Match the column I with column II and mark the appropriate choice.

	Column I (Structure)		Column II (Packing efficiency)
a.	Simple cubic structure	1.	68%
b.	Face centred cubic structure	2.	74%
c.	Body centred cubic structure	3.	52%

Solution :
$$\eqalign{ & {\text{For}}\,\,{\text{simple}}\,\,{\text{cubic}}\,\,{\text{structure,}} \cr & V = {a^3}\left( {{\text{volume}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{unit}}\,\,{\text{cell}}} \right) \cr & V' = \frac{4}{3}\pi {r^3}\left( {{\text{volume}}\,\,{\text{of}}\,\,{\text{one}}\,\,{\text{atom}}} \right) \cr & = \frac{4}{3}\pi {\left( {\frac{a}{2}} \right)^3} \cr & = \frac{{\pi {a^3}}}{6} \cr & {\text{Packing}}\,\,{\text{efficiency}} \cr & = \frac{{V'}}{V} \cr & = \frac{{\frac{{\pi {a^3}}}{6}}}{{{a^3}}} \cr & = \frac{\pi }{6} \cr & = 0.52\,\,{\text{or}}\,\,52\% \cr & {\text{For}}\,\,fcc\,\,{\text{structure,}} \cr & V' = 4 \times \frac{4}{3}\pi {r^3}\left( {{\text{4}}\,\,{\text{atoms}}\,\,{\text{per}}\,\,{\text{unit}}\,\,{\text{cell}}} \right) \cr & r = \frac{a}{{2\sqrt 2 }} \cr & V' = \frac{{16}}{3}\pi {\left( {\frac{a}{{2\sqrt 2 }}} \right)^3} \cr & \,\,\,\,\,\,\,\,\, = \frac{\pi }{{3\sqrt 2 }}{a^3} \cr & {\text{Volume}}\,\,{\text{of}}\,\,{\text{unit}}\,\,{\text{cell}} = V = {a^3} \cr & {\text{Packing}}\,\,{\text{efficiency}} \cr & = \frac{{V'}}{V} \cr & = \frac{{\pi {a^3}}}{{3\sqrt 2 {a^3}}} \cr & = \frac{\pi }{{3\sqrt 2 }} \cr & = 0.74\,\,{\text{or}}\,\,{\text{74% }} \cr & {\text{For}}\,\,bcc\,\,{\text{structure,}} \cr & V' = 2 \times \frac{4}{3}\pi {r^3}\left( {{\text{2}}\,\,{\text{atoms}}\,\,{\text{per}}\,\,{\text{unit}}\,\,{\text{cell}}} \right) \cr & r = \frac{{\sqrt 3 }}{4}a \cr & V' = 2 \times \frac{4}{3}\pi {\left( {\frac{{\sqrt 3 }}{4}a} \right)^3} \cr & \,\,\,\,\,\,\,\, = \frac{{\sqrt 3 \pi {a^3}}}{8} \cr & V = {a^3} \cr & {\text{Packing}}\,\,{\text{efficiency}} \cr & = \frac{{V'}}{V} \cr & = \frac{{\sqrt 3 \pi {a^3}}}{{8{a^3}}} \cr & = \frac{{\sqrt 3 }}{8}\pi \cr & = 0.68\,\,{\text{or}}\,\,{\text{68% }} \cr} $$