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To determine which set of quantum numbers is not valid, we need to review the rules that constrain the quantum numbers:
1. The principal quantum number [tex]\( n \)[/tex] must be a positive integer ([tex]\( n \geq 1 \)[/tex]).
2. The angular momentum quantum number [tex]\( l \)[/tex] must be an integer that satisfies [tex]\( 0 \leq l < n \)[/tex].
3. The magnetic quantum number [tex]\( m \)[/tex] must be an integer that satisfies [tex]\( -l \leq m \leq l \)[/tex].
Let's check each set according to these rules:
1. Set 1: [tex]\( n = 2, l = 1, m = 0 \)[/tex]
- The principal quantum number [tex]\( n = 2 \)[/tex] is a positive integer, which is valid.
- The angular momentum quantum number [tex]\( l = 1 \)[/tex] satisfies [tex]\( 0 \leq l < n \)[/tex] (0 ≤ 1 < 2), which is valid.
- The magnetic quantum number [tex]\( m = 0 \)[/tex] satisfies [tex]\( -l \leq m \leq l \)[/tex] (-1 ≤ 0 ≤ 1), which is valid.
- Therefore, this set of quantum numbers is valid.
2. Set 2: [tex]\( n = 1, l = 0, m = 0 \)[/tex]
- The principal quantum number [tex]\( n = 1 \)[/tex] is a positive integer, which is valid.
- The angular momentum quantum number [tex]\( l = 0 \)[/tex] satisfies [tex]\( 0 \leq l < n \)[/tex] (0 ≤ 0 < 1), which is valid.
- The magnetic quantum number [tex]\( m = 0 \)[/tex] satisfies [tex]\( -l \leq m \leq l \)[/tex] (-0 ≤ 0 ≤ 0), which is valid.
- Therefore, this set of quantum numbers is valid.
3. Set 3: [tex]\( n = 3, l = 3, m = 3 \)[/tex]
- The principal quantum number [tex]\( n = 3 \)[/tex] is a positive integer, which is valid.
- The angular momentum quantum number [tex]\( l = 3 \)[/tex] does not satisfy [tex]\( 0 \leq l < n \)[/tex]; it should be in the range 0 to 2 (but here l = 3, which is not less than n = 3). This is invalid.
- The magnetic quantum number [tex]\( m = 3 \)[/tex] cannot be checked as the angular momentum quantum number [tex]\( l = 3 \)[/tex] itself is already invalid.
Thus, the third set [tex]\( n = 3, l = 3, m = 3 \)[/tex] is not a valid set of quantum numbers. The number 3 is the invalid set.
1. The principal quantum number [tex]\( n \)[/tex] must be a positive integer ([tex]\( n \geq 1 \)[/tex]).
2. The angular momentum quantum number [tex]\( l \)[/tex] must be an integer that satisfies [tex]\( 0 \leq l < n \)[/tex].
3. The magnetic quantum number [tex]\( m \)[/tex] must be an integer that satisfies [tex]\( -l \leq m \leq l \)[/tex].
Let's check each set according to these rules:
1. Set 1: [tex]\( n = 2, l = 1, m = 0 \)[/tex]
- The principal quantum number [tex]\( n = 2 \)[/tex] is a positive integer, which is valid.
- The angular momentum quantum number [tex]\( l = 1 \)[/tex] satisfies [tex]\( 0 \leq l < n \)[/tex] (0 ≤ 1 < 2), which is valid.
- The magnetic quantum number [tex]\( m = 0 \)[/tex] satisfies [tex]\( -l \leq m \leq l \)[/tex] (-1 ≤ 0 ≤ 1), which is valid.
- Therefore, this set of quantum numbers is valid.
2. Set 2: [tex]\( n = 1, l = 0, m = 0 \)[/tex]
- The principal quantum number [tex]\( n = 1 \)[/tex] is a positive integer, which is valid.
- The angular momentum quantum number [tex]\( l = 0 \)[/tex] satisfies [tex]\( 0 \leq l < n \)[/tex] (0 ≤ 0 < 1), which is valid.
- The magnetic quantum number [tex]\( m = 0 \)[/tex] satisfies [tex]\( -l \leq m \leq l \)[/tex] (-0 ≤ 0 ≤ 0), which is valid.
- Therefore, this set of quantum numbers is valid.
3. Set 3: [tex]\( n = 3, l = 3, m = 3 \)[/tex]
- The principal quantum number [tex]\( n = 3 \)[/tex] is a positive integer, which is valid.
- The angular momentum quantum number [tex]\( l = 3 \)[/tex] does not satisfy [tex]\( 0 \leq l < n \)[/tex]; it should be in the range 0 to 2 (but here l = 3, which is not less than n = 3). This is invalid.
- The magnetic quantum number [tex]\( m = 3 \)[/tex] cannot be checked as the angular momentum quantum number [tex]\( l = 3 \)[/tex] itself is already invalid.
Thus, the third set [tex]\( n = 3, l = 3, m = 3 \)[/tex] is not a valid set of quantum numbers. The number 3 is the invalid set.
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