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

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5.6 Specification of Composition • 139

percent. If we represent alloy density and atomic weight byρaveandAave, respectively,

then

ρave=

100

C 1

ρ 1

+

C 2

ρ 2

(5.13a)

Computation of

density (for a

two-element metal

alloy)

ρave=

C 1 ′A 1 +C 2 ′A 2

C 1 ′A 1

ρ 1

+

C 2 ′A 2

ρ 2

(5.13b)

Aave=

100

C 1

A 1

+

C 2

A 2

(5.14a)

Computation of

atomic weight (for a

two-element metal

alloy)

Aave=

C 1 ′A 1 +C 2 ′A 2

100

(5.14b)

It should be noted that Equations 5.12 and 5.14 are not always exact. In their

derivations, it is assumed that total alloy volume is exactly equal to the sum of the

volumes of the individual elements. This normally is not the case for most alloys;

however, it is a reasonably valid assumption and does not lead to significant errors

for dilute solutions and over composition ranges where solid solutions exist.

EXAMPLE PROBLEM 5.4

Derivation of Composition-Conversion Equation

Derive Equation 5.9a.

Solution

To simplify this derivation, we will 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 ′ (5.15)

Using the definition ofC′ 1 (Equation 5.8) and incorporating the expression

fornm 1 , Equation 5.7, and the analogous expression fornm 2 yields

C 1 ′=

nm 1

nm 1 +nm 2

× 100

=

m 1 ′

A 1

m 1 ′

A 1

+

m 2 ′

A 2

× 100

(5.16)

Rearrangement of the mass-in-grams equivalent of Equation 5.6 leads to

m 1 ′=

C 1 M′

100

(5.17)