Which of following species has the magnetic moment value of 3.87 BM? 

Options

A. `Fe^(3+)`
B. `Cr^(2+)`
C. `Co^(2+)`
D. `Au^(3+)`

Answer

To determine which species has a magnetic moment value of 3.87 Bohr magneton (BM), we will follow these steps:

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

### Step 2: Analyze Each Species
We will analyze the magnetic moment for each species given in the options.

#### Species 1: Fe³⁺
1. **Electron Configuration**: Iron (Fe) has an atomic number of 26. The electron configuration is [Ar] 3d⁶ 4s². For Fe³⁺, we lose 3 electrons, resulting in:
\[ \text{Fe}^{3+}: [Ar] 3d^5 \]
2. **Unpaired Electrons**: In 3d⁵, all 5 electrons are unpaired.
3. **Calculate Magnetic Moment**:
\[ n = 5 \implies \mu = \sqrt{5(5 + 2)} = \sqrt{35} \approx 5.92 \text{ BM} \]
This is not 3.87 BM.

#### Species 2: Cr²⁺
1. **Electron Configuration**: Chromium (Cr) has an atomic number of 24. The electron configuration is [Ar] 3d⁵ 4s¹. For Cr²⁺, we lose 2 electrons, resulting in:
\[ \text{Cr}^{2+}: [Ar] 3d^4 \]
2. **Unpaired Electrons**: In 3d⁴, there are 4 unpaired electrons.
3. **Calculate Magnetic Moment**:
\[ n = 4 \implies \mu = \sqrt{4(4 + 2)} = \sqrt{24} \approx 4.9 \text{ BM} \]
This is not 3.87 BM.

#### Species 3: Cr³⁺
1. **Electron Configuration**: For Cr³⁺, we lose 3 electrons, resulting in:
\[ \text{Cr}^{3+}: [Ar] 3d^3 \]
2. **Unpaired Electrons**: In 3d³, there are 3 unpaired electrons.
3. **Calculate Magnetic Moment**:
\[ n = 3 \implies \mu = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \text{ BM} \]
This matches 3.87 BM.

#### Species 4: Au³⁺
1. **Electron Configuration**: Gold (Au) has an atomic number of 79. The electron configuration is [Xe] 4f¹⁴ 5d¹⁰ 6s². For Au³⁺, we lose 3 electrons, resulting in:
\[ \text{Au}^{3+}: [Xe] 4f^{14} 5d^8 \]
2. **Unpaired Electrons**: In 5d⁸, there are 2 unpaired electrons.
3. **Calculate Magnetic Moment**:
\[ n = 2 \implies \mu = \sqrt{2(2 + 2)} = \sqrt{8} \approx 2.83 \text{ BM} \]
This is not 3.87 BM.

### Conclusion
The species with a magnetic moment value of 3.87 BM is **Cr³⁺**.

---