A: The energy gap `Delta_(t)=(4)/(9)Delta_(0)`
R: `Delta_(0)` is always greater than `Delta_(t)`, for same metal ion and ligand.

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion, then mark (1)
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion, then mark (2).
C. ) If Assertion is true statement but Reason is false, then mark (3).
D. If both Assertion and Reason are false statements, then mark (4)

Answer

To solve the question, we need to analyze the statements given:

**Statement A:** The energy gap \( \Delta_t = \frac{4}{9} \Delta_0 \)

**Statement R:** \( \Delta_0 \) is always greater than \( \Delta_t \) for the same metal ion and ligand.

### Step-by-Step Solution:

1. **Understanding the Terms**:
- \( \Delta_t \) refers to the splitting energy in a tetrahedral field.
- \( \Delta_0 \) refers to the splitting energy in an octahedral field.

2. **Energy Splitting in Different Geometries**:
- In an octahedral field, there are 6 ligands surrounding the metal ion, which leads to a greater repulsion among the d-orbitals. This results in a larger energy gap \( \Delta_0 \).
- In a tetrahedral field, there are only 4 ligands. The arrangement leads to less repulsion among the d-orbitals, resulting in a smaller energy gap \( \Delta_t \).

3. **Mathematical Relationship**:
- The relationship \( \Delta_t = \frac{4}{9} \Delta_0 \) indicates that the energy gap in a tetrahedral field is indeed a fraction of the energy gap in an octahedral field. This confirms that \( \Delta_t \) is less than \( \Delta_0 \).

4. **Conclusion on the Statements**:
- Since \( \Delta_t \) is calculated as \( \frac{4}{9} \Delta_0 \), it is clear that \( \Delta_0 \) is greater than \( \Delta_t \). Thus, both statements A and R are correct.

### Final Answer:
Both statements are correct. Statement A is true because \( \Delta_t \) is indeed \( \frac{4}{9} \Delta_0 \), and Statement R is also true since \( \Delta_0 \) is always greater than \( \Delta_t \) for the same metal ion and ligand.

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