## Gibbs Phase Rule

### Quick Start

This is an expert-level app to make the point that phase diagrams follow (of course!) Gibbs Phase Rule. If you've not heard of this fundamental law then a quick read should provide an instant education.

### Gibbs Phase Rule

In the 2-Phase and 3-Phase apps we had multiple phases doing complex things. Although phase diagrams can get a lot more complicated, there are limits. So what is the upper limit of phases to expect? Can there be a 5-phase region? What are our expectations?

Thankfully scientists have worried about this already, and the Gibbs Phase Rule captures the interplay amongst interacting Components (C), the maximum number of Phases (P), and the number of variables that affect phase formation (cryptically called Degrees of Freedom, F):

Gibbs Phase Rule: F=C-P+2

This applies to all phase diagrams which are thermodynamically based; you may note that most diagrams have axes in terms of temperature, pressure, and composition. Let us consider our ternary example (S,X,W). In this case, Temperature and Pressure are not documented, so they are out of the scope of concern. Effectively, therefore,

Gibbs Phase Rule: F=C-P

Composition is graphed for a particular, constant Temperature and Pressure condition, so our Degrees of Freedom will be entirely based on ratios of components (i.e. mole fractions). Our ternary diagram has three vertices (S,X,W), and defining any 2 of these defines the last. Hence the phases in the diagram only have 2 Degrees of Freedom. In practice, the Degrees of Freedoms is = (number of axes plotted – 1).

So for our ternary phase diagram, F=2, C=3. Our maximum number of Phases is P=3.

What does this mean? If you are creating vials of mixtures and observe 4 phases, then either you have (a) impure components … or a chemical reaction created new species (i.e. C>3); or (b) the components have not reached a thermodynamically equilibrated state.

We can apply this phase rule to a sub-region of the phase diagram. Let use zoom-in on the 2-phase region; specifically on a single tie-line.

P=2, C=3 , so F= 1

This means that by specifying only one component (of S,X,W) the amounts of the others two components are instantly fixed. We can move freely through the 2-phase region changing just the one component and the proportions of the other two components change as we do so. The end points of the tie lines (indicated with yellow dots) indicate the compositions of each of the two phases dictated by the single variable. There is an
*apparent* degree of freedom still available - we can move (mouse or pointer) up and down the tie line and the relative proportions of the two phases will change but that's not a phase rule problem as the
*compositions* of the phases at each end are not affected by this change in
*ratio* of the phases. This point is seen even more clearly when we come to 3 phases.

In the 3-phase region P=3, C=3 so F=0. The 3-phase region is fixed - it's the 3 points at the vertices of the grey triangle, again shown with yellow dots. But as we move around inside the triangle the ratios of the 3 phases change whilst the yellow dots (unlike in the 2-phase region) remain fixed. This is the same apparent degree of freedom mentioned for the 2-phase region, but again it is the
*ratios* which are changing, not the *compositions*.

Here is the 3-phase model again with the extra dots and, as discussed below, the Plait Points. The readout gives you F,C and P values.

### For the experts...

In the 2-phase regions the tie-lines get shorter and shorter and meet at a critical point where the volumes and compositions of the two phases become equal. These are called Plait Points (or isothermal critical points) and are shown in red.

There is fascinating Alternation Rule that emerges from the Gibbs Phase Rule that tells us adjacent phases regions can only decrement/increments in units of “1”. So you can have a 1-phase next to a 2-phase and a 2-phase next to a 3-phase ... but you can't have two 2-phases next to each other and, as you can see in PDE3, you can't go from a 3-phase to a 1-phase region directly. Another name for it is the Boundary Rule: "Any p-phase region can be bounded only by regions containing p +/- 1 phases, where p denotes the number of phases".

The Law of Adjoining Phases says that:

R_{1}=R-D^{-}-D^{+}>=0

Where R=2 for our ternary diagram, D^{-} is the number of phases that disappear when we cross a phase boundary, D^{+} is the number of phases that appear when we cross a phase boundary.