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Question

A body centered cubic lattice is made up of hollow spheres of B. Spheres of solid A are present in hollow spheres of B. Radius A is half of radius of B. What is the ratio of total volume of spheres of B unoccupied by A in a unit cell and volume of unit cell?

A
73π64
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B
73128
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C
7π24
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D
7π 364
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Solution

The correct option is D 7π 364
In a bcc unit cell, the spheres are present at the corners as well as at the centre, the spheres at corners does not touch each other. But the spheres at body centre touches all the spheres at the corners of the unit cell.
Let us consider, the radius of sphere B be 'R' and the radius of sphere A be 'r'.

r=R2

The total number of spheres B present in the unit cell will be 2. Similarly, the total number of spheres A present in the unit cell will also be 2.
In order to find out the volume of sphere B not occupied by the sphere A, we need to substact the total volume of the spheres A from the total volume of spheres B.
Volume of spheres of B unoccupied by A =Total volume of sphere B Total volume of sphere A

Total volume of B unoccupied by A
in a unit cell
=(2×43 πR3)(2×43 πr3)
Since, r=R2
Total volume of B unoccupied by A
in a unit cell is
=2×43(R3R38)×π=7πR33

For bcc,
3a=4R
a=4R3
Volume of unit cell =a3=6433R3
Required ratio =7πR3/36433R3=7π 364

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