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

                (ii) body-centred cubic

Answers (1)

(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 :             

                                                       = 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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