The ionic radii of the ions \(S^{2-}, C l^{-},\) and \(K^{+}\) are 184,181,138 pm respectively. Explain why these ions have different sizes even though they contain the same number of electrons.

Ionic radii refer to the measured size of an atom's ion in a crystal lattice. It reflects the distance from the nucleus to the outer boundary of the electron cloud surrounding the ion. But why do ions that have the same number of electrons, known as isoelectronic species, exhibit different ionic radii? The simple answer lies in the effective nuclear charge and the number of electron shells.

For isoelectronic ions like \(S^{2-}, Cl^{-},\) and \(K^{+}\), each has 18 electrons. However, due to differences in nuclear charge, the electron clouds are pulled toward the nucleus to varying degrees. Sulfur, with fewer protons (a lower nuclear charge), cannot pull the electron cloud as close as potassium can, leading to a larger ionic radius. Conversely, with its higher nuclear charge, potassium draws the electrons in tighter, resulting in a smaller ionic radius.

When explaining ionic radii, it's crucial to consider not only the repulsion among the electrons but also the attraction of these electrons to the nucleus, which is modulated by the number of protons present. This balance ultimately determines the observed size of an ion.

The nuclear charge is fundamentally the total charge of the nucleus, which is determined by the number of protons it contains. This charge influences how tightly electrons are held within an atom or ion. A greater nuclear charge typically results in a stronger attraction of electrons toward the nucleus, causing a decrease in the size of the electron cloud, hence reducing the ionic radius.

As each of the isoelectronic ions \(S^{2-}, Cl^{-},\) and \(K^{+}\) has an identical electron configuration, the difference in their ionic sizes is a direct consequence of their varying nuclear charges. Potassium, being the element with the highest proton count among the three, exerts the strongest pull on its electrons, therefore, it has the smallest ionic radius.

To truly comprehend the impact of nuclear charge on ionic radii, it is essential to appreciate how it competes with electron-electron repulsion. This competition between forces is pivotal in determining the spatial distribution of electrons, which ultimately dictates the size of the ion.

Electron configuration describes the distribution of electrons in an atom or ion across different energy levels and subshells. It provides a framework for understanding the chemical and physical properties of an element. In the context of isoelectronic ions, despite a similar electron configuration, variances in the ionic radii are observed due to the differences in nuclear charge as explained earlier.

The electron configurations of isoelectronic ions such as \(S^{2-}, Cl^{-},\) and \(K^{+}\) are identical because they all have the same number of electrons, filling similar orbitals. However, their behavior is influenced by effective nuclear charge, which alters their atomic or ionic sizes. This concept is crucial for explaining not only the properties of isolated ions but also their interactions when forming compounds.

It is important to realize that while electron configuration determines the potential energy and stability of an electron cloud, it is the combination of this arrangement with the actual effective nuclear charge that ultimately shapes the physical size of the ion.