Final answer:
The ions O₂⁻, N₃⁻, F⁻, Na⁺, and Mg²⁺ are arranged in increasing order of ionic radius based on their charges and position in the periodic table: Mg²⁺ < Na⁺ < F⁻ < N₃⁻ < O₂⁻.
Step-by-step explanation:
Order of Increasing Ionic Radius
To arrange the ions O₂⁻, N₃⁻, F⁻, Na⁺, and Mg²⁺ in order of increasing ionic radius, one must consider their positions in the periodic table and their charges. As a rule of thumb, ions with the same charge will increase in size as you move down a group in the periodic table because they have more electron shells. Also, within the same period, ionic radius decreases with increasing positive charge due to greater nuclear charge causing a stronger pull on the electrons.
Here is the correct order based on these guidelines: Mg²⁺ < Na⁺ < F⁻ < N₃⁻ < O₂⁻. This order is because Mg²⁺ and Na⁺ are cations with Mg²⁺ being smaller due to its greater charge and being in the same period as Na⁺. F⁻, N₃⁻, and O₂⁻ are anions where radius increases as the nuclear charge decreases without a corresponding increase in positive charge, with O₂⁻ being the largest due to having more electrons than N₃⁻ and F⁻, despite all having negative charges.