Page 359 - Materials Science and Engineering An Introduction

9.17 The Gibbs Phase Rule • 331

be specified to define the state of the system completely. Expressed another way, F
is the number of these variables that can be changed independently without altering
the number of phases that coexist at equilibrium. The parameter C in Equation 9.16
represents the number of components in the system. Components are normally ele-
ments or stable compounds and, in the case of phase diagrams, are the materials at the
two extremes of the horizontal compositional axis (e.g., H 2 O and C 12 H 22 O 11 , and Cu
and Ni for the phase diagrams shown in Figures 9.1 and 9.3a, respectively). Finally, N
in Equation 9.16 is the number of noncompositional variables (e.g., temperature and
pressure).
Let us demonstrate the phase rule by applying it to binary temperature–composition
phase diagrams, specifically the copper–silver system, Figure 9.7. Because pressure is
constant (1 atm), the parameter N is 1—temperature is the only noncompositional vari-
able. Equation 9.16 now takes the form

P + F = C + 1 (9.17)
The number of components C is 2 (namely, Cu and Ag), and
P + F = 2 + 1 = 3

or
F = 3 - P
Consider the case of single-phase fields on the phase diagram (e.g., a, b, and liquid
regions). Because only one phase is present, P 1 and

F = 3 - P
= 3 - 1 = 2
This means that to completely describe the characteristics of any alloy that exists within
one of these phase fields, we must specify two parameters—composition and tempera-
ture, which locate, respectively, the horizontal and vertical positions of the alloy on the
phase diagram.
For the situation in which two phases coexist—for example, a + L, b + L, and a + b
phase regions (Figure 9.7)—the phase rule stipulates that we have but one degree of
freedom because

F = 3 - P
= 3 - 2 = 1
Thus, it is necessary to specify either temperature or the composition of one of the
phases to completely define the system. For example, suppose that we decide to spec-
ify temperature for the a L phase region, say, T 1 in Figure 9.23. The compositions
of the a and liquid phases (C a and C L ) are thus dictated by the extremes of the tie
line constructed at T 1 across the a L field. Note that only the nature of the phases
is important in this treatment and not the relative phase amounts. This is to say that
the overall alloy composition could lie anywhere along this tie line constructed at
temperature T 1 and still give C a and C L compositions for the respective a and liquid
phases.
The second alternative is to stipulate the composition of one of the phases for this
two-phase situation, which thereby fixes completely the state of the system. For exam-
ple, if we specified C a as the composition of the a phase that is in equilibrium with the
liquid (Figure 9.23), then both the temperature of the alloy (T 1 ) and the composition of
the liquid phase (C L ) are established, again by the tie line drawn across the a + L phase
field so as to give this C a composition.