Question

The red color of ruby is due to the presence of Cr(III) ions at octahedral sites in the close-packed oxide lattice of \(\mathrm{Al}_{2} \mathrm{O}_{3} .\) Draw the crystal-field splitting diagram for 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.

Short answer

-y^2). #tag_title# Step 2: Drawing the Crystal-Field Splitting Diagram for Cr(III) #tag_content# In an octahedral environment, the Cr(III) ion has a 3d^3 electronic configuration. After the splitting of energy levels, the three lower energy orbitals (t2g) are filled, and the two higher energy orbitals (eg) are unoccupied. Below is the crystal-field splitting diagram: ``` eg: ↑ | | t2g: ↑(dxy) ↑(dyz) ↑(dxz) ``` #tag_title# Step 3: Predicting Variation in Wavelength of Absorption with Pressure #tag_content# When pressure is increased, the size of the Al_2O_3 lattice is reduced. This causes the Cr(III) ion to experience a stronger crystal field where the energy difference between t2g and eg orbitals increases. According to the equation \(\Delta E = h c / \lambda\), where \(\Delta E\) is the energy difference, h is Planck's constant, c is the speed of light, and \(\lambda\) is the wavelength, we observe that as the energy difference increases, the wavelength of absorption decreases. Therefore, upon increasing pressure, the wavelength of absorption of the ruby will decrease.

Step-by-step solution

Understanding Crystal Field Theory

Crystal field theory helps in understanding the behavior of transition metal ions in complex compounds. Due to the presence of ligands around the metal ion, the degenerate energy levels of the metal ion split into different energy levels. In the case of octahedral environment, the five degenerate d orbitals of the metal ion split into two groups: three lower energy orbitals (dxy, dyz, and dxz) and two higher energy orbitals (dz^2 and dx^2

Key concepts

Understanding Transition Metal Complexes

Transition metal complexes are fascinating entities where central metal atoms or ions are surrounded by molecules or ions known as ligands. These ligands donate pairs of electrons to the metal, creating a coordination complex. Transition metals are known for their ability to form a variety of complexes due to their unique electronic configurations and the presence of vacant d-orbitals.

In a typical complex, the metal and ligands interact through what's known as a coordination bond, a type of chemical bond where ligands provide both electrons to be shared with the metal atom. The coordination number, which is the number of ligand atoms that directly interact with the central metal atom, determines the overall geometry of the complex.

One of the most crucial aspects of studying transition metal complexes is to understand their electronic structure, as it plays a key role in determining the physical and chemical properties of these complexes, including color, magnetic behavior, and reactivity.

Exploring d Orbital Splitting

d orbital splitting is a fundamental concept in understanding the electronic structure of transition metal complexes. When a transition metal forms a complex, the degenerate d orbitals (which have the same energy in the isolated atom) split into separate energy levels due to the varying extent of interaction with the ligands. This phenomenon is elegantly explained by the crystal field theory.

In an octahedral complex, for example, the six ligands approach the metal ion along the axes of the octahedron, causing the d orbitals to experience different electrostatic forces. As a result, the energy levels split into two sets: the lower energy triply degenerate set (t2g) comprising the dxy, dyz, and dxz orbitals, and the higher energy doubly degenerate set (eg) comprising the dz^2 and dx^2-y^2 orbitals. The energy gap between these sets, known as crystal field splitting energy \(\Delta_o\), is crucial as it determines the absorption of light and, consequently, the color of the complex. The extent of this splitting is influenced by factors such as the nature of the ligands and the geometric arrangement around the metal ion.

The Ruby Crystal Structure

The exquisite red color of a ruby crystal is a direct result of its complex structure: a form of aluminum oxide (\(\mathrm{Al}_2 \mathrm{O}_3\)) known as corundum, infused with trace amounts of chromium (III) ions - a prime example of a transition metal complex. In this structure, chromium ions substitute for some of the aluminum ions in the lattice, occupying octahedral sites and forming a distorted octahedral complex with the surrounding oxygen atoms.

Under high pressure, we can expect changes to occur within this structured environment. Pressure would cause a distortion in the ruby's crystal lattice, altering the distances between ions and, therefore, changing the energy levels of the d orbitals of the chromium ions. As pressure is applied, the crystal field splitting \(\Delta_o\) can either increase or decrease, depending on how the crystal structure adapts to the stress, often leading to a shift in the wavelength of light absorbed. This essentially leads to a change in color perception, which can be observed as a shift in the ruby's red hue with variations in pressure.