How many of the following ions have the same magnetic moments ?
`Fe^(2+)Mn^(2+)Cr^(2+)Ni^(2+)`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To determine how many of the given ions have the same magnetic moments, we need to analyze the number of unpaired electrons in each ion. The magnetic moment can be calculated using the formula:

\[ \mu = \sqrt{n(n + 2)} \, \text{BM} \]

where \( n \) is the number of unpaired electrons.

Let's analyze each ion step by step:

### Step 1: Determine the electronic configuration of each ion.

1. **Fe²⁺ (Iron ion)**
- Atomic number of Fe = 26
- Electronic configuration of Fe = [Ar] 4s² 3d⁶
- For Fe²⁺, we remove 2 electrons (from 4s):
- Electronic configuration of Fe²⁺ = [Ar] 3d⁶

2. **Mn²⁺ (Manganese ion)**
- Atomic number of Mn = 25
- Electronic configuration of Mn = [Ar] 4s² 3d⁵
- For Mn²⁺, we remove 2 electrons (from 4s):
- Electronic configuration of Mn²⁺ = [Ar] 3d⁵

3. **Cr²⁺ (Chromium ion)**
- Atomic number of Cr = 24
- Electronic configuration of Cr = [Ar] 4s¹ 3d⁵
- For Cr²⁺, we remove 2 electrons (1 from 4s and 1 from 3d):
- Electronic configuration of Cr²⁺ = [Ar] 3d⁴

4. **Ni²⁺ (Nickel ion)**
- Atomic number of Ni = 28
- Electronic configuration of Ni = [Ar] 4s² 3d⁸
- For Ni²⁺, we remove 2 electrons (from 4s):
- Electronic configuration of Ni²⁺ = [Ar] 3d⁸

### Step 2: Count the number of unpaired electrons in each ion.

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

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

3. **Cr²⁺ (3d⁴)**
- Configuration: ↑↑↑↑↓
- Unpaired electrons = 4

4. **Ni²⁺ (3d⁸)**
- Configuration: ↑↑↑↑↑↓ ↓
- Unpaired electrons = 2

### Step 3: Compare the number of unpaired electrons.

- **Fe²⁺**: 4 unpaired electrons
- **Mn²⁺**: 5 unpaired electrons
- **Cr²⁺**: 4 unpaired electrons
- **Ni²⁺**: 2 unpaired electrons

### Conclusion:

- Fe²⁺ and Cr²⁺ both have 4 unpaired electrons, which means they have the same magnetic moment.
- Mn²⁺ has 5 unpaired electrons, and Ni²⁺ has 2 unpaired electrons, which are different from each other and from Fe²⁺ and Cr²⁺.

Thus, the ions that have the same magnetic moments are **Fe²⁺ and Cr²⁺**.

### Final Answer:
**2 ions (Fe²⁺ and Cr²⁺) have the same magnetic moment.**

---