Page 88 - Materials Science and Engineering An Introduction

60 • Chapter 3 / The Structure of Crystalline Solids

3.5 DENSITY COMPUTATIONS
A knowledge of the crystal structure of a metallic solid permits computation of its theo-
retical density r through the relationship
Theoretical density nA
for metals r = (3.8)
V C N A
where
n = number of atoms associated with each unit cell
A = atomic weight
V C = volume of the unit cell
23
N A = Avogadro>s number (6.022 * 10 atoms/mol)

EXAMPLE PROBLEM 3.4

Theoretical Density Computation for Copper
Copper has an atomic radius of 0.128 nm, an FCC crystal structure, and an atomic weight of
63.5 g/mol. Compute its theoretical density, and compare the answer with its measured density.

Solution
Equation 3.8 is employed in the solution of this problem. Because the crystal structure is
FCC, n, the number of atoms per unit cell, is 4. Furthermore, the atomic weight A Cu is given
as 63.5 g/mol. The unit cell volume V C for FCC was determined in Example Problem 3.1 as
16R 12, where R, the atomic radius, is 0.128 nm.
3
Substitution for the various parameters into Equation 3.8 yields
r = nA Cu = nA Cu
3
V C N A (16R 12)N A
(4 atoms/unit cell)(63.5 g/mol)
=
23
3
-8
[1612(1.28 * 10 cm) /unit cell](6.022 * 10 atoms/mol)
= 8.89 g/cm 3
3
The literature value for the density of copper is 8.94 g/cm , which is in very close agreement
with the foregoing result.

3.6 POLYMORPHISM AND ALLOTROPY
Some metals, as well as nonmetals, may have more than one crystal structure, a
polymorphism phenomenon known as polymorphism. When found in elemental solids, the condi-
allotropy tion is often termed allotropy. The prevailing crystal structure depends on both the
temperature and the external pressure. One familiar example is found in carbon:
graphite is the stable polymorph at ambient conditions, whereas diamond is formed
at extremely high pressures. Also, pure iron has a BCC crystal structure at room
temperature, which changes to FCC iron at 912 C (1674 F). Most often a modifica-
tion of the density and other physical properties accompanies a polymorphic trans-
formation.