| Column I (Structure) | Column II (Packing efficiency) | ||
| [A] | Simple cubic structure | (i) | 68% |
| [B] | Face centred cubic structure | (ii) | 74% |
| [C] | Body centred cubic structure | (iii) | 52% |
A) [A] \[\to \] (iii), [B] \[\to \] (ii), [C] \[\to \] (i)
B) [A] \[\to \] (i), [B] \[\to \] (ii), [C] \[\to \] (iii)
C) [A] \[\to \] (ii), [B] \[\to \] (i), [C] \[\to \] (iii)
D) [A] \[\to \] (iii), [B] \[\to \] (i), [C] \[\to \] (ii)
Correct Answer: A
Solution :
| For simplexubic structure, |
| \[V={{a}^{3}}\](volume of the unit cell) |
| \[V=\frac{4}{3}\pi {{r}^{3}}\] (volume of one atom) |
| \[=\frac{4}{3}\pi {{\left( \frac{a}{2} \right)}^{3}}=\frac{\pi {{a}^{3}}}{6}\] |
| Packing efficiency \[=\frac{V'}{V}=\frac{\pi {{a}^{3}}/6}{{{a}^{3}}}=\frac{\pi }{6}=0.52\] = 0.52 or 52% |
| For fcc structure, |
| \[{{V}^{'}}=4\times \frac{4}{3}\pi {{r}^{3}}\](4 atoms per unit cell) |
| \[r=\frac{a}{2\sqrt{2}}\] |
| \[V'=\frac{16}{3}\pi {{\left( \frac{a}{2\sqrt{2}} \right)}^{3}}=\frac{\pi }{3\sqrt{2}}{{a}^{3}}\] |
| Volume of unit cell \[=V={{a}^{3}}\] |
| Packing efficiency |
| \[=\frac{V'}{V}=\frac{\pi {{a}^{3}}}{3\sqrt{2}{{a}^{3}}}=\frac{\pi }{3\sqrt{2}}=0.74\] or 74% |
| For bee structure, |
| \[V'=2\times \frac{4}{3}\pi {{r}^{3}}\] (2 atoms per unit cell) |
| \[r=\frac{\sqrt{3}}{4}a\] |
| \[V'=2\times \frac{4}{3}\pi {{\left( \frac{\sqrt{3}}{4}a \right)}^{3}}=\frac{\sqrt{3}\pi {{a}^{3}}}{8}\] |
| \[V={{a}^{3}}\] |
| Packing efficiency |
| \[=\frac{V'}{V}=\frac{\sqrt{3}\pi {{a}^{3}}}{8{{a}^{3}}}=\frac{\sqrt{3}}{8}\pi =0.68\] or 68% |
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