JEE Main 2019 — Solid State Question with Solution

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 = $$\frac{Volumeofparticles}{VolumeofUnitcell}$$

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

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

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

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

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

$$\therefore$$ Packing efficiency = 90 %