Question

The complexes \(\left[\mathrm{V}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\) and \(\left[\mathrm{VF}_{6}\right]^{3-}\) are both known. (a) Draw the \(d\) -orbital energy-level diagram for \(\mathrm{V}(\mathrm{III})\) octahedral complexes. (b) What gives rise to the colors of these complexes? (c) Which of the two complexes would you expect to absorb light of higher energy? Explain.

Short answer

(a) The d-orbital energy-level diagram for V(III) octahedral complexes is as follows: ``` e_g: - - t_{2g}: ↑↓ ↑↓ ↑ ``` (b) The colors of these complexes arise due to electronic transitions between the d-orbital energy levels, specifically from the \(t_{2g}\) orbitals to the \(e_g\) orbitals. (c) The \([\text{VF}_6]^{3-}\) complex will absorb light of higher energy compared to \([\text{V}(\text{H}_2\text{O})_6]^{3+}\) due to the stronger ligand field created by F^-, which results in a larger energy difference between the \(t_{2g}\) and \(e_g\) orbitals.

Step-by-step solution

(a) Draw the d-orbital energy-level diagram for V(III) octahedral complexes)

For V(III) in an octahedral complex, the electron configuration is 3d^2, i.e., there are two electrons in the d-orbitals. In an octahedral crystal field, the d-orbitals split into two energy levels, the lower energy \(e_g\) and the higher energy \(t_{2g}\), due to interactions between the metal ion and ligands. There are two types of d-orbitals: the \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\) orbitals (in \(t_{2g}\)) and the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals (in \(e_g\)). In this case, the two electrons fill the \(t_{2g}\) orbitals according to Hund's rule and the filled d-orbital diagram looks like: ``` e_g: - - t_{2g}: ↑↓ ↑↓ ↑ ```

(b) Explain what gives rise to the colors of these complexes)

The colors of the complexes arise due to electronic transitions between the d-orbital energy levels. When the complex absorbs energy in the form of light, the electrons can be excited from the \(t_{2g}\) orbitals to the \(e_g\) orbitals. The absorbed light corresponds to a specific energy difference between these two levels, which is manifested in terms of color. Since the colors of absorbed light and the light transmitted/reflected are complementary, we perceive the color of the complexes accordingly.

(c) Determine which of the two complexes would absorb light of higher energy and explain the reasoning)

To determine which complex absorbs higher energy light, we can analyze the ligands involved in the complexes using the spectrochemical series. In the spectrochemical series, ligands are ordered according to their ability to split the d-orbitals of the central metal ion. Larger splitting corresponds to a stronger ligand field. The order is: \(I^{-} < Br^{-} < Cl^{-} < F^{-} < OH^{-} < H_{2}O\) In this case, we have two complexes: 1. \([\text{V}(\text{H}_2\text{O})_6]^{3+}\) - where water (\(H_2O\)) is the ligand. 2. \([\text{VF}_6]^{3-}\) - where fluoride (\(F^{-}\)) is the ligand. Since F^- is higher in the spectrochemical series than H2O, it represents a stronger ligand field and will cause larger splitting of the V(III) d-orbitals energy levels. Consequently, a larger energy difference between the \(t_{2g}\) and \(e_g\) orbitals will be required for electronic transitions in \([\text{VF}_6]^{3-}\). Thus, we can expect \([\text{VF}_6]^{3-}\) to absorb light of higher energy compared to \([\text{V}(\text{H}_2\text{O})_6]^{3+}\).

Key concepts

Octahedral Complexes

Octahedral complexes are formed when a central metal ion is surrounded by six ligands arranged in a geometric octahedron shape. This arrangement is common for transition metal complexes and is crucial for understanding their properties and behaviors. The configuration leads to a specific pattern of electron interactions between the metal and ligands, resulting in a unique arrangement and splitting of the d-orbitals. When the metal is in an octahedral field, these interactions cause the d-orbitals to split into two distinct sets differing in energy: the lower energy set known as the \(t_{2g}\) orbitals and the higher energy \(e_g\) orbitals. This energy level difference is what gives rise to several interesting properties, such as color and magnetic behaviors, in the complexes.

Spectrochemical Series

The spectrochemical series is a sequence that ranks ligands based on their ability to split d-orbital energies in metal complexes. The stronger the field generated by the ligand, the greater the splitting it causes in the d-orbitals of the metal ion. This sequence is crucial for predicting the relative strength of different ligands and therefore the color and reactivity of the complex. Ligands such as iodide \(I^-\), chloride \(Cl^-\), and water \(H_2O\) are weak field ligands and cause minimal splitting. On the other hand, ligands such as cyanide \(CN^-\) and carbon monoxide \(CO\) are strong field, causing significant splitting. Understanding this series helps in predicting which complex will absorb more energy and hence appear as different colors to the viewer.

Electronic Transitions

Electronic transitions refer to the movement of electrons between different energy levels or orbitals within a molecule or complex. In the context of octahedral complexes, these transitions are between the \(t_{2g}\) and \(e_g\) orbitals. When light is absorbed, an electron can be excited from a lower energy \(t_{2g}\) orbital to a higher energy \(e_g\) orbital. The energy required for such transitions corresponds to the color of light absorbed by the complex. The unabsorbed light is what is transmitted or reflected, giving the complex its apparent color. These transitions are not just critical for color properties but also affect the chemical stability and reactivity of complexes.

Ligand Field Theory

Ligand field theory (LFT) builds upon crystal field theory, incorporating elements of molecular orbital theory to explain the splitting of d-orbitals in transition metal complexes. LFT explains how the different ligand fields, created by the ligands surrounding the central metal ion, affect the distribution of electrons among the d-orbitals. This theory offers a more nuanced view than its predecessor, allowing for a better understanding of covalent interactions between the ligands and the metal ion. It accounts not only for d-orbital splitting but also for complex stability, magnetic properties, and the influence of various ligand types on the energy spectrum of the orbitals. Ligand field theory thus provides the framework for predicting and explaining the spectroscopic and magnetic behavior of coordination compounds.