Question

(a) Sketch a diagram that shows the definition of the crystal-field splitting energy \((\Delta)\) for an octahedral crystal-field. (b) What is the relationship between the magnitude of \(\Delta\) and the energy of the \(d\)-\(d\) transition for a \(d^{1}\) complex? (c) Calculate \(\Delta\) in \(\mathrm{k} J / \mathrm{mol}\) if a \(d^{1}\) complex has an absorption maximum at 545 \(\mathrm{nm} .\)

Short answer

(a) The diagram for an octahedral crystal-field shows that the \(d\) orbitals split into two energy levels, with \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals at higher energy \(e_g\) and \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals at lower energy \(t_{2g}\). (b) The energy of the \(d\)-\(d\) transition for a \(d^{1}\) complex is equal to the crystal-field splitting energy, \(\Delta\). (c) For a \(d^{1}\) complex with an absorption maximum at 545 \(\mathrm{nm}\), the crystal-field splitting energy, \(\Delta\), is \(218.7\,\mathrm{kJ/mol}\).

Step-by-step solution

Sketch the diagram for an octahedral crystal-field

To draw the diagram, we should first understand that in an octahedral complex, there are 6 ligands surrounding the central metal ion. The ligands approach along the \(x\), \(y\), and \(z\) axes, causing the \(d\) orbitals to split into two energy levels. The \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals have higher energy (collectively called \(e_g\) orbitals) and the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals have lower energy (collectively called \(t_{2g}\) orbitals). Sketch the diagram accordingly.

Relationship between \(\Delta\) and \(d\)-\(d\) transition energy

The crystal-field splitting energy, represented as \(\Delta\), is the difference in energy between the lower-energy \(t_{2g}\) orbitals and the higher-energy \(e_g\) orbitals. In a \(d^{1}\) complex, there is only one electron in the \(d\) orbitals, which occupies the lower-energy \(t_{2g}\) orbitals. The energy of the \(d\)-\(d\) transition corresponds to the energy required for the electron to transition from the \(t_{2g}\) orbitals to the \(e_g\) orbitals. Therefore, the energy of the \(d\)-\(d\) transition is equal to the crystal-field splitting energy, \(\Delta\).

Calculate \(\Delta\) in \(\mathrm{k} J / \mathrm{mol}\) for a \(d^{1}\) complex with an absorption maximum at 545 \(\mathrm{nm}\)

To calculate \(\Delta\), we need to convert the absorption wavelength to energy using the equation: Energy = \(\frac{hc}{\lambda}\) where \(h\) is the Planck constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), \(c\) is the speed of light (\(3 \times 10^8 \mathrm{m/s}\)), and \(\lambda\) is the wavelength (545 \(\mathrm{nm}\)). First, convert the wavelength from nanometers to meters: \(\lambda = 545\,\mathrm{nm} \times \frac{1\,\mathrm{m}}{1 \times 10^9\,\mathrm{nm}} = 545 \times 10^{-9}\,\mathrm{m}\) Now, calculate the energy using the converted wavelength: Energy = \(\frac{(6.626 \times 10^{-34}\, \mathrm{Js})(3 \times 10^8\, \mathrm{m/s})}{545 \times 10^{-9}\, \mathrm{m}} = 3.64 \times 10^{-19}\,\mathrm{J}\) Now we have the energy in joules. We will convert it to kJ/mol: \(\Delta = \frac{3.64 \times 10^{-19} \mathrm{J}}{1 \times 10^{-3}\,\mathrm{kJ/J}} \times \frac{6.022 \times 10^{23}\,\mathrm{mol^{-1}}}{1\,\mathrm{mol}} = 218.7\,\mathrm{kJ/mol}\) So, the crystal-field splitting energy, \(\Delta\), is \(218.7\,\mathrm{kJ/mol}\).

Key concepts

Understanding Octahedral Complex

In an octahedral complex, a central metal ion is surrounded symmetrically by six ligands. These ligands align along the x, y, and z axes. As they interact with the metal's d orbitals, they cause a splitting of energy levels. Instead of the five d orbitals being at the same energy, they split into two groups due to this approach.
The higher energy orbitals, known as \(e_g\) orbitals, include \(d_{z^2}\) and \(d_{x^2-y^2}\). On the other hand, the lower energy orbitals are the \(t_{2g}\) orbitals, including \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\).
This difference in energy between the \(t_{2g}\) and \(e_g\) orbitals is essential to understanding how these complexes absorb and color light.

The Dynamics of d-d Transition

The d-d transition involves the movement of an electron from a lower energy \(t_{2g}\) orbital to a higher energy \(e_g\) orbital. In a \(d^1\) complex, there is one electron in the \(d\) orbitals, which initially resides in one of the \(t_{2g}\) orbitals because they are of lower energy.
This transition is important because it corresponds to the absorption of light. When the electron absorbs the exact amount of energy equal to the crystal-field splitting energy \(\Delta\), it "jumps" to the higher energy \(e_g\) orbital.

• This transition is crucial because it is directly linked to the color that the complex exhibits.
• Spectroscopy often measures this transition energy to gain insights into the complex’s properties.

Exploring Crystal-Field Splitting Energy

Crystal-field splitting energy \(\Delta\) is the measure of the energy difference between the \(t_{2g}\) and \(e_g\) orbitals in an octahedral complex. This value is vital because it affects the physical and chemical properties of the complex, including its color and magnetic behavior.
To calculate \(\Delta\), we utilize the wavelength of light absorbed by the complex. Given an absorption maximum at 545 nm, one can find the energy using the formula:
\[\text{Energy} = \frac{hc}{\lambda}\]

• Where \(h\) is Planck's constant \( (6.626 \times 10^{-34}\, \mathrm{Js}) \).
• \(c\) is the speed of light \( (3 \times 10^8\, \mathrm{m/s}) \).
• \(\lambda\) is the wavelength \(545 \times 10^{-9}\, \mathrm{m}\).

Upon calculation:
\[\Delta = 218.7\, \mathrm{kJ/mol}\]
This value signifies how much energy is needed for the electron transition, playing a pivotal role in the complex’s behavior and its interaction with light.