Chemistry - Solid State Question with Solution | TestHub

If the given radius ratio $\left(\frac{{\mathrm{r}}_{{\mathrm{A}}^{+}}}{{\mathrm{r}}_{{\mathrm{B}}^{-}}}\right)=(\sqrt{2}-1)$,

this implies that cations are present in octahedral voids made by anions.

${\mathrm{B}}^{-}$forms fcc lattice and ${\mathrm{A}}^{+}$are present in their octahedral voids.

Now we can say -

Anions are in contact with each other

Cations are NOT in contact with each other.

By face diagonal we can say.

$\sqrt{2}\mathrm{a}=4{\mathrm{r}}_{{\mathrm{B}}^{-}}$ $\Rightarrow \mathrm{a}=2\sqrt{2}{\mathrm{r}}_{{\mathrm{B}}^{-}}$

Total volume occupied by all ions $=4\left[\frac{4}{3}\pi r_{{\mathrm{A}}^{+}}^3+\frac{4}{3}\pi r_{{\mathrm{B}}^{+}}^3\right]$ $=\frac{6}{3}\pi \left(r_{{\mathrm{A}}^{+}}^3+r_{{\mathrm{B}}^{-}}^3\right)=\frac{16\pi }{3}[5\sqrt{2}-6]$ Packing fraction $=\frac{\frac{16\pi }{3}(5\sqrt{2}-6){\mathrm{r}}_{{\mathrm{B}}^{-}}^3}{16\sqrt{2}{\mathrm{r}}_{{\mathrm{B}}^{-}}^3}$ $=\frac{\pi }{3\sqrt{2}}(5\sqrt{2}-6)=0.79$