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 V(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?

Short answer

The d-orbital energy-level diagram for V(III) octahedral complexes shows 2 electrons in the lower energy \(t_{2g}\) orbitals and empty \(e_g\) orbitals. The colors of these complexes arise from the electron transitions between \(t_{2g}\) and \(e_g\) orbitals. The \(\mathrm{[VF_6]^3-}\) complex absorbs higher energy light, as F⁻ is a stronger ligand, causing a greater energy splitting between the d-orbitals than in the \(\mathrm{[V(H_2O)_6]^3+}\) complex.

Step-by-step solution

a. Drawing d-orbital energy-level diagram for V(III) octahedral complexes.

To draw the d-orbital energy-level diagram, we need to consider the position of Vanadium (V) in the periodic table. The electronic configuration of Vanadium is \([Ar]4s^23d^3\), which means V(III) has an electronic configuration of \([Ar]3d^2\). In an octahedral complex, the d-orbitals are split into two groups: \(t_{2g}\) (lower in energy) and \(e_g\) (higher in energy). So, for the V(III) ion, we will fill \(t_{2g}\) orbitals with 2 electrons in accordance with Hund's rule, leaving \(e_g\) orbitals empty. Now, we can draw the d-orbital energy-level diagram like this: e_g ___|___ | | | | ---------|--------- t_2g ___|___ | | | ↑ ↑

b. Source of the colors of the complexes.

The colors of the complexes are due to the transition of electrons between d-orbitals. In the case of these octahedral complexes, the transition mainly occurs between the \(t_{2g}\) and \(e_g\) orbitals. When an electron is excited from a lower energy level (\(t_{2g}\)) to a higher energy level (\(e_g\)), they absorb a certain amount of energy, which corresponds to a specific wavelength, or color, of light. The complementary color of the absorbed light will be the observed color of the complex.

c. Identifying the complex that absorbs higher energy light and explaining the reason.

To determine which of the \(\mathrm{[V(H_2O)_6]^3+}\) and \(\mathrm{[VF_6]^3-}\) complexes absorb higher energy light, we need to look at the ligand field strengths. In general, the wavelength of absorbing light is inversely related to the energy difference between \(t_{2g}\) and \(e_g\) orbitals. In our case, F¯ is a stronger ligand than H₂O, meaning it creates a larger splitting of the d-orbitals, placing \(t_{2g}\) orbitals at lower energy and \(e_g\) orbitals at higher energy. Thus, in the \(\mathrm{[VF_6]^3-}\) complex, the energy difference between these orbitals is greater compared to the \(\mathrm{[V(H_2O)_6]^3+}\) complex. Consequently, the \(\mathrm{[VF_6]^3-}\) complex would absorb light of higher energy because of the larger energy gap between its d-orbitals.

Key concepts

D-Orbital Energy-Level Diagrams

D-orbital energy-level diagrams are visual representations used to show the distribution of electrons among the five d orbitals in transition metal ions. In an octahedral complex, the presence of six ligands symmetrically arranged around the central metal ion creates an electrostatic field that influences the energies of the d orbitals. The d orbitals split into two distinct energy levels: the lower-energy \(t_{2g}\) set and the higher-energy \(e_g\) set.

The \(t_{2g}\) orbitals consist of the dxy, dxz, and dyz orbitals, while the \(e_g\) orbitals are made up of the dx2-y2 and dz2 orbitals. Hund's rules govern how electrons occupy these orbitals: electrons fill the lower-energy \(t_{2g}\) orbitals first and are distributed to minimize electron-electron repulsions, leading to the maximum number of unpaired electrons. For example, Vanadium (III), which has three electrons in its d orbitals, would fill the \(t_{2g}\) orbitals with two electrons, one in each of two of the orbitals, in accordance with Hund's rule.

The diagram thus serves a dual purpose: it simplifies our understanding of the electron configurations in transition metal ions and assists in predicting magnetic properties and possible electronic transitions that give rise to colors in coordination compounds.

Ligand Field Theory

Ligand Field Theory (LFT) is an extension of crystal field theory that considers the covalent, as well as the ionic aspects of coordination. According to LFT, ligands are treated as point charges in the case of weak-field ligands, or as overlapping orbitals in the case of strong-field ligands, influencing their interaction with the metal d orbitals. The strength of the ligand, therefore, has a marked effect on the extent of the splitting of the d orbitals.

When we consider an octahedral complex, the ligands approach along the axes, directly interacting with the \(e_g\) orbitals, which are oriented along the x, y, and z axes. The \(t_{2g}\) orbitals, having a lobed structure positioned between these axes, are less affected. Strong-field ligands, such as F⁻ in the \(\mathrm{[VF_6]^{3-}}\) complex, exert a greater splitting effect than weak-field ligands like H2O in the \(\mathrm{[V(H_2O)_6]^{3+}}\) complex. This difference in splitting can dictate the color of the complex, its magnetic properties, and the absorbance of light at certain energies, making it a crucial concept in inorganic chemistry.

Electronic Transitions in Coordination Compounds

The vibrant colors exhibited by coordination compounds, such as \(\mathrm{[V(H_2O)_6]^{3+}}\) and \(\mathrm{[VF_6]^{3-}}\), are a result of electronic transitions within the d orbitals induced by light energy. When a complex absorbs light, it gains energy that allows electrons to transition from a lower-energy state (typically the \(t_{2g}\) orbitals) to a higher-energy state (the \(e_g\) orbitals). This jump in energy levels falls within the visible spectrum and corresponds to a specific color. The energy gap between \(t_{2g}\) and \(e_g\) dictates the wavelength of light absorbed, which follows the equation \(\Delta E = hu\), where \(\Delta E\) is the difference in energy between the two orbitals, \(h\) is Planck's constant, and \(u\) is the frequency of light.

As ligands like F⁻ produce a larger gap between the \(t_{2g}\) and \(e_g\) orbitals compared to H2O, the \(\mathrm{[VF_6]^{3-}}\) complex absorbs light of higher energy (shorter wavelength) and thus displays a color corresponding to the complementary color of the absorbed light. Understanding electronic transitions is crucial for explaining the color and spectral properties of coordination compounds and is fundamental to fields such as spectroscopy and photochemistry.