General Chemistry
General Chemistry
7th Edition
ISBN: 9780073402758
Author: Chang, Raymond/ Goldsby
Publisher: McGraw-Hill College
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Chapter 12, Problem 12.121SP
Interpretation Introduction

Interpretation:

Packing efficiency of a simple cubic cell, body-centred cubic and face-centred cubic unit cells, have to be calculated.

Concept Introduction:

The major types of cubic unit cells are:

  • Simple cubic unit cell.
  • Face-centered cubic unit cell.
  • Body-centered cubic unit cell.

Packing efficiency, is the percentage of the cell space occupied by the spheres in a cubic unit cell of any type. The atoms in the unit cells are considered as spheres.

Packingefficiency=VolumeofatomsinunitcellVolumeofunitcell×100%

Expert Solution & Answer
Check Mark

Answer to Problem 12.121SP

The packing efficiency of a simple cubic unit cell is 52.3%.

The packing efficiency of a body-centred cubic unit cell is 67.98%.

The packing efficiency of a face-centered cubic unit cell is 74.01%

Explanation of Solution

Calculating the packing efficiency for a simple cubic unit cell:

Edge length is a=2r

The relation between edge length and volume is V=a3

So the volume of the unit cell is V=(2r)3

The number of atoms in the simple cubic unit cell is 1.

An atom is considered as sphere.

The volume of sphere is 4πr33.

So, the volume of atoms in the simple cubic unit cell can be expressed as:

Volumeofatomsinunitcell =(4πr33)

Packingefficiency=VolumeofatomsinunitcellVolumeofunitcell×100%

Packingefficiency=1×(4πr33)(2r)3×100%

=4πr338r3×100%=14πr33×128r3=π2×3×100%

=π6×100%=3.146×100%π=3.14=52.3%

Thus, the packing efficiency of a simple cubic unit cell is 52.3%.

Calculating the packing efficiency for a body-centered cubic unit cell:

Edge length is a=4r3

The relation between edge length and volume is V=a3

So the volume of the unit cell is V=(4r3)3

The number of atoms in the body-centered cubic unit cell is 2.

An atom is considered as sphere.

The volume of sphere is 4πr33.

So, the volume of atoms in the body-centered cubic unit cell can be expressed as:

Volumeofatomsinunitcell =(4πr33)

Packingefficiency=VolumeofatomsinunitcellVolumeofunitcell×100%

Packingefficiency=2×(4πr33)(4r3)3×100%

=8πr3364r333×100%=18 πr33×33864r3

=33π8×3×100%=1.7321×3.148×100%3=1.7321;π=3.14=0.6798×100%=67.98%

Thus, the packing efficiency of a body-centered cubic unit cell is 67.98%.

Calculating the packing efficiency for a face-centered cubic unit cell:

Edge length is a=8r

The relation between edge length and volume is V=a3

So the volume of the unit cell is V=(8r)3

The number of atoms in the body-centered cubic unit cell is 4.

An atom is considered as sphere.

The volume of sphere is 4πr33.

So, the volume of atoms in the body-centered cubic unit cell can be expressed as:

Volumeofatomsinunitcell =(4πr33)

Packingefficiency=VolumeofatomsinunitcellVolumeofunitcell×100%

Packingefficiency=4×(4πr33)(8r)3×100%

=16πr3388r3×100%=216πr33×1188r3×100%=2π38×100%

=2×3.143×2.8284×100%8=2.8284;π=3.14=6.288.4852×100%=0.7401×100%=74.01%

Thus, the packing efficiency of a face-centered cubic unit cell is 74.01%

Conclusion

Using figure 11.2 as reference, the packing efficiency for the different types of cubic unit cells was calculated.

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Chapter 12 Solutions

General Chemistry

Ch. 12.7 - Prob. 1RCCh. 12 - Prob. 12.1QPCh. 12 - Prob. 12.2QPCh. 12 - Prob. 12.3QPCh. 12 - Prob. 12.4QPCh. 12 - Prob. 12.5QPCh. 12 - Prob. 12.6QPCh. 12 - Prob. 12.7QPCh. 12 - Prob. 12.8QPCh. 12 - Prob. 12.9QPCh. 12 - Prob. 12.10QPCh. 12 - Prob. 12.11QPCh. 12 - Prob. 12.12QPCh. 12 - Prob. 12.13QPCh. 12 - 12.14 Diethyl ether has a boiling point of 34.5°C,...Ch. 12 - Prob. 12.15QPCh. 12 - Prob. 12.16QPCh. 12 - Prob. 12.17QPCh. 12 - 12.18 What kind of attractive forces must be...Ch. 12 - Prob. 12.19QPCh. 12 - Prob. 12.20QPCh. 12 - Prob. 12.21QPCh. 12 - Prob. 12.22QPCh. 12 - Prob. 12.23QPCh. 12 - Prob. 12.24QPCh. 12 - Prob. 12.25QPCh. 12 - Prob. 12.26QPCh. 12 - Prob. 12.27QPCh. 12 - Prob. 12.28QPCh. 12 - Prob. 12.29QPCh. 12 - Prob. 12.30QPCh. 12 - Prob. 12.31QPCh. 12 - Prob. 12.32QPCh. 12 - Prob. 12.33QPCh. 12 - Prob. 12.34QPCh. 12 - Prob. 12.35QPCh. 12 - 12.36 A solid is hard, brittle, and electrically...Ch. 12 - Prob. 12.37QPCh. 12 - Prob. 12.38QPCh. 12 - Prob. 12.39QPCh. 12 - Prob. 12.40QPCh. 12 - Prob. 12.41QPCh. 12 - Prob. 12.42QPCh. 12 - Prob. 12.43QPCh. 12 - Prob. 12.44QPCh. 12 - Prob. 12.45QPCh. 12 - Prob. 12.46QPCh. 12 - Prob. 12.47QPCh. 12 - Prob. 12.48QPCh. 12 - Prob. 12.49QPCh. 12 - Prob. 12.50QPCh. 12 - Prob. 12.51QPCh. 12 - Prob. 12.52QPCh. 12 - Prob. 12.53QPCh. 12 - Prob. 12.54QPCh. 12 - Prob. 12.55QPCh. 12 - Prob. 12.56QPCh. 12 - Prob. 12.57QPCh. 12 - Prob. 12.58QPCh. 12 - Prob. 12.59QPCh. 12 - Prob. 12.60QPCh. 12 - Prob. 12.61QPCh. 12 - Prob. 12.62QPCh. 12 - 12.63 What is the relationship between...Ch. 12 - Prob. 12.64QPCh. 12 - 12.65 Why is solid carbon dioxide called dry ice? Ch. 12 - Prob. 12.66QPCh. 12 - 12.67 Referring to Figure 12.28, estimate the...Ch. 12 - Prob. 12.68QPCh. 12 - Prob. 12.69QPCh. 12 - Prob. 12.70QPCh. 12 - Prob. 12.71QPCh. 12 - Prob. 12.72QPCh. 12 - Prob. 12.73QPCh. 12 - Prob. 12.74QPCh. 12 - 12.75 These compounds are liquid at −10°C; their...Ch. 12 - 12.76 Freeze-dried coffee is prepared by freezing...Ch. 12 - Prob. 12.77QPCh. 12 - 12.78 Steam at 100°C causes more serious burns...Ch. 12 - 12.79 Vapor pressure measurements at several...Ch. 12 - Prob. 12.80QPCh. 12 - Prob. 12.81QPCh. 12 - Prob. 12.82QPCh. 12 - Prob. 12.83QPCh. 12 - Prob. 12.84QPCh. 12 - Prob. 12.85QPCh. 12 - Prob. 12.86QPCh. 12 - Prob. 12.87QPCh. 12 - Prob. 12.88QPCh. 12 - Prob. 12.89QPCh. 12 - 12.90 Name the kinds of attractive forces that...Ch. 12 - Prob. 12.91QPCh. 12 - Prob. 12.92QPCh. 12 - Prob. 12.93QPCh. 12 - Prob. 12.94QPCh. 12 - Prob. 12.95QPCh. 12 - Prob. 12.96QPCh. 12 - Prob. 12.97QPCh. 12 - Prob. 12.98QPCh. 12 - 12.99 The liquid-vapor boundary line in the phase...Ch. 12 - Prob. 12.100QPCh. 12 - Prob. 12.101QPCh. 12 - Prob. 12.102QPCh. 12 - Prob. 12.103QPCh. 12 - Prob. 12.104QPCh. 12 - Prob. 12.105QPCh. 12 - Prob. 12.106QPCh. 12 - 12.107 The following graph shows approximate plots...Ch. 12 - Prob. 12.108QPCh. 12 - Prob. 12.109QPCh. 12 - Prob. 12.110QPCh. 12 - Prob. 12.111QPCh. 12 - Prob. 12.112QPCh. 12 - Prob. 12.113QPCh. 12 - Prob. 12.114QPCh. 12 - 12.115 Use the concept of intermolecular forces to...Ch. 12 - Prob. 12.116QPCh. 12 - 12.117 What is the origin of dark spots on the...Ch. 12 - Prob. 12.118QPCh. 12 - 12.119 The electrical conductance of copper metal...Ch. 12 - Prob. 12.120SPCh. 12 - Prob. 12.121SPCh. 12 - Prob. 12.122SPCh. 12 - Prob. 12.123SPCh. 12 - Prob. 12.124SPCh. 12 - 12.125 The boiling point of methanol is 65.0°C and...Ch. 12 - Prob. 12.126SPCh. 12 - Prob. 12.127SPCh. 12 - Prob. 12.128SPCh. 12 - Prob. 12.129SP
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