Which of the following combinations of quantum numbers is possible for a `4p` orbital?

Options

A. `n = 4,l = 1, m_(l) = 0, +-1`
B. `m= 4, l = 1, m_(l) =- 1`
C. `n = 4, l = 1,m_(l) = 0`
D. `n = 4, l = 1, m_(l) = +1`

Explanation

To determine which combinations of quantum numbers are possible for a `4p` orbital, we need to analyze the quantum numbers involved:

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

2. **Azimuthal Quantum Number (l)**: This quantum number determines the shape of the orbital. For `p` orbitals, \( l \) is always 1. The possible values for \( l \) range from 0 to \( n-1 \). Since \( n = 4 \), the possible values for \( l \) are 0 (s), 1 (p), 2 (d), and 3 (f). Therefore, for a `4p` orbital, \( l = 1 \).

3. **Magnetic Quantum Number (m_l)**: This quantum number specifies the orientation of the orbital in space. The possible values for \( m_l \) range from \(-l\) to \(+l\). Since \( l = 1 \) for a `p` orbital, the possible values for \( m_l \) are \(-1, 0, +1\).

4. **Spin Quantum Number (m_s)**: This quantum number describes the spin of the electron and can take values of \(-\frac{1}{2}\) or \(+\frac{1}{2}\).

Now, let's summarize the possible quantum numbers for a `4p` orbital:

- \( n = 4 \)
- \( l = 1 \)
- \( m_l = -1, 0, +1 \)
- \( m_s = -\frac{1}{2} \) or \( +\frac{1}{2} \)

Based on this analysis, any combination of quantum numbers that includes \( n = 4 \), \( l = 1 \), \( m_l \) in the range of \(-1, 0, +1\), and \( m_s \) as either \(-\frac{1}{2}\) or \(+\frac{1}{2}\) is valid for a `4p` orbital.

### Summary of Possible Combinations:
- \( (n=4, l=1, m_l=-1, m_s=\pm\frac{1}{2}) \)
- \( (n=4, l=1, m_l=0, m_s=\pm\frac{1}{2}) \)
- \( (n=4, l=1, m_l=+1, m_s=\pm\frac{1}{2}) \)