Question

Percentage of free space in a body centered cubic unit cell is:
A. $30$%
B. $32$%
C. $34$%
D. $28$%

Answer 1

Hint:In a body-centered cubic unit cell, the radius is one-fourth of diagonal length. The diagonal edge length of the body-centered cubic unit cell is$a\sqrt{3}$. By this, we will determine the volume of the cube. Then divide the volume of two spheres by the volume of the cube to determine the packing efficiency.

Complete step by step solution:
The total space occupied by the particles in percentage is defined as the packing efficiency.
The formula to determine packing efficiency is as follows:
$\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}={\displaystyle \frac{\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{o}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}}\times 100$
The volume of the sphere is, ${\displaystyle \frac{4}{3}}\pi {\mathrm{r}}^3$
The formula of the volume of the cube is, $a^3$
So, the packing efficiency is,
$\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}={\displaystyle \frac{2\times {\displaystyle \frac{4}{3}}\pi {\mathrm{r}}^3}{{\mathrm{a}}^3}}\times 100$
The volume of the body-centred cubic unit cell ${\mathrm{a}}^3$ is as follows:
In the body-centred cubic unit cell, the relation between atomic radius edge length is as follows:
$r={\displaystyle \frac{a\sqrt{3}}{4}}$
Where,
$r$ is the atomic radius.
$a$ is the edge length of the unit cell.
Rearrange for edge length, $a={\displaystyle \frac{4r}{\sqrt{3}}}$
So, the volume body-centred cubic unit cell is, $a^3={\left({\displaystyle \frac{4r}{\sqrt{3}}}\right)}^3$
Substitute ${\left({\displaystyle \frac{4r}{\sqrt{3}}}\right)}^3$for $a^3$ in packing efficiency formula.
$\Rightarrow \mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}={\displaystyle \frac{2\times {\displaystyle \frac{4}{3}}\pi {\mathrm{r}}^3}{{\left({\displaystyle \frac{4r}{\sqrt{3}}}\right)}^3}}\times 100$
$$\Rightarrow \mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}={\displaystyle \frac{8}{3}}\pi {\mathrm{r}}^3\times {\displaystyle \frac{3\sqrt{3}}{64r^3}}\times 100$$
$$\Rightarrow \mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}=68$$
So, the packing efficiency of a body-centred cubic unit cell is $$68$$%.
The total volume of the body centred cubic unit cell is $$100$$% out of which $$68$$% is occupied so, the free space is,
$$100-68=32$$
So, the percentage of free space in a body-centered cubic unit cell is $$32$$%.

Therefore, option (B) $$32$$% is correct.

Note:
The packing efficiency of the face-centered cubic unit cell which is found in hcp and ccp is $$78$$% and the percentage of free space is $$22$$%. The packing efficiency of the simple cubic unit cell is $$52.4$$% and the percentage of free space is $$47.6$$%. The maximum packing efficiency is of the face-centered cubic unit cell. In the face-centered cubic lattice, the radius is one-fourth of the diagonal length. The diagonal edge length of the face-centered cubic unit cell is $a\sqrt{2}$. In a simple cubic unit cell, the edge length is double the radius of the unit cell.