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# Calculate the efficiency of packing in case of a metal crystal for (ii) body centred cubic

1.10   Calculate the efficiency of packing in case of a metal crystal for

(ii) body-centred cubic

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(ii) Body centred cubic:-  In body centred cubic, we have atoms at all corners and at body centre.

Clearly, the atom at the centre will be in touch with the other two atoms diagonally arranged.

$b = \sqrt{2}a$ ;  and  $c = \sqrt{3}a$

Also the length of body diagonal is equal to 4r.

Thus,                              $\sqrt{3}a = 4r$

$a = \frac{4r}{ \sqrt{3}}$

The volume of a cube :

$\dpi{80} = a^3 = \left ( \frac{4r}{ \sqrt{3}} \right )^3$

In BCC, the total number of atoms is 2.

$Packing\ efficiency = \frac{Volume\ of\ one\ atom}{Volume\ of\ cubic\ unit\ cell}\times100 \%$

$= \frac{2\times(\frac{4}{3}\Pi r^3)}{(\frac{4}{\sqrt{3}}r)^3}\times100 \%$

$= 68\%$

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