Among `V(Z=23)`, `Cr(Z=24)`, `Mn(Z=25)` which will have the highest magnetic moment?

Options

A. V
B. Cr
C. Mn
D. Fe

Answer

To determine which element among Vanadium (V, Z=23), Chromium (Cr, Z=24), and Manganese (Mn, Z=25) has the highest magnetic moment, we need to calculate the number of unpaired electrons for each element and then use the formula for magnetic moment.

### Step 1: Determine the electronic configuration and number of unpaired electrons for Vanadium (V, Z=23)

1. **Electronic Configuration**:
- The atomic number of Vanadium (Z=23) means it has 23 electrons.
- The electronic configuration is:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^3 \]
2. **Unpaired Electrons**:
- In the 3d subshell, there are 3 electrons. All three electrons are unpaired.
- Therefore, \( n = 3 \) (number of unpaired electrons).

3. **Magnetic Moment Calculation**:
- Using the formula for magnetic moment:
\[ \mu = \sqrt{n(n + 2)} = \sqrt{3(3 + 2)} = \sqrt{15} \, \text{BM} \]

### Step 2: Determine the electronic configuration and number of unpaired electrons for Chromium (Cr, Z=24)

1. **Electronic Configuration**:
- The atomic number of Chromium (Z=24) means it has 24 electrons.
- The electronic configuration is:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^1 \, 3d^5 \]
- Note: Chromium has an exceptional configuration where one electron from the 4s subshell is promoted to the 3d subshell.
2. **Unpaired Electrons**:
- In the 3d subshell, there are 5 electrons, all of which are unpaired.
- Therefore, \( n = 6 \) (5 from 3d and 1 from 4s).

3. **Magnetic Moment Calculation**:
- Using the formula for magnetic moment:
\[ \mu = \sqrt{n(n + 2)} = \sqrt{6(6 + 2)} = \sqrt{48} \, \text{BM} \]

### Step 3: Determine the electronic configuration and number of unpaired electrons for Manganese (Mn, Z=25)

1. **Electronic Configuration**:
- The atomic number of Manganese (Z=25) means it has 25 electrons.
- The electronic configuration is:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^5 \]
2. **Unpaired Electrons**:
- In the 3d subshell, there are 5 electrons, and all are unpaired.
- Therefore, \( n = 5 \).

3. **Magnetic Moment Calculation**:
- Using the formula for magnetic moment:
\[ \mu = \sqrt{n(n + 2)} = \sqrt{5(5 + 2)} = \sqrt{35} \, \text{BM} \]

### Step 4: Compare the magnetic moments

- Vanadium: \( \mu = \sqrt{15} \, \text{BM} \)
- Chromium: \( \mu = \sqrt{48} \, \text{BM} \)
- Manganese: \( \mu = \sqrt{35} \, \text{BM} \)

### Conclusion:
The highest magnetic moment is for Chromium (Cr), as \( \sqrt{48} \) is greater than \( \sqrt{35} \) and \( \sqrt{15} \).

### Final Answer:
**Chromium (Cr) has the highest magnetic moment.**

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