Question

If the radius of $B{r^ - }$ ion is 0.182 nm, how large can a cation be fit in its tetrahedral holes?
A) 0.414 nm
B) 0.0753 nm
C) 0.091nm
D) 0.225nm

Answer 1

Hint: Radius ratio is the ratio of cation to anion radius. The equation used for determining the radius ratio is $\rho = \dfrac{{r + }}{{r - }}$ . The rules help us determine the arrangement of ions in the crystal structure. The prediction of coordination number is also possible with the help of the radius ratio. It also helps to determine the stability of the ionic crystal.

Complete answer:
The stability limit is when the cation is touching all the anions and the anions are touching at their edges. The radius ratio is greater than 0.155, the compound can be stable. The table given below shows the relationship between radius ratio and coordination number.

Radius Ratio	Coordination Number	Type of Void
<0.155	2	Linear
0.155-0.225	3	Trigonal Planar
0.225-0.414	4	Tetrahedral
0.414-0.732	6	Octahedral
0.732-1	8	Cubic

The information given to us is that the radius of $B{r^ - }$ ion is 0.182 nm and the geometry is tetrahedral. From the table above we can say that, $\dfrac{{r + }}{{r - }} = 0.414{\text{ }}to{\text{ }}0.225$
$\dfrac{{r + }}{{r - }} = 0.414$
$r + = 0.414 \times r - \to 0.414 \times 0.182nm$
$r + = 0.0753nm$
Therefore, the cation of radius 0.0753nm (maximum) can fit into the tetrahedral void.

Note:
Radius Ratio plays an important role in determining a stable structure in the ionic crystal. The arrangement of ions in the crystal structure is also determined. The crystal structures can be predicted by the radius ratio only if it includes lengths, lattice parameters, etc. Thus, prediction is possible when the values of the radius of the ions are taken from the same reference or origin. The correct relative sizes can be only achieved by this. The ionic radii are approximate one and precisely can be found by using X-Ray Crystallography.