Compare both the numbers of protons and electrons present in each to rank the following ions in order of increasing radius: \(\mathrm{As}^{3-}, \mathrm{Br}^{-}, \mathrm{K}^{+}, \mathrm{Mg}^{2+}\)

When studying elements and ions, one may encounter groups known as isoelectronic series. These are sets of atoms or ions that possess the same number of electrons, thus the same electronic structure. The notion of an isoelectronic series is vital to understanding properties such as ionic radius as it provides a way to compare sizes of such species when the number of electrons remains constant.

For example, in the exercise, ions like \( \mathrm{As}^{3-} \) and \( \mathrm{K}^{+} \) are part of an isoelectronic series since both have gained or lost electrons to end up with the same total number of them. This equalizing of electron quantity facilitates the direct comparison of ionic sizes based on nuclear charge, rather than the number of electrons. Remember, as they are in the same series, the ion with the higher nuclear charge will have a smaller radius due to greater electron attraction.

The term nuclear charge refers to the total charge from the protons in the nucleus of an atom. It has a significant impact on the atom's radius. As the number of protons in an atom's nucleus increases, so does the positive charge, resulting in a stronger pull on the electrons orbiting the nucleus. This force of attraction draws the electrons closer, which generally results in a smaller atomic or ionic radius.

In addressing the problem given, it becomes clear why \( \mathrm{Mg}^{2+} \) would have the smallest radius. Despite having the same number of electrons as the others in the series, magnesium's 12 protons exert a less forceful pull than, say, arsenic's 33 protons, leading to a less compact atomic structure. A helpful way to visualize nuclear charge and its effects could be to imagine a game of tug of war—more protons mean a stronger team, pulling the electrons nearer to the nucleus.

The periodic table is a systematic arrangement of elements, providing valuable insights into their properties and relations. It allows prediction of atom behavior based on position. When dealing with ionic radius, the periodic table plays a crucial role, particularly as it houses elemental atomic numbers which indicate the number of protons in an element’s nucleus.

As noticed in the given exercise, locating each element on the periodic table instantly informs us about their atomic number. These atomic numbers (like 33 for arsenic or 19 for potassium) help determine the nuclear charge: a cornerstone for comparing ionic radii in our isoelectronic series. By understanding how to read the periodic table and utilizing it as a reference, one can discern the relative ionic size of different elements and ions systematically.