Question

Three elements A,B and C crystalize in a cubic lattice with A atoms at the corners, B atoms at the cube center and C atoms at the centre of the face of the cube.
When all the atoms from two different body diagonals are removed, then find the ratio of effective number of particles $\left(Z_{eff}\right)$ initially (before removal of atoms) to the effective number of particles finally (after removal of atoms).

• A. $5:3$
• B. $3:5$
• C. $7:10$
• D. $10:7$

Answer 1

The correct option is D $10:7$

Three elements A, B and C crystalize in a cubic solid lattice
A atoms are present at the corners of the cube.
$\text{Number of A atoms}=8\times \frac{1}{8}=1$


B atoms are present at the cubic center.
There is only one cubic center.
Each atom contributes 1 to the cube.
$\text{Number of B atoms}=1$


C atoms are present at the center of the face of the cube.
$\text{Number of C atoms}=6\times \frac{1}{2}=3$

$(Z_{eff}{)}_{\text{initially}}=1+1+3=5$

When atoms from two body diagonals are removed :
4 corner atoms and 1 body centre atom is removed
$\text{Number of A atoms}=4\times \frac{1}{8}=\frac{1}{2}$

$\text{Number of B atoms}=1-1=0$
$\text{Number of C atoms}=6\times \frac{1}{2}=3$

$(Z_{eff}{)}_{\text{finally}}=0.5+0+3=3.5$

$\frac{(Z_{eff}{)}_{\text{initially}}}{(Z_{eff}{)}_{\text{finally}}}=\frac{5}{3.5}=\frac{10}{7}$