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Question: The ratio of number of atoms present in a simple cubic, body centered cubic and face centered cubic ...

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

Explanation

Solution

. 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 =18\dfrac{1}{8} and the contribution of each atom present at the center of face =12\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=18\dfrac{1}{8}
The contribution of eight atoms at the corners=18×8\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=18\dfrac{1}{8}
The contribution of eight atoms at the corners=18×8\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=18\dfrac{1}{8}
The contribution of eight atoms at the corners=18×8\dfrac{1}{8}\times 8= 1
The contribution of each atom present at the center of face =12\dfrac{1}{2}
The contribution of six atoms present at the center of each face =12×6\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 = 52.452.4%
In face centered cube = 7474%
In body centered cube= 6868%