A transition metal complex shows a magnetic moment of 5.9 B.M. at room temperature. The number of unpaired electron on the metal is

Options

A. 3
B. 4
C. 5
D. 2

Answer

To determine the number of unpaired electrons in a transition metal complex that shows a magnetic moment of 5.9 Bohr magnetons (B.M.), we can use the formula for magnetic moment:

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

where \( \mu \) is the magnetic moment in Bohr magnetons and \( n \) is the number of unpaired electrons.

### Step-by-Step Solution:

1. **Set up the equation**:
We know the magnetic moment \( \mu = 5.9 \) B.M. Therefore, we can set up the equation:
\[ 5.9 = \sqrt{n(n + 2)} \]

2. **Square both sides**:
To eliminate the square root, we square both sides of the equation:
\[ (5.9)^2 = n(n + 2) \]
\[ 34.81 = n(n + 2) \]

3. **Rearrange the equation**:
Rearranging gives us a standard quadratic equation:
\[ n^2 + 2n - 34.81 = 0 \]

4. **Use the quadratic formula**:
The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 2 \), and \( c = -34.81 \).

5. **Calculate the discriminant**:
First, we calculate the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-34.81) = 4 + 139.24 = 143.24 \]

6. **Calculate the roots**:
Now substituting back into the quadratic formula:
\[ n = \frac{-2 \pm \sqrt{143.24}}{2} \]
\[ n = \frac{-2 \pm 11.99}{2} \]

This gives us two potential solutions:
\[ n_1 = \frac{-2 + 11.99}{2} \approx 4.995 \quad \text{(approximately 5)} \]
\[ n_2 = \frac{-2 - 11.99}{2} \quad \text{(this will yield a negative value, which is not possible)} \]

7. **Conclusion**:
The only feasible solution is \( n \approx 5 \). Therefore, the number of unpaired electrons in the transition metal complex is **5**.

### Final Answer:
The number of unpaired electrons on the metal is **5**.