Question

Tetrahedral complexes of \(\mathrm{Co}^{2+}\) are quite common. Use a \(d\) -orbital splitting diagram to rationalize the stability of \(\mathrm{Co}^{2+}\) tetrahedral complex ions.

Short answer

In tetrahedral complexes of \(\mathrm{Co}^{2+}\), the d-orbitals are split into two groups: \(e_g\) and \(t_{2g}\). The 3d electron configuration of \(\mathrm{Co}^{2+}\) is \(t_{2g}^5\, e_g^2\), in which 5 electrons stabilize in the \(t_{2g}\) orbitals and 2 electrons destabilize in the \(e_g\) orbitals. The stabilization of electrons in the \(t_{2g}\) orbitals outweighs the destabilization in the \(e_g\) orbitals, resulting in an overall stable complex. Thus, the d-orbital splitting diagram helps to rationalize the stability of tetrahedral complexes of \(\mathrm{Co}^{2+}\).

Step-by-step solution

Determine the electron configuration of \(\mathrm{Co}^{2+}\)

Cobalt (Co) has an atomic number of 27, which gives it an electron configuration of \([\mathrm{Ar}]\, 3d^7\, 4s^2\). In the +2 oxidation state, Co loses 2 electrons from its 4s orbital, resulting in the electron configuration of \(\mathrm{Co}^{2+}\) as \([\mathrm{Ar}]\, 3d^7\).

Understand the d-orbital splitting in a tetrahedral crystal field

In a tetrahedral crystal field, the d-orbitals are split into two groups: \(e_g\) and \(t_{2g}\) (please note that this naming convention differs from the one in octahedral complexes). The \(e_g\) group consists of the d_xy, d_xz, and d_yz orbitals. The \(t_{2_g}\) group contains the d_x^2-y^2 and d_z^2 orbitals. The splitting energy in tetrahedral complexes is labeled as \(-\frac{2}{5}\Delta_\textrm{t}\), where \(\Delta_\textrm{t}\) is the tetrahedral splitting energy. The \(t_{2g}\) orbitals are lower in energy and are stabilized, while the \(e_g\) orbitals are raised in energy and are destabilized.

Fill the 3d orbitals with the electrons of \(\mathrm{Co}^{2+}\)

As the electron configuration of \(\mathrm{Co}^{2+}\) is \([\mathrm{Ar}]\, 3d^7\), we have 7 electrons to fill the 3d orbitals. Following Hund's rule and the Aufbau principle, we fill the orbitals in the following order: \(t_{2g}\) (lower energy) before \(e_g\) (higher energy). As there are only 5 d-orbitals, we will fill the \(t_{2g}\) completely with 5 electrons, leaving 2 electrons to be filled in the \(e_g\) orbitals.

Analyze the electron configuration in the d-orbital splitting diagram

On the tetrahedral d-orbital splitting diagram, the electron configuration of \(\mathrm{Co}^{2+}\) will be filled as follows: \[t_{2_g}^5\, e_g^2\] This indicates that 5 electrons stabilize in the \(t_{2g}\) orbitals and 2 electrons destabilize in the \(e_g\) orbitals. The stabilization of the \(t_{2g}\) electrons overcompensates the destabilization of the \(e_g\) electrons, making the overall complex stable. In conclusion, the tetrahedral complexes of \(\mathrm{Co}^{2+}\) are stable, mainly due to the stabilization of electrons in the \(t_{2g}\) orbitals, which outweighs the destabilization of electrons in the \(e_g\) orbitals. The d-orbital splitting diagram helps to rationalize this stability.

Key concepts

Tetrahedral Crystal Field Theory

Tetrahedral crystal field theory is a part of crystal field theory specifically focused on explaining the behaviors of metal ions in tetrahedral coordination complexes. In the tetrahedral environment, a central metal ion is surrounded asymmetrically by four ligands located at the corners of a tetrahedron. This arrangement has a significant effect on the energy levels of the metal ion's d-orbitals. Unlike in an octahedral field where the d-orbitals split into two groups with the higher energy set being called the eg and the lower energy set being called the t2g, the energy splitting occurs in reverse for a tetrahedral field.

The two orbitals that point directly towards the corners of the tetrahedron (dx^2-y^2 and dz^2) are now part of the t2g set, and they experience less repulsion and thus are lower in energy. Conversely, the three orbitals that lie between the axes (dxy, dxz, and dyz), now called the eg set, point away from the ligands and experience more repulsion, which raises their energy. The difference in energy between these two sets is represented by the tetrahedral splitting energy, denoted as \(\Delta_t\). The relative energy decrease is about \(\frac{-2}{5}\) of the octahedral crystal field splitting. This concept is essential for predicting the stability and color of the tetrahedral metal complexes.

To sum up, the tetrahedral crystal field theory helps us understand why certain orbitals occupy lower energy levels, which directly influences electron configuration and, ultimately, the stability and properties of complexes.

3d Orbitals Electron Configuration

The 3d orbitals are a set of five orbitals - dxy, dxz, dyz, dx^2-y^2, and dz^2 - that are found in the third shell of a transition element and can contain a maximum of ten electrons in total. For cobalt (Co), which is a transition metal, the electron configuration is vital for interpreting its chemical behavior, especially when it becomes a part of a complex.

When electrons occupy these d-orbitals, they must follow certain principles, such as the Aufbau principle, stating that electrons fill orbitals starting with the lowest available energy levels before moving to higher ones. Additionally, Hund's rule mandates that electrons with the same spin occupy different orbitals within the same energy level before they pair up. These principles guide how electrons are distributed among the 3d orbitals in a metal ion.

For a Co^2+ ion, with seven electrons in its 3d orbitals after losing two from 4s, the filling of these orbitals will follow the observed splitting pattern in a tetrahedral field, thus affecting its properties and behavior in a complex. These principles and an understanding of d-orbital splitting underpin the electron configuration in such complexes and are fundamental to predicting their chemical behavior and stability.

Co2+ Complex Stability

The stability of the Co^2+ complex can be reasoned using the d-orbital splitting diagram and the principles of crystal field theory. When forming a complex, the cobalt ion is influenced by the field created by the surrounding ligands. In a tetrahedral complex, stabilizing and destabilizing forces come into play, altering the distribution and energy of the d-electrons.

As we've learned from the principles of d-orbital splitting, in a tetrahedral complex, the t2g orbitals (dx^2-y^2 and dz^2) contain fewer electrons because they have lower energy compared to the eg orbitals (dxy, dxz, and dyz). With Co^2+ (which has a 3d^7 electron configuration), all three lower energy t2g orbitals are filled with five electrons, and the remaining two electrons occupy the higher energy eg orbitals. The result is that the overall energy of the system is lowered because the total energy saved by filling the three lower-energy orbitals is greater than the energy cost of placing two electrons in the higher-energy orbitals.

Moreover, the inherent pairing energy also plays a role in the complex's stability. When additional electrons are added to already half-filled orbitals, they pair up with existing electrons, increasing the total energy of the system. The Co^2+ with its specific electron configuration in a tetrahedral environment manages to minimize this pairing energy, which contributes to the overall stability of the complex. Thus, the tetrahedral Co^2+ complexes are stable due to the particular way the d-electrons are distributed according to the splitting diagram provided by the crystal field theory.