To determine whether each set of quantum numbers describes a valid orbital, we must check the following conditions for the principal quantum number [tex]\(n\)[/tex], angular momentum quantum number [tex]\(l\)[/tex], and magnetic quantum number [tex]\(m\)[/tex]:
1. [tex]\(n > 0\)[/tex]
2. [tex]\(0 \leq l < n\)[/tex]
3. [tex]\(-l \leq m \leq l\)[/tex]
Let's evaluate each set of quantum numbers one by one:
1. Set [tex]\( (n=1, l=0, m=0) \)[/tex]:
- [tex]\( n = 1 \)[/tex]: [tex]\(n > 0\)[/tex], so this is valid.
- [tex]\( l = 0 \)[/tex]: [tex]\(0 \leq l < n\)[/tex], so this is valid.
- [tex]\( m = 0 \)[/tex]: [tex]\(-l \leq m \leq l\)[/tex], so this is valid.
- Conclusion: This is a valid set of quantum numbers.
2. Set [tex]\( (n=2, l=1, m=3) \)[/tex]:
- [tex]\( n = 2 \)[/tex]: [tex]\(n > 0\)[/tex], so this is valid.
- [tex]\( l = 1 \)[/tex]: [tex]\(0 \leq l < n\)[/tex], so this is valid.
- [tex]\( m = 3 \)[/tex]: [tex]\(-l \leq m \leq l\)[/tex], [tex]\(-1 \leq 3 \leq 1\)[/tex] is not satisfied.
- Conclusion: This is not a valid set of quantum numbers.
3. Set [tex]\( (n=2, l=2, m=2) \)[/tex]:
- [tex]\( n = 2 \)[/tex]: [tex]\(n > 0\)[/tex], so this is valid.
- [tex]\( l = 2 \)[/tex]: [tex]\(0 \leq l < n\)[/tex], [tex]\(0 \leq 2 < 2\)[/tex] is not satisfied.
- [tex]\( m = 2 \)[/tex]: This does not need to be checked further since [tex]\(l\)[/tex] is not valid.
- Conclusion: This is not a valid set of quantum numbers.
4. Set [tex]\( (n=3, l=0, m=0) \)[/tex]:
- [tex]\( n = 3 \)[/tex]: [tex]\(n > 0\)[/tex], so this is valid.
- [tex]\( l = 0 \)[/tex]: [tex]\(0 \leq l < n\)[/tex], so this is valid.
- [tex]\( m = 0 \)[/tex]: [tex]\(-l \leq m \leq l\)[/tex], so this is valid.
- Conclusion: This is a valid set of quantum numbers.
5. Set [tex]\( (n=5, l=4, m=-3) \)[/tex]:
- [tex]\( n = 5 \)[/tex]: [tex]\(n > 0\)[/tex], so this is valid.
- [tex]\( l = 4 \)[/tex]: [tex]\(0 \leq l < n\)[/tex], so this is valid.
- [tex]\( m = -3 \)[/tex]: [tex]\(-l \leq m \leq l\)[/tex], [tex]\(-4 \leq -3 \leq 4\)[/tex] is satisfied.
- Conclusion: This is a valid set of quantum numbers.
6. Set [tex]\( (n=4, l=-2, m=2) \)[/tex]:
- [tex]\( n = 4 \)[/tex]: [tex]\(n > 0\)[/tex], so this is valid.
- [tex]\( l = -2 \)[/tex]: [tex]\(0 \leq l < n\)[/tex], [tex]\(0 \leq -2 < 4\)[/tex] is not satisfied.
- [tex]\( m = 2 \)[/tex]: This does not need to be checked further since [tex]\(l\)[/tex] is not valid.
- Conclusion: This is not a valid set of quantum numbers.
Final Conclusion: The sets of quantum numbers that describe valid orbitals are:
- [tex]\( (n=1, l=0, m=0) \)[/tex]
- [tex]\( (n=3, l=0, m=0) \)[/tex]
- [tex]\( (n=5, l=4, m=-3) \)[/tex]
