#### A metal crystallizes in a body-centered cubic lattice with an edge length of $3.5\AA.$ If the metal atoms have a radius of $1.2\AA$, what is the packing efficiency of the crystal?Option: 1 34%Option: 2 48%Option: 3 68%Option: 4 74%

The packing efficiency of a body-centered cubic BCC structure is given by 68%.

To arrive at the answer, we can use the formula for the packing efficiency of a BCC structure:

Packing efficiency

$=\frac{\text { number of atoms in the unit cell } \times \text { volume of each atom }}{\text { volume of the unit cell }}$

In a BCC structure, there are two atoms per unit cell - one at each of the eight corners and one at the center of the cube. The volume of each atom can be calculated using the formula for the volume of a sphere:

Volume of each atom$=\frac{4}{3} \times \pi \times r^3$

Substituting the given values, we get:

Volume of each atom $=\frac{4}{3} \times \pi \times(1.2 \AA)^3=7.2384 \AA^3$

The volume of the unit cell can be calculated as:

Volume of the unit cell$(\text { edge length })^3$

Substituting the given value, we get:

Volume of the unit cell $(3.5 A)^3=42.875 A^3$

Putting everything back in the formula for packing efficiency, we get:

Packing efficiency $=\frac{2 \times 7.2384 A^3}{42.875 A^3}=0.68 \text { or } 68 \%$

Therefore, the correct answer is (c) 68%.