Chemistry
Chemistry
4th Edition
ISBN: 9780078021527
Author: Julia Burdge
Publisher: McGraw-Hill Education
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Chapter 11, Problem 136AP
Interpretation Introduction

Interpretation:

The packing efficiencies of a simple cubic, body-centered, and face-centered cell are to be calculated

Concept introduction:

A crystal lattice is made of small repeating unit cells.

A unit cell is of primitive or centered type. The spheres in a unit cell can represent an atom, a molecule, or an ion.

Each sphere is shared by adjacent cells. A sphere at corners is shared by eight unit cells. A sphere in the center of each face is shared by two unit cells.

The volume of the unit cell is calculated as:

V=a3;

Here, a

is the edge length.

The formula to calculate the volume of a sphere is as:

V=43πr3

Here, r

is the radius of the sphere.

Packing efficiency is the ratio of the volume of atoms in a unit cell to the volume of the unit cell. Its expression is:

Packing efficiency=volume of atoms in unit cellvolume of unit cell×100%

Expert Solution & Answer
Check Mark

Answer to Problem 136AP

Solution: 52.4%; 68.0 %;74%

Explanation of Solution

Given information: For SCC, a=2r; for BCC, a=4r3; for FCC, a=8r.

In a simple cubic cell, spheres are present only at the eight corners of the unit cell.

The contribution of 8 spheres at the corners to one unit cell is:

18×8=1

Thus, one sphere is present in the unit cell.

The formula to calculate the volume of a unit cell is as follows:

V=a3 ……(1)

Substitute 2r

for a

in the above expression as:

V=(2r)3=8r3

The volume of a sphere is 43πr3

The formula to calculate packing efficiency is as:

Packing efficiency=volume of atoms in unit cellvolume of unit cell×100% …… (2)

Substitute 43πr3

for the volume of atoms in the unit cell and 8r3

for the volume of the unit cell in the above expression as:

Packing efficiency=43πr38r3×100%=52.4%

In BCC, there are spheres at the eight corners of the unit cell and one sphere is present in the center of the unit cell.

The contribution of 8 spheres at the corners to one unit cell is:

18×8=1

One sphere is in the center of the unit cell.

Thus, two spheres are present in a body centered unit cell.

The volume of a sphere is 43πr3.

Thus, the volume of two spheres will be as:

V=2×43πr3=83πr3

Substitute 4r3 for a

in equation (1) as:

V=(4r3)3=64r333

Substitute 83πr3

for the volume of atoms in the unit cell and 64r333

for the volume of the unit cell in equation (2) as follows:

Packing efficiency=83πr36433r3×100%=83π×3364×100%=67.9%68.0%

In FCC cubic cell, there are spheres at the eight corners of the unit cell and one sphere in the center of six faces.

The contribution of 8 spheres at the corners to one-unit cell is:

18×8=1

The contribution of 6

spheres present in the center of each face to one-unit cell is:

12×6=3

Thus, four spheres are present in a face-centered unit cell.

The volume of a sphere is 43πr3.

Thus, the volume of four spheres is:

4×43πr3=163πr3

Substitute 8r

for a

in equation (1):

V=(8r)3

Substitute 163πr3

for the volume of atoms in the unit cell and (8r)3

for the volume of the unit cell in equation (2) as follows:

Packing efficiency=163πr3(8r)3×100%=74%

Conclusion

The packing efficiencies of simple cubic, body-centered, and face-centered cell are 52.4%, 68.0%,

and 74%, respectively.

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Chapter 11 Solutions

Chemistry

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