Question

Which of the following expression is correct for packing fraction of NaCl if $a^{-}$ ions from one face and one diagonal are removed?

• A. $\frac{\frac{13}{3} \pi r_{-}^{3}+\frac{16}{3} \pi r_{+}^{3}}{8\left(r_{+}+r_{-}\right)^{3}}$
• B. $\frac{\frac{13}{3} \pi r_{-}^{3}+\frac{4}{3} \pi r_{+}^{3}}{8\left(r_{+}+r_{-}\right)^{3}}$
• C. $\frac{\frac{16}{3} \pi r_{-}^{3}+\frac{13}{3} \pi r_{+}^{3}}{8\left(r_{+}+r_{-}\right)^{3}}$
• D. $\frac{\frac{4}{3} \pi r_{-}^{3}+\frac{13}{3} \pi r_{+}^{3}}{8\left(r_{+}+r_{-}\right)^{3}}$

Answer 1

Answer: (A)

Effective number of remaining
$Cl ^{-}=4-2 \times \frac{1}{8}+\frac{1}{2}=\frac{13}{4}$
Effective number of $Na ^{+}=4$
$\text { Packing fraction }=\frac{\text { Occupied volume }}{\text { Total volume }} $
$=\frac{\frac{13}{4} \times \frac{4}{3} \pi r_{-}^3+4 \times \frac{4}{3} \pi r_{+}^3}{8\left(r_{+}+r_{-}\right)^3} $
$\frac{\frac{13}{3} \pi r_{-}^3+\frac{16}{3} \pi r_{+}^3}{8\left(r_{+}+r_{-}\right)^3}$