if element `,_(25)X+y` has spain magnetric moment 1.732 B.M then:

Options

A. number of unpaired electron `=1`
B. number of unparired electron `=2`
C. `Y=4`
D. `Y=6`

Answer

To solve the problem, we need to determine the number of unpaired electrons in the element \( _{25}X^y \) given that its spin magnetic moment is 1.732 Bohr magnetons (B.M).

### Step-by-Step Solution:

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

2. **Setting Up the Equation**:
We know that the magnetic moment is 1.732 B.M. Therefore, we can set up the equation:
\[ \sqrt{n(n + 2)} = 1.732 \]

3. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ n(n + 2) = (1.732)^2 \]
Calculating \( (1.732)^2 \):
\[ (1.732)^2 = 3 \]
Hence, we have:
\[ n(n + 2) = 3 \]

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

5. **Factoring the Quadratic**:
We can factor this quadratic equation:
\[ (n + 3)(n - 1) = 0 \]
This gives us two possible solutions for \( n \):
\[ n + 3 = 0 \quad \Rightarrow \quad n = -3 \quad (\text{not valid since } n \text{ cannot be negative}) \]
\[ n - 1 = 0 \quad \Rightarrow \quad n = 1 \]

6. **Conclusion on Unpaired Electrons**:
Thus, the number of unpaired electrons \( n \) is 1.

7. **Identifying the Element**:
The atomic number 25 corresponds to the element Manganese (Mn). The electronic configuration of Mn is:
\[ \text{Mn: } 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^5 \]
Manganese has 5 unpaired electrons in its ground state.

8. **Determining the Value of \( y \)**:
To achieve 1 unpaired electron, Mn must lose 4 electrons. The electrons will be removed from the 3d subshell first due to its higher energy compared to the 4s subshell. Thus, the new configuration after losing 4 electrons will be:
\[ 4s^2 3d^1 \]
This configuration has 1 unpaired electron.

9. **Final Answer**:
Therefore, the value of \( y \) is 4.

### Summary:
The number of unpaired electrons is 1, and the value of \( y \) is 4.