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 \(\mathrm{kJ} / \mathrm{mol}\) if a \(d^{1}\) complex has an absorption maximum at \(590 \mathrm{~nm}\).

Short answer

(a) An octahedral crystal-field splitting diagram splits the d orbitals into two energy levels: lower level with d_xy, d_xz, and d_yz orbitals, and higher level with d_x2-y2 and d_z2 orbitals. The energy difference between these levels is Δ. (b) In a d^1 complex, the d-d transition energy is equal to the crystal-field splitting energy Δ. (c) Using Planck's equation and given absorption maximum wavelength (590 nm), we calculate Δ to be approximately 203.2 kJ/mol.

Step-by-step solution

(a) Draw the crystal-field splitting diagram for an octahedral field.

An octahedral crystal-field splitting diagram shows the energy levels of the d-orbitals in a metal complex under the influence of an octahedral ligand arrangement. It splits the d orbitals into two energy levels, with three orbitals (d_xy, d_xz, and d_yz) lying at a lower energy level than the other two (d_x2-y2 and d_z2). The energy difference between these two levels is called the crystal-field splitting energy, represented as Δ. To draw the diagram: 1. Draw a horizontal axis to represent energy levels. 2. Draw two horizontal lines, one higher than the other, representing the two energy levels. 3. Label the two lines according to their corresponding d-orbitals 4. Label the energy difference between the two energy levels as Δ.

(b) Relationship between crystal-field splitting energy and d-d transition energy.

In a d^1 complex, there is one electron in the lower-energy orbitals (d_xy, d_xz, and d_yz). When a d-d transition occurs, this electron is excited into one of the higher-energy orbitals (d_x2-y2 or d_z2). The energy required for the d-d transition is equal to the crystal-field splitting energy, Δ. Thus, for a d^1 complex: d-d transition energy = Δ

(c) Calculate the crystal-field splitting energy Δ.

We are given the absorption maximum wavelength (λ) for a d^1 complex at 590 nm. To calculate the crystal-field splitting energy (Δ), we'll first find the energy of the d-d transition using Planck's equation and then convert this energy to kJ/mol. Planck's equation: E = h * c / λ Where: E is the energy of the d-d transition h is Planck's constant (6.626 x 10^-34 J·s) c is the speed of light (3.00 x 10^8 m/s) λ is the wavelength of maximum absorption (590 nm) Step 1: Convert wavelength to meters 590 nm = 590 x 10^-9 m Step 2: Calculate the d-d transition energy E = (6.626 x 10^-34 J·s) * (3.00 x 10^8 m/s) / (590 x 10^-9 m) E = 3.376 x 10^-19 J Step 3: Convert energy to kJ/mol Δ = (3.376 x 10^-19 J) * (1 kJ / 1000 J) * (6.022 x 10^23 mol^-1) Δ = 203.2 kJ/mol Thus, the crystal-field splitting energy (Δ) for the d^1 complex is approximately 203.2 kJ/mol.