(a) Sketch a diagram that shows the definition of the...

(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 $\mathrm{kJ} / \mathrm{mol}$ if a $d^{1}$ complex has an absorption maximum at $545 \mathrm{nm}$.

(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 $\mathrm{kJ} / \mathrm{mol}$ if a $d^{1}$ complex has an absorption maximum at $545 \mathrm{nm}$.

Key Concepts

Crystal Field Theory

This concept explains how the degenerate d orbitals in a transition metal ion split into different energy levels when surrounded by a set of ligands. The nature and strength of the interaction between the metal ion and the ligands determine the energy separation, known as the crystal-field splitting energy (?), which is central to understanding the electronic structure and properties of coordination complexes.

Octahedral Crystal Field Splitting

In an octahedral coordination environment, the five d orbitals split into two groups: the lower-energy t?g set and the higher-energy e_g set. The crystal-field splitting energy (?) quantifies the energy difference between these sets. This splitting pattern is key to predicting the electronic transitions, absorption spectra, and magnetic properties of octahedral complexes.

d-d Transitions

d-d transitions occur when an electron in a d orbital is excited from a lower-energy orbital to a higher-energy orbital within the split d orbital set. In a d1 complex, the absorption of light corresponds directly to the energy gap, ?, making the wavelength of the absorption maximum a direct measure of the crystal-field splitting energy.

Energy-Wavelength Relationship

This relationship is based on the equation E = hc/?, where E is the energy of the photon, h is Planck’s constant, c is the speed of light, and ? is the wavelength. This equation is used to relate the absorption maximum observed in the spectrum to the energy of the electronic transition in the complex.

Molar Energy Conversion

Once the energy per photon (in Joules) is calculated using the energy-wavelength relationship, it is necessary to convert this energy to a per-mole basis. This is accomplished by multiplying by Avogadro’s number and then converting the result into appropriate units, such as kilojoules per mole (kJ/mol), to provide a meaningful comparison with thermodynamic data.

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