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Which of the following sets of quantum numbers describe valid orbitals? Check all that apply.

A. [tex]\(n=1, \ l=0, \ m=0\)[/tex]
B. [tex]\(n=2, \ l=1, \ m=3\)[/tex]
C. [tex]\(n=2, \ l=2, \ m=2\)[/tex]
D. [tex]\(n=3, \ l=0, \ m=0\)[/tex]
E. [tex]\(n=5, \ l=4, \ m=-3\)[/tex]
F. [tex]\(n=4, \ l=-2, \ m=2\)[/tex]

Sagot :

GinnyAnswer
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]
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