Consider the solids : body-centered cubic (bee), face-centered cubic (fcc), simple cubic (sc) (or primitive), constructed of spheres of the same size. When they are arranged in increasing order of the percentage of free space in a unit cell. which order is correct?

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sc, fcc. bcc

bcc, fcc, sc

fcc bcc, sc

bcc, sc, fcc

answer is A.

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Detailed Solution

$n = \frac{1}{8} \times 8 = 1$

$\begin{array}{l} 2 r = a \\ r = a / 2 \end{array}$

$\begin{array}{l} = \frac{8 r^{3} - 4 / 3 π r^{3}}{8 r^{2}} \times 100 \\ = \frac{4 r^{3} (2 - \frac{π}{3})}{8 r^{3}} \end{array}$

$\begin{array}{l} = \frac{0.96}{2} \times 100 \\ = 48 % \end{array}$

For FCC unit cell

$\begin{aligned} \sqrt{2} a = 4 r \\ a = 2 \sqrt{2} r \end{aligned}$

$% free space = \frac{a^{3} - 2 4 / 3 π r^{3}}{a^{3}}$

$= \frac{16 \sqrt{2} r^{3} - \frac{16}{3} π r^{3}}{16 \sqrt{2} r^{3}} x 100$

$\begin{array}{l} = (\sqrt{2} - \frac{r}{3}) \times 100 \\ = 37.4 % \end{array}$

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