The Gibbs Phase Rule

$$\input{equations/gibbs-phase-rule}$$ (26-1)

The Gibbs phase rule is a very useful equation because it put precise limits on the number of phases $f$ that can be simultaneously in equilibrium for a given number of components.

What does Equation 26-1 mean? Consider the following example of a single component (pure) phase diagram $C=1$.

Figure: A single component phase diagram. On the right figure, the color represents a molar extensive quantities (i.e., blue is a low value of $\ensuremath{\overline{V}}$ and red is a large value of $\ensuremath{\overline{V}}$) that apply to each phase at that particular $P$ and $T$.

Consider a single-phase region:
$D= 2-f + C = 2-1+1 =2$

This implies that two variables ( $P$ and $T$) can be changed independently (i.e., pick any $dP$ and $dT$) and a single phase remains in equilibrium.

Consider where two phases are in equilibrium:
$D= 2-f+C = 2-2+1=1$,
There is only one degree of freedom-for the two phases to remain in equilibrium, one variable can be changed freely (for instance, $dP$) but then the change in the other variable (i.e., $dT$) must depend on the change of the free variables:

$$\frac{dP}{dT} = f(P,T)$$

Finally, consider where three phases are in equilibrium then:
$D=2-3+1=0$.

There can be no change any variable that maintains three phase equilibrium.