Question

# The species Ar, ${{K}^{+}}$ and $C{{a}^{2+}}$ contain the same number of electrons. In which order do their radii increase?(a)- $C{{a}^{2+}}$ < ${{K}^{+}}$ < $Ar$ (b)- ${{K}^{+}}$ < $Ar$ < $C{{a}^{2+}}$ (c)- $Ar$ < ${{K}^{+}}$ < $C{{a}^{2+}}$ (d)- $C{{a}^{2+}}$ < $Ar$ < ${{K}^{+}}$

Hint: In isoelectronic species, as the nuclear charge increases, the force of attraction by the nucleus increases on the electrons also increases due to which the radii decrease.

Isoelectronic species or ions are the ions of different elements that have the same number of electrons but differ in the magnitude of the nuclear charge.
Besides ions, a neutral atom may also have the same number of electrons hence it is also an isoelectronic species.
Example, sulphide ion (${{S}^{2-}}$ ), chloride ion ($C{{l}^{-}}$), argon (Ar), and potassium ion (${{K}^{+}}$ ) are isoelectronic species.
Variation of size among isoelectronic species: As the nuclear charge increases, the force of attraction by the nucleus on the electrons also increases. As a result, the ionic radius decreases.
The nuclear charge is calculated by the number of electrons + charge on the ion.
In Ar,${{K}^{+}}$ and$C{{a}^{2+}}$ the number of electrons is 18.
The nuclear charge of Ar is 18 because there is no charge on the ion.
The nuclear charge of the ${{K}^{+}}$ion is 19 because potassium ion has a +1 charge.
The nuclear charge of the $C{{a}^{2+}}$ion is 20 because calcium ion has a +2 charge.
So, the calcium ion has the most charge, hence it will be the smallest and Argon has the least charge so it will be the largest.
The size range will be: $C{{a}^{2+}}$ < ${{K}^{+}}$ < $Ar$
So, the correct answer is “Option A”.

Note: With the help of calculation the nuclear charge we can find the radii range of a large number of ions. Example, nitrogen ion (${{N}^{3-}}$ ), oxygen ion (${{O}^{2-}}$ ), fluorine ion (${{F}^{-}}$ ), sodium ion ($N{{a}^{+}}$), magnesium ion ($M{{g}^{2+}}$ ) and aluminium ion ($A{{l}^{3+}}$ ). They have 10 electrons. They have nuclear charge +7, +8, +9, +11, +12 and +13. Hence, the radii will be:
$A{{l}^{3+}}$ < $M{{g}^{2+}}$ < $N{{a}^{+}}$ < ${{F}^{-}}$ < ${{O}^{2-}}$ < ${{N}^{3-}}$