Question

As shown in Figure 23.26, the \(d-d\) transition of \(\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\) produces an absorption maximum at a wavelength of about 500 \(\mathrm{nm}\) . (a) What is the magnitude of \(\Delta\) for \(\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\) in \(\mathrm{kJ} / \mathrm{mol} ?\) (b) How would the magnitude of \(\Delta\)change if the \(\mathrm{H}_{2} \mathrm{O}\) ligands in \(\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\) were replaced with \(\mathrm{NH}_{3}\) ligands?

Short answer

The magnitude of \(\Delta\) for \(\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\) is 660.8 kJ/mol. When the \(\mathrm{H}_{2} \mathrm{O}\) ligands are replaced with \(\mathrm{NH}_{3}\) ligands, the magnitude of Δ will increase due to the stronger field strength of \(\mathrm{NH}_{3}\).

Step-by-step solution

Calculate the energy difference (Δ)

We are given the transition occurs at a wavelength of 500 nm. We can find the energy of the absorbed light using the equation: \(E = \dfrac{hc}{\lambda}\) where E is the energy, h is Planck's constant (\(6.626 \times 10^{-34}\ \text{J s}\)), c is the speed of light (\(3.00 \times 10^{8}\ \text{m/s}\)), and λ is the wavelength (500 nm or \(5.00 \times 10^{-7}\ \text{m}\)). Let's calculate the energy: \(E = \dfrac{(6.626 \times 10^{-34}\, \text{J s})(3.00 \times 10^{8}\, \text{m/s})}{5.00 \times 10^{-7}\, \text{m}}\) Solve for E: \(E = 3.976 \times 10^{-19}\, \text{J}\) Now, we'll convert this energy to \(\mathrm{kJ} / \mathrm{mol}\) using Avogadro's number (\(6.022 \times 10^{23}\, \text{mol}^{-1}\)): \(\Delta = \dfrac{3.976 \times 10^{-19}\, \text{J}}{6.022 \times 10^{23}\, \text{mol}^{-1}} \times 10^3\, \text{kJ / J}\) Solve for Δ: \(\Delta = 660.8\, \text{kJ/mol}\) So, the magnitude of Δ for \(\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\) is 660.8 kJ/mol.

Predict the change in Δ when ligands are changed

We now want to predict the change in the magnitude of Δ when the \(\mathrm{H}_{2} \mathrm{O}\) ligands are replaced with \(\mathrm{NH}_{3}\) ligands. The difference in Δ values will be due to the difference in the ligand field strength between \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{NH}_{3}\). In general, we can use the spectrochemical series to compare the field strength of different ligands. This series ranks ligands from weak-field to strong-field. Ammonia (\(\mathrm{NH}_{3}\)) is a stronger field ligand than water (\(\mathrm{H}_{2} \mathrm{O}\)). When a stronger field ligand is introduced, the energy gap Δ between the \(d\) orbitals increases because of greater crystal field splitting. This is because greater ligand field strength induces a larger splitting of the \(d\) orbitals. Therefore, when the \(\mathrm{H}_{2} \mathrm{O}\) ligands are replaced with \(\mathrm{NH}_{3}\) ligands, the magnitude of Δ will increase.

Key concepts

d-d Transition

The d-d transition refers to the absorption of light that promotes an electron from a lower energy d-orbital to a higher energy d-orbital within a transition metal complex. This type of electronic transition is responsible for the vivid colors often seen in coordination compounds.

The process occurs when a photon with just the right energy hits the complex, and this energy value corresponds to the specific wavelength of light absorbed. The remaining wavelengths are reflected or transmitted, and these give the compound its observed color. For instance, the complex mentioned in the exercise absorbs light with a wavelength of about 500 nm, which lies within the visible spectrum and gives rise to a complementary color perceived by the observer.

The magnitude of the energy difference between the d-orbitals, denoted by Δ (also called the crystal field splitting energy), can be calculated using the formula:
\[E = \frac{hc}{\lambda}\]
where E is the energy of the light absorbed, h is Planck's constant, c is the speed of light, and \(\lambda\) is the wavelength. The energy of the d-d transition obtained from this calculation provides insights into the electronic arrangement and properties of the metal complex.

Ligand Field Strength

Ligand field strength is a measure of the ability of a ligand to split the d-orbitals of the central metal ion in a coordination compound. This strength determines the energy required to move an electron between the different energy levels of the d-orbitals, directly influencing the color and chemical behavior of the complex.

For instance, in the given exercise, we are asked to consider how the energy difference Δ would change if the water (\(\mathrm{H}_2 \mathrm{O}\)) ligands in the \(\left[\mathrm{Ti}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{3+}\) complex were replaced with ammonia (\(\mathrm{NH}_3\)) ligands. Since ammonia is known to be a strong-field ligand compared to water, it would cause a greater separation of the d-orbital energy levels, leading to a change in the color of the complex and the value of Δ, indicating the varying ligand field strengths of these molecules.

Spectrochemical Series

The spectrochemical series is a list that ranks ligands based on their field strength, from weak-field to strong-field ligands. Ligands at the low end of the series, such as iodide and bromide, cause small degrees of splitting in the d-orbitals of the metal ion, whereas ligands at the high end, like cyanide and carbon monoxide, cause larger splittings.

This series is extremely useful in predicting and explaining the color, magnetic properties, and reactivity of coordination complexes. When a ligand is replaced with another that is higher in the spectrochemical series, the energy gap (Δ) increases. Conversely, replacing the ligand with one that is lower in the series will decrease the gap. As a result, the energy of the absorbed light, hence the color of the complex, can change dramatically. In the exercise provided, this is exemplified by the difference in field strength between water and ammonia — where ammonia, ranking higher in the series, will produce a larger value of Δ when it serves as a ligand.

Crystal Field Splitting

The concept of crystal field splitting concerns how the energy levels of the d-orbitals in a transition metal ion are affected by the surrounding ligands. In an isolated metal ion, the five d-orbitals have the same energy. However, when ligands approach the metal ion and form a coordination complex, these energies are no longer equal due to the ligand's electric field.

In an octahedral complex such as the one discussed in the exercise, ligands coordinate along the axes, causing the d-orbitals to split into two groups: the lower energy 't2g' set (comprising the dxy, dxz, and dyz orbitals) and the higher energy 'eg' set (comprising the dx2-y2 and dz2 orbitals). The energy gap between these two sets is the Δ mentioned above. The size of this gap is crucial in determining many properties of the complex, including the d-d transitions that can occur. A larger crystal field splitting can significantly affect the color and magnetic properties of the complex, as well as its stability and reactivity. The exercise illustrates how the replacement of ligands can alter Δ and thereby modify the optical and chemical characteristics of a complex.