Which ion of each of the following pairs has the larger radius: (a) \(\mathrm{Ca}^{2+}, \mathrm{Ba}^{2+} ;\) (b) \(\mathrm{As}^{3-}, \mathrm{Se}^{2-} ;\) (c) \(\mathrm{Sn}^{2+}, \mathrm{Sn}^{4+}\) ?

Understanding the structure of the periodic table is essential in predicting the properties of elements, including ionic radii. As we move down a group in the periodic table, the ionic radius generally increases. This is because with each additional period, a new shell of electrons is added, increasing the distance between the nucleus and the outermost electrons. This effectively makes the ion larger.

On the other hand, moving from left to right across a period, the ionic radius tends to decrease. Even though electrons are being added to the same principal energy level, the number of protons also increases. This stronger nuclear charge pulls the electrons closer, reducing the size of the ion. Hence, for ions of the same charge, we expect the ones from elements further down a group to have larger radii due to the increase in electron shells.

When comparing the sizes of ions, we have to consider several factors, such as the charge of the ion and the number of electrons. In general, cations (positively charged ions) are smaller than their neutral atoms because the loss of electrons reduces electron-electron repulsion and allows the remaining electrons to be pulled closer to the nucleus by the unchanged nuclear charge.

For anions (negatively charged ions), the addition of electrons increases electron-electron repulsion, which can cause the ion to expand and thus be larger than the neutral atom. However, if we're comparing cations to anions, anions are typically larger due to the overall increase in electron repulsion outpacing the effect of the relatively smaller increase in nuclear charge.

Isoelectronic ions have the same number of electrons but different nuclear charges. In this scenario, we see an inverse relationship between nuclear charge and ionic size. Ions with a higher nuclear charge (more protons) will pull the same electron cloud closer to the nucleus, resulting in a smaller radius. For example, among isoelectronic ions, those with more positive charges (higher oxidation states) will be smaller compared to those with fewer positive charges.

Thus, the concept of isoelectric series helps us understand how even though ions may share the same electronic configuration, the net charge and the atomic number play a key role in determining their size. As a learning tip, it's beneficial to remember the correlation between higher positive nuclear charge and smaller ionic size within an isoelectronic series.