Problem 70 - Chemistry: The Central Science

Question

Suppose that a transition-metal ion was in a lattice in which it was in contact with just two nearby anions, located on opposite sides of the metal. Diagram the splitting of the metal \(d\) orbitals that would result from such a crystal field. Assuming a strong field, how many unpaired electrons would you expect for a metal ion with six \(d\) electrons? (Hint: Consider the linear axis to be the z-axis).

Step-by-Step Solution

Answer

In a strong linear crystal field, the d orbitals split into two energy levels: d_z^2 being the highest in energy, and the remaining d orbitals (d_xy, d_xz, d_yz, d_x^2 - y^2) being lower in energy. For a metal ion with six d electrons, they will fill lower energy orbitals first, resulting in four unpaired electrons in d_xy, d_xz, d_yz, and d_x^2 - y^2 orbitals, and two paired electrons in d_z^2 orbital. Therefore, there will be four unpaired electrons.

Step 1: Understand crystal field splitting in a linear axis.

In a linear crystal field, the anions are located along the axis connecting the ligands, which we consider to be the z-axis. In this case, a transition-metal ion has two ligands on opposite sides along the z-axis.

Step 2: Analyze how d orbitals are affected by the splitting.

There are five d orbitals, which are d_xy, d_xz, d_yz, d_x^2 - y^2, and d_z^2. In a linear crystal field, the anions affect d orbitals according to their orientation with respect to the z-axis. The d_z^2 is directly on the z-axis, whereas d_x^2 - y^2, d_xy, d_xz, and d_yz are perpendicular to the z-axis.

Step 3: Determine the new energy levels of the d orbitals after the splitting.

Due to the crystal field splitting, the d_z^2 orbital will move to a higher energy level because it is directly along the z-axis, where the anions are located. The other four d orbitals will remain at the same energy level because they are not directly aligned with the z-axis. After the splitting, the energy levels will look like this: the d_z^2 orbital is the highest in energy, and the other d orbitals (d_xy, d_xz, d_yz, d_x^2 - y^2) are lower in energy.

Step 4: Distribute the six d electrons considering a strong crystal field.

In a strong crystal field, electrons will first fill in the lower energy d orbitals before moving to the higher energy d_z^2 orbital. Since there are four lower energy d orbitals (d_xy, d_xz, d_yz, d_x^2 - y^2), the six d electrons will be distributed among them as follows: 1st d electron -> d_xy 2nd d electron -> d_xz 3rd d electron -> d_yz 4th d electron -> d_x^2 - y^2 5th d electron -> d_z^2 6th d electron -> d_z^2

Step 5: Count the unpaired electrons.

To determine the number of unpaired electrons, we need to count the number of d orbitals with only one electron. In this case, there are four unpaired electrons: one each in d_xy, d_xz, d_yz, and d_x^2 - y^2 orbitals. The d_z^2 orbital contains two electrons, so they are paired. So, for a transition-metal ion with six d electrons in a strong linear crystal field, there would be four unpaired electrons.

Key Concepts

d orbital splitting

In transition-metal complexes, the phenomenon of d orbital splitting occurs due to the interaction between the metal ion and surrounding ligands. When ligands approach an ion, they exert an electrostatic force that affects the energy levels of the d orbitals. These five d orbitals, typically degenerate (having the same energy), split into different energy levels depending on their orientation relative to the ligands.
In the case of a linear crystal field, where ligands are along the z-axis as stated in the exercise, the d orbitals experience a unique splitting pattern. The orbital oriented along the z-axis, namely the \(d_{z^2}\) orbital, is pushed to a higher energy level since it directly interacts with the ligands. The other orbitals such as \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), and \(d_{x^2 - y^2}\) remain at a lower energy level as they are not lined up along the z-axis.
Understanding d orbital splitting helps in predicting the behavior and properties of transition-metal complexes. This splitting is crucial for determining magnetic properties and the color of the complexes.

transition-metal ions

Transition-metal ions are elements found in the d-block of the periodic table. They possess partially filled d orbitals and are known for their ability to form various complex ions. A unique feature of transition-metal ions is their ability to exhibit different oxidation states, which allows them to participate in a wide range of chemical reactions.
These ions typically engage in coordination chemistry, where they bind to molecules or ions called ligands, forming a coordination complex. The typical atom found in the center of these complexes often has a coordinate bond with the ligands due to its d orbital. This leads to various properties such as magnetic behavior and distinct color changes in solutions.
Transition-metal ions like to make life interesting! Their complex chemistry is due to their ability to adapt their oxidation states and coordinate numbers, allowing them to form a variety of structures and functionalities in solutions.

unpaired electrons

Unpaired electrons in transition-metal ions are significant because they determine the magnetic properties of a compound. Generally, the number of unpaired electrons in a complex influences whether it is paramagnetic or diamagnetic.
Paramagnetic materials have one or more unpaired electrons, which means they are attracted to an external magnetic field. On the other hand, diamagnetic materials have all electrons paired up, and hence, they are weakly repelled by magnetic fields.
For example, consider the exercise’s metal ion with six d electrons subjected to a strong crystal field. The electrons initially fill the lower energy d orbitals first. If there are degenerate orbitals and less energy barrier due to field splitting, these electrons will pair up, reducing the overall number of unpaired electrons. In a strong field, this results in a maximum of four unpaired electrons out of six d electrons.
Understanding the number and behavior of unpaired electrons in transition-metal complexes allows chemists to predict and explain the magnetic nature of compounds, which is central to numerous applications in modern science.