Ncert Exercises & Solutions

1.14 If the radius of the octahedral void is r and radius of the atoms in close packing is R, derive a relation between r and R.

A sphere with centre O, is fitted into the octahedral void as shown in the above figure. It
can be observed from the figure that ∆POQ is right-angled

∠

POQ = 90

°

Now, applying Pythagoras theorem, we can write:

{PQ}^{2} = {PO}^{2} + {OQ}^{2}

\Rightarrow {(2 R)}^{2} = {(R + r)}^{2} + {(R + r)}^{2}

\Rightarrow {(2 R)}^{2} = 2 {(R + r)}^{2}

\Rightarrow 2 R^{2} = {(R + r)}^{2}

\Rightarrow \sqrt{2} R = R + r

\Rightarrow r = \sqrt{2} R - R

\Rightarrow r = (\sqrt{2} - 1) R

\Rightarrow r = 0.414 R