Question

The ratio of number of atoms present in a simple cubic, body centered cubic and face centered cubic structures are, respectively:
(a) 1 : 2 : 4
(b) 8 : 1 : 6
(c) 4 : 2 : 1
(d) 4 : 2 : 3

Answer 1

Hint:. The simple cube consists of 8 atoms at the corners, body centered cube consists of 8 atoms at the corners and 1 atom in the body center and face centered consists of 8 atoms at the corner, 6 atoms at the centers of each face. The contribution of each atom at the corner is =$\dfrac{1}{8}$ and the contribution of each atom present at the center of face =$\dfrac{1}{2}$. Now you can easily calculate the ratio of a simple cube, body centered cubic and face centered cube.

Complete step by step answer:
Simple cube consists of eight atoms at the corners of the cube.
The contribution of each atom at the corner=$\dfrac{1}{8}$
The contribution of eight atoms at the corners=$\dfrac{1}{8}\times 8$= 1
Therefore, no of atoms present in simple cubic=1
Body centered cube consists of eight atoms at the corners of the cube and one atom in the center of the cube.
The contribution of each atom at the corner=$\dfrac{1}{8}$
The contribution of eight atoms at the corners=$\dfrac{1}{8}\times 8$= 1
Atom present in the body center= 1
Therefore, no of atoms present in body centered cube=1+1 =2

Face centered cube consists of eight atoms at the corners of the cube and six atoms at the center of each face.
The contribution of each atom at the corner=$\dfrac{1}{8}$
The contribution of eight atoms at the corners=$\dfrac{1}{8}\times 8$= 1
The contribution of each atom present at the center of face =$\dfrac{1}{2}$
The contribution of six atoms present at the center of each face =$\dfrac{1}{2}\times 6$ = 3
Therefore, no of atoms present in body centered cube=1 + 3 = 4
Hence, the ratio of number of atoms present in a simple cubic, body centered cubic and face centered cubic structures is 1 : 2 : 4.
So, the correct answer is “Option A”.

Note: The packing efficiency i.e. the volume occupied
In the simple cube = $52.4%$
In face centered cube = $74%$
In body centered cube= $68%$