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