Problem 37 - Chemistry: The Central Science

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 \(590 \mathrm{~nm}\).

Step-by-Step Solution

Answer

The crystal field splitting energy diagram for an octahedral field consists of two sets of orbitals: \(3\) lower-energy \(d_{t_{2g}}\) orbitals, and \(2\) higher-energy \(d_{e_{g}}\) orbitals. For a \(d^{1}\) complex, the energy of the \(d-d\) transition is equal to the crystal field splitting energy, \(\Delta\). Given an absorption maximum at \(590 \mathrm{~nm}\), the value of \(\Delta\) is calculated to be \(5.597 \mathrm{kJ/mol}\).

Step 1: 1. Sketch the crystal field splitting energy diagram for an octahedral field

: An octahedral crystal field is formed when the central metal ion is surrounded by six ligands in an octahedral arrangement. In this configuration, the \(d\) orbital energies experience different electrostatic interactions with the ligands, causing the \(d\) orbital energies to split into two different energy levels. The five \(d\) orbitals split into two sets as follows: - \(3\) lower-energy orbitals: \(d_{t_{2g}}\) - which includes \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals. - \(2\) higher-energy orbitals: \(d_{e_{g}}\) - which includes \(d_{x^{2}-y^{2}}\) and \(d_{z^{2}}\) orbitals. The energy difference between these sets of orbitals is termed as the Crystal Field splitting energy, represented as \(\Delta\). The diagram below shows the splitting of the energy levels in an octahedral crystal field. ``` Δ ____/¯¯¯¯¯¯\____ | d(e_g) Orbitals | |_________________| \ __________________/ | d(t_2g) Orbitals | ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ```

Step 2: 2. Define the relationship between the magnitude of Δ and the energy of the d-d transition for a d1 complex

: A \(d^{1}\) complex has only one electron in its \(d\) orbitals, which will occupy the lower-energy orbitals. A \(d-d\) transition occurs when the \(d\) electron is excited to a higher-energy orbital. In an octahedral field, if the electron transitions from a \(t_{2g}\) orbital to an \(e_{g}\) orbital, the energy of the transition is equal to the crystal field splitting energy, \(\Delta\). Therefore, in a \(d^{1}\) complex, the energy of the \(d-d\) transition is equal to \(\Delta\). In other words, for the \(d^{1}\) complex, Energy of the \(d-d\) transition (\(E_{transition}\)) = \(\Delta\)

Step 3: 3. Calculate Δ in kJ/mol for the d1 complex given its absorption maximum

: If a \(d^{1}\) complex has an absorption maximum at \(590 \mathrm{~nm}\), we can calculate the energy of the transition from the wavelength (\(\lambda\)) using the formula: \(E_{transition} = \cfrac{hc}{\lambda}\), where \(h\) is the Planck constant (\(6.626 \times 10^{-34} \mathrm{J \cdot s}\)) and \(c\) is the speed of light (\(2.998 \times 10^{8} \mathrm{m/s}\)). First, convert the wavelength to meters: \(590 \times 10^{-9}\mathrm{m}\) Now, substitute the known values to calculate the energy: \(E_{transition} = (\cfrac{6.626 \times 10^{-34} \mathrm{J \cdot s} \times 2.998 \times 10^{8} \mathrm{m/s}}{590 \times 10^{-9} \mathrm{m}})\) \(E_{transition} = 3.367 \times 10^{-19} \mathrm{J}\) Since \(E_{transition} = \Delta\), we have \(\Delta = 3.367 \times 10^{-19} \mathrm{J}\). To convert this energy to kJ/mol, we use: \(\Delta \cfrac{1000 \mathrm{J}}{1 \mathrm{kJ}} \times \cfrac{1 \mathrm{mol}}{6.022 \times 10^{23}} = \cfrac{3.367 \times 10^{-19} \mathrm{J}\times 1000 }{6.022 \times 10^{23}}\) \(\Delta = 5.597 \mathrm{kJ/mol}\) Hence, the crystal-field splitting energy, \(\Delta\), for the given \(d^{1}\) complex is \(5.597 \mathrm{kJ/mol}\).

Key Concepts

Octahedral Crystal Field

In an octahedral crystal field, a metal ion is surrounded by six ligands. These ligands are positioned in an octahedral shape around the central ion. This arrangement affects the energy levels of the metal's d orbitals.
The d orbitals are split into two groups: three lower energy orbitals, known as the t_2g orbitals (
d_xy, d_xz, and d_yz), and two higher energy orbitals, known as the e_g orbitals (
d_x²-y² and d_z²). This splitting occurs due to the different electrostatic interactions each orbital experiences from the surrounding ligands.
The energy separation between these two sets of orbitals is known as the Crystal Field Splitting Energy, represented by the Greek letter \(\Delta\). This concept is crucial for predicting the magnetic and optical properties of transition metal complexes.

d-d Transition

d-d transitions are electronic transitions that occur in coordination complexes. These transitions involve the movement of an electron from one d orbital to another, usually from a lower energy t_2g orbital to a higher energy e_g orbital in octahedral complexes.
In a d¹ complex, a single electron is found in one of these d orbitals. When the complex absorbs energy, typically in the form of visible light, the electron can be excited to move to the higher energy orbital.
This type of electron transition significantly affects the color of the complex. The energy required for an electron to transition between these orbitals is related to the crystal field splitting energy \(\Delta\),
and is equal to the energy of the absorbed light: E_transition = \(\Delta\). This relationship explains why such complexes often exhibit vibrant colors.

Crystal Field Splitting Energy

The crystal field splitting energy, \(\Delta\), is a key component in solid-state chemistry. It describes the energy difference between the t_2g and e_g orbitals in an octahedral crystal field.
To calculate \(\Delta\) for a \(d^1\) complex, one can use the wavelength of light it absorbs. Specifically, by measuring the absorption maximum, the \(\Delta\) value can be calculated using the formula \[E_{transition} = \frac{hc}{\lambda}\], where \(h\) is Planck's constant and \(c\) is the speed of light.
This formula allows the calculation of the energy in joules, which can then be converted to more practical units such as kJ/mol. For example, if the absorption maximum occurs at 590 nm, substituting these values can yield an energy transition of about 5.597 kJ/mol. Understanding \(\Delta\) helps predict how different ligands will affect a complex's properties, influencing both its magnetism and color.