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 \(d^{2}\) complex? (c) Calculate \(\Delta\) in \(\mathrm{kJ} / \mathrm{mol}\) if a \(d^{1}\) complex has an absorption maximum at \(545 \mathrm{~nm}\).

Short answer

For an octahedral crystal field, the \(d\) orbitals split into two sets, \(t_{2g}\) and \(e_g\), with the energy difference between them being \(\Delta\). In a \(d^2\) complex, the relationship between \(\Delta\) and \(d-d\) transition energy is \(E(\Delta) = \Delta\). For a \(d^1\) complex with an absorption maximum at \(545\, \mathrm{nm}\), we can calculate \(\Delta\) using \(E = \frac{hc}{\lambda}\), to find that \(\Delta\) in \(\mathrm{kJ/mol}\) is approximately \(2.17 \mathrm{kJ/mol}\).

Step-by-step solution

Sketching the Crystal-Field Splitting Energy Diagram for an Octahedral Field

For an octahedral crystal field, the \(d\) orbitals split into two sets with different energy levels. There are three orbitals with lower energy, known as \(t_{2g}\) orbitals, while the other two orbitals, \(e_g\) orbitals, have higher energy. The energy difference between these two sets of orbitals represents the crystal-field splitting energy, \(\Delta\), which is also referred to as \(\Delta_\text{o}\) for an octahedral field. Draw a vertical axis representing orbital energy, and label two horizontal lines: one for \(t_{2g}\) orbitals and another for \(e_g\) orbitals. Mark the difference between them as \(\Delta\).

Relation between the Magnitude of \(\Delta\) and the Energy of \(d-d\) Transition for a \(d^{2}\) Complex

The relationship between the magnitude of \(\Delta\) and the energy of \(d-d\) transition for a \(d^{2}\) complex is given by the formula \(E(\Delta)=\Delta/n\), where \(n\) is the total number of unpaired electrons in the \(d\) orbitals that are excited to another orbital. In this case, a \(d^{2}\) complex indicates that the transition occurs when one of the two unpaired electrons in the \(t_{2g}\) orbitals transfers to the remaining unoccupied \(e_g\) orbital. Therefore, the relationship is: \(E(\Delta) = \Delta / 1 = \Delta\).

Calculating \(\Delta\) in \(\mathrm{kJ} / \mathrm{mol}\) for a \(d^{1}\) Complex

To calculate the energy difference in an octahedral \(d^{1}\) complex with an absorption maximum at \(545 \mathrm{~nm}\), we first need to determine the energy of the \(d-d\) transition. The energy can be calculated using the equation: \(E = \dfrac{hc}{\lambda}\), where \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), \(c\) is the speed of light (\(2.998 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the wavelength in meters. Since \(\Delta = E(\Delta)\) for a \(d^{1}\) complex: 1. Convert the wavelength from nanometers to meters: \(\lambda_{\text{m}} = 545 \times 10^{-9} \mathrm{m}\) 2. Calculate the energy in joules: \(E = \dfrac{(6.626 \times 10^{-34} \mathrm{Js}) (2.998 \times 10^{8} \mathrm{m/s})}{545 \times 10^{-9} \mathrm{m}}\) 3. Convert the energy from joules to kilojoules per mole: \(\Delta = E \times (1 / 1000) \times \dfrac{1 \mathrm{mol}}{6.022 \times 10^{23}}\) Following these steps, we obtain the crystal-field splitting energy, \(\Delta\), for the \(d^{1}\) complex in \(\mathrm{kJ} / \mathrm{mol}\).

Key concepts

Octahedral Crystal Field

Understanding the concept of an octahedral crystal field is fundamental in coordination chemistry. Imagine the orbital patterns of an atom's d-electron cloud intersecting with the point-charge electric fields from a set of six ligands arranged at the corners of an octahedron. This interaction modifies the energy levels of the d-electrons.

In an undisturbed state without a crystal field, the five d-orbitals on a transition metal have the same energy. When a metal ion is surrounded by the ligands to form an octahedral complex, this symmetry is broken. The d-orbitals split into two distinct energy levels due to the electrostatic interaction with the ligands. The three lower-energy orbitals are called t_2g (d_xy, d_xz, d_yz), and the two higher-energy orbitals are termed e_g (d_z², d_x²-y²).

This splitting occurs because the e_g orbitals point directly at the ligands, leading to greater repulsion and thus higher energy, while the t_2g orbitals point between the ligands, experiencing less repulsion. The energy gap between these sets of orbitals is referred to as the crystal-field splitting energy, or \(\Delta_o\). It's this gap that governs many of the physical properties and reactions of coordination complexes.

d-d Transition

The d-d transition is a type of electronic transition that occurs in coordination complexes. It entails the movement of an electron from one d-orbital to another, specifically between the t_2g and e_g levels in an octahedral complex. Visible light contains the right amount of energy for these transitions, leading to the colorful nature of many complexes.

The relationship between the magnitude of the crystal-field splitting energy (\(\Delta\)) and the energy required for the d-d transition is crucial. The energy of the d-d transition essentially matches the crystal-field splitting energy. When an electron jumps from a lower-energy t_2g orbital to a higher-energy e_g orbital, it absorbs a photon of light. The wavelength of the absorbed light is inversely proportional to the energy of the photon, which corresponds to the energy gap (\(\Delta\)).

Going a level deeper, the absorption of light by a complex results in its characteristic color, which is the complementary color of the light absorbed. If a complex absorbs green light, for instance, it will appear red. It is this understanding of the d-d transition that not only explains the vibrant hues of these complexes but also allows for their analysis using spectroscopy.

Energy Calculation in Coordination Complexes

The energy of d-d transitions in coordination complexes can be calculated with precision, which is essential in the fields of material science and coordination chemistry. To determine the crystal-field splitting energy (\(\Delta\)) using the wavelength of absorbed light, we employ the relationship between energy (E) and wavelength (\(\lambda\)) described by Planck's equation \(E = \frac{hc}{\lambda}\), where h is Planck's constant and c is the speed of light.

In an exercise where we are given the wavelength at which maximum absorption occurs, such as 545 nm for a d¹ complex, we start by converting the wavelength to meters and then apply Planck's equation to find E. Since the energy of the absorbed photon is equivalent to the energy gap, we know that \(\Delta = E\). Finally, for practical applications, we convert this energy into kilojoules per mole, which is the common unit used in chemistry to describe the amount of energy per mole of substance.

By conducting such calculations, we can determine the precise energy levels involved in a complex's electronic transitions, which can further inform the properties of the complex, how it interacts with light, and its potential applications. It's a powerful illustration of how quantum mechanics dovetails with observable phenomena like color and adsorption in the macroscopic world of chemistry.