Problem 76 - Chemistry The Molecular Science

Question

Use crystal-field theory to explain why some \(\mathrm{Co}^{3+}\) octahedral complexes are diamagnetic and others are paramagnetic.

Step-by-Step Solution

Answer

Co³⁺ complexes are diamagnetic with strong field ligands and paramagnetic with weak field ligands.

Step 1: Understand Crystal Field Theory

Crystal Field Theory (CFT) explains the electronic structure of metal complexes by considering the effect of ligand electric fields on the metal's d-orbitals. In octahedral complexes, the d-orbitals split into two sets: the lower-energy t₂g set and the higher-energy eₙ set.

Step 2: Identify Co3+ Electron Configuration

For Co³⁺, the electron configuration is \[\text{Co} = [\text{Ar}]3d^6\text{. For } \text{Co}^{3+}, \text{remove three electrons: } 3d^6.\] This gives us a \(3d^6\) configuration after ionization.

Step 3: Determine Splitting in Octahedral Field

In an octahedral field, the 3d orbitals split into two groups: three lower-energy t₂g orbitals and two higher-energy eₙ orbitals. The extent of splitting is represented by the crystal field splitting energy (Δ).

Step 4: Differentiate High-Spin and Low-Spin States

If Δ is small, electrons prefer to occupy higher energy eₙ orbitals ('high-spin'), resulting in unpaired electrons and thus paramagnetism. If Δ is large, electrons pair within t₂g orbitals ('low-spin'), leading to all electrons pair, resulting in diamagnetism.

Step 5: Apply to Co3+ Complexes

For a \(\text{Co}^{3+} 3d^6\) complex, in a high-spin case, the arrangement is \[t_{2g}^4 e_{g}^2\text{ with four unpaired electrons, making it paramagnetic.}\] In a low-spin case, the configuration is \[t_{2g}^6 e_{g}^0\text{ with all paired electrons, making it diamagnetic.}\]

Step 6: Conclusion

The nature of the ligands influences the strength of Δ. Strong field ligands cause low-spin configurations (diamagnetic), while weak field ligands lead to high-spin configurations (paramagnetic).

Key Concepts

Co3+ octahedral complexes

In coordination chemistry, the study of complex compounds involving transition metals, Co\(^{3+}\) octahedral complexes are quite prominent. These complexes are formed when cobalt ions bond with six surrounding ligands in a specific geometrical shape, commonly recognized as an octahedral arrangement. Understanding the electronic configuration of cobalt is crucial here. Cobalt in its neutral state has an electron configuration of \([ ext{Ar}] 4s^2 3d^7\). When it loses three electrons to become Co\(^{3+}\), the configuration becomes \(3d^6\).

In an octahedral complex, the Co\(^{3+}\) ion interacts with ligand electric fields due to the surrounding six ligands. Crystal field theory (CFT) explains how these interactions affect the electronic states of the d-orbitals of the Co\(^{3+}\) ion by causing them to split into two distinct energy levels:

• The lower-energy \(t_{2g}\) orbitals
• The higher-energy \(e_{g}\) orbitals

high-spin and low-spin states

In octahedral complexes, the concept of high-spin and low-spin states stems from the splitting of the d-orbitals due to ligand interactions. The extent of this splitting, or the crystal field splitting energy (\(\Delta\)), determines the arrangement of electrons.

• If \(\Delta\) is small (common with weak field ligands), electrons tend to occupy the higher-energy \(e_{g}\) orbitals instead of pairing in the \(t_{2g}\) orbitals, leading to a high-spin state. This results in several unpaired electrons, as the electrons fill up singly before they pair.
• Conversely, if \(\Delta\) is large (favored by strong field ligands), electrons will pair within the lower-energy \(t_{2g}\) orbitals, resulting in a low-spin state with fewer or no unpaired electrons.

For Co\(^{3+}\) complexes, if the complex is high-spin, the \(3d^6\) electrons might distribute as \(t_{2g}^4 e_{g}^2\), leading to four unpaired electrons. In contrast, a low-spin arrangement would be \(t_{2g}^6 e_{g}^0\), with all electrons paired.

paramagnetism and diamagnetism

The terms paramagnetism and diamagnetism pertain to the magnetic properties of substances, particularly influenced by their electronic configurations. In Co\(^{3+}\) octahedral complexes, these properties depend significantly on the electron distribution in the d-orbitals.

• Paramagnetic substances have unpaired electrons that cause them to be attracted to magnetic fields. For high-spin Co\(^{3+}\) complexes with \(t_{2g}^4 e_{g}^2\) configurations, four unpaired electrons create a strong paramagnetic behavior.
• Diamagnetic substances, on the other hand, have all their electrons paired. They are repelled by magnetic fields. When a Co\(^{3+}\) complex is in a low-spin state with a \(t_{2g}^6 e_{g}^0\) configuration, it is diamagnetic, as all of its electrons are paired.

These magnetic properties depend on the nature of the ligands surrounding the cobalt ion. Strong field ligands promote low-spin and therefore diamagnetic characteristics, whereas weak field ligands lead to high-spin and paramagnetic properties. Understanding these differences helps us design and predict the behavior of metal complexes in various applications.