Consider the bcc unit cells of the solids 1 and 2 with the position of atoms as shown below. The radius of atom B is twice that of atom A. The unit cell edge length is 50% more in solid 2 than in 1. What is the approximate packing efficiency in solid 2?

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45%

75%

90%

65%

answer is D.

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Detailed Solution

Solid 2's packing fraction is:
$\begin{array}{l} p.f. = \frac{{(z_{eff} \times \frac{4}{3} π r_{A}^{3})}_{A} + {(z_{eff} \times \frac{4}{3} π r_{B}^{3})}_{B}}{a^{3}} \\ 2 (r_{A} + r_{B}) = \sqrt{3} a \\ \Rightarrow 2 (r_{A} + 2 r_{A}) = \sqrt{3} a \\ \Rightarrow 2 \sqrt{3} r_{A} = a \end{array} p.f. = \frac{1 \times \frac{4}{3} π r_{A}^{3} + \frac{4}{3} π (8 r_{A}^{3})}{8 \times 3 \sqrt{3} r_{A}^{3}} = \frac{9 \times \frac{4}{3} π}{8 \times 3 \sqrt{3}} = \frac{π}{2 \sqrt{3}} p. efficiency = \frac{π}{2 \sqrt{3}} \times 100 ≃ 90 %$

Therefore, the correct option is D 90%

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