A metal ion from the first transition series has a magnetic moment `(`calculated`)` of `2.83 BM`. How many unpaired electrons are expected to be presetn in the ion?

Options

A. 6
B. 4
C. 3
D. 2

Answer

To determine the number of unpaired electrons in a metal ion from the first transition series with a magnetic moment of 2.83 BM, we can follow these steps:

### Step 1: Understand the formula for magnetic moment
The magnetic moment (μ) is related to the number of unpaired electrons (n) in a metal ion by the formula:
\[ \mu = \sqrt{n(n + 2)} \]

### Step 2: Substitute the given magnetic moment into the formula
We know that the magnetic moment is 2.83 BM. Therefore, we can set up the equation:
\[ 2.83 = \sqrt{n(n + 2)} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives us:
\[ (2.83)^2 = n(n + 2) \]
Calculating \( (2.83)^2 \):
\[ 2.83^2 \approx 8.0089 \]
So, we have:
\[ 8.0089 = n(n + 2) \]

### Step 4: Rearrange the equation
This can be rearranged to:
\[ n^2 + 2n - 8.0089 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2, c = -8.0089 \):
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-8.0089) = 4 + 32.0356 = 36.0356 \]
Now, substituting into the quadratic formula:
\[ n = \frac{-2 \pm \sqrt{36.0356}}{2 \cdot 1} \]
Calculating \( \sqrt{36.0356} \approx 6 \):
\[ n = \frac{-2 \pm 6}{2} \]

### Step 6: Calculate the possible values for n
Calculating the two possible values:
1. \( n = \frac{4}{2} = 2 \)
2. \( n = \frac{-8}{2} = -4 \) (not a valid solution since n cannot be negative)

### Conclusion
Thus, the number of unpaired electrons (n) is:
\[ \boxed{2} \]