Problem 80 - Chemistry The Central Science

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?

Step-by-Step Solution

Answer

The d-orbital energy-level diagram for \(\mathrm{V}(\mathrm{III})\) octahedral complexes has 2 electrons in the lower-energy \(t_{2g}\) orbitals and 0 electrons in the higher-energy \(e_g\) orbitals. The colors of these complexes arise from electronic transitions between the \(t_{2g}\) and \(e_g\) orbitals when they absorb light. Among the two complexes, \(\left[\mathrm{VF}_{6}\right]^{3-}\) has a higher crystal field splitting energy due to the stronger field ligand F-, and hence will absorb light of higher energy.

Step 1: (a) Draw the d-orbital energy-level diagram for \(\mathrm{V}(\mathrm{III})\) octahedral complexes.

First, we need to find the electronic configuration of \(\mathrm{V}(\mathrm{III})\). The atomic number of Vanadium(V) is 23, which has an electronic configuration of \([Ar] 3d^3 4s^2\). Thus, \(\mathrm{V}(\mathrm{III})\) will have an electronic configuration of \([Ar] 3d^2\). In an octahedral complex, the five d-orbitals split into two sets of energy levels: the lower-energy \(t_{2g}\) set consists of \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals, while the higher-energy \(e_g\) set consists of \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals. The energy gap between the two sets is called the crystal field splitting energy, denoted as \(\Delta_o\). In the case of \(\mathrm{V}(\mathrm{III})\), it has two d-electrons, and they will fill the lower-energy \(t_{2g}\) orbitals first. The d-orbital energy-level diagram for \(\mathrm{V}(\mathrm{III})\) octahedral complexes will have the following configuration: - \(t_{2g}\): 2 electrons - \(e_g\): 0 electrons

Step 2: (b) What gives rise to the colors of these complexes?

The colors of these complexes arise from the absorption of light, causing electronic transitions between the \(t_{2g}\) and \(e_g\) sets of d-orbitals. When the incident light excites an electron in the lower-energy \(t_{2g}\) orbitals to the higher-energy \(e_g\) orbitals, the absorbed light's energy corresponds to the crystal field splitting energy, \(\Delta_o\). The light not absorbed by the complex will be transmitted or reflected, and that is the color we observe.

Step 3: (c) Which of the two complexes would you expect to absorb light of higher energy?

The energy of the absorbed light depends on the value of the crystal field splitting energy, \(\Delta_o\). Ligands that cause a larger splitting energy are considered strong-field ligands, while those causing smaller splitting energy are considered weak-field ligands. In the spectrochemical series, we know that F- is a stronger field ligand compared to H2O. Hence, the crystal field splitting energy, \(\Delta_o\), in the complex \(\left[\mathrm{VF}_{6}\right]^{3-}\) will be greater than that in the complex \(\left[\mathrm{V}\left(\mathrm{H}_{2}\mathrm{O}\right)_{6}\right]^{3+}\). As a result, the electronic transitions in \(\left[\mathrm{VF}_{6}\right]^{3-}\) will require higher energy; thus, this complex will absorb light of higher energy.

Key Concepts

Octahedral Complexes

In the world of coordination chemistry, octahedral complexes are fascinating entities with a central metal atom or ion surrounded symmetrically by six ligands at the corners of an octahedron. The importance of octahedral complexes in chemistry cannot be overstated, as they are among the most common coordination geometries found in metal complexes.

Understanding the structure of octahedral complexes is key to studying their properties and reactions. When the ligands coordinate to a metal ion, they can affect the energy levels of the metal's d-orbitals, which leads us to the concept of crystal field theory. This theory provides insights into the electronic transitions responsible for the colors we see in these complexes, and helps predict the magnetic and spectroscopic properties of the complex.

Let's consider an example featuring Vanadium(III). If we were to visualize the setup, imagine positioning six ligands (which could be molecules like water or ions like fluoride) around a central Vanadium ion, forming a shape like two square-based pyramids stuck together at their bases. This arrangement greatly influences the electronic structure of the Vanadium ion, as will be elucidated in the following section on d-orbital energy-level diagrams.

D-Orbital Energy-Level Diagram

The d-orbital energy-level diagram is a visual representation illustrating the energy separation of metal d-orbitals when subjected to the electric field produced by surrounding ligands. In an octahedral complex, this separation results in specific d-orbitals having varying energies due to their spatial orientation relative to the ligands.

In isolation, the five d-orbitals of a transition metal have the same energy, but when the octahedral crystal field is introduced, we observe a split into two distinct energy levels: the higher-energy 'e_g' set and the lower-energy 't_{2g}' set. The 'e_g' set consists of the d_{x^2-y^2} and d_{z^2} orbitals, while the 't_{2g}' set is composed of the d_{xy}, d_{xz}, and d_{yz} orbitals.

Electron Configuration and Filling

Considering Vanadium(III) with its two electrons in the d-orbitals, the electrons will first occupy the lower-energy 't_{2g}' orbitals, according to the Aufbau principle and Hund's rule of maximum multiplicity. This arrangement results in a stable electronic configuration, with both electrons paired in the 't_{2g}' set, leaving the 'e_g' set empty.

It's essential to note that the pattern of orbital filling and the crystal field splitting energy, denoted as \( \Delta_o \), play a critical role in the color and chemical behavior of the complex, as they determine the absorption of light and the possible electronic transitions within the d-orbitals.

Spectrochemical Series

The spectrochemical series is a vital tool used in crystal field theory that ranks ligands based on their ability to split the d-orbital energies in transition metal complexes. This series allows chemists to predict the color of complexes by correlating the ligands' field strength with the energy required for electronic transitions. It ranges from weak-field ligands, like water, to strong-field ligands, like the fluoride ion.

In the spectrochemical series, ligands that produce a larger energy gap, \( \Delta_o \), are known to cause high-frequency light (with more energy) to be absorbed. As an example, in the Vanadium(III) complexes discussed earlier, fluoride ions ([F]^-) are strong-field ligands, leading to a larger splitting in the d-orbital energy levels compared to water molecules ([H2O]).

Impact on Light Absorption

Due to the difference in field strength, the absorption of light will be different for the \(\left[\mathrm{VF}_{6}\right]^{3-}\) complex compared to the \(\left[\mathrm{V}\left(\mathrm{H}_{2}\mathrm{O}\right)_{6}\right]^{3+}\) complex. The stronger field generated by the fluoride ligands in the former requires higher energy for electronic transitions, which means this complex will absorb light of a higher energy, often resulting in a color change that can be useful in various chemical applications such as reaction monitoring or material design.