Question

For the given close packed structure of a salt made of cation $X$ and anion $Y$ shown below (ions of only one face are shown for clarity), the packing fraction is approximately
$\text{packing fraction =}\frac{\text{packing efficiency}}{100}$

• A. $0.52$
• B. $0.48$
• C. $0.63$
• D. $0.74$

Answer 1

The correct option is C $0.63$

Packing fraction $\left(f\right)=\frac{3\times \frac{4}{3}\pi r_{+}^3+1\times \frac{4}{3}\pi r_{-}^3}{a^3}$
$=\frac{1\times \frac{4}{3}\pi [3{\left(\frac{r_{+}}{r_{-}}\right)}^3+1]}{{\left(\frac{a}{r_{-}}\right)}^3}$
Now $2r_{-}=a$
$\therefore \frac{a}{r_{-}}=2$
Also. $\frac{r_{+}}{r_{-}}=0.414$
So, $f=\frac{1\times \frac{4}{3}\times 3.14[3\times (0.414{)}^3+1]}{(2{)}^3}$
$=0.634\approx 0.63$