The pair having the same magnetic moment is
[at. No. `Cr = 24, Mn = 25, Fe = 26 " and " Co = 27]`

Options

A. `[Cr(H_(2)O)_(6)]^(2+) and [Fe(H_(2)O)_(6)]^(2+)`
B. `[Mn(H_(2)O)_(6)]^(2+) and [Cr(H_(2)O)_(6)]^(2+`
C. `["CoCl"_(4)]^(2-) and [Fe(H_(2)O)_(6)]^(2+)`
D. `[Cr(H_(2)O)_(6)]^(2+) and ["CoCl"_(4)]^(2-)`

Answer

To solve the problem of identifying the pair of metal ions with the same magnetic moment, we need to follow these steps:

### Step 1: Identify the oxidation states and electronic configurations
We are given four metal ions: Cr²⁺, Mn²⁺, Fe²⁺, and Co²⁺. First, we need to determine their electronic configurations in the +2 oxidation state.

1. **Chromium (Cr)**: Atomic number 24
- Ground state configuration: [Ar] 3d⁵ 4s¹
- For Cr²⁺: [Ar] 3d⁴ (loses one 4s and one 3d electron)

2. **Manganese (Mn)**: Atomic number 25
- Ground state configuration: [Ar] 3d⁵ 4s²
- For Mn²⁺: [Ar] 3d⁵ (loses two 4s electrons)

3. **Iron (Fe)**: Atomic number 26
- Ground state configuration: [Ar] 3d⁶ 4s²
- For Fe²⁺: [Ar] 3d⁶ (loses two 4s electrons)

4. **Cobalt (Co)**: Atomic number 27
- Ground state configuration: [Ar] 3d⁷ 4s²
- For Co²⁺: [Ar] 3d⁷ (loses two 4s electrons)

### Step 2: Count the number of unpaired electrons (n)
Next, we need to determine the number of unpaired electrons (n) for each ion:

1. **Cr²⁺ (3d⁴)**:
- Configuration: ↑ ↑ ↑ ↑ (4 unpaired electrons)
- n = 4

2. **Mn²⁺ (3d⁵)**:
- Configuration: ↑ ↑ ↑ ↑ ↑ (5 unpaired electrons)
- n = 5

3. **Fe²⁺ (3d⁶)**:
- Configuration: ↑ ↑ ↑ ↑ ↓ (4 unpaired electrons)
- n = 4

4. **Co²⁺ (3d⁷)**:
- Configuration: ↑ ↑ ↑ ↑ ↓ ↓ (3 unpaired electrons)
- n = 3

### Step 3: Calculate the magnetic moment
The magnetic moment (µ) can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]

Now, we can calculate the magnetic moment for each ion:

1. **Cr²⁺**:
\[ \mu = \sqrt{4(4 + 2)} = \sqrt{4 \times 6} = \sqrt{24} \approx 4.9 \, \mu_B \]

2. **Mn²⁺**:
\[ \mu = \sqrt{5(5 + 2)} = \sqrt{5 \times 7} = \sqrt{35} \approx 5.9 \, \mu_B \]

3. **Fe²⁺**:
\[ \mu = \sqrt{4(4 + 2)} = \sqrt{4 \times 6} = \sqrt{24} \approx 4.9 \, \mu_B \]

4. **Co²⁺**:
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{3 \times 5} = \sqrt{15} \approx 3.9 \, \mu_B \]

### Step 4: Identify the pair with the same magnetic moment
From the calculations:
- Cr²⁺ and Fe²⁺ both have a magnetic moment of approximately 4.9 µ_B.
- Mn²⁺ has a magnetic moment of approximately 5.9 µ_B.
- Co²⁺ has a magnetic moment of approximately 3.9 µ_B.

Thus, the pair having the same magnetic moment is **Cr²⁺ and Fe²⁺**.

### Final Answer:
The pair having the same magnetic moment is **Cr²⁺ and Fe²⁺**.