Final answer:
The elements Mg, Ba, F, and S are arranged in order of increasing atomic radius based on their positions in the periodic table as F < S < Mg < Ba. Moving down a group increases atomic radius, and moving from left to right across a period decreases it.
Step-by-step explanation:
To arrange the elements Mg, Ba, F, and S in order of increasing atomic radius, we must look at their positions on the periodic table. As we move down a group, the atomic radius increases because additional electron shells are added, causing the electron cloud to extend further from the nucleus. Conversely, as we move from left to right across a period, the atomic radius decreases because the number of protons increases, which pulls the electron cloud closer due to the increased positive charge of the nucleus.
Thus, the order of increasing atomic radius for the given elements is as follows: F < S < Mg < Ba. Fluorine (F) is to the right of sulfur (S) in the same period and thus has a smaller radius. Magnesium (Mg) is to the left of both F and S in the same period, and as we move down to barium (Ba), which is in the same group as Mg but in a lower period, the radius increases significantly.