The magnetic moment of a transition metal ion is found to be 4.90 BM. The number of unpaired electrons present in the ion is

Explanation

To determine the number of unpaired electrons in a transition metal ion based on its magnetic moment, we can use the formula for magnetic moment:

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

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

### Step 1: Set up the equation
Given that the magnetic moment \( \mu \) is 4.90 BM, we can set up the equation:

\[ 4.90 = \sqrt{n(n + 2)} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:

\[ (4.90)^2 = n(n + 2) \]

Calculating \( (4.90)^2 \):

\[ 24.01 = n(n + 2) \]

### Step 3: Rearrange the equation
Now we can rearrange the equation to form a quadratic equation:

\[ n^2 + 2n - 24.01 = 0 \]

### Step 4: Apply the quadratic formula
To solve for \( n \), we can use the quadratic formula:

\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, \( a = 1 \), \( b = 2 \), and \( c = -24.01 \). Plugging in these values:

\[ n = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-24.01)}}{2 \cdot 1} \]

Calculating the discriminant:

\[ n = \frac{-2 \pm \sqrt{4 + 96.04}}{2} \]
\[ n = \frac{-2 \pm \sqrt{100.04}}{2} \]
\[ n = \frac{-2 \pm 10.002}{2} \]

### Step 5: Calculate the possible values for \( n \)
Calculating the two possible values for \( n \):

1. \( n = \frac{-2 + 10.002}{2} = \frac{8.002}{2} = 4.001 \) (approximately 4)
2. \( n = \frac{-2 - 10.002}{2} = \frac{-12.002}{2} = -6.001 \) (not a valid solution since \( n \) cannot be negative)

### Step 6: Conclusion
Thus, the number of unpaired electrons \( n \) is approximately 4.

**Final Answer:** The number of unpaired electrons present in the ion is 4.

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