Gibb's Phase Rule
For a system in equilibrium, the phase rule relates the number of
components (substances), variables (temperature, pressure) and phases to
something called the degrees of freedom:
F=C+N-P
F = degrees of freedom; the number of independent variables that must be
arbitrarily fixed to establish the intensive state of a system
C = number of components
P = number of phases
N = number of noncompositional variables
This is a very important relationship. It describes the intensive state of a
system of components. Recall that an intensive variable is one that is
independent of the size of the system. (Recall, also, that
an extensive variable depends on the system size.) The temperature at which
pure water boils at one atmosphere is 100°C, regardless of whether there is
10 ml or 10,000 gallons. The Gibbs Phase Rule will be used extensively in
the coming sections, so it is important that you understand it completely and
utilize it effectively. Let's look at each of these variables in a little more
detail.
The number of components, C, is fairly straightforward. If I mix oil and
water, I have two components. If I put some tin into molten lead, I have two
components. Water is an example of a one-component system. If I put salt in
water, I once again have two components. You might argue that when I
dissolve salt in water I have only one component: salt water. What I have is
one phase; there are still two components. What if I put in too much salt? It
begins to precipitate out and I get two phases, but in both instances I have
two components: the salt and the water.
As you can see in the saltwater example, the number of phases, P, is
something that is not always known. Sometimes this is the quantity we
desire, but more often than not, we can determine the number of phases in a
system by inspecting the system, either visually or with a microscope. If we
keep in mind that a phase is a homogeneous region of matter, we can usually
identify how many of these regions are present. Remember that these
regions need not be continuous. Bubbles in a liquid represent only two
phases, even though the bubbles are separated from each other by the liquid.
The number of noncompositional variables, N, is simply the process
variables we wish to consider other than composition. Usually, these are
temperature and pressure, so that N=2, but for condensed systems; e.g., two
solids, the effect of pressure is often times negligible, so that N can also be
1.
The degrees of freedom, F, are difficult to conceptualize. They don't have
any physical significance, that is, they don't represent a certain phase, or a
process variable. Just as with the other variable in the Gibbs Phase Rule,
though, they count the number of something. You can think of the degrees
of freedom as counting whatever is left from adding and subtracting the
other variables (C, N and P). If F=0, we say the system is invariant.
Similarly, we can have monovariant, divariant or trivariant systems
for F=1, 2 or 3, respectively.
Often times, we wish to determine the degrees of freedom. We know how
many components we have, we know how many process variables
(noncompositional variables) we wish to vary, and we sometimes know how
many phases we have. By determining F, we find out how many of the
process variables can be changed independently without affecting the
number of phases present. The best way to strengthen your grasp on this
concept is to forge ahead with phase diagrams. As the complexity of the
diagrams increases, you should see why it is important to know the degrees
of freedom for a given system.
Types of Phase Diagrams
Phase diagrams are a graphical representation of three types of information:
1. number of phases present
2. composition of each phase
3. quantity of each phase
Degrees of freedom
Consider a system in an equilibrium state. In this state, the system has one or more phases; each
phase contains one or more species; and intensive properties such as T, p, and the mole fraction of a
species in a phase have definite values. Starting with the system in this state, we can make changes
that place the system in a new equilibrium state having the same kinds of phases and the same species,
but different values of some of the intensive properties. The number of different independent intensive
variables that we may change in this way is the number of degrees of freedom or variance, F, of the
system.
Clearly, the system remains in equilibrium if we change the amount of a phase without changing its
temperature, pressure, or composition. This, however, is the change of an extensive variable and is not
counted as a degree of freedom.
The phase rule, in the form to be derived, applies to a system that continues to have complete
thermal, mechanical, and transfer equilibrium as intensive variables change. This means different
phases are not separated by adiabatic or rigid partitions, or by semipermeable or impermeable
membranes. Furthermore, every conceivable reaction among the species is either at reaction
equilibrium or else is frozen at a fixed advancement during the time period we observe the system.
The number of degrees of freedom is the maximum number of intensive properties of the equilibrium
system we may independently vary, or fix at arbitrary values, without causing a change in the number
and kinds of phases and species. We cannot, of course, change one of these properties to just any
value whatever. We are able to vary the value only within a certain finite (sometimes quite narrow)
range before a phase disappears or a new one appears.
The number of degrees of freedom is also the number of independent intensive variables needed to
specify the equilibrium state in all necessary completeness, aside from the amount of each phase. In
other words, when we specify values of F different independent intensive variables, then the values of
all other intensive variables of the equilibrium state have definite values determined by the physical
nature of the system.