To determine the increasing order of ionic radii for the isoelectronic species Cl⁻, Ca²⁺, K⁺, S²⁻, we must first understand that all these ions have the same number of electrons (18 electrons) because they are isoelectronic with argon (Ar). However, their ionic radii differ due to variations in nuclear charge (number of protons).

Step 1: Identify the number of protons (atomic number) for each species:

• Cl⁻: Chlorine has atomic number 17, so number of protons = 17
• Ca²⁺: Calcium has atomic number 20, so number of protons = 20
• K⁺: Potassium has atomic number 19, so number of protons = 19
• S²⁻: Sulfur has atomic number 16, so number of protons = 16

Step 2: Understand the effect of nuclear charge: For isoelectronic species, the ion with a higher nuclear charge (more protons) will have a stronger attraction on the same number of electrons, pulling them closer to the nucleus, resulting in a smaller ionic radius. Conversely, an ion with lower nuclear charge has weaker attraction and thus a larger radius.

Step 3: Compare the nuclear charges:

Order of number of protons: S²⁻ (16) < Cl⁻ (17) < K⁺ (19) < Ca²⁺ (20)

Therefore, the order of ionic radii (which is opposite to the order of nuclear charge) is: Ca²⁺ < K⁺ < Cl⁻ < S²⁻

Final Answer: The correct increasing order of ionic radii is Ca²⁺ < K⁺ < Cl⁻ < S²⁻, which corresponds to the option: ${\mathrm{Ca}}^{2+},K^{+},{\mathrm{Cl}}^{-},S^{2-}$

Related Topics and Formulae

Isoelectronic Species: Ions or atoms that have the same number of electrons. Example: N³⁻, O²⁻, F⁻, Ne, Na⁺, Mg²⁺, Al³⁺ all have 10 electrons.

Ionic Radius Trend for Isoelectronic Ions: For a series of isoelectronic ions, the ionic radius decreases as the nuclear charge (atomic number) increases. Mathematically, the radius is inversely proportional to the effective nuclear charge.

Formula: Although there is no simple formula, the size can be compared using the concept: $r\propto \frac{1}{Z_{\text{eff}}}$, where Z_eff is the effective nuclear charge.