The crystal field splitting energy for octahedral`(Delta_(0))` and tetrahedral `(Delta_(t))` complexes is related as .

Options

A. `Delta_(t)=(4)/(9)Delta_(0)`
B. `Delta_(t)=(1)/(2)Delta_(0)`
C. `Delta_(0) =-2Delta_(t)`
D. `Delta_(0)= - (4)/(9)Delta_(t)`

Answer

To solve the question regarding the relationship between the crystal field splitting energies for octahedral (\( \Delta_0 \)) and tetrahedral (\( \Delta_t \)) complexes, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Crystal Field Splitting Energy (CFSE)**:
- Crystal field splitting energy refers to the energy difference between the split d-orbitals in a coordination complex. In octahedral complexes, the d-orbitals split into two sets: \( t_{2g} \) (lower energy) and \( e_g \) (higher energy). In tetrahedral complexes, the splitting is reversed, with \( e \) (lower energy) and \( t_2 \) (higher energy).

2. **Identify the Number of Ligands**:
- In octahedral complexes, there are 6 ligands surrounding the central metal ion, while in tetrahedral complexes, there are 4 ligands. The number of ligands affects the strength of the electrostatic interactions and, consequently, the extent of splitting.

3. **Compare the Splitting Energies**:
- The presence of 6 ligands in octahedral complexes leads to a greater electrostatic interaction compared to the 4 ligands in tetrahedral complexes. This results in a larger crystal field splitting energy (\( \Delta_0 \)) for octahedral complexes than for tetrahedral complexes (\( \Delta_t \)).
- Mathematically, we can express this relationship as:
\[ \Delta_0 > \Delta_t \]

4. **Establish the Relationship**:
- It is known from crystal field theory that the relationship between the two splitting energies can be expressed as:
\[ \Delta_t = \frac{4}{9} \Delta_0 \]
- This equation shows that the crystal field splitting energy for tetrahedral complexes is less than that for octahedral complexes and quantifies the difference.

5. **Conclusion**:
- Therefore, the relationship between the crystal field splitting energies for octahedral and tetrahedral complexes is given by:
\[ \Delta_t = \frac{4}{9} \Delta_0 \]