Question

In the fluorite structure, if the radius ratio is \[\sqrt {\frac{3}{2}} - 1\], how many ions does each cation touch?
A. 4 anions
B. 12 cations
C. 8 anions
D. 0 cations

Answer 1

Hint: The ratio of the radius to that of the anion is called the radius ratio.
The chemical formula of fluorite is \[Ca{F_2}\]. It forms a cubic close packing arrangement. Four calcium ions are in the unit cell and fluoride ions occupy the tetrahedral voids.

Complete Step by Step Solution:
For the stability of an ionic compound, each cation should be surrounded by the maximum number of anions, and anions in turn should be surrounded by cations. This is to ensure the maximum electrostatic force of attraction.

The number of oppositely ions surrounding each ion is called its coordination number. Since the ionic bond is non-directional, the arrangement of ions within the ionic crystal is determined by the sizes of the ions.

The ratio of the radius to that of the anion is called the radius ratio.
\[Radius \ ratio = \frac{{radius \ of \ cation\left( {{r_ + }} \right)}}{{radius \ of \ the \ anion\left( {{r_ - }} \right)}}\]

Most ionic compounds have a close-packed arrangement in which one type of ion typically larger ions, form the close-packed arrangement and the other type of ions i.e., smaller ones occupy the voids.

Fluorite has a ccp arrangement. In fluorite or CaF₂, there are four \[C{a^{2 + }}\]ions in the unit cell. Therefore, there are eight tetrahedral voids all of which are occupied by \[{F^ - }\] ions. Hence, there are 4\[Ca{F_2}\] units per unit cell.

Here we are given the radius ratio \[\sqrt {\frac{3}{2}} - 1\].
\[\sqrt {\frac{3}{2}} - 1 = \sqrt {1.5} - 1\]
\[ = 1.2247 - 1\]
\[ = 0.2247\]
This value is nearly similar to 0.225.
This value is in the range of 0.225-0.414.
So, the coordination number of anions is 4.
We see that in this case the cation in the void of the fcc structure is not deformed.
So, each anion is enclosed by 4 cations.
There are 8 anions.

Accordingly, the no.of cations enclosing the particular anion are 12 and 8 anions in the tetrahedral voids that touch the individual cation.
So, options B and C are correct.

Note: Higher the radius ratio, the larger the size of the cation and hence the greater the coordination number. Fluorite has a cubic close packing arrangement in which cations form fcc with each cation surrounded by eight anions and each anion surrounded by four cations.