Which of the following sets of quantum numbers is correct for an electron in 4f-orbtial ?

Options

A. `n=3,l=2,m=-2,s=+1/2`
B. `n=4,l=4,m=-2,s=-1/2`
C. `n=4,l=3,m=+1,s=+1/2`
D. `n=4,l=3,m=+4,s=+1/2`

Answer

To determine the correct set of quantum numbers for an electron in a 4f orbital, we need to understand the definitions and constraints of quantum numbers:

1. **Principal Quantum Number (n)**: This number indicates the energy level of the electron. For a 4f orbital, the principal quantum number \( n \) must be 4.

2. **Azimuthal Quantum Number (l)**: This number defines the shape of the orbital. The value of \( l \) can range from 0 to \( n-1 \). For f orbitals, \( l \) is equal to 3. Therefore, for a 4f orbital, \( l \) must be 3.

3. **Magnetic Quantum Number (m_l)**: This number indicates the orientation of the orbital in space. The values of \( m_l \) can range from \(-l\) to \(+l\). For \( l = 3 \), \( m_l \) can take values of -3, -2, -1, 0, +1, +2, or +3.

4. **Spin Quantum Number (m_s)**: This number describes the spin of the electron. It can either be +1/2 or -1/2.

Now, let's summarize the quantum numbers for an electron in a 4f orbital:

- \( n = 4 \)
- \( l = 3 \)
- \( m_l = -3, -2, -1, 0, +1, +2, +3 \) (any of these values is acceptable)
- \( m_s = +1/2 \) or \( m_s = -1/2 \) (either value is acceptable)

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- For a 4f orbital, \( n = 4 \).

2. **Determine the Azimuthal Quantum Number (l)**:
- For f orbitals, \( l = 3 \).

3. **Find the Possible Values for Magnetic Quantum Number (m_l)**:
- Since \( l = 3 \), the possible values for \( m_l \) are -3, -2, -1, 0, +1, +2, +3.

4. **Choose a Value for Spin Quantum Number (m_s)**:
- The spin quantum number can be either +1/2 or -1/2.

5. **Combine the Quantum Numbers**:
- A valid set of quantum numbers for an electron in a 4f orbital could be \( (n, l, m_l, m_s) = (4, 3, +1, +1/2) \) or any combination where \( m_l \) is one of the acceptable values and \( m_s \) is either +1/2 or -1/2.

### Conclusion:
The correct set of quantum numbers for an electron in a 4f orbital is \( n = 4, l = 3, m_l = \text{(any value from -3 to +3)}, m_s = \pm 1/2 \).