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

The stability of the tetrahedral Co^2+ complex can be rationalized using a d-orbital splitting diagram. The Co^2+ ion has 7 d-electrons, filling 3 electrons in the lower-energy e orbitals, and 4 electrons in the higher-energy t2 orbitals. The Crystal Field Stabilization Energy (CFSE) for the Co^2+ tetrahedral complex is calculated to be \(0.6\Delta_{tet}\), which is positive, indicating that the complex is energetically favored and stable.

Step-by-step solution

Obtain the electron configuration of Co^2+

Cobalt, Co, has an atomic number of 27, and its electron configuration is [Ar] 3d^7 4s^2. When Co loses two electrons to form Co^2+, its electron configuration becomes [Ar] 3d^7.

Understand d-orbital splitting in a tetrahedral complex

In a tetrahedral complex, the d-orbitals split into two groups due to the ligands' effect. The d_xy, d_xz, and d_yz orbitals form a lower-energy group (called e set), while the d_z^2 and d_x^2-y^2 orbitals form a higher-energy group (called t2 set). The energy difference between these two groups is denoted as Δ_tet.

Calculate the Crystal Field Stabilization Energy (CFSE)

Crystal Field Stabilization Energy (CFSE) is the energy difference between the electron configuration situation in a free metal ion and the respective complex ion. For the tetrahedral complex, the stabilization energy per electron in the e orbitals is \(-\frac{2}{5}\Delta_{tet}\), and in the t2 orbitals, it is \(+\frac{3}{5}\Delta_{tet}\). To compute the CFSE, we will multiply the stabilization energies with the number of electrons in each orbital group and sum them up.

Create the d-orbital splitting diagram for Co^2+ tetrahedral complex

Since the Co^2+ ion has 7 d-electrons, we will fill in the available d-orbitals following Hund's rule and the Aufbau principle: - 3 electrons in the e orbitals (lower-energy group) - 4 electrons in the t2 orbitals (higher-energy group)

Calculate CFSE for the Co^2+ tetrahedral complex

We can now compute the CFSE for the Co^2+ tetrahedral complex using the stabilization energies and the number of electrons in each group: CFSE=\((-3\cdot \frac{2}{5} + 4\cdot \frac{3}{5})\Delta_{tet}\) = \(0.6\Delta_{tet}\) Since the CFSE value is positive, it indicates that the tetrahedral Co^2+ complex is stabilized relative to the free Co^2+ ion.

Conclusion

The stability of the tetrahedral Co^2+ complex can be rationalized using a d-orbital splitting diagram. The calculated CFSE for the complex is positive, which suggests that it is energetically favored and leads to the complex's stability.