Problem 53 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Sketch a diagram that shows the definition of the crystalfield splitting energy \((\Delta)\) for an octahedral crystal-field. \((\mathbf{b})\) 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 \(\mathrm{kJ} / \mathrm{mol}\) if a \(d^{1}\) complex has an absorption maximum at \(545 \mathrm{nm}\).

Step-by-Step Solution

Answer

(a) In an octahedral crystal field, the five degenerate d-orbitals are split into two levels: three lower-energy orbitals (T₂g: dxy, dyz, and dxz) and two higher-energy orbitals (Eg: dx²-y² and d³z²-r²). The energy difference between these levels is the crystal field splitting energy (∆). (b) The energy of the d-d transition for a d¹ complex is equal to the crystal field splitting energy (∆). (c) Using the given absorption maximum (545 nm), we can calculate the energy of the d-d transition and the value of ∆ as follows: \( E = \dfrac{hc}{\lambda} \) Substitute the values \( h = 6.626 \times 10^{-34} \, \text{Js} \), \( c = 2.998 \times 10^8 \, \text{m/s} \), and \( \lambda = 545 \times 10^{-9} \, \text{m} \) to calculate E. Next, convert E to kJ/mol using Avogadro's number (NA): \( \Delta (\text{kJ/mol}) = \dfrac{E (\text{J}) \times N_\text{A}}{10^3} \) Substitute \( N_\text{A} = 6.022 \times 10^{23} \, \text{mol}^{-1} \) and the calculated E value to find the crystal field splitting energy (∆) in kJ/mol.

Step 1: Part (a): Sketch the diagram

Draw an energy diagram showing the d-orbitals in the absence and presence of the crystal field splitting for an octahedral crystal field. We start with five degenerate d-orbitals (all having equal energy). The crystal field interaction splits these orbitals into two energy levels: three lower-energy orbitals (dxy, dyz, and dxz) and two higher-energy orbitals (dx²-y² and d³z²-r²). The energy difference between the two levels (∆) represents the crystal field splitting energy.

Step 2: Part (b): Relationship between ∆ and d-d transition energy

In an octahedral crystal-field, an electron can be excited from the lower-energy orbitals (T₂g) to the higher-energy orbitals (Eg). The energy required for this transition is equal to the crystal field splitting energy (∆). Thus, the energy of the d-d transition for a d¹ complex is: \( \Delta \)

Step 3: Part (c): Calculate ∆ in kJ/mol for given absorption maximum

We are given that the absorption maximum is at 545 nm. We can use this to calculate the energy of the d-d transition and, in turn, the value of ∆. We'll use the relationship between energy (E), Planck's constant (h), and the speed of light (c): \( E = \dfrac{hc}{\lambda} \) Where λ is the wavelength of the absorption maximum (545 nm). We'll get the energy in the unit of Joules and then convert it to kJ/mol using Avogadro's number (NA): \( \Delta (\text{kJ/mol}) = \dfrac{E (\text{J}) \times N_\text{A}}{10^3} \) Plugging in the values: \( h = 6.626 \times 10^{-34} \, \text{Js} \) \( c = 2.998 \times 10^8 \, \text{m/s} \) \( \lambda = 545 \times 10^{-9} \, \text{m} \) \( N_\text{A} = 6.022 \times 10^{23} \, \text{mol}^{-1} \) Calculate the energy, perform the necessary conversions, and determine the value of crystal field splitting energy (∆) in kJ/mol.

Key Concepts

Crystal Field Splitting Energy

Crystal Field Splitting Energy, often denoted as \( \Delta \), refers to the energy difference created by the interaction between metal d orbitals and the surrounding ligands in a crystal field. Visualizing this concept requires us to imagine, initially, five d-orbitals at an equal energy state, also known as degenerate. However, upon interaction with the ligands in an octahedral setup, this degeneracy is lifted.

In an octahedral crystal field, the d-orbitals split into two distinct sets:

• Three lower-energy orbitals: \( d_{xy}, d_{yz}, \text{and} \, d_{xz} \) (collectively referred to as \( T_{2g} \)).
• Two higher-energy orbitals: \( d_{x^2-y^2} \text{and} \; d_{3z^2-r^2} \) (known as \( E_g \)).

This splitting is crucial because it determines the electronic transitions that can occur. The magnitude of \( \Delta \) is influenced by the nature of the ligand, the oxidation state of the metal, and the metal's place in the periodic table.

Octahedral Complexes

Octahedral complexes are among the most common and stable types found in coordination chemistry. They comprise a central metal atom or ion surrounded symmetrically by six ligands positioned at the vertices of an octahedron. This geometry significantly affects the distribution of the d-orbitals' energy, leading to crystal field splitting.

In octahedral complexes, ligands approach the metal ion along the axis. This increases electrostatic repulsions with the d-orbitals pointing directly along these axes (\( d_{x^2-y^2} \text{and} \; d_{3z^2-r^2} \)), causing them to have higher energy than their counterparts. The other three d-orbitals (\( d_{xy}, d_{yz}, \text{and} \, d_{xz} \)) are oriented between the axes, experiencing less repulsion and thus have lower energy.

Octahedral complexes are not only important due to their prevalence but also because they offer insight into how coordination affects the properties of a metal, including color, magnetism, and reactivity.

d-d Transition

A \( d-d \) transition refers to the movement of an electron between d-orbitals of differing energy levels within a transition metal complex. In an octahedral complex, the electron transitions typically occur from the lower-energy \( T_{2g} \) orbitals to the higher-energy \( E_g \) orbitals. This electronic transition is what gives rise to the characteristic colors of many transition metal complexes.

The frequency and wavelength of the light absorbed during a \( d-d \) transition correspond to the energy difference or crystal field splitting energy, \( \Delta \). For example, if a complex absorbs light in the visible region, the complement of that absorbed color is perceived by the human eye.

Understanding \( d-d \) transitions helps explain why coordination complexes exhibit such vibrant colors and is fundamental to comprehending more complex topics in inorganic chemistry, including spectrochemical series and ligand field theory.