Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-10 GTBL042-Callister-v2 August 13, 2007 18:16

378 • Chapter 10 / Phase Diagrams

Concept Check 10.7

(a) For the SiO 2 –Al 2 O 3 system, what is the maximum temperature that is possible

without the formation of a liquid phase?(b)At what composition or over what range

of compositions will this maximum temperature be achieved?

[The answer may be found at http://www.wiley.com/college/callister (Student Companion Site).]

10.17 TERNARY PHASE DIAGRAMS

Phase diagrams have also been determined for metallic (as well as ceramic) systems

containing more than two components; however, their representation and interpre-

tation may be exceedingly complex. For example, a ternary, or three-component,

composition–temperature phase diagram in its entirety is depicted by a three-

dimensional model. Portrayal of features of the diagram or model in two dimensions

is possible but somewhat difficult.

10.18 THE GIBBS PHASE RULE

The construction of phase diagrams as well as some of the principles governing the

conditions for phase equilibria are dictated by laws of thermodynamics. One of these

Gibbs phase rule is theGibbs phase rule,proposed by the nineteenth-century physicist J. Willard Gibbs.

This rule represents a criterion for the number of phases that will coexist within a

system at equilibrium, and is expressed by the simple equation

General form of the P+F=C+N (10.16)

Gibbs phase rule

wherePis the number of phases present (the phase concept is discussed in Section

10.3). The parameterFis termed thenumber of degrees of freedomor the num-

ber of externally controlled variables (e.g., temperature, pressure, composition) that

must be specified to completely define the state of the system. Expressed another

way,Fis the number of these variables that can be changed independently with-

out altering the number of phases that coexist at equilibrium. The parameterCin

Equation 10.16 represents the number of components in the system. Components

are normally elements or stable compounds and, in the case of phase diagrams, are

the materials at the two extremities 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 10.1 and

10.3a, respectively). Finally,Nin Equation 10.16 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 10.7. Since

pressure is constant (1 atm), the parameterNis 1—temperature is the only noncom-

positional variable. Equation 10.16 now takes the form

P+F=C+ 1 (10.17)

Furthermore, the number of componentsCis 2 (viz. Cu and Ag), and

P+F= 2 + 1 = 3

or

F= 3 −P