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.

$\text{Packing}\text{efficiency}\text{=}\frac{\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}}{\text{Volume}\text{of}\text{unit}\text{cell}}\text{×}\text{100\%}$

Answer to Problem 12.121SP

The packing efficiency of a simple cubic unit cell is $\text{52}\text{.3}\text{\%}$.

The packing efficiency of a body-centred cubic unit cell is $67.98\text{\%}$.

The packing efficiency of a face-centered cubic unit cell is $74.01\text{\%}$

Explanation of Solution

Calculating the packing efficiency for a simple cubic unit cell:

Edge length is $\text{a}\text{=}\text{2r}$

The relation between edge length and volume is $V=a^3$

So the volume of the unit cell is $\text{V}\text{=}{\left(\text{2r}\right)}^3$

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

An atom is considered as sphere.

The volume of sphere is $\frac{{\text{4πr}}^{\text{3}}}{\text{3}}$.

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

$\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}\text{ =}\text{1×}\left(\frac{{\text{4πr}}^{\text{3}}}{\text{3}}\right)$

$\text{Packing}\text{efficiency}\text{=}\frac{\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}}{\text{Volume}\text{of}\text{unit}\text{cell}}\text{×}\text{100\%}$

$\text{Packing}\text{efficiency}\text{=}\frac{1\times \left(\frac{{\text{4πr}}^{\text{3}}}{\text{3}}\right)}{{\left(\text{2r}\right)}^{\text{3}}}\text{×100\%}$

$\begin{array}{c}=\frac{\frac{{\text{4πr}}^{\text{3}}}{\text{3}}}{\text{8}{\text{r}}^{\text{3}}}\text{×100\%}\\ =\frac{{}^1\bcancel{\text{4}}\text{π}{\bcancel{\text{r}}}^{\text{3}}}{\text{3}}\times \frac{1}{{}^2\bcancel{\text{8}}{\bcancel{\text{r}}}^{\text{3}}}\\ \text{=}\frac{\text{π}}{\text{2}\times 3}\text{×100\%}\end{array}$

$\begin{array}{c}\text{=}\frac{\text{π}}{6}\text{×100\%}\\ \text{=}\frac{3.14}{6}\times \text{100\%}\because \text{π}=3.14\\ =\text{52}\text{.3}\text{\%}\end{array}$

Thus, the packing efficiency of a simple cubic unit cell is $\text{52}\text{.3}\text{\%}$.

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

Edge length is $\text{a}\text{=}\frac{\text{4r}}{\sqrt{3}}$

The relation between edge length and volume is $V=a^3$

So the volume of the unit cell is $\text{V}\text{=}{\left(\frac{\text{4r}}{\sqrt{3}}\right)}^3$

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

An atom is considered as sphere.

The volume of sphere is $\frac{{\text{4πr}}^{\text{3}}}{\text{3}}$.

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

$\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}\text{ =}\text{2×}\left(\frac{{\text{4πr}}^{\text{3}}}{\text{3}}\right)$

$\text{Packing}\text{efficiency}\text{=}\frac{\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}}{\text{Volume}\text{of}\text{unit}\text{cell}}\text{×}\text{100\%}$

$\text{Packing}\text{efficiency}\text{=}\frac{2\times \left(\frac{{\text{4πr}}^{\text{3}}}{\text{3}}\right)}{{\left(\frac{\text{4r}}{\sqrt{3}}\right)}^3}\text{×100\%}$

$\begin{array}{c}=\frac{\frac{{\text{8πr}}^{\text{3}}}{\text{3}}}{\frac{64{\text{r}}^{\text{3}}}{3\sqrt{3}}}\text{×100\%}\\ =\frac{{}^1\bcancel{\text{8}}\text{ π}{\bcancel{\text{r}}}^{\text{3}}}{\text{3}}\times \frac{3\sqrt{3}}{{}^{8}6\bcancel{4}{\bcancel{\text{r}}}^{\text{3}}}\end{array}$

$\begin{array}{c}\text{=}\frac{\bcancel{3}\sqrt{3}\text{π}}{8\times \bcancel{3}}\text{×100\%}\\ \text{=}\frac{1.7321\times \text{3}\text{.14}}{8}\text{×100\%}\because \sqrt{3}=1.7321;\text{π}=3.14\\ =0.6798\text{×100\%}\\ =67.98\text{\%}\end{array}$

Thus, the packing efficiency of a body-centered cubic unit cell is $67.98\text{\%}$.

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

Edge length is $\text{a}\text{=}\sqrt{\text{8r}}$

The relation between edge length and volume is $V=a^3$

So the volume of the unit cell is $\text{V}\text{=}{\left(\sqrt{\text{8r}}\right)}^3$

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

An atom is considered as sphere.

The volume of sphere is $\frac{{\text{4πr}}^{\text{3}}}{\text{3}}$.

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

$\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}\text{ =}\text{4×}\left(\frac{{\text{4πr}}^{\text{3}}}{\text{3}}\right)$

$\text{Packing}\text{efficiency}\text{=}\frac{\text{Volume}\text{of}\text{atoms}\text{in}\text{unit}\text{cell}}{\text{Volume}\text{of}\text{unit}\text{cell}}\text{×}\text{100\%}$

$\text{Packing}\text{efficiency}\text{=}\frac{4\times \left(\frac{{\text{4πr}}^{\text{3}}}{\text{3}}\right)}{{\left(\sqrt{\text{8r}}\right)}^3}\text{×100\%}$

$\begin{array}{c}=\frac{\frac{{\text{16πr}}^{\text{3}}}{\text{3}}}{8\sqrt{8}{\text{r}}^{\text{3}}}\text{×100\%}\\ =\frac{{}^2\text{1}\bcancel{\text{6}}\text{π}{\bcancel{\text{r}}}^{\text{3}}}{\text{3}}\times \frac{1}{{}^1\bcancel{8}\sqrt{8}{\bcancel{\text{r}}}^{\text{3}}}\text{×100\%}\\ =\frac{2\text{π}}{\text{3}\sqrt{8}}\text{×100\%}\end{array}$

$\begin{array}{c}\text{=}\frac{2\times 3.14}{3\times 2.8284}\text{×100\%}\because \sqrt{8}=2.8284;\text{π}=3.14\\ \text{=}\frac{6.28}{8.4852}\text{×100\%}\\ =0.7401\text{×100\%}\\ =74.01\text{\%}\end{array}$

Thus, the packing efficiency of a face-centered cubic unit cell is $74.01\text{\%}$

Conclusion

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