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PROFESSOR CIMA: All right, now this always tricks up students.
And I struggled with this last week trying to figure out how was I going
to explain this. So, we know that, as we've been talking about,
that these functions, these Gibbs energy functions-- so this is
a Gibbs energy function for the liquid.
Right? And it depends on temperature and pressure,
as we've said. It can depend on composition, because this
far we've been talking about just single component, or pure substance.
But it can depend on a lot of things. And the same is true for the solid.
So these are two different functions, but to have the equilibrium they have
to be equal. So let's go back to the pure substance for
a second. It says for something that's at equilibrium
between, let's say, a liquid and a solid.
So, here's a pure substance. I can throw away the composition for a second.
No matter how complicated these functions are, if I choose a
temperature, I can calculate a P, right? In these simple cases these are just lines,
and it's easy. But even if we've a complicated equation,
in principle, because I've set these two equal, that means that, if I
choose a temperature, there has to be a P.
If I just have a single phase, there's no requirement, right?
So I can have any temperature and pressure within that phase region.
So in other words, when I have a single phase, I have two degrees of
freedom, two things I can change. If I have two phases, I can only have one
thing I can change. The degrees of freedom have been reduced.
And that's what the phase rule is about. It's just as simple as that.
It's just the mathematics of saying, well, if I've got set two functions
equal that one another, and they each of them had depended on two variables,
If we put them equal to one another, that means if I choose one, I should
be able to calculate the other. That's all it is.
And so the phase rule is if F is the degrees of freedom, and P is the
number of phases. Not pressure now.
Number of phases. And C is the number of components.
So here I've just added another x there. Then you have that F = C - P + 2.
Where T and P are allowed to change. Let's see how it works.
So let's say I have-- oh I did it over here.
Let's make a little table. So let's say I have a single component, C
equals 1. I'll put the number of phases here, and the
number of degrees of freedom. So, if I have one phase, I put it in there,
it says P is 1. I have one component, one minus 1 is zero,
plus 2. I have two degrees of freedom.
Let's go to our single component phase diagram here.
Like this one. That means I'm out here.
One phase, it means I can change the temperature and pressure, no problem.
Let's say I have two phases. Well, I put in a 2 there.
It's 1 minus 2, that's minus 1 plus 2. That's 1.
I only have one degree of freedom. So that's like a point on one of these lines.
A point on one of these lines means if I change the temperature if I still
maintain two phases, the pressure changes. Only one degree of freedom.
And the last one for a single component is three phases.
If I have three phases, well, that means 1 minus 3, that's
minus 2 plus 2. That means there's zero degrees of freedom.
What's that about? Well, of course, you go right here.
And the point that I purposely left out in my discussion was this one.
It's called the triple point. It's the only point in the phase diagram,
where you have three phases in equilibrium.
Well, there could be other ones. Yep?
STUDENT: What do you mean by component? PROFESSOR CIMA: Component?
So we had it. Oh, it might have gotten erased.
A component is a molecule or an atom, if it's an atomic solution, that goes
into making this phase. Remember, this phase can be a solution.
It can have a mixture of atoms, or it can be a mixture of molecules.
And so, if I have two components in a phase, that means that gives me a
degree of freedom. Right?
I can make a different concentration. OK, so if I have a triple point, that means
I can't change either temperature or pressure and still have three
phases there. OK, now, that's when T and P, are variables.
Lots of times, and we'll get to this on Friday, we'll talk about examples,
where you can't change the pressure. I mean, we live on Earth and just out here
it's always one atmosphere. So a lot of times we consider pressure fixed.
And then the phase rule becomes-- I've got to make sure I use the same notation
right? Because I've reduced the number of degrees
of freedom.