# Why are temperature and pressure represented as the number 2 in Gibbs' Phase Rule?

In our class we are learning about Gibbs' Phase Rule; $F=C-P+2$, in which $F$ stands for degree of freedom, $C$ for number of components, and $P$ for number of phases in the system. Our professor told us that the number $2$ stood for Pressure and Temperature, but I don't understand why they are treated as constants when Composition isn't.

I must be missing something obvious, but right now I can't see what it might be...

• If the number of components is 1 and the number of phases is 1, then the number of degrees of freedom is 2: you can vary pressure and temperature independently. Jun 4, 2018 at 11:49

The underlying ah-ha moment is that the Gibbs phase rule works best in (constructing) a phase diagram. And those are usually 2-dimensional - with pressure along one axis and temperature the other. So the "2" simply reflects that 2 dimensionality. At 2 degrees of freedom, there are no constraints and the system can change either pressure or temperature and be stable without a phase transition. If you have 1 degree of freedom, then you either have a constraint on $P = k$ or $T = k$ or a "mixture" $f(P,T) = k$ (If P is changed, T must be a specific value) . At 0 degrees of freedom you are at a triple point $(P =k, T=k)$ or similar (there are a handful of named "points") - any movement in either direction means the universe is going to change the phases present or the components present.

So they are not really constants, but if the degrees of freedom are 0, then they are... "fixed" - or better yet "constrained".

The archetypical example is Ice-Water-Steam. There will be at least 3 lines in this diagram from 2 phases in equilibrium and the 1 component - and 1 point.

Out of the 3 components, lets take 2 and see what we get.

$$DoF = 1-2+2 = 1$$.

So we see that for 2 phases to be in equilibrium we have 1 constraint eating away at our DoF and we know then that there will be a fixed relationship between T and P here - we can change P and still have the equilibrium but then we have to change T as well with a specific amount. This creates a line in our T-P diagram. Since there are 3 ways to arrange this we get 3 lines (I+W) + (S+W) + (I+S)

So lets see what happens if we force all three phases to be at equilibrium:

$$DoF = 1-3+2 = 0$$.

So that is no wiggle room. There will be a point where this happens, a specific combination of P and T - where this equilibrium exists.

Now lets say I invent a new phase, and I call it Ice-9. Lets see if we can find a place where this coexists with all the others.

$$DoF = 1-4+2 = -1$$.

Nope. This is not stable. No equilibrium can exist which permits my Ice-9, regular ice, water and steam at the same time. So now we know that Kurt Vonnegut is just making stuff up, what a hack. The result can be seen here... 3 lines and a point. Don't mind the dotted stuff, that is because water is more complicated than what it is at first glance

Gibbs rule applies to systems in equillibrium. The composition is no real degree of freedom: Whatever you add (of one pure component) will partially (or completely) turn into sth else, unless it's a solid and the system contains the same solid phase at the given temperature and pressure.