Problem 84 - Chemistry: The Central Science

Question

The red color of ruby is due to the presence of \(\mathrm{Cr}(\mathrm{III})\) ions at octahedral sites in the dose-packed exide lattice of \(\mathrm{Al}_{2} \mathrm{O}_{2}\). Draw the crystal-field splitting diagram for \(\mathrm{Cr}\) (III) in this environment. Suppose that the ruby crystal is subjected to high pressure. What do you predict for the variation in the wavelength of absorption of the ruby as a function of pressure? Explain.

Step-by-Step Solution

Answer

The crystal-field splitting diagram for Cr(III) in an octahedral environment has three electrons occupying the t₂g orbitals. When subjected to high pressure, the crystal lattice size decreases, increasing the octahedral crystal field splitting energy Δ₀. As a result, the wavelength of absorption in ruby shifts from longer wavelengths (red) towards shorter wavelengths (blue) as pressure increases.

Step 1: 1. Draw the crystal-field splitting diagram for Cr(III) ion in an octahedral environment.#

First, let's list the electron configuration of Cr(III). The atomic number of Chromium (Cr) is 24, so its ground state electron configuration is [Ar] 3d5 4s1. When Cr becomes a +3 ion (Cr(III)), it loses 3 electrons (2 from 4s and 1 from 3d orbitals) and its electron configuration becomes [Ar] 3d3. Now, we will draw the crystal-field splitting diagram for Cr(III) in an octahedral environment. In an octahedral field, the d orbitals split into two sets: the lower-energy t₂g set (\(d_{xy}, d_{xz}, d_{yz}\)) and the higher-energy e_g set (\(d_{z^2}, d_{x^2-y^2}\)). The energy difference between these two sets is referred to as Δ₀ (in the octahedral field). Since Cr(III) has 3d3 configuration, the three electrons will occupy the t₂g orbitals following Hund's rule. The crystal-field splitting diagram will appear like this: t₂g: ↑↓ ______ ↑ ______ ↑ d(xy) d(xz) d(yz) eₙ: ______ ______ d(z²) d(x²-y²)

Step 2: 2. Analyze the effect of high pressure on the crystal field splitting diagram.#

When the ruby crystal is subjected to high pressure, the overall size of the crystal lattice will decrease, which leads to a decrease in the distance between the Cr(III) ions and the surrounding ligands (i.e., O²⁻ ions in the Al₂O₃ lattice). This change in distance will increase the electrostatic interactions between the central ion (Cr(III)) and the surrounding ligands, raising the energy of the eₙ orbital and decreasing that of the t₂g orbital. The result is an increase in the octahedral crystal field splitting energy Δ₀.

Step 3: 3. Predict the variation in the wavelength of absorption of ruby as a function of pressure.#

The variation in the wavelength of the absorbed light by ruby depends on the size of the energy gap between the t₂g and eₙ orbitals (Δ₀). According to the equation: \(E = \dfrac{hc}{\lambda}\) where E is the energy difference between the two orbitals, h is the Planck's constant, c is the speed of light, and λ is the wavelength, we have that: \(\Delta_0 = \dfrac{hc}{\lambda}\) As the pressure increases, Δ₀ increases. Therefore, λ, the wavelength of absorbed light, will decrease. This means that as pressure increases, the wavelength of absorption in ruby will shift from longer wavelengths (red color) towards shorter wavelengths (towards blue color).

Key Concepts

Octahedral Complex

In chemistry, coordination complexes are formed when a central metal atom or ion is bonded to surrounding molecules or ions, known as ligands. An octahedral complex has a coordination number of six, typically involving six ligands arranged around the central metal atom in a symmetrical geometry. This arrangement results in the ligands being positioned at the corners of an imaginary octahedron.
The octahedral geometry is one of the most common structural types for transition metal complexes. It greatly influences the electronic properties of the central metal ion. Within an octahedral field, the five degenerate d-orbitals of the metal ion split into two sets of orbitals due to ligand interactions.
- The lower energy set, known as the t₂g orbitals, includes the d_xy, d_xz, and d_yz orbitals. - The higher energy set, called the e_g orbitals, includes the d_z² and d_x²-y² orbitals. This separation between energy levels is referred to as the crystal field splitting energy ( Δ₀ ). Understanding this splitting is crucial for predicting the color and magnetic properties of octahedral complexes.

Spectrochemical Series

The spectrochemical series is an empirical list that ranks ligands based on their ability to split the d-orbitals of a metal ion in a coordination complex. This splitting is known as the crystal field splitting, which can affect the color and other properties of the complex.
Ligands are ranked from those that cause smaller splitting energies (weak-field ligands) to those causing larger splitting energies (strong-field ligands). Here are some examples:

• Weak-field ligands: I^-, Br^-, Cl^-, F^-
• Intermediate-field ligands: H₂O, NH₃
• Strong-field ligands: CN^-, CO

Higher splitting energies, caused by strong-field ligands, can lead to low-spin complexes, where electrons pair up in the lower energy orbitals before occupying higher ones. Conversely, weak-field ligands often form high-spin complexes, resulting in maximal unpaired electrons.
The spectrochemical series aids in predicting and rationalizing the magnetic and optical properties of coordination complexes, as well as their stability and reactivity.

Pressure Effect on Crystal Lattice

Applied pressure can significantly affect the crystal lattice of a solid, such as crystalline materials like ruby. When pressure is applied, it forces the atoms or ions in the crystal lattice closer together. This change affects the interaction between the central metal ion and the surrounding ligands in a coordination complex.
In the specific case of rubies, containing Cr(III) ions in an octahedral arrangement:

• The decrease in the metal-ligand bond distance leads to stronger electrostatic interactions.
• This increases the crystal field splitting energy (Δ₀), making the energy gap between the t₂g and e_g orbitals larger.

As Δ₀ increases with pressure, the wavelength (λ) of light absorbed by the material decreases, shifting toward shorter wavelengths according to the relationship Δ₀ = \(\dfrac{hc}{\lambda}\). This results in a possible color shift observable in the absorption spectra of the material, commonly toward blue as pressure increases. Understanding these shifts is essential for applications in material sciences and related fields.