Question

Sodium has the body-centered cubic structure, and its lattice parameter is$\mathbf{4}\mathbf{.}\mathbf{28}\stackrel{\mathbf{°}}{\mathbf{A}}$.

(a) How many Na atoms does a unit cell contain?

(b) What fraction of the volume of the unit cell is occupied by Na atoms, if they are represented by spheres in contact with one another?

Short answer

a) The number of Na atoms present in the unit cell is 2.

b) The fraction of the volume of the unit cell that is occupied by Na atoms when they are represented by spheres in contact with one another is 0.68.

Step-by-step solution

Subpart (a) The numbers of Na atoms present in the unit cell.

In the case of bcc,

There are 8 corners in one cube and the contribution of one atom to the unit cell is $\frac{1}{8}$; as each atom contributes to 4 unit cells up and 4 unit cells down.

In the center contribution of one atom is completely 1 to the unit cell.

Therefore, the number of Na atoms present in the unit cell is;

$1+1=2$

Subpart (b) The fraction of the volume of the unit cell that is occupied by Na atoms when they are represented by spheres in contact with one another.

In bcc, the total number of atoms present is 2.

Therefore, the volume occupied is;

$V_1=2\times \frac{4}{3}\pi {\text{r}}^{\text{3}}$

Also (bcc);

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

As;

${\text{Volume (V}}_2)={\text{a}}^3$

Therefore, the fraction of the volume;