Question

Which of the following lattices has the highest packing efficiency (i) simple cubic (ii) body-centred cubic and (iii) hexagonal close-packed lattice?

Answer 1

Hey there!

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Explanation :

→ In simple cubic, a = 2r

∴ Packing efficiency = $\frac{ \text{Volume occupied by all the spheres in unit cell} }{ \text{Volume of the unit cell} }$ × 100%

⇒ $\frac{1 × \frac{4}{3}πr^{3} × 100}{(2r)^{3}}$ %

⇒ $\frac{π}{6}$ × 100%

⇒ $\frac{22}{7 × 6}$ × 100%

= 52.4%

→ In bcc, 4r = $\sqrt{3}$ a

⇒ a = $\frac{4}{\sqrt{3}} r$

∴ Packing efficiency = $\frac{ \text{Volume occupied by all the spheres in unit cell} }{ \text{Volume of the unit cell} }$ × 100%

= $\frac{2 × \frac{4}{3}πr^{3}}{(\frac{4}{\sqrt{3}}r)^{3}}$ × 100%

= 68%

→ In fcc,

4r = $\sqrt{2}$ a

⇒ a = $\frac{4}{\sqrt{2}}$ r

∴ Packing efficiency = $\frac{ \text{Volume occupied by all the spheres in unit cell} }{ \text{Volume of the unit cell} }$ × 100%

= Packing efficiency = $\frac{4 × \frac{4}{3}πr^{3}}{( \frac{4}{\sqrt{2}}r)^{3}}$ × 100%

= 74%

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Answer :

Hexagonal closed-packed lattice has maximum efficiency with 74% whereas body-centred cubic has 68% and simple cubic has 52.4%.