Answer :
To determine which sets of quantum numbers describe valid orbitals, we need to verify if each set satisfies the rules for quantum numbers:
1. Principal quantum number (n): This must be a positive integer.
2. Angular momentum quantum number (l): This must be an integer within the range [tex]\(0 \leq l < n\)[/tex].
3. Magnetic quantum number (m): This must be an integer within the range [tex]\(-l \leq m \leq l\)[/tex].
Let's evaluate each set based on these rules:
1. [tex]\(n=1, l=0, m=0\)[/tex]
- [tex]\(n = 1\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 0\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 0 < 1\)[/tex])
- [tex]\(m = 0\)[/tex]: valid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-0 \leq 0 \leq 0\)[/tex])
- This set is valid.
2. [tex]\(n=2, l=1, m=3\)[/tex]
- [tex]\(n = 2\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 1\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 1 < 2\)[/tex])
- [tex]\(m = 3\)[/tex]: invalid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-1 \leq 3 \leq 1\)[/tex] does not hold)
- This set is not valid.
3. [tex]\(n=2, l=2, m=2\)[/tex]
- [tex]\(n = 2\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 2\)[/tex]: invalid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 2 < 2\)[/tex] does not hold)
- [tex]\(m = 2\)[/tex]: is conditional on [tex]\(l\)[/tex] being valid which it is not
- This set is not valid.
4. [tex]\(n=3, l=0, m=0\)[/tex]
- [tex]\(n = 3\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 0\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 0 < 3\)[/tex])
- [tex]\(m = 0\)[/tex]: valid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-0 \leq 0 \leq 0\)[/tex])
- This set is valid.
5. [tex]\(n=5, l=4, m=-3\)[/tex]
- [tex]\(n = 5\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 4\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 4 < 5\)[/tex])
- [tex]\(m = -3\)[/tex]: valid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-4 \leq -3 \leq 4\)[/tex])
- This set is valid.
6. [tex]\(n=4, l=-2, m=2\)[/tex]
- [tex]\(n = 4\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = -2\)[/tex]: invalid since [tex]\(l\)[/tex] must be a non-negative integer within [tex]\(0 \leq l < n\)[/tex]
- [tex]\(m = 2\)[/tex]: is conditional on [tex]\(l\)[/tex] being valid which it is not
- This set is not valid.
Therefore, the valid sets of quantum numbers are:
[tex]\[ (1, 0, 0), (3, 0, 0), (5, 4, -3) \][/tex]
1. Principal quantum number (n): This must be a positive integer.
2. Angular momentum quantum number (l): This must be an integer within the range [tex]\(0 \leq l < n\)[/tex].
3. Magnetic quantum number (m): This must be an integer within the range [tex]\(-l \leq m \leq l\)[/tex].
Let's evaluate each set based on these rules:
1. [tex]\(n=1, l=0, m=0\)[/tex]
- [tex]\(n = 1\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 0\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 0 < 1\)[/tex])
- [tex]\(m = 0\)[/tex]: valid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-0 \leq 0 \leq 0\)[/tex])
- This set is valid.
2. [tex]\(n=2, l=1, m=3\)[/tex]
- [tex]\(n = 2\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 1\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 1 < 2\)[/tex])
- [tex]\(m = 3\)[/tex]: invalid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-1 \leq 3 \leq 1\)[/tex] does not hold)
- This set is not valid.
3. [tex]\(n=2, l=2, m=2\)[/tex]
- [tex]\(n = 2\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 2\)[/tex]: invalid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 2 < 2\)[/tex] does not hold)
- [tex]\(m = 2\)[/tex]: is conditional on [tex]\(l\)[/tex] being valid which it is not
- This set is not valid.
4. [tex]\(n=3, l=0, m=0\)[/tex]
- [tex]\(n = 3\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 0\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 0 < 3\)[/tex])
- [tex]\(m = 0\)[/tex]: valid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-0 \leq 0 \leq 0\)[/tex])
- This set is valid.
5. [tex]\(n=5, l=4, m=-3\)[/tex]
- [tex]\(n = 5\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = 4\)[/tex]: valid since [tex]\(0 \leq l < n\)[/tex] (here [tex]\(0 \leq 4 < 5\)[/tex])
- [tex]\(m = -3\)[/tex]: valid since [tex]\(-l \leq m \leq l\)[/tex] (here [tex]\(-4 \leq -3 \leq 4\)[/tex])
- This set is valid.
6. [tex]\(n=4, l=-2, m=2\)[/tex]
- [tex]\(n = 4\)[/tex]: principal quantum number, valid (positive integer)
- [tex]\(l = -2\)[/tex]: invalid since [tex]\(l\)[/tex] must be a non-negative integer within [tex]\(0 \leq l < n\)[/tex]
- [tex]\(m = 2\)[/tex]: is conditional on [tex]\(l\)[/tex] being valid which it is not
- This set is not valid.
Therefore, the valid sets of quantum numbers are:
[tex]\[ (1, 0, 0), (3, 0, 0), (5, 4, -3) \][/tex]
