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Class 10 Class 12
Calculate the efficiency of packing in case of a metal crystal for
(i) simple cubic
(ii) body- centred cubic
(iii) face - centred cubic . (With the assumptions that atoms are touching each other).

Solution:

(i) Simple Cubic: A simple cubic unit cell has one sphere (or atom) per unit cell. If r is the radius of the sphere, then volume occupied by one sphere present in unit cell = $\frac{4}{3}{\mathrm{\pi r}}^{3}$

Edge length of unit cell (a) = r + r = 2r
Volume of cubic
Volume of occupied by sphere  = $\frac{4}{3}{\mathrm{\pi r}}^{3}$
Percentage volume occupied = percentage of efficiency of packing

For simple cubic metal crystal the efficiency of packing = 52.4%.

(b) Body centred cubic: From the figure it is clear that the atom at the centre will be in touch with other two atoms diagonally arranged and shown with solid boundaries.

The length of the body diagonal c is equal to 4r where r is the radius of the sphere (atom). But c = 4r, as all the three spheres along the diagonal touch each other

∴

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

or                    $\mathrm{r}=\frac{\sqrt{3}}{4}\mathrm{a}.$

Fig. BCC unit cell. In this sort of structure total number of atoms is two and their volume is $2×\left(\frac{4}{3}\right){\mathrm{\pi r}}^{3}.$

Volume of the cube, a
3 will be equal to

Therefore, Percentage of efficiency Volume occupied by four - spheres

(c) Face centred cubic: A face centred cubic cell (fcc) contains four spheres (or atoms) per unit volume occupied by one sphere of radius

r = 4/3 $\mathrm{\pi }$r3

Volume occupied by four spheres present in the unit cell

r = 4/3 $\mathrm{\pi }$r3 x 4 = 16/3 $\mathrm{\pi }$r3

Fig.  Face centred cubic unit cell. Figure indicates that spheres placed at the corners touches a face centred sphere. Length of the face diagonal
= r + 2r + r = 4r

Hence, for face centred cubic, efficiency of packing = 74%.

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