To arrange the ions Cl^-, S^2-, Ca²⁺, and the neutral atom Ar in order of increasing size, we need to consider several factors including nuclear charge, number of electrons, and electron-electron interactions in the electron shells.

1. Atomic and Ionic Sizes: In general, neutral atoms and ions with more electrons and similar nuclear charge have larger sizes. This is due to increased electron-electron repulsion within the outer shells.

2. Isoelectronic Species: Cl^-, Ca²⁺, and Ar are isoelectronic, meaning they each have the same number of electrons (18 electrons), but different nuclear charges. A higher nuclear charge (more protons in the nucleus) pulls the electron cloud closer, decreasing the size of the ion or atom. In this case, Ar (18 protons) has a higher nuclear charge than Cl^- (17 protons) and Ca²⁺ (20 protons) has the highest.

3. Comparing a Non-Isoelectronic Species: S^2- has 18 electrons, the same as Cl^-, Ca²⁺, and Ar, but sulfur typically has fewer protons (16 protons) than Argon. However, the addition of two electrons to form S^2- significantly increases its radius compared to the neutral sulfur atom due to increased electron-electron repulsion outweighing the nuclear charge.

Ordered by Size: Smaller ions/atoms have higher nuclear charge relative to the number of electrons. Here’s the order:

• Ca²⁺ (smallest): higher nuclear charge tightly holds the electrons.


• Ar: similar electron count but slightly less nuclear charge than Ca²⁺.


• Cl^-: one fewer proton than Ar, leading to less pull on the same number of electrons, resulting in a slightly larger size.


• S^2- (largest): despite having an equal number of electrons as the others, the significantly lower nuclear charge (16 protons) allows more spread out electron cloud.

In conclusion, the order of increasing size will be: $$ Ca^{2+} < Ar < Cl^{-} < S^{2-} $$. S^2- is the largest due to its lower nuclear charge compared to its electron count, resulting in greater electron cloud expansion.