An electron having the quantum numbers n=4, l=3 , m=0 , `s=-1/2` would be in the orbital

Options

A. 3s
B. 3p
C. 4d
D. 4f

Explanation

To determine the orbital in which an electron with the quantum numbers \( n=4 \), \( l=3 \), \( m=0 \), and \( s=-\frac{1}{2} \) resides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n \) indicates the energy level or shell of the electron. Here, \( n = 4 \) means the electron is in the fourth energy level.

2. **Identify the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l \) indicates the subshell or type of orbital. The value of \( l \) can range from \( 0 \) to \( n-1 \).
- For \( l = 3 \), the corresponding subshell is \( f \) (where \( l=0 \) is \( s \), \( l=1 \) is \( p \), \( l=2 \) is \( d \), and \( l=3 \) is \( f \)).

3. **Identify the Magnetic Quantum Number (m)**:
- The magnetic quantum number \( m \) indicates the specific orbital within the subshell. It can take values from \( -l \) to \( +l \).
- For \( l = 3 \), \( m \) can be \( -3, -2, -1, 0, +1, +2, +3 \). Here, \( m = 0 \) indicates that the electron is in the \( f \) orbital that is oriented along the z-axis.

4. **Identify the Spin Quantum Number (s)**:
- The spin quantum number \( s \) indicates the spin of the electron. It can be either \( +\frac{1}{2} \) (spin up) or \( -\frac{1}{2} \) (spin down).
- Here, \( s = -\frac{1}{2} \) indicates that the electron has a spin down.

5. **Conclusion**:
- Combining all the information, the electron with quantum numbers \( n=4 \), \( l=3 \), \( m=0 \), and \( s=-\frac{1}{2} \) is in the \( 4f \) subshell, specifically in the \( 4f \) orbital with spin down.
- Since \( m=0 \) corresponds to the central orbital in the \( f \) subshell, we can denote this as \( 4f_{0} \).

### Final Answer:
The electron is in the \( 4f \) orbital.

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