The number of species among the following,having magnetic moment value of 2.84 BM is_______.
`Fe^(2+), Cr, Cr^(3+), Ti^(2+), Mn^(2+), V^(3+)`

Answer

To determine the number of species among the given ions that have a magnetic moment value of 2.84 Bohr Magneton (BM), we need to analyze the electronic configurations and the number of unpaired electrons for each species.

### Step-by-Step Solution:

1. **Understanding Magnetic Moment**:
The magnetic moment (μ) is given by the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons. We need to find \( n \) such that \( \mu = 2.84 \) BM.

2. **Setting Up the Equation**:
Plugging in the magnetic moment value:
\[ 2.84 = \sqrt{n(n + 2)} \]
Squaring both sides gives:
\[ 2.84^2 = n(n + 2) \]
\[ 8.0656 = n^2 + 2n \]
Rearranging gives:
\[ n^2 + 2n - 8.0656 = 0 \]

3. **Solving the Quadratic Equation**:
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 2, c = -8.0656 \).
\[ n = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8.0656)}}{2 \cdot 1} \]
\[ n = \frac{-2 \pm \sqrt{4 + 32.2624}}{2} \]
\[ n = \frac{-2 \pm \sqrt{36.2624}}{2} \]
\[ n = \frac{-2 \pm 6.02}{2} \]
This gives us \( n \approx 2 \) (since \( n \) must be a non-negative integer).

4. **Analyzing Each Species**:
Now we will analyze each species to find the number of unpaired electrons:

- **Fe²⁺**:
- Electron configuration: [Ar] 4s² 3d⁶ → Fe²⁺: [Ar] 3d⁶
- Unpaired electrons: 4 (not suitable).

- **Cr**:
- Electron configuration: [Ar] 4s² 3d⁵
- Unpaired electrons: 6 (not suitable).

- **Cr³⁺**:
- Electron configuration: [Ar] 4s² 3d⁵ → Cr³⁺: [Ar] 3d³
- Unpaired electrons: 3 (not suitable).

- **Ti²⁺**:
- Electron configuration: [Ar] 4s² 3d² → Ti²⁺: [Ar] 3d²
- Unpaired electrons: 2 (suitable).

- **Mn²⁺**:
- Electron configuration: [Ar] 4s² 3d⁵ → Mn²⁺: [Ar] 3d⁵
- Unpaired electrons: 5 (not suitable).

- **V³⁺**:
- Electron configuration: [Ar] 4s² 3d³ → V³⁺: [Ar] 3d²
- Unpaired electrons: 2 (suitable).

5. **Conclusion**:
The species with 2 unpaired electrons are Ti²⁺ and V³⁺. Therefore, the number of species having a magnetic moment value of 2.84 BM is **2**.

### Final Answer:
The number of species among the following having a magnetic moment value of 2.84 BM is **2** (Ti²⁺ and V³⁺).