Which of the following pair of transition metal ions, have the same calculated values of magnetic moment?

Options

A. `Ti^(2+) and V^(2+)`
B. `Fe^(2+) and Cu^(2+)`
C. `Cr^(2+) and Fe^(2+) `
D. `Co^(2+) and and Ti^(2+)`

Explanation

To solve the question of which pair of transition metal ions have the same calculated values of magnetic moment, we need to follow these steps:

### Step-by-Step Solution:

1. **Understand the Magnetic Moment Formula**:
The magnetic moment (μ) can be calculated using the formula:
\[ μ = \sqrt{n(n + 2)} \text{ Bohr Magneton} \]
where \( n \) is the number of unpaired electrons.

2. **Identify the Transition Metal Ions**:
The pairs of transition metal ions we need to analyze are:
- Ti²⁺ and V²⁺
- Fe²⁺ and Cu²⁺
- Cr²⁺ and Fe²⁺
- Co²⁺ and Ti²⁺

3. **Determine the Electronic Configurations**:
We will write the electronic configurations for each ion to find the number of unpaired electrons.

- **Ti²⁺ (Titanium)**:
- Atomic number: 22
- Configuration: [Ar] 3d² 4s²
- For Ti²⁺, remove 2 electrons from 4s:
- Configuration: [Ar] 3d²
- Unpaired electrons: 2

- **V²⁺ (Vanadium)**:
- Atomic number: 23
- Configuration: [Ar] 3d³ 4s²
- For V²⁺, remove 2 electrons from 4s:
- Configuration: [Ar] 3d³
- Unpaired electrons: 3

- **Fe²⁺ (Iron)**:
- Atomic number: 26
- Configuration: [Ar] 3d⁶ 4s²
- For Fe²⁺, remove 2 electrons from 4s:
- Configuration: [Ar] 3d⁶
- Unpaired electrons: 4

- **Cu²⁺ (Copper)**:
- Atomic number: 29
- Configuration: [Ar] 3d¹⁰ 4s²
- For Cu²⁺, remove 2 electrons from 4s:
- Configuration: [Ar] 3d⁹
- Unpaired electrons: 1

- **Cr²⁺ (Chromium)**:
- Atomic number: 24
- Configuration: [Ar] 3d⁵ 4s²
- For Cr²⁺, remove 2 electrons from 4s:
- Configuration: [Ar] 3d⁵
- Unpaired electrons: 5

- **Co²⁺ (Cobalt)**:
- Atomic number: 27
- Configuration: [Ar] 3d⁷ 4s²
- For Co²⁺, remove 2 electrons from 4s:
- Configuration: [Ar] 3d⁷
- Unpaired electrons: 3

4. **Compare the Number of Unpaired Electrons**:
Now we compare the number of unpaired electrons for each pair:
- Ti²⁺ (2 unpaired) vs V²⁺ (3 unpaired) - Different
- Fe²⁺ (4 unpaired) vs Cu²⁺ (1 unpaired) - Different
- Cr²⁺ (5 unpaired) vs Fe²⁺ (4 unpaired) - Different
- Co²⁺ (3 unpaired) vs Ti²⁺ (2 unpaired) - Different

5. **Identify the Correct Pair**:
The only pair that has the same number of unpaired electrons is:
- **Cr²⁺ (4 unpaired) and Fe²⁺ (4 unpaired)**

### Conclusion:
The correct answer is **Cr²⁺ and Fe²⁺**, as they both have the same number of unpaired electrons, leading to the same calculated values of magnetic moment.