Chapter 11, Problem 11.134QP

A quantitative measure of how efficiently spheres pack into unit cells is called packing efficiency, which is the percentage of the cell space occupied by the spheres. Calculate the packing efficiencies of a simple cubic cell, a body-centered cubic cell, and a face-centered cubic cell. (Hint: Refer to Figure 11.22 and use the relationship that the volume of a sphere is $\frac{4}{3}\pi r^3$, where r is the radius of the sphere.)

Figure 11.22 The relationship between the edge length (a) and radius (r) of atoms in the simple cubic cell, body-centered cubic cell, and face-centered cubic cell.

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 11.134QP

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 have been calculated.