Page 142 - Materials Science and Engineering An Introduction

114 • Chapter 4 / Imperfections in Solids

reasonably valid assumption and does not lead to significant errors for dilute solutions
and over composition ranges where solid solutions exist.

EXAMPLE PROBLEM 4.3

Derivation of Composition-Conversion Equation
Derive Equation 4.6a.

Solution
To simplify this derivation, we assume that masses are expressed in units of grams and denoted
with a prime (e.g., m 1 ). Furthermore, the total alloy mass (in grams) M is
M = m 1 + m 2 (4.12)

(Equation 4.5a) and incorporating the expression for n m1 ,
Using the definition of C 1
Equation 4.4, and the analogous expression for n m2 yields
n m1
C 1 = * 100
n m1 + n m2
m 1
Tutorial Video A 1
= * 100 (4.13)
m 1 m 2
+
A 1 A 2
Rearrangement of the mass-in-grams equivalent of Equation 4.3a leads to

C 1 M
m 1 = (4.14)
100
Substitution of this expression and its m 2 equivalent into Equation 4.13 gives

C 1 M
C 1 = 100A 1 * 100 (4.15)
C 1 M C 2 M
+
100A 1 100A 2
Upon simplification, we have

C 1 A 2
C 1 = * 100
C 1 A 2 + C 2 A 1
which is identical to Equation 4.6a.

EXAMPLE PROBLEM 4.4

Composition Conversion—From Weight Percent to Atom Percent
Determine the composition, in atom percent, of an alloy that consists of 97 wt% aluminum and
3 wt% copper.

Solution
If we denote the respective weight percent compositions as C Al 97 and C Cu 3, substitution
into Equations 4.6a and 4.6b yields