Magnetic moment of `X^(3+)` ion of 3d series is `2sqrt6` BM. What is the atomic number of X ?

Options

A. 24
B. 26
C. 25
D. 28

Answer

To solve the problem, we need to find the atomic number of the element \( X \) given that the magnetic moment of the \( X^{3+} \) ion is \( 2\sqrt{6} \) BM.

### Step-by-Step Solution:

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

2. **Setting Up the Equation**:
We know that:
\[ \mu = 2\sqrt{6} \text{ BM} \]
Therefore, we can set up the equation:
\[ 2\sqrt{6} = \sqrt{n(n + 2)} \]

3. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ (2\sqrt{6})^2 = n(n + 2) \]
This simplifies to:
\[ 24 = n(n + 2) \]

4. **Rearranging the Equation**:
Rearranging gives us a quadratic equation:
\[ n^2 + 2n - 24 = 0 \]

5. **Factoring the Quadratic**:
We can factor this equation:
\[ (n + 6)(n - 4) = 0 \]
Thus, the solutions for \( n \) are:
\[ n = -6 \quad \text{or} \quad n = 4 \]

6. **Selecting the Valid Solution**:
Since \( n \) represents the number of unpaired electrons, it cannot be negative. Therefore, we take:
\[ n = 4 \]

7. **Analyzing the \( X^{3+} \) Ion**:
The \( X^{3+} \) ion means that the element \( X \) has lost 3 electrons. In the case of transition metals, electrons are removed first from the outermost \( s \) orbital and then from the \( d \) orbital.

8. **Determining the Electron Configuration**:
Since \( n = 4 \) indicates there are 4 unpaired electrons, we can deduce that:
- 2 electrons will be removed from the \( 4s \) subshell.
- 1 electron will be removed from the \( 3d \) subshell.

This suggests that the electron configuration of \( X \) before losing electrons is:
\[ [Ar] 4s^2 3d^5 \]

9. **Counting Total Electrons**:
The total number of electrons in this configuration is:
- \( 18 \) from \( [Ar] \)
- \( 2 \) from \( 4s^2 \)
- \( 5 \) from \( 3d^5 \)

Therefore, the total is:
\[ 18 + 2 + 5 = 25 \]

10. **Conclusion**:
The atomic number of \( X \) is \( 25 \).

### Final Answer:
The atomic number of \( X \) is \( 25 \).