Problem 44 - Chemistry: An Atoms-Focused Approach

Question

For which \(d^{\text {n }}\) electron configurations in a tetrahedral geometry are high spin and low spin configurations theoretically possible?

Step-by-Step Solution

Answer

Answer: High spin and low spin configurations are theoretically possible for \(d^2\) and \(d^4\) configurations in a tetrahedral geometry.

Step 1: Review the Tetrahedral Crystal Field Splitting

In a tetrahedral complex, the d orbital energy levels are split into 2 different energy levels: \(e\) and \(t_2\). The \(t_2\) orbital set, which consists of 3 orbitals (\(d_{xy}\), \(d_{xz}\), and \(d_{yz}\)), is lower in energy while the \(e\) orbital set, consisting of 2 orbitals (\(d_{z^2}\) and \(d_{x^2-y^2}\)), is higher in energy. The energy difference between these two sets is denoted by \(\Delta_t\).

Step 2: Define High Spin and Low Spin Configurations

High spin configurations occur when all orbitals are singly occupied before any is doubly occupied. This results in a higher total spin state. On the other hand, low spin configurations occur when electrons pair up in lower energy orbitals before occupying the higher energy orbitals. This results in a lower total spin state.

Step 3: Identify When High Spin and Low Spin Configurations are Possible

Tetrahedral complexes generally have a smaller energy difference between orbitals, so high spin configurations are more common. The only cases where low spin configurations can occur is when the pairing energy (the energy required to pair two electrons in the same orbital) is less than \(\Delta_t\), the energy difference between the two sets of orbitals. Let's explore the possible \(d^{\text {n }}\) electron configurations: \(d^1: e^1 t_2^0\) Both high spin and low spin configurations are the same. \(d^2: e^2 t_2^0 (LS) \text{ or } e^1 t_2^1(HS)\) Low spin and high spin configurations are possible. \(d^3: e^2 t_2^1\) Both high spin and low spin configurations are the same. \(d^4: e^2 t_2^2 (LS)\text{ or } e^3 t_2^1(HS)\) Low spin and high spin configurations are possible. \(d^5: e^2 t_2^3\) Both high spin and low spin configurations are the same. \(d^6: e^4 t_2^2\) Only high spin configuration is possible. \(d^7: e^4 t_2^3\) Only high spin configuration is possible. \(d^8: e^4 t_2^4\) Only high spin configuration is possible. \(d^9: e^5 t_2^4\) Only high spin configuration is possible. \(d^{10}: e^5 t_2^5\) Both high spin and low spin configurations are the same.

Step 4: State the Results

Based on our analysis, tetrahedral \(d^{\text {n }}\) electron configurations with high spin and low spin configurations are theoretically possible for \(d^2\) and \(d^4\).

Key Concepts

Crystal Field Splitting

In a tetrahedral geometry, crystal field splitting is a crucial concept that describes how the d orbitals are affected by interacting with the surrounding ligands. When ligands approach a metal ion in a tetrahedral pattern, they influence the energy levels of the d orbitals. Typically, the 5 d orbitals split into two energy levels:

• The lower energy level containing three orbitals, known as the \(t_2\) set \((d_{xy}, d_{xz}, d_{yz})\).
• The higher energy level containing two orbitals, known as the \(e\) set \((d_{z^2}, d_{x^2-y^2})\).

The energy gap between the \(t_2\) and \(e\) orbital sets in tetrahedral complexes is denoted by \(\Delta_t\). It's important to note that this splitting is smaller than the one in octahedral complexes due to less direct ligand interaction, making tetrahedral complexes typically have high spin configurations.

High Spin Configuration

High spin configurations are characteristic in situations where electrons fill available orbitals singly before pairing up. This behavior arises because the energy required to pair electrons in the same orbital, known as pairing energy, is greater than the energy gap \(\Delta_t\) between the \(t_2\) and \(e\) orbitals in tetrahedral complexes.

When electrons do not pair up until all orbitals are singly occupied, it results in a higher spin state. For instance, in a \(d^4\) configuration, the high spin state \(e^3 t_2^1\) has one of the electrons in the higher \(e\) orbital rather than pairing in the lower \(t_2\) orbitals.

• This configuration maximizes the number of unpaired electrons.
• High spin states are typically observed in tetrahedral complexes since \(\Delta_t\) is relatively small.

Low Spin Configuration

Low spin configurations occur when electrons pair in lower energy \(t_2\) orbitals before filling the higher \(e\) orbitals. In this scenario, the pairing energy is less than the crystal field splitting energy \(\Delta_t\).

For example, in a \(d^2\) configuration, a low spin state \(e^2 t_2^0\) would mean both electrons are paired in the \(t_2\) orbitals, resulting in fewer unpaired electrons compared to the high spin state. However, in tetrahedral complexes, the \(\Delta_t\) is often smaller than the pairing energy, typically making low spin configurations less favorable unless specific conditions affect the stability of pairing.

• Low spin configurations have fewer unpaired electrons.
• They are more common in octahedral rather than tetrahedral complexes.

Pairing Energy

Pairing energy is the energy required to pair two electrons in the same orbital. In the context of crystal field theory, it plays a pivotal role in determining whether a complex will have a high spin or low spin state.

In tetrahedral complexes, the small splitting \(\Delta_t\) between \(t_2\) and \(e\) orbitals means that pairing energy often exceeds this gap, fostering a preference for high spin configurations. Simply put, it's more energetically favorable to place electrons in separate orbitals rather than pair them. Only when the pairing energy is lower than \(\Delta_t\) will a low spin configuration become favorable, but this situation is rare in tetrahedral arrangements.

• A high pairing energy makes high spin configurations more likely.
• A low pairing energy can lead to low spin configurations, though uncommon in tetrahedral complexes.