Question

If the radius of an octahedral void is $r$ and the radius of atoms in close packing is $R$, then the relation between $r$ and $R$ is :

• A. $r=0.414R$
• B. $R=0.414r$
• C. $r=2R$
• D. $r=\sqrt{2}R$

Answer 1

The correct option is A $r=0.414R$

A sphere fitted into the octahedral void is shown by shaded circle in the figure. The spheres present in other layers are not shown in the figure. Joining the centre of 4 spheres form a square as shown.

Now,

$\Delta ABC$ is a right angled triangle, right angle at $\angle B$.

$AB=R+r$, $BC=R+r$ and $AC=2R$.

where,

$R=$ radius of the atom in close packing.

$r=$ radius of the octahedral void.

Now, applying Pythagoras theorem in $\Delta ABC$.

$A{B}^2+B{C}^2=A{C}^2$ -------( 1)

Substituting the value of AB, BC and AC in (1) equation we have

$(R+r{)}^2+(R+r{)}^2=(2R{)}^2$

$\Rightarrow 2(R+r{)}^2=4R^2$

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

Taking square root on both side

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

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

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

$\Rightarrow r=(1.414-1)\times R=0.414\times R$

$\therefore$ (A) option is correct.