Question

The available space occupied by spheres of equal size in three dimensions in both hcp and ccp arrangement is:

• A. 74 %
• B. 70 %
• C. 60.4 %
• D. 52.4 %

Answer 1

The correct option is A 74 %

$\text{Solution:}-(z=\text{no. of atoms per unit cell)}$

$\text{[Percentage of occupied volume}=z\times \frac{\text{Volume of sphere}}{\text{volume of cube /hexagon}}]$

$z=4,$

$4\pi =a\sqrt{2}$

$r=a\sqrt{2}$

$\text{percentage}=4\times \frac{4/3\pi {\gamma }^3}{a^3}\times 100\%$

$=\frac{16}{3}\times \frac{\pi {\gamma }^3}{a^3}$

$=\frac{\pi }{3\sqrt{2}}\times 100$

$\text{(\%of occupied Vol.}=74\%\text{)}$

$\text{for hcp,}$

$z=62r=a$

$r=9/2$

$\text{percentage}=6\times \frac{4/3\pi {\gamma }^3}{3\sqrt{2}{a}^3}$

$

$\text{(Volume of hexagon}=3\sqrt{2}{a}^3)$

$=\frac{\pi {\gamma }^3}{3\sqrt{2}{a}^3}\times 100$

$=\frac{\pi }{3\sqrt{2}}\times 100$

$\text{(\%of occupied Vol.}=74\%\text{)}$

$\text{Hence correct ans is}\left(A\right)$