now divided into two parts.

Some of the material goes into the liquid phase,

some of the material goes into the solid phase.

So that's what the second expression on the visual gives you.

Now once we know that the fraction of the liquid and

the solid have to sum to 1, what we're able to do is to substitute for

the fraction of the liquid, 1 minus the fraction of the solid.

When we do that, and substitute it into that second equation,

what we're then able to do is to write an equation in which we have

eliminated the fraction of liquid.

And so now everything is written in terms of fraction of solid.

And we carry this through.

What we're then able to do is to determine what is the fraction of solid.

If you look at the line the way we're describing the fraction of solid

is the distance between XB naught and

XB with respect to the liquid phase.

So that's going to give us the fraction

of the liquid phase divided by the total distance

between the composition of the solid minus the composition of the liquid.

So, when you look at the picture, what we have is, the solid is on the right hand

side, but what determines the fraction of the solid phase is the distance

between XB0 and the composition of the liquid at that given temperature.

Now, conversely, what we can do is rearrange the equation as we had done

above by just simply then calculating what the fraction of liquid is.

And when we look at the fraction of liquid,

it is just simply given by the expression the distance,

as determined from the composition of the solid minus the composition of the alloy,

then divided by the entire composition range inside of that two phase field.

Although, once you have the composition, or the fraction of solid,

all you then need to do is to simply take the fraction of solid and subtract

that from 1, and of course, that's going to give you the fraction of liquid.