If the value of principal quantum number is 4 , the possible values for magnetic quantum number will be :-

Options

A. 4
B. 16
C. 32
D. 9

Answer

To determine the possible values for the magnetic quantum number (m_l) when the principal quantum number (n) is 4, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- Given that \( n = 4 \).

2. **Determine the Possible Values of the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \).
- Therefore, for \( n = 4 \), the possible values of \( l \) are:
\[ l = 0, 1, 2, 3 \]

3. **Determine the Possible Values of the Magnetic Quantum Number (m_l)**:
- The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \), including zero.
- For each value of \( l \):
- If \( l = 0 \): \( m_l = 0 \) (1 value)
- If \( l = 1 \): \( m_l = -1, 0, +1 \) (3 values)
- If \( l = 2 \): \( m_l = -2, -1, 0, +1, +2 \) (5 values)
- If \( l = 3 \): \( m_l = -3, -2, -1, 0, +1, +2, +3 \) (7 values)

4. **Count the Total Possible Values of m_l**:
- Now, we sum the number of possible values for each \( l \):
\[ 1 \text{ (for } l=0\text{)} + 3 \text{ (for } l=1\text{)} + 5 \text{ (for } l=2\text{)} + 7 \text{ (for } l=3\text{)} = 1 + 3 + 5 + 7 = 16 \]

5. **Conclusion**:
- Therefore, the total number of possible values for the magnetic quantum number \( m_l \) when \( n = 4 \) is **16**.