Question

For an ionic crystal of general formula AX and coordination number 6, the value of radius ratio will be:
(a) Greater than 0.73
(b) In between 0.73 and 0.40
(c) In between 0.41 and 0.22
(d) Less than 0.22

Answer 1

The no of oppositely charged ions surrounding each ion is called its coordination number and for octahedral void i.e. the void which is surrounded by six spheres, it is 6 and knowing its radius ratio i.e. the ratio of the radius of the cation to the radius of anion, you can easily answer it.

Complete step by step answer:
In ionic solids, the anions are present in closed packing arrangement and the cations occupy the voids i.e. either tetrahedral (formed at the center of four spheres) or octahedral (i.e. formed at the center of six spheres). The number of oppositely charged ions which surrounds each ion is called its coordination number. Therefore, the relation between the size of the void and the sphere in the closed packed arrangement is expressed in terms of cation to that of anion.
So, radius ratio may be defined as the ratio of the cation to the radius of the anion. i.e.:
Radius ratio =$\dfrac {\text {Radius of the cation}} {\text {Radius of the anion}}$

    = $\dfrac{{{r}^ {+}}} {{{r}^ {-}}}$

The possible coordination numbers and the structural arrangements of anions around the cations for different values of $\dfrac{{{r}^ {+}}} {{{r}^ {-}}}$ areas:

Radius ratio$\dfrac{{{r}^ {+}}} {{{r}^ {-}}}$	Possible coordination number	Structural arrangement
0.155-0.225	3	Trigonal planar
0.225-0.414	4	Tetrahedral
0.414-0.732	6	Octahedral
0.732-1.0	8	cubic

So, it is clear from the table that the cations to occupy tetrahedral void , thew limiting lowest value of $\dfrac{{{r}^{+}}}{{{r}^{-}}}$is 0.225 and the range is 0.225-0.414 and that to occupy the octahedral void is 0.414 and the range is 0.414-0.732 and if the radius ratio is above 0.732, it occupies cubic void.
Hence, the option (b) is correct.

Note: The radius ratio of cation and the radius ratio of anion i.e. radius ratio $\dfrac{{{r}^{+}}}{{{r}^{-}}}$plays an important role in determining the structures of ionic solids and the coordination number of the ions.i.e. we know the radius ratio$\dfrac{{{r}^{+}}}{{{r}^{-}}}$, we can easily find its structure and hence its coordination number.