(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 $590 \mathrm{~nm}$.

Key Concepts

Crystal Field Splitting

This concept refers to the energy difference created when a metal ion's degenerate d orbitals interact with the electric field generated by surrounding ligands. In an octahedral environment, the d orbitals split into two energy levels, with three lower-energy orbitals (t2g) and two higher-energy orbitals (eg), and the difference in energy between these groups is denoted by ?.

Octahedral Coordination Geometry

Octahedral geometry is a common spatial arrangement of six ligands around a central metal atom. The way these ligands approach the metal determines the extent of the splitting of the d orbitals. A diagram typically shows the relative orientations of the orbitals, emphasizing how the eg orbitals experience greater repulsion from the ligands than the t2g orbitals.

d-d Transitions

d-d transitions involve the movement of electrons between the split d orbitals in a metal complex. The energy absorbed or emitted during these transitions corresponds directly to the crystal-field splitting energy (?), meaning that the magnitude of ? is equivalent to the energy required to promote an electron from a lower-energy orbital to a higher-energy one.

Energy-Wavelength Conversion

This key concept involves converting the measured wavelength of an absorbed photon into its corresponding energy using the equation E = hc/?, where h is Planck’s constant and c is the speed of light. The result is then converted into energy per mole (typically kJ/mol) by multiplying by Avogadro's number and performing unit adjustments, which is standard practice in quantifying the energy of absorption in spectroscopic studies.

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