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
(packing fraction $=\frac{\text{ packing efficiency }}{100}$ )

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

Answer 1

Answer: (B)

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