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