Question

The complex \(\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}\) has five unpaired electrons, whereas \(\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{4-}\) has only one. Using the ligand field model, depict the electron configuration for each ion. What can you conclude about the effects of the different ligands on the magnitude of \(\Delta_{0} ?\)

Short answer

Cyanide causes larger splitting \\(\Delta_0\\) due to its strong field nature, unlike water, a weak field ligand.

Step-by-step solution

Identify central metal and electronic configuration

Both complexes have manganese as the central metal. Manganese in \(\left[\mathrm{Mn}\left(\mathrm{H}_{2}\mathrm{O}\right)_{6}\right]^{2+}\) is in the +2 oxidation state, corresponding to an electronic configuration of \[\mathrm{Mn}^{2+}: \ [\mathrm{Ar}]\,3d^5\]. Similarly, in \(\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{4-}\), manganese is in the +4 oxidation state, with an electronic configuration of \[\mathrm{Mn}^{4+}: \ [\mathrm{Ar}]\,3d^3\].

Analyze the octahedral field splitting

For both complexes, manganese is surrounded by six ligands forming an octahedral field. In an octahedral field, the five d-orbitals of the 3d shell split into two groups: the lower energy \(t_{2g}\) (dxy, dyz, dxz) and higher energy \(e_g\) (dx^2-y^2, dz^2) orbitals.

Electron configuration of \\(\left[\mathrm{Mn}\left(\mathrm{H}_{2}\mathrm{O}\right)_{6}\right]^{2+}\\)

Water is a weak field ligand, hence the 5 unpaired electrons in the \(3d^5\) configuration of \(\mathrm{Mn}^{2+}\) do not pair up, remaining as high spin. The electron configuration is \(t_{2g}^3 e_g^2\), with each electron in a separate d-orbital, resulting in five unpaired electrons.

Electron configuration of \\(\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{4-}\\)

Cyanide is a strong field ligand, causing a larger splitting \(\Delta_0\) in the d-orbitals. As a result, the 3 electrons in the \(3d^3\) configuration for \(\mathrm{Mn}^{4+}\) pair up in the lower energy \(t_{2g}\) orbitals as \(t_{2g}^3 e_g^0\), leaving one unpaired electron.

Conclusion about ligand effects on \\(\Delta_0\\)

The significant difference in the number of unpaired electrons indicates that cyanide, acting as a strong field ligand, leads to larger splitting \(\Delta_0\), causing pairing of 3d electrons. Water, as a weak field ligand, results in a smaller \(\Delta_0\) with no pairing, hence more unpaired electrons.

Key concepts

Electron Configuration

Understanding electron configuration is crucial when studying transition metal complexes. It helps to predict properties such as magnetism and color. In transition metals, electrons are arranged in shells, subshells, and orbitals. Each transition metal element has a unique electron configuration that defines how many electrons occupy each orbital.

For manganese, found in the complexes like \([\mathrm{Mn}(\mathrm{H}_2\mathrm{O})_6]^{2+}\) and \([\mathrm{Mn}(\mathrm{CN})_6]^{4-}\), the electron configuration changes with the oxidation state. For example, manganese with a +2 oxidation state has the configuration \[\mathrm{Mn}^{2+}: \ [\mathrm{Ar}]\,3d^5\]. This means the electrons from its 3d orbitals are occupying five different places since no electrons have paired up.

In contrast, for the +4 oxidation state, the configuration changes to \[\mathrm{Mn}^{4+}: \ [\mathrm{Ar}]\,3d^3\]. Only three electrons remain, altering the possible configurations and influences the chemical behavior of the molecule.

Oxidation State

The oxidation state of a metal in a complex provides insight into its electron distribution. It affects both the overall charge of the complex and the electron configuration of the metal.

In \([\mathrm{Mn}(\mathrm{H}_2\mathrm{O})_6]^{2+}\), the manganese has a +2 oxidation state. This means manganese has lost two electrons beyond what it holds in its neutral state. Understanding oxidation states allows us to determine the number of valence electrons and predict bonding and structure.

On the other hand, \([\mathrm{Mn}(\mathrm{CN})_6]^{4-}\) involves manganese in a +4 oxidation state. Thus, it has given away four electrons compared to its elemental state. Each increase in oxidation state typically corresponds to a decrease in the number of d-electrons, affecting the metal's physical and chemical properties.

Octahedral Complexes

In coordination chemistry, octahedral complexes are a common geometry, featuring six ligand atoms equally spaced around a central metal atom, like manganese in our examples. The ligands donate electron pairs to the metal, forming coordinate bonds.

The octahedral shape influences the splitting of d-orbitals. \([\mathrm{Mn}(\mathrm{H}_2\mathrm{O})_6]^{2+}\) and \([\mathrm{Mn}(\mathrm{CN})_6]^{4-}\) each exhibit octahedral structure, yet their ligand types significantly affect their properties. The spatial arrangement determines how ligands interact with the d-orbitals, leading to different levels of stabilization and energy changes.

• For weak field ligands like water, the energy difference between d-orbitals is small, which often results in a high spin arrangement.
• Strong field ligands like cyanide cause a greater energy difference, leading to low spin configurations.

d-Orbital Splitting

The phenomenon of d-orbital splitting is central to the formation of color and magnetic properties in metal complexes. In an octahedral crystal field, the approach of ligands causes the degeneration of the five d-orbitals.

These d-orbitals split into two different energy levels:

• The lower energy set, known as \( t_{2g} \), includes the d_{xy}, d_{yz}, and d_{xz} orbitals.
• The higher energy set, the \( e_g \) level, includes the d_{x^2-y^2} and d_{z^2} orbitals.

The difference in energy between these two sets is denoted as \( \Delta_0 \).

Whether the electrons occupy the high energy \( e_g \) orbitals or not depends on the ligand strength:

• Weak field ligands, like water in \([\mathrm{Mn}(\mathrm{H}_2\mathrm{O})_6]^{2+}\), result in smaller \( \Delta_0 \), leaving unpaired electrons in different d-orbitals due to high spin condition.
• Strong field ligands, such as cyanide in \([\mathrm{Mn}(\mathrm{CN})_6]^{4-}\), create a larger \( \Delta_0 \), leading to the pairing of electrons in the \( t_{2g} \) orbitals, hence fewer unpaired electrons.