Page 335 - Materials Science and Engineering An Introduction

9.8 Interpretation of Phase Diagrams • 307

Of course, identical answers are obtained if compositions are expressed in weight per-
cent copper instead of nickel.
Thus, the lever rule may be employed to determine the relative amounts or frac-
tions of phases in any two-phase region for a binary alloy if the temperature and com-
position are known and if equilibrium has been established. Its derivation is presented
as an example problem.

It is easy to confuse the foregoing procedures for the determination of phase
compositions and fractional phase amounts; thus, a brief summary is warranted.
Compositions of phases are expressed in terms of weight percents of the components
(e.g., wt% Cu, wt% Ni). For any alloy consisting of a single phase, the composition
of that phase is the same as the total alloy composition. If two phases are present,
the tie line must be employed, the extremes of which determine the compositions of
the respective phases. With regard to fractional phase amounts (e.g., mass fraction
of the a or liquid phase), when a single phase exists, the alloy is completely that
phase. For a two-phase alloy, the lever rule is used, in which a ratio of tie line seg-
ment lengths is taken.

Concept Check 9.3 A copper–nickel alloy of composition 70 wt% Ni–30 wt% Cu is
slowly heated from a temperature of 1300 C (2370 F).
(a) At what temperature does the first liquid phase form?
(b) What is the composition of this liquid phase?
(c) At what temperature does complete melting of the alloy occur?
(d) What is the composition of the last solid remaining prior to complete melting?
Concept Check 9.4 Is it possible to have a copper–nickel alloy that, at equilibrium, con-
sists of an a phase of composition 37 wt% Ni–63 wt% Cu and also a liquid phase of composition
20 wt% Ni–80 wt% Cu? If so, what will be the approximate temperature of the alloy? If this is
not possible, explain why.
[The answers may be found at www.wiley.com/college/callister (Student Companion Site).]

EXAMPLE PROBLEM 9.1

Lever Rule Derivation
Derive the lever rule.

Solution
Consider the phase diagram for copper and nickel (Figure 9.3b) and alloy of composition C 0 at
1250 C, and let C a , C L , W a , and W L represent the same parameters as given earlier. This deri-
vation is accomplished through two conservation-of-mass expressions. With the first, because
only two phases are present, the sum of their mass fractions must be equal to unity; that is,
W a + W L = 1 (9.3)

For the second, the mass of one of the components (either Cu or Ni) that is present in both of
the phases must be equal to the mass of that component in the total alloy, or
W a C a + W L C L = C 0 (9.4)