Question

For a tetrahedral complex of a metal in the first transition series, which of the following statements concerning energies of the \(3 d\) orbitals is correct? (a) The five \(d\) orbitals have the same energy. (b) The \(d_{x^{2}-y^{2}}\) and \(d_{z^{2}}\) orbitals are higher in energy than the \(d_{x z}, d_{y z},\) and \(d_{x y}\) orbitals. (c) The \(d_{x z}, d_{y z},\) and \(d_{x y}\) orbitals are higher in energy than the \(d_{x^{2} y^{2}}\) and \(d_{z^{2}}\) orbitals. (d) The \(d\) orbitals all have different energies.

Short answer

Option (c) is correct: \(d_{x z}, d_{y z},\) and \(d_{x y}\) orbitals are higher in energy.

Step-by-step solution

Understand Tetrahedral Splitting

In a tetrahedral complex, the crystal field splitting of the metal's 3d orbitals means that they are grouped into two sets with different energies, due to the symmetry of the tetrahedral field.

Identify the d-Orbital Groupings

The three orbitals, \(d_{x z}, d_{y z},\) and \(d_{x y}\), are known as the t\(_2\) orbitals, and the two orbitals \(d_{x^{2} - y^{2}}\) and \(d_{z^{2}}\) are known as the e orbitals.

Determine Energies of the d-Orbitals

In a tetrahedral field, the t\(_2\) orbitals have higher energy than the e orbitals, as the e orbitals are destabilized less by the ligand field.

Match Statement with Correct Energy Ordering

According to the energy splitting described, option (c) is correct: The \(d_{x z}, d_{y z},\) and \(d_{x y}\) orbitals are higher in energy than the \(d_{x^{2} y^{2}}\) and \(d_{z^{2}}\) orbitals.

Key concepts

Crystal Field Splitting

Crystal Field Splitting is a crucial concept in understanding the behavior of transition metal complexes. When transitioning metals form complexes with ligands, they create fields that affect the energies of the metal's d orbitals. These changes in energy levels are referred to as crystal field splitting.
In a tetrahedral complex, ligands approach the central metal from directions between the principal axes. This orientation causes the crystal field splitting. In this scenario, the d orbitals split into two sets: t\(_2\) and e orbitals.
The t\(_2\) set—comprised of orbitals such as \(d_{xz}\), \(d_{yz}\), and \(d_{xy}\)—experiences a higher increase in energy compared to the e set, which includes \(d_{x^2-y^2}\) and \(d_{z^2}\).
This energy difference is less pronounced than in octahedral complexes, thus leading to the particular characteristics observed in many tetrahedral complex reactions and properties.

3d Orbitals

The 3d orbitals are a set of five orbitals found in transition metals, labeled as \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{x^2-y^2}\), and \(d_{z^2}\). These orbitals play significant roles because they directly participate in bonding and determine the properties of the complex.
In transition metals, the 3d orbitals are partially filled. This partial filling allows for a wide range of possibilities in terms of electron configurations. Each orbital can hold up to two electrons, distinguished by their spin. The energy and configuration of these orbitals are affected by the ligand field created in a complex.

• The t\(_2\) orbitals—\(d_{xy}\), \(d_{yz}\), and \(d_{xz}\)—have lobes orientated between the axes.
• In contrast, the e orbitals—\(d_{x^2-y^2}\) and \(d_{z^2}\)—point along the axes.

These orientations impact how the orbitals interact with the incoming ligands, dictating the energy level changes.

Transition Metals

Transition Metals are elements that have partially filled d orbitals. They occupy the central block of the periodic table and include metals like iron, copper, and nickel. Unique properties such as color, magnetism, and multiple oxidation states can be attributed to their d electrons.
Their ability to form complex ions is due to these d orbitals. Transition metals can participate in a variety of coordination environments because of the different spatial orientations and energies of these orbitals.

• They often show catalytic behavior due to the presence of unpaired d electrons.
• Transition metals can form either tetrahedral or octahedral complexes, depending on the ligand environment and the metal's electronic structure.

These properties make them incredibly useful in industrial and biological processes.

Orbital Energy Levels

Orbital Energy Levels refer to the specific energies associated with an atom's orbitals. When a transition metal forms a complex, these energy levels are split due to interaction with the ligands.
In the context of a tetrahedral complex, the splitting pattern shows that the t\(_2\) orbitals increase in energy more than the e orbitals. This splitting effectively changes the way electrons are distributed among these orbitals.
Understanding this pattern is essential because it influences various properties of the complex, such as stability and color. The difference between the energy levels of these orbitals is called the crystal field splitting energy \(\Delta_t\).
This energy difference is crucial in determining the electronic transitions, which can be observed spectroscopically, providing insights into the complex's structure and reactivity.