The magnetic moment of an ion in its `+3` oxidation state is 3.85 BM. The number of unpaired d-electrons present in it are

Options

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

Answer

To determine the number of unpaired d-electrons in an ion with a magnetic moment of 3.85 BM in its +3 oxidation state, we can follow these steps:

### Step 1: Understand the relationship between magnetic moment and unpaired electrons
The magnetic moment (μ) of a transition metal ion can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Set up the equation
Given that the magnetic moment \( \mu \) is 3.85 BM, we can set up the equation:
\[ 3.85 = \sqrt{n(n + 2)} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (3.85)^2 = n(n + 2) \]
Calculating \( (3.85)^2 \):
\[ 14.8225 = n(n + 2) \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ n^2 + 2n - 14.8225 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 2, c = -14.8225 \):
\[ b^2 - 4ac = 2^2 - 4(1)(-14.8225) = 4 + 59.29 = 63.29 \]
Now substituting into the quadratic formula:
\[ n = \frac{-2 \pm \sqrt{63.29}}{2} \]
Calculating \( \sqrt{63.29} \):
\[ n = \frac{-2 \pm 7.96}{2} \]
Calculating the two possible values for \( n \):
1. \( n = \frac{5.96}{2} = 2.98 \) (approximately 3)
2. \( n = \frac{-9.96}{2} \) (not a valid solution since \( n \) cannot be negative)

### Step 6: Conclusion
Since \( n \) must be a whole number, we round \( 2.98 \) to \( 3 \). Thus, the number of unpaired d-electrons present in the ion is:
\[ \boxed{3} \]