Problem 19 - General Chemistry: Principles and Modern Applications

Question

If the ion \(\mathrm{Co}^{2+}\) is linked with strong-field ligands to produce an octahedral complex, the complex has one unpaired electron. If \(\mathrm{Co}^{2+}\) is linked with weak-field ligands, the complex has three unpaired electrons. How do you account for this difference?

Step-by-Step Solution

Answer

The difference in the number of unpaired electrons when \(\mathrm{Co}^{2+}\) is linked with strong-field versus weak-field ligands is due to the relative energy gap (\(\Delta\)) between the five d-orbitals created by these ligands. In a strong-field case, the energy gap is large and the electrons prefer to occupy higher energy orbitals rather than pair up, leading to one unpaired electron. In a weak-field case, the energy gap is less, hence, the fourth, fifth and sixth electrons occupy lower energy orbitals, resulting in three unpaired electrons.

Step 1: Explanation for Complex with Strong-Field Ligands

When a cobalt ion (\(\mathrm{Co}^{2+}\)) forms an octahedral complex with strong-field ligands, the ligands cause a large splitting of the d-orbitals. As a consequence, the energy difference (\(\Delta\)) between the d-orbitals is large. In the case of \(\mathrm{Co}^{2+}\), which has 7 d-electrons ([Ar] 3d^7 electron configuration), after 3 electrons fill the lower energy orbitals (t_2g), the next electron occupies one of the higher energy orbitals (e_g) to minimize total spin. Therefore, there will be only one unpaired electron.

Step 2: Explanation for Complex with Weak-Field Ligands

When the \(\mathrm{Co}^{2+}\) ion is bonded with weak-field ligands, the splitting of the d-orbitals is relatively small. This scenario is also known as 'high-spin' due to the greater number of unpaired electrons. Here, the two high-energy orbitals (e_g) are not much higher in energy than the three low-energy orbitals (t_2g), so the fourth electron also fills into the t_2g set. Thus, the \(\mathrm{Co}^{2+}\) ion in a weak-field complex has three unpaired electrons.

Step 3: Conclusion

Therefore, the number of unpaired electrons in an octahedral complex of \(\mathrm{Co}^{2+}\) depends on the type of ligands (strong or weak field) the ion is bonded with. This is in accordance with Crystal Field Theory, which suggests that the distribution of the d-electrons and consequently, the number of unpaired electrons, is influenced by the energy difference between the d-orbitals in a metal complex.

Key Concepts

Strong-Field Ligands

In the realm of crystal field theory, strong-field ligands play a pivotal role in determining the electron configuration of a metal complex. These ligands induce a significant splitting of the d-orbitals within a metal ion. When a cobalt ion, specifically \(\mathrm{Co}^{2+}\), is surrounded by strong-field ligands in an octahedral arrangement, it experiences a large energy difference (\(\Delta\)) between the lower energy t_{2g} orbitals and the higher energy e_{g} orbitals.

Strong-field ligands have the capacity to pair electrons within the t_{2g} set before any electron occupies the e_{g} orbitals. For \(\mathrm{Co}^{2+}\) with its \(3d^7\) electron configuration, three electrons occupy the t_{2g} orbitals, causing only one electron to remain unpaired. Thus, when linked with strong-field ligands, the complex ends up having just a single unpaired electron.

• Strong-field ligands cause large d-orbital splitting.
• Electrons pair in lower d-orbitals minimizing unpaired electrons.

Weak-Field Ligands

Weak-field ligands, on the other hand, result in much smaller splitting of the d-orbitals in a metal complex. This induced splitting is slight enough that the energy difference between the low energy t_{2g} orbitals and the high energy e_{g} orbitals is not as pronounced. Using the case of \(\mathrm{Co}^{2+}\) again, its electron pairing operates differently when weak-field ligands are present.

For weak-field ligands, electrons occupy the orbitals in such a manner to reduce the number of electron pairs and thus maintain higher spins. Consequently, in the octahedral complex of \(\mathrm{Co}^{2+}\), more electrons remain unpaired. With three unpaired electrons, such complexes are often described as 'high-spin' complexes.

• Weak-field ligands cause small d-orbital splitting.
• Result in more unpaired electrons due to less electron pairing.
• Typically form high-spin complexes.

Octahedral Complex

An octahedral complex is a common structural arrangement in coordination chemistry, characterized by a central metal ion surrounded by six ligands positioned at the vertices of an octahedron. This geometry affects how the d-orbitals of the metal ion split under the influence of ligand fields. Specifically, it results in the splitting of the d-orbitals into two distinct sets - the lower energy t_{2g} orbitals and the higher energy e_{g} orbitals.

In octahedral complexes, the arrangement and type of ligands determine whether the complex will be high-spin or low-spin. Strong-field ligands tend to form low-spin complexes by pairing electrons in the t_{2g} orbitals. Conversely, weak-field ligands allow electrons to occupy the e_{g} orbitals, often leading to high-spin configurations with increased unpaired electrons.

• Central metal ion surrounded by six ligands in an octahedral shape.
• Results in t_{2g} and e_{g} orbital splitting.
• Ligand type influences whether the complex is high-spin or low-spin.