The species Ar, K⁺ and Ca²⁺ contain the same number of electrons. In which order do their radii increase?

All three species—argon (Ar), potassium ion (K⁺), and calcium ion (Ca²⁺)—contain the same number of electrons (18). This makes them an isoelectronic series.

However, their nuclear charges differ because they have different atomic numbers:

• Ar has atomic number Z = 18.
• K has atomic number Z = 19, and K⁺ has lost one electron, so its effective nuclear charge is still Z = 19.
• Ca has atomic number Z = 20, and Ca²⁺ has lost two electrons, so its effective nuclear charge is Z = 20.

The shielding from the inner‑electron cloud is essentially the same for all three, because the electron configuration is identical (1s² 2s² 2p⁶ 3s² 3p⁶). Therefore the only factor that changes the size of the electron cloud is the increase in the attractive force from the nucleus.

The ionic radius r is inversely related to the effective nuclear charge Z_eff:

$r\; \propto \; \backslash frac\{1\}\{Z\_\{\backslash text\{eff\}\}\}$

Since Z_eff increases from Ar (18) to K⁺ (19) to Ca²⁺ (20), the radius decreases in the same order.

Consequently, the radii increase as follows:

Ca²⁺ < K⁺ < Ar