The magnetic moment of a transition metal ion is 3.87BM. The number of unpaired electrons present in it is

Answer

To determine the number of unpaired electrons in a transition metal ion with a magnetic moment of 3.87 BM, we can use the formula for magnetic moment:

### Step-by-Step Solution:

1. **Understand the Formula**: The magnetic moment (μ) is given by the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

2. **Substitute the Given Value**: We know the magnetic moment is 3.87 BM. Therefore, we can set up the equation:
\[ 3.87 = \sqrt{n(n + 2)} \]

3. **Square Both Sides**: To eliminate the square root, we square both sides:
\[ (3.87)^2 = n(n + 2) \]
Calculating \( (3.87)^2 \):
\[ 14.9769 \approx 15 \]
So we can approximate:
\[ 15 = n(n + 2) \]

4. **Rearrange the Equation**: Rearranging gives us:
\[ n(n + 2) - 15 = 0 \]
This simplifies to:
\[ n^2 + 2n - 15 = 0 \]

5. **Use the Quadratic Formula**: We can solve this quadratic equation using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 2 \), and \( c = -15 \).

6. **Calculate the Discriminant**: First, calculate the discriminant:
\[ b^2 - 4ac = 2^2 - 4(1)(-15) = 4 + 60 = 64 \]

7. **Substitute into the Quadratic Formula**:
\[ n = \frac{-2 \pm \sqrt{64}}{2(1)} = \frac{-2 \pm 8}{2} \]

8. **Find the Roots**: This gives us two possible solutions:
\[ n = \frac{6}{2} = 3 \quad \text{and} \quad n = \frac{-10}{2} = -5 \]
Since the number of unpaired electrons cannot be negative, we take:
\[ n = 3 \]

9. **Conclusion**: Therefore, the number of unpaired electrons present in the transition metal ion is:
\[ \boxed{3} \]