Which of these ions cannot form both high- and low-spin octahedral complexes: (a) \(\mathrm{Mn}^{3+} ;\) (b) \(\mathrm{Nb}^{3+} ;\) (c) \(\mathrm{Ru}^{3+}\) (d) \(\mathrm{Ni}^{2+} ?\)

When metal ions form complexes, especially octahedral ones, their d-orbitals split into different energy levels. This happens because the ligands, which are the molecules or ions surrounding the central metal ion, interact differently with the metal ion's d-orbitals. In an octahedral field, the d-orbitals split into two groups with different energies. Three of the orbitals (called \( t_{2g} \) orbitals) have lower energy, and the other two (called \( e_g \) orbitals) have higher energy. This splitting is crucial for understanding how metal complexes absorb light and how their electrons are distributed. The difference between the energy levels of the \( t_{2g} \) and \( e_g \) orbitals is called the crystal field splitting energy, denoted as \( \Delta_0 \). When studying high-spin and low-spin complexes, recognizing this splitting helps explain why metals may prefer different electron configurations depending on the strength of the ligand field.

Ligand field theory is an extension of crystal field theory that explains the behavior of complexes involving transition metals and their ligands. It combines ideas from crystal field theory and molecular orbital theory. This theory looks at how ligands affect the energy levels of the metal ion's d-orbitals. Depending on the field strength of the ligands, they can either increase or decrease the energy gap between the split d-orbitals. Ligands that cause a large splitting are called strong-field ligands, and they often lead to low-spin complexes because the crystal field splitting energy \( \Delta_0 \) is large enough to favor pairing of electrons in lower energy orbitals. On the other hand, weak-field ligands cause smaller energy splitting and tend to form high-spin complexes, where electrons occupy higher energy orbitals to maximize unpaired spins. This theory helps in predicting and explaining the magnetic properties, colors, and stability of metal complexes.

Electron configuration is a way of describing the distribution of electrons in an atom or ion. It follows a specific order based on the energy levels of orbitals. For transition metals, the d-orbital configuration is key to understanding their chemical behavior, especially in complex formation. Each metal ion has a unique electron configuration depending on its oxidation state. For example, \( \mathrm{Mn}^{3+} \) has the electron configuration \( [Ar] 3d^4 \), meaning it has four electrons in its 3d orbitals after losing three electrons from its neutral state. When forming complexes, the arrangement of these d-electrons can vary based on the ligand field strength, leading to either high-spin or low-spin configurations. High-spin complexes occur when electrons occupy higher energy d-orbitals to minimize electron pairing, often seen with weak-field ligands. Low-spin complexes, favored by strong-field ligands, result when electrons pair up in lower energy d-orbitals to minimize the total energy. Understanding electron configuration allows us to predict the geometry, reactivity, and other properties of metal complexes.