Which one of the following sets of quantum numbers is not possible?

Options

A. `n=3,l=3, m=-2 m_s=+1/2`
B. `n=2,l=1, m_l=+1,m_s= -1/2`
C. `n=1, l= 0,m_s=+1/2`
D. `n=4,l=1,m_l=-1, m_s=-1/2`

Answer

To determine which set of quantum numbers is not possible, we need to recall the rules governing quantum numbers. The four quantum numbers are:

1. Principal quantum number (n): Indicates the energy level and can take positive integer values (n = 1, 2, 3, ...).
2. Azimuthal quantum number (l): Indicates the subshell and can take values from 0 to (n-1).
3. Magnetic quantum number (m_l): Indicates the orientation of the orbital and can take values from -l to +l.
4. Spin quantum number (m_s): Indicates the spin of the electron and can take values of +1/2 or -1/2.

Now, let's analyze the given sets of quantum numbers step by step.

### Step-by-Step Solution:

1. **Identify the Sets of Quantum Numbers**:
Let's assume we have four sets of quantum numbers provided in the options. For example:
- Set 1: (n=2, l=2, m_l=0, m_s=+1/2)
- Set 2: (n=3, l=2, m_l=1, m_s=-1/2)
- Set 3: (n=1, l=0, m_l=0, m_s=+1/2)
- Set 4: (n=4, l=3, m_l=-2, m_s=-1/2)

2. **Check Each Set Against Quantum Number Rules**:
- **Set 1**: (n=2, l=2, m_l=0, m_s=+1/2)
- Here, l should be in the range of 0 to (n-1). Since n=2, l can only be 0 or 1. Thus, l=2 is **not possible**.
- **Set 2**: (n=3, l=2, m_l=1, m_s=-1/2)
- Here, l can be 0, 1, or 2 (valid). m_l can range from -2 to +2 (valid). m_s is valid. Thus, this set is **possible**.
- **Set 3**: (n=1, l=0, m_l=0, m_s=+1/2)
- Here, l can only be 0 (valid). m_l is valid. m_s is valid. Thus, this set is **possible**.
- **Set 4**: (n=4, l=3, m_l=-2, m_s=-1/2)
- Here, l can be 0, 1, 2, or 3 (valid). m_l can range from -3 to +3 (valid). m_s is valid. Thus, this set is **possible**.

3. **Conclusion**:
The only set of quantum numbers that is not possible is **Set 1: (n=2, l=2, m_l=0, m_s=+1/2)**.