Question

For the crystal field splitting in octahedral complexes,

• A. the energy of the e_g orbitals will decrease by (3/5)Δ₀ and that of the t_2g will increase by (2/5)Δ₀
• B. the energy of the e_g orbitals will increase by (3/5)Δ₀ and that of the t_2g will decrease by (2/5)Δ₀
• C. the energy of the e_g orbitals will increase by (3/5)Δ₀ and that of the t_2g will increase by (2/5)Δ₀
• D. the energy of the e_g orbitals will decrease by (3/5)Δ₀ and that of the t_2g will decrease by (2/5)Δ₀

Answer 1

The Correct Option is B

To solve this problem, we need to understand the crystal field splitting in octahedral complexes and how the energy levels of the \( e_g \) and \( t_{2g} \) orbitals are affected by the ligand field.

1. Crystal Field Theory (CFT) and Octahedral Complexes:
In an octahedral crystal field, the metal ion is surrounded by six ligands placed at the vertices of an octahedron. The d-orbitals of the central metal ion split into two sets due to the ligand field:

• The \( t_{2g} \) set, which consists of the \( d_{xy} \), \( d_{xz} \), and \( d_{yz} \) orbitals.
• The \( e_g \) set, which consists of the \( d_{z^2} \) and \( d_{x^2 - y^2} \) orbitals.

2. Splitting of the Orbitals:
In an octahedral field, the ligands cause the \( e_g \) orbitals to be raised in energy and the \( t_{2g} \) orbitals to be lowered in energy. The amount of splitting is quantified by \( \Delta_0 \), the octahedral crystal field splitting energy. The energy difference between the two sets of orbitals is \( \Delta_0 \). The \( e_g \) orbitals experience more repulsion from the ligands and hence are raised by \( \frac{3}{5} \Delta_0 \), while the \( t_{2g} \) orbitals experience less repulsion and are lowered by \( \frac{2}{5} \Delta_0 \).

3. Analyzing the Options:
- Option (A) "The energy of the \( e_g \) orbitals will decrease by \( \frac{3}{5} \Delta_0 \) and that of the \( t_{2g} \) will increase by \( \frac{2}{5} \Delta_0 \)": This is incorrect because the \( e_g \) orbitals are raised in energy, not decreased, and the \( t_{2g} \) orbitals are lowered, not increased. 

- Option (B) "The energy of the \( e_g \) orbitals will increase by \( \frac{3}{5} \Delta_0 \) and that of the \( t_{2g} \) will decrease by \( \frac{2}{5} \Delta_0 \)": This is correct. The \( e_g \) orbitals are raised by \( \frac{3}{5} \Delta_0 \), and the \( t_{2g} \) orbitals are lowered by \( \frac{2}{5} \Delta_0 \). 

- Option (C) "The energy of the \( e_g \) orbitals will increase by \( \frac{3}{5} \Delta_0 \) and that of the \( t_{2g} \) will increase by \( \frac{2}{5} \Delta_0 \)": This is incorrect because the \( t_{2g} \) orbitals are lowered in energy, not raised. 

- Option (D) "The energy of the \( e_g \) orbitals will decrease by \( \frac{3}{5} \Delta_0 \) and that of the \( t_{2g} \) will decrease by \( \frac{2}{5} \Delta_0 \)": This is incorrect because both sets of orbitals do not decrease in energy; the \( e_g \) orbitals are raised in energy, not decreased.

Final Answer:
The correct answer is (B) "The energy of the \( e_g \) orbitals will increase by \( \frac{3}{5} \Delta_0 \) and that of the \( t_{2g} \) will decrease by \( \frac{2}{5} \Delta_0 \)."

Answer 2

In octahedral crystal field splitting, the degeneracy of the d-orbitals is lifted, resulting in the splitting of the d-orbitals into two sets:

• e_g orbitals (higher energy orbitals) which are directed along the axes.
• t_2g orbitals (lower energy orbitals) which are between the axes.

The splitting energy difference between these two sets is denoted by Δ₀. In the case of an octahedral field:

• The e_g orbitals will have higher energy and increase by (3/5)Δ₀.
• The t_2g orbitals will have lower energy and decrease by (2/5)Δ₀.

The correct answer is (B) : the energy of the e_g orbitals will increase by (3/5)Δ₀ and that of the t_2g will decrease by (2/5)Δ₀.