Question

Explain why \(\mathrm{Cr}^{2+}\) forms high-spin and low-spin octahedral complexes, but \(\mathrm{Cr}^{3+}\) does not.

Short answer

\( \mathrm{Cr}^{2+} \) can have variable spin states; \( \mathrm{Cr}^{3+} \) cannot. Thus, \( \mathrm{Cr}^{2+} \) forms both high- and low-spin complexes, while \( \mathrm{Cr}^{3+} \) does not.

Step-by-step solution

Understanding High-Spin and Low-Spin Complexes

High-spin and low-spin complexes refer to the arrangement of electrons in the d-orbitals when a metal ion such as chromium forms a complex with ligands. In octahedral complexes, these d-orbitals split into a set of three lower-energy orbitals (t2g) and a set of two higher-energy orbitals (eg). Whether a complex is high-spin or low-spin depends on the relative strengths of the crystal field splitting energy (Δ_oct) and the pairing energy.

Analyze the Electron Configuration of \\( \mathrm{Cr}^{2+}\\)

Chromium(II) ion, \(\mathrm{Cr}^{2+}, \) has the electron configuration \[ \left[ \mathrm{Ar} \right] 3d^4. \] In an octahedral field, these four electrons can either pair up in the lower-energy t2g orbitals (low-spin) or fill the higher-energy eg orbitals singly as well (high-spin) when the crystal field splitting energy is not very large. Therefore, both high-spin and low-spin states are possible for \(\mathrm{Cr}^{2+}\) based on ligand field strength.

Analyze the Electron Configuration of \\( \mathrm{Cr}^{3+}\\)

Chromium(III) ion, \( \mathrm{Cr}^{3+},\) has the electron configuration \[ \left[ \mathrm{Ar} \right] 3d^3. \] With only three electrons to place, they all occupy the lower-energy t2g orbitals. There is no electron pairing involved, and adding more electrons to the higher-energy eg orbitals doesn't occur because all electrons are unpaired in the t2g set. Thus, \( \mathrm{Cr}^{3+}\) doesn't face a high-spin/low-spin situation since it only forms a single configuration.

Conclusion from Analysis

The ability to form both high-spin and low-spin complexes for \(\mathrm{Cr}^{2+}\) arises from having the choice between filling or pairing electrons in the t2g or eg orbitals. In contrast, \(\mathrm{Cr}^{3+}\) automatically populates the t2g orbitals without any oppportunity for variation, thus only allowing one configuration.

Key concepts

Crystal Field Theory

Crystal Field Theory is a model used to understand the behavior of electrons in transition metal complexes. When a metal ion forms a complex, the surrounding ligands generate an electric field. This field influences the distribution of the d-electrons in the metal ion. In octahedral complexes, this field causes the five d-orbitals to split into two levels: the lower-energy set known as t2g and the higher-energy set called eg. The energy difference between these two sets is known as the crystal field splitting energy, denoted by \( \Delta_{\text{oct}} \). Whether a complex is high-spin or low-spin depends significantly on \( \Delta_{\text{oct}} \).

• If ligands are weak field, they cause a small \( \Delta_{\text{oct}} \), leading to high-spin complexes where electrons remain unpaired in the t2g and eg orbitals, minimizing pairing.
• Conversely, strong field ligands induce a larger \( \Delta_{\text{oct}} \), resulting in low-spin complexes where electrons pair up in the t2g orbitals to lower energy.

Understanding this splitting is crucial for predicting the magnetic properties and color of coordination complexes.

Electron Configuration

The electron configuration of an ion reflects its distribution of electrons among the various atomic orbitals. For transition metal ions like chromium, this distribution particularly concerns the d-orbitals due to their involvement in complex formation. The \( \text{Cr}^{2+} \) ion, for instance, has an electron configuration of \[ \left[ \text{Ar} \right] 3d^4. \] In an octahedral complex, these four electrons need to be distributed among the two energy levels formed by crystal field splitting. Depending on the ligand field strength, these electrons may pair up in the lower-energy t2g orbitals or occupy the higher-energy eg orbitals separately. In contrast, the \( \text{Cr}^{3+} \) ion has an electron configuration of \[ \left[ \text{Ar} \right] 3d^3. \] With only three electrons, they occupy the t2g orbitals entirely without any pairing in eg, since there's no need for electron accommodation in the higher energy set. This difference highlights how varying electron configurations influence the formation of high-spin and low-spin complexes.

High-spin and Low-spin Complexes

High-spin and low-spin complexes are determined by how electrons are arranged in the d-orbitals under the influence of ligands. Whether a complex assumes a high-spin or low-spin configuration relies on the balance between the crystal field splitting energy (\( \Delta_{\text{oct}} \)) and the pairing energy needed to pair electrons within orbitals.

• In high-spin complexes, weak field ligands cause smaller crystal field splitting. Electrons remain unpaired in the t2g and eg levels, minimizing energy used in electron pairing. This typically results in complexes with more unpaired electrons and higher magnetic properties.
• Low-spin complexes occur with strong field ligands that have large splitting energy. Electrons pair up in the lower-energy t2g orbitals to save energy, resulting in fewer unpaired electrons and lower magnetic properties.

The distinct behaviors of \( \text{Cr}^{2+} \) and \( \text{Cr}^{3+} \) in forming high-spin and low-spin complexes can be understood through these principles. While \( \text{Cr}^{2+} \) has the variability of filling or pairing its d-electrons, \( \text{Cr}^{3+} \) does not need this balance assessment since its electrons naturally fit within the low-energy t2g set without the possibility of rearrangement.