Question

The two ions ${A^ + }$ and ${B^ - }$ have radii 88 pm and 200 pm respectively. In the closed packed crystal of compound $AB$ predict the coordination number of ${A^ + }$

Answer 1

For Solid crystals, the arrangement is fixed and specific. Coordination number is the number of atoms or ions nearest immediately surrounding to the central atom in the ionic solid or crystal.
In case if we know the size of ions in an ionic crystal, then we can take the ratio of cation to anion, this value will give us some value and then we can compare that value from the table that we know corresponds to coordination number and ratio of ions.

Complete step by step answer:
We know that the given ions are ${A^ + }$ and ${B^ - }$ which are monovalent ions, so their arrangement in 3 dimensional space is given as a closed packed structure.
Unit cell is the smallest repeating unit in the crystal lattice. ${A^ + }$ and ${B^ - }$ forms together the unit cell.
Now, let us calculate the ratio of cation to anion.
We know that ${A^ + }$ is cation and radius is 88 pm.
${r_{{A^ + }}} = 88pm$ We know that ${B^ - }$ is anion and radius is 200 pm.
${r_{{B^ - }}} = 200pm$
We know the formula for radius ratio.
${\text{Radius Ratio = }}\dfrac{{{\text{Cation radius}}}}{{{\text{Anion radius}}}}$
Now, rewrite equation using ${A^ + }$ and ${B^ - }$
${\text{Radius Ratio = }}\dfrac{{{r_{{A^ + }}}}}{{{r_{{B^ - }}}}}$
Now, substitute the radius values
${\text{Radius Ratio = }}\dfrac{{88}}{{200}}$
$\therefore {\text{Radius Ratio = 0}}{\text{.44}}$
Now compare this value with table given below:

Serial Number	Radius Ratio	Coordination Number
1	0-0.155	2
2	0.225-0.155	3
3	0.414-0.225	4
4	0.732-0.414	6
5	1.0-0.732	8
6	1.00	12

Now, when we compare the ratio, our value that we got is 0.44, which lies at Serial Number 4 in value range 0.732-0.414, which corresponds to coordination number of 6.
Thus the structure with ${A^ + }$ and ${B^ - }$ has coordination number of 6 for ${A^ + }$.

Note:
The structure which has coordination number of 6 has octahedral arrangement.
There are 3 types of arrangement in 3-dimensional structure. The 3 types of unit cell:
Simple cubic structure – atoms are arranged at corners of the cube.
Body-centred cubic structure - atoms are arranged at corners of the cube along with one atom at centre of the cube.
Face-centred cubic structure - atoms are arranged at corners of the cube, and at centre of 6 faces.