Question

The name of the $NaCl$ crystal unit cell and its packing efficiency are respectively:

• A. $FCC$ and $74$ %
• B. $BCC$ and $74$ %
• C. $FCC$ and $68$ %
• D. $FCC$ and $54$ %

Answer 1

The correct option is A $FCC$ and $74$ %

$NaCl$ has $FCC$ crystal unit cell.

$FCC$(Face Centered Cubic) lattice each ion is $6-$ coordinate and has a local octahedral geometry.

$FCC$ unit cell contains $4$ number of atoms.

In $△ABC$,

$A{C}^2=A{B}^2+B{C}^2=b^2$

$\Rightarrow a^2+a^2=b^2$

$\Rightarrow 2a^2=(4\pi {)}^2$

$\Rightarrow a=\sqrt{\frac{16}{2}}r$

$\Rightarrow a=\frac{4}{\sqrt{2}}r=2\sqrt{2}r$

Volume of one sphere$=\frac{4}{3}\pi r^3$

$\because 4$ atoms , so $4$ spheres

Volume of $4$ sphere $=4\times \frac{4}{3}\pi r^3$

and volume of cube $=a^3=(2\sqrt{2}r{)}^3$

Therefore, $Packingefficiency=\frac{Volumeoccupiedby4sphereinunitcell\ast 100}{Volumeofunitcell}$

$\Rightarrow Packingefficiency=\frac{4\times \frac{4}{3}\pi r^3}{\left(2\sqrt{2}{)}^3\right(r{)}^3}\times 100=74\%$