The absorption maximum for the complex ion \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) occurs at \(470 \mathrm{nm}\). (a) Predict the color of the complex and (b) calculate the crystal field splitting in kilojoules per mole.

In the study of optics, complementary colors are pairs of colors that, when combined, cancel each other out by producing a grayscale color like white or black. When light with a specific color is absorbed by a substance, the color that is perceived by our eyes is the complementary color, since the absorbing material reflects or transmits the other colors.

For instance, the complex ion with an absorption maximum at 470 nm behaves like a filter that removes the violet light from the spectrum because it absorbs that particular wavelength. According to the color wheel, the complementary color of violet is yellowish-orange. Thus, the substance will appear orange to us because it reflects the orange spectrum while absorbing the violet end.

Planck's equation is pivotal in quantum mechanics, and it helps us understand the relationship between the energy of a photon and its electromagnetic wavelength. This equation is given by: \[E = \frac{h \cdot c}{\lambda}\] where \(E\) is the energy of the photon, \(h\) is Planck’s constant (\(6.626 \times 10^{-34}\ Js\)), \(c\) is the speed of light in a vacuum (\(3.0 \times 10^8\ m/s\)), and \(\lambda\) is the wavelength of the light.

Knowing the wavelength of the absorbed light allows us to calculate the exact energy of the photons the ion absorbs. This energy corresponds to the difference in energy levels that the electrons in the complex ion occupy before and after the absorption—also known as crystal field splitting.

Absorption maximum refers to the wavelength at which a substance absorbs the maximum amount of light. This peak absorption is indicative of the specific electronic transitions occurring within the substance. Each substance has its unique absorption spectrum based on its molecular or atomic structure, and complex ions have spectra influenced by their crystal field interactions.

When a complex ion like \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) absorbs light at its absorption maximum of 470 nm, it suggests that the energy of the photons at this wavelength is just right to lift the electrons to a higher energy state. This energy lift or jump is directly related to the crystal field splitting energy for the compound.

Complex ions consist of a central metal ion surrounded by a group of molecules or ions called ligands. The ligands are bound to the metal ion through coordinate covalent bonds. The arrangement of these ligands around the central ion, and the nature of the ligands themselves, significantly influence the complex's absorption properties.

The configuration of these components dictates the crystal field splitting, which refers to the difference in energy between sets of d orbitals in a metal ion in a crystal field provided by surrounding ligands. Such crystal field splitting can lead to specific absorption bands within the visible spectrum. For transition metal complexes, the splitting plays a crucial role in their distinctive coloration, as manifested in the color we perceive for \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\).