Compounds containing the \(\mathrm{Sc}^{3+}\) ion are colorless, whereas those containing the \(\mathrm{Ti}^{3+}\) ion are colored. Explain.

Understanding the electronic configuration of transition metal ions is crucial to explain their physical and chemical properties. In the context of understanding color, the electronic configuration tells us how electrons are distributed in an atom's orbitals, especially the d-orbitals. For Scandium (Sc) and Titanium (Ti), this configuration is altered when they form ions.

Scandium, when it loses three electrons to form Sc³⁺, ends up with an electron configuration that mirrors that of Argon, with no electrons in the d-orbital ([Ar]). In contrast, Titanium loses three electrons to become Ti³⁺, resulting in one electron remaining in the d-orbitals ([Ar] 3d^{1}). This difference in electronic configuration is pivotal because the presence or absence of electrons in these d-orbitals allows us to predict whether a compound can exhibit color through the process of d-d transitions.

Transition metal compounds showcase an array of colors due to an interesting phenomenon known as d-d transitions. When light strikes these compounds, some wavelengths are absorbed and others are transmitted or reflected. The reflected light is what we actually perceive as color. The specific wavelengths absorbed depend on the differences in energy between the d-orbitals, which in turn are influenced by the electronic configuration of the metal ion.

For example, the Sc³⁺ ion does not have any electrons in its d-orbitals after ionization, meaning it can't absorb visible light through d-d transitions, resulting in a colorless appearance. Conversely, Ti³⁺ compounds are colored because the singular unpaired electron in the d-orbital of Ti³⁺ can shift between different energy levels within the visible spectrum, absorbing some light and imparting a color to the compound.

The visible spectrum is a portion of the electromagnetic spectrum that the human eye can detect, ranging approximately from 400 to 700 nanometers (nm) in wavelength. The absorption of specific wavelengths within this range is what leads to the colored appearance of many compounds.

For instance, if a Ti³⁺ compound absorbs light near the blue end of the spectrum, the light transmitted or reflected might appear to be red or orange, complementary colors to blue. This absorption corresponds directly to the energy required for the d-d transition—an electron moving from a lower energy d-orbital to a higher one. Since Sc³⁺ lacks d-electrons altogether, there is no possibility of absorbing visible light through this mechanism, thus no color is exhibited.

Unpaired d-electrons play a vital role in determining whether a transition metal ion will exhibit color. These electrons are not paired with another electron in the same orbital, and they possess enough energy to undergo transitions to different energy levels within the d-orbitals. It is this movement, or jump, from one d-orbital to another that can result in the absorption of light within the visible spectrum.

The Ti³⁺ ion, with one unpaired d-electron, will show color due to such transitions, as the unpaired electron has the ability to absorb visible light and jump to a higher energy state. In contrast, the Sc³⁺ ion, which loses all of its d-electrons upon forming the ion, does not have any unpaired electrons available for d-d transitions, rendering the Sc³⁺ compounds colorless.