Which of the following sets of quantum numbers are impossible for electrons? Explain why in each case.
`|{:("Set",n,l,m,s),((i),1,0,1,+1/2),((ii),3,0,0,-1/2),((iii),1,2,2,+1/2),((iv),4,3,-3,+1/2),((v),5,2,1,-1/2),((vi),3,2,1,0):}|`

Answer

To determine which sets of quantum numbers are impossible for electrons, we need to recall the rules governing quantum numbers:

1. **Principal Quantum Number (n)**: This can take any positive integer value (1, 2, 3,...).
2. **Azimuthal Quantum Number (l)**: This can take integer values from 0 to (n-1). Therefore, for a given n, l can be 0, 1, 2, ..., (n-1).
3. **Magnetic Quantum Number (m)**: This can take integer values from -l to +l, including zero.
4. **Spin Quantum Number (s)**: This can only be +1/2 or -1/2.

Now, let's analyze each set of quantum numbers provided in the question:

### Set (i): (n=1, l=0, m=1, s=+1/2)
- **n = 1**: Valid
- **l = 0**: Valid (l can be 0 when n=1)
- **m = 1**: Invalid (m must be in the range -l to +l, which for l=0 is only 0)
- **s = +1/2**: Valid
- **Conclusion**: This set is impossible due to m.

### Set (ii): (n=3, l=0, m=0, s=-1/2)
- **n = 3**: Valid
- **l = 0**: Valid (l can be 0 when n=3)
- **m = 0**: Valid (m can be 0 when l=0)
- **s = -1/2**: Valid
- **Conclusion**: This set is possible.

### Set (iii): (n=1, l=2, m=2, s=+1/2)
- **n = 1**: Valid
- **l = 2**: Invalid (l cannot be 2 when n=1; l can only be 0)
- **m = 2**: Invalid (m cannot be 2 when l=2)
- **s = +1/2**: Valid
- **Conclusion**: This set is impossible due to l.

### Set (iv): (n=4, l=3, m=-3, s=+1/2)
- **n = 4**: Valid
- **l = 3**: Valid (l can be 0, 1, 2, or 3 when n=4)
- **m = -3**: Valid (m can be -3, -2, -1, 0, +1, +2, +3 when l=3)
- **s = +1/2**: Valid
- **Conclusion**: This set is possible.

### Set (v): (n=5, l=2, m=1, s=-1/2)
- **n = 5**: Valid
- **l = 2**: Valid (l can be 0, 1, 2, 3, or 4 when n=5)
- **m = 1**: Valid (m can be -2, -1, 0, +1, +2 when l=2)
- **s = -1/2**: Valid
- **Conclusion**: This set is possible.

### Set (vi): (n=3, l=2, m=1, s=0)
- **n = 3**: Valid
- **l = 2**: Valid (l can be 0, 1, or 2 when n=3)
- **m = 1**: Valid (m can be -2, -1, 0, +1, +2 when l=2)
- **s = 0**: Invalid (s can only be +1/2 or -1/2)
- **Conclusion**: This set is impossible due to s.

### Summary of Impossible Sets:
- Set (i): Impossible due to m.
- Set (iii): Impossible due to l.
- Set (vi): Impossible due to s.

### Final Answer:
The impossible sets of quantum numbers are:
1. Set (i): (1, 0, 1, +1/2)
2. Set (iii): (1, 2, 2, +1/2)
3. Set (vi): (3, 2, 1, 0)