QJEEM19Ch1a15: JEE Main 2019 Chemistry - Class 12 JEEM19Ch1a15 Solution: Consider the bcc unit cells of the solids 1 and 2 with the…

Given that, The radius of atom B is twice that of atom A.

Let r = radius of atom A then radius of atom B = 2r

Let edge length of the cube = $a$

We know length of body diagonal = $a\sqrt 3$

Here length of body diagonal in solid 2 = r + r + 2(2r) = 6r

$\therefore$ $a\sqrt 3$ = 6r

$\Rightarrow$ $a$ = $2\sqrt 3$ r

We know,

Packing fraction = ${{Volume\,of\,particles} \over {Volume\,of\,Unit\,cell}}$

Here total number of particles in one unit cell = $\left( {{1 \over 8} \times 8} \right)$ + 1

= 1(atom A) + 1(atom B)

$\therefore$ Packing fraction = ${{{4 \over 3}\pi {r^3} + {4 \over 3}\pi {{\left( {2r} \right)}^2}} \over {{a^3}}}$

= ${{{4 \over 3}\pi 9{r^3}} \over {{{\left( {2\sqrt 3 r} \right)}^3}}}$

= ${{12\pi {r^3}} \over {24\sqrt 3 {r^3}}}$ = 0.906

$\therefore$ Packing efficiency = 90 %