Problem 53 - Chemistry The Central Science In Si Units

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

The crystal field splitting energy is approximately 219 kJ/mol.

Step 1: Understand the Octahedral Crystal Field Splitting

In an octahedral crystal field, the degeneracy of the five d orbitals in a transition metal ion is broken due to the presence of ligands. This results in two groups of orbitals: the higher energy \(e_g\) set (consisting of the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals) and the lower energy \(t_{2g}\) set (consisting of the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals). The energy difference between these two sets is known as the crystal field splitting energy, denoted as \(\Delta\).

Step 2: Relationship Between Δ and d-d Transition Energy

In a \(d^1\) complex, the single electron occupies one of the lower-energy \(t_{2g}\) orbitals. A \(d-d\) transition occurs when this electron absorbs enough energy to transition to a higher energy \(e_g\) orbital. The energy required for this transition is equal to \(\Delta\), the crystal field splitting energy.

Step 3: Calculate the Energy of the Absorbed Photon

The energy of a photon absorbed during a transition can be calculated using the formula \(E = \frac{{hc}}{{\lambda}}\), where \(E\) is the energy, \(h\) is Planck's constant \(6.626 \times 10^{-34} \, \text{J} \, \text{s}\), \(c\) is the speed of light \(3.00 \times 10^8 \, \text{m/s}\), and \(\lambda\) is the wavelength in meters. For 545 nm, convert this to meters: \(545 \, \text{nm} = 545 \times 10^{-9} \, \text{m}\).

Step 4: Plug in the Values to Calculate E

Substituting the given values into the equation: \[E = \frac{{(6.626 \times 10^{-34}) \times (3.00 \times 10^8)}}{{545 \times 10^{-9}}}\]Calculating this gives the energy in joules: \(E \approx 3.65 \times 10^{-19} \, \text{J}\).

Step 5: Convert Energy to kJ/mol

To convert the energy to \(\text{kJ/mol}\), multiply the energy per photon by Avogadro's number (\(6.022 \times 10^{23} \, \text{mol}^{-1}\)) and convert joules to kilojoules. \[E_{\text{kJ/mol}} = 3.65 \times 10^{-19} \, \text{J/photon} \times 6.022 \times 10^{23} \, \text{mol}^{-1} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} \] This gives \(E \approx 219 \, \text{kJ/mol}\).

Key Concepts

crystal field splitting energy

In the world of chemistry, especially when it comes to understanding transition metals, crystal field splitting energy, symbolized as \( \Delta \), plays a crucial role. This energy arises in coordination complexes where metal ions interact with surrounding ligand fields. In an octahedral arrangement, these ligands cause the usually fivefold degenerate d orbitals to split into two different energy sets.

Picture the d orbitals as a team of five players originally on the same level. When the ligands enter the scene, they create an energy field that forces these players into two groups. The first pair, known as the \( e_g \) orbitals (\( d_{x^2 - y^2} \) and \( d_{z^2} \)), moves to a higher energy level. Meanwhile, the triplet \( t_{2g} \) orbitals (\( d_{xy} \), \( d_{xz} \), and \( d_{yz} \)) stay put at a lower level.

This energy difference between \( e_g \) and \( t_{2g} \) levels is the crystal field splitting energy \( \Delta \). It's a measure of how strong the ligand field is influencing the metal ion. Factors like the type of ligands and the nature of the metal ion affect \( \Delta \), making it larger or smaller. A higher \( \Delta \) means that more energy is needed to promote an electron among these levels.

d-d transitions

"d-d transitions" are a fascinating phenomenon in the realm of transition metal complexes. They refer to the movement of electrons between the split d orbitals due to the crystal field. When a photon of light matches the crystal field splitting energy \( \Delta \), an electron from the lower energy \( t_{2g} \) orbitals can jump to the higher energy \( e_g \) orbitals.

This process is called a \( d-d \) transition. It's a form of electronic excitation pivotal in determining the colors of transition metal complexes. As different wavelengths (or colors) of light are absorbed during this transition, the light that is not absorbed gets reflected or transmitted. This gives the compound its characteristic color.

• For a \( d^1 \) complex, the single electron sits in a \( t_{2g} \) orbital until a photon of energy equal to \( \Delta \) strikes it.
• Once excited, the electron hops to one of the \( e_g \) orbitals, absorbing that specific wavelength of light.
• The color we perceive is complementary to the absorbed wavelength.

Thus, \( d-d \) transitions are integral to the visual characteristics of these compounds.

octahedral complexes

Octahedral complexes are a key topic when studying crystal field theory. These complexes occur when a central metal ion is surrounded by six ligands positioned at the corners of an octahedron. This symmetrical arrangement of ligands plays a crucial role in the splitting of the metal's d orbitals.

In such a geometric setup, the six ligands create an electric field that significantly impacts the energies of the metal's d orbitals. Specifically, it causes the orbital reformation into two distinct sets, \( t_{2g} \) and \( e_g \), setting the stage for crystal field splitting. This octahedral arrangement is one of the most common and studied due to its relevance and simplicity.

• Six ligands around the metal ion ensure a symmetric field.
• The d orbitals respond by splitting, creating \( \Delta \).
• The relationship between this splitting and the ligands' electron-donating abilities influences properties like magnetic behavior and absorption spectra.

Understanding octahedral complexes helps predict and explain various chemical and physical properties in transition metal chemistry, making it an essential concept for students and chemists alike.