The value of the 'spin only magnetic moment for one of the following configuration is 2.84 BM. The correct one is

Options

A. `d^(5)` (in strong ligand field)
B. `d^(3)` (in weak as well as in strong fields)
C. `d^(4)` (in weak ligand field)
D. `d^(4)` (in strong ligand field)

Explanation

To determine the correct electronic configuration that corresponds to a spin-only magnetic moment of 2.84 Bohr Magneton (BM), we will follow these steps:

### Step 1: Understand the Formula for Magnetic Moment
The spin-only magnetic moment (μ) can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Set Up the Equation
Given that the magnetic moment is 2.84 BM, we can set up the equation:
\[ 2.84 = \sqrt{n(n + 2)} \]

### Step 3: Square Both Sides
Squaring both sides of the equation to eliminate the square root gives:
\[ (2.84)^2 = n(n + 2) \]
Calculating \( (2.84)^2 \):
\[ 8.0656 \approx n(n + 2) \]

### Step 4: Rearrange the Equation
This leads to the quadratic equation:
\[ n^2 + 2n - 8.0656 = 0 \]

### Step 5: Apply the Quadratic Formula
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 2, c = -8.0656 \):
\[ n = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8.0656)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ b^2 - 4ac = 4 + 32.2624 = 36.2624 \]
Now substituting back:
\[ n = \frac{-2 \pm \sqrt{36.2624}}{2} \]
Calculating \( \sqrt{36.2624} \approx 6.02 \):
\[ n = \frac{-2 \pm 6.02}{2} \]

### Step 6: Solve for n
Calculating the two possible values for \( n \):
1. \( n = \frac{4.02}{2} = 2.01 \) (approximately 2)
2. \( n = \frac{-8.02}{2} = -4.01 \) (not possible as \( n \) cannot be negative)

Thus, \( n \approx 2 \).

### Step 7: Identify the Configuration
Now, we need to find which electronic configuration has 2 unpaired electrons.

- **d5 in strong field ligand**: 0 unpaired electrons (all paired).
- **d3 in weak field ligand**: 3 unpaired electrons.
- **d4 in weak field ligand**: 4 unpaired electrons.
- **d4 in strong field ligand**: 2 unpaired electrons.

### Conclusion
The correct configuration that corresponds to a spin-only magnetic moment of 2.84 BM is **d4 in a strong field ligand**.