The Gibbs phase rule is very important in thermodynamics so let us look at what the
actual phase rule is. This phase rule results by looking at how many degrees of freedom
do we have and how many equations can we can write down and therefore how many degrees
of freedom are left when we do that type of calculation. We are doing this for a non
reactive - no chemical reactions - otherwise we have to take chemical reactions into account when we look
at the phase rule. This is the number of degrees of freedom. This is the number of components,
the number of different chemical compounds in the system. Then P is the number of phases. It is important
to keep in mind we are only talking about intensive variables. The phase rule tells
us nothing about how much of a given phase we have when we have more than one phase. We will
look at a phase diagram to look at how we apply this phase rule. In this case we are
looking at partially miscible liquids. Pressure on this axis and the mole fraction of component
1 on this axis. At the low pressures in this case we have the vapor phase and as we increase
the pressure we have liquids. Alpha phase is completely immiscible liquid and beta is
a completely immiscible liquid. Beta is enriched in component 1 and alpha is enriched in component
2. We have alpha and vapor so this liquid phase in equilibrium with vapor and this is
beta in equilibrium with vapor. Then alpha plus beta means we have 2 phases in the liquid
that do not mix with each other but each phase has both component. When we pick a point within
here it just corresponds to what fraction is alpha and what fraction is beta at equilibrium.
Let's look at a point in the alpha phase region. In the alpha phase region we want to apply
phase rule. In this case we have 2 components. Component1 and component 2. We have 1 phase
and so the phase rule says we have 3 degrees of freedom. In our diagram we have already
set the temperature so this diagram is for a given temperature. Each diagram of course
is going to be different for different temperatures. We have 2 other intensive variables so for
example pressure would be 1 of those. If we look at this we can say at this pressure.
We still do not know the composition of the alpha phase so the other would be the mole
fraction of component 1. Now we have specified the system completely. Temperature, pressure
and the mole fraction. Temperature is fixed by the diagram, pressure here, and then this
particular composition. All of the other intensive variables are now fixed. Look at a different
example. Let's say the beta phase plus vapor. We will look at the region where we have vapor
in equilibrium with the beta phase. We are going to apply again the phase rule. Number of degrees of freedom: we have
2 components and now we have 2 phases. So we have two degrees of freedom. 1 is temperature
and we have already set that in the diagram so we can set 1 more. If we set pressure then
we have completely specified the system. If we look at the diagram it says that if we
set this pressure and we are somewhere in here once we set the pressure. Which means
here is the composition of the vapor phase. Here is the composition of the liquid phase.
The phase rule does not tell us how much of each phase. It just says we have a mixture
and these compositions are fixed once we fix the pressure. Just to reemphasize we are talking
about intensive variables when we are applying the phase rule and how many intensive variable
can we specify before this system is completely specified.