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

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3.7 Density Computations—Ceramics • 53

EXAMPLE PROBLEM 3.6

Theoretical Density Calculation for Sodium Chloride

On the basis of crystal structure, compute the theoretical density for sodium

chloride. How does this compare with its measured density?

Solution

The theoretical density may be determined using Equation 3.6 wheren′, the

number of NaCl units per unit cell, is 4 because both sodium and chloride ions

form FCC lattices. Furthermore,

∑

AC=ANa= 22 .99 g/mol

∑

AA=ACl= 35 .45 g/mol

Since the unit cell is cubic,VC=a^3 ,abeing the unit cell edge length. For the

face of the cubic unit cell shown below,

a

2(rNa+ + rCl–)

rNa+

rCl–

Na+ Cl–

a= 2 rNa++ 2 rCl−

rNa+andrCl−being the sodium and chlorine ionic radii, given in Table 3.4 as

0.102 and 0.181 nm, respectively. Thus,

VC=a^3 =(2rNa++ 2 rCl−)^3

And, finally,

ρ=

n′(ANa+ACl)

(2rNa++ 2 rCl−)^3 NA

=

4(22. 99 + 35 .45)

[2(0. 102 × 10 −^7 )+2(0. 181 × 10 −^7 )]

3

(6. 02 × 1023 )

= 2 .14 g/cm^3

This result compares very favorably with the experimental value of 2.16 g/cm^3.