`{:(,"Column I",,"Column II"),(,"(Metal ion configuration in weak ligand field)",,("CFSE, "Delta_(0)" value")),(("A"),d^(4),(p),0.0),(("B"),d^(5),(q),-0.4),(("C"),d^(6),(r),-0.6),(("D"),d^(7),(s),-0.8):}`

Options

A. `A-r, B-s, C-p, D-q`
B. `A-r, B-p, C-q, D-s`
C. `A-p, B-r, C-q, D-s`
D. `A-q, B-r, C-s, D-p`

Answer

To solve the problem, we need to match the metal ion configurations in a weak ligand field (Column I) with their corresponding CFSE (Crystal Field Stabilization Energy) and Δ₀ values (Column II). Let's go through each metal ion configuration step by step.

### Step 1: Understand the Crystal Field Theory
In octahedral complexes, the d-orbitals split into two groups: the lower energy t₂g orbitals and the higher energy eg orbitals. The energy difference between these two sets of orbitals is represented by Δ₀.

### Step 2: Identify the CFSE Formula
The CFSE can be calculated using the formula:
\[ \text{CFSE} = (n_{t2g} \times -\frac{2}{5} \Delta_0) + (n_{eg} \times \frac{3}{5} \Delta_0) \]
where \( n_{t2g} \) is the number of electrons in the t₂g orbitals and \( n_{eg} \) is the number of electrons in the eg orbitals.

### Step 3: Analyze Each Metal Ion Configuration

#### A. For d⁴ Configuration
- In a weak field, there is no pairing.
- Filling: 1, 2, 3 (t₂g), 4 (eg).
- \( n_{t2g} = 3 \), \( n_{eg} = 1 \).
- CFSE Calculation:
\[ \text{CFSE} = (3 \times -\frac{2}{5} \Delta_0) + (1 \times \frac{3}{5} \Delta_0) = -\frac{6}{5} \Delta_0 + \frac{3}{5} \Delta_0 = -\frac{3}{5} \Delta_0 \]
Thus, for A, CFSE = -0.6 Δ₀.

#### B. For d⁵ Configuration
- In a weak field, there is no pairing.
- Filling: 1, 2, 3 (t₂g), 4, 5 (eg).
- \( n_{t2g} = 3 \), \( n_{eg} = 2 \).
- CFSE Calculation:
\[ \text{CFSE} = (3 \times -\frac{2}{5} \Delta_0) + (2 \times \frac{3}{5} \Delta_0) = -\frac{6}{5} \Delta_0 + \frac{6}{5} \Delta_0 = 0 \]
Thus, for B, CFSE = 0.

#### C. For d⁶ Configuration
- In a weak field, there is no pairing.
- Filling: 1, 2, 3 (t₂g), 4, 5 (eg), 6 (eg).
- \( n_{t2g} = 4 \), \( n_{eg} = 2 \).
- CFSE Calculation:
\[ \text{CFSE} = (4 \times -\frac{2}{5} \Delta_0) + (2 \times \frac{3}{5} \Delta_0) = -\frac{8}{5} \Delta_0 + \frac{6}{5} \Delta_0 = -\frac{2}{5} \Delta_0 \]
Thus, for C, CFSE = -0.4 Δ₀.

#### D. For d⁷ Configuration
- In a weak field, there is no pairing.
- Filling: 1, 2, 3 (t₂g), 4, 5 (eg), 6, 7 (eg).
- \( n_{t2g} = 5 \), \( n_{eg} = 2 \).
- CFSE Calculation:
\[ \text{CFSE} = (5 \times -\frac{2}{5} \Delta_0) + (2 \times \frac{3}{5} \Delta_0) = -2\Delta_0 + \frac{6}{5} \Delta_0 = -\frac{10}{5} \Delta_0 + \frac{6}{5} \Delta_0 = -\frac{4}{5} \Delta_0 \]
Thus, for D, CFSE = -0.8 Δ₀.

### Final Matching
Now we can match the configurations with their CFSE values:
- A (d⁴) → R (-0.6 Δ₀)
- B (d⁵) → P (0.0)
- C (d⁶) → Q (-0.4 Δ₀)
- D (d⁷) → S (-0.8 Δ₀)

### Summary of Matches
- A → R
- B → P
- C → Q
- D → S