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To determine the validity of the given quantum numbers for the electron shell with [tex]\( n = 3 \)[/tex]:
First, we must review the constraints on the quantum numbers:
1. The azimuthal quantum number ([tex]\( I \)[/tex]) must satisfy [tex]\( 0 \leq I \leq n - 1 \)[/tex].
For [tex]\( n = 3 \)[/tex], this means [tex]\( I \)[/tex] can be: [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].
2. The magnetic quantum number ([tex]\( m \)[/tex]) must satisfy [tex]\( -I \leq m \leq I \)[/tex].
For each valid [tex]\( I \)[/tex], let's identify the possible values for [tex]\( m \)[/tex]:
- If [tex]\( I = 0 \)[/tex], then [tex]\( m \)[/tex] can only be [tex]\( 0 \)[/tex].
- If [tex]\( I = 1 \)[/tex], then [tex]\( m \)[/tex] can be [tex]\( -1, 0, \)[/tex] or [tex]\( 1 \)[/tex].
- If [tex]\( I = 2 \)[/tex], then [tex]\( m \)[/tex] can be [tex]\( -2, -1, 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].
Now, let's check each of the provided quantum numbers:
1. [tex]\( I = 3 \)[/tex]
- This is not valid because it does not lie within the range [tex]\( 0 \leq I \leq 2 \)[/tex].
2. [tex]\( m = 3 \)[/tex]
- For any [tex]\( I \)[/tex], [tex]\( m \)[/tex] values range from [tex]\( -I \)[/tex] to [tex]\( I \)[/tex]. Since the highest [tex]\( I \)[/tex] can be [tex]\( 2 \)[/tex], [tex]\( m = 3 \)[/tex] is not valid.
3. [tex]\( I = 0 \)[/tex]
- This is valid because [tex]\( 0 \leq I \leq 2 \)[/tex].
4. [tex]\( m = -2 \)[/tex]
- For [tex]\( I = 2 \)[/tex], [tex]\( m = -2 \)[/tex] is valid as it lies within [tex]\( -2 \leq m \leq 2 \)[/tex].
5. [tex]\( I = -1 \)[/tex]
- This is not valid because [tex]\( I \)[/tex] must be non-negative and [tex]\( -1 \)[/tex] is less than [tex]\( 0 \)[/tex].
6. [tex]\( m = 2 \)[/tex]
- For [tex]\( I = 2 \)[/tex], [tex]\( m = 2 \)[/tex] is valid as it lies within [tex]\( -2 \leq m \leq 2 \)[/tex].
Summarizing our findings, the valid quantum numbers from the given options are:
[tex]\[ I = 0 \][/tex]
[tex]\[ m = -2 \][/tex]
[tex]\[ m = 2 \][/tex]
Thus, the valid quantum numbers for the electron shell [tex]\( n = 3 \)[/tex] from the given list are:
[tex]\[ ([0], [-2, 2]). \][/tex]
First, we must review the constraints on the quantum numbers:
1. The azimuthal quantum number ([tex]\( I \)[/tex]) must satisfy [tex]\( 0 \leq I \leq n - 1 \)[/tex].
For [tex]\( n = 3 \)[/tex], this means [tex]\( I \)[/tex] can be: [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].
2. The magnetic quantum number ([tex]\( m \)[/tex]) must satisfy [tex]\( -I \leq m \leq I \)[/tex].
For each valid [tex]\( I \)[/tex], let's identify the possible values for [tex]\( m \)[/tex]:
- If [tex]\( I = 0 \)[/tex], then [tex]\( m \)[/tex] can only be [tex]\( 0 \)[/tex].
- If [tex]\( I = 1 \)[/tex], then [tex]\( m \)[/tex] can be [tex]\( -1, 0, \)[/tex] or [tex]\( 1 \)[/tex].
- If [tex]\( I = 2 \)[/tex], then [tex]\( m \)[/tex] can be [tex]\( -2, -1, 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].
Now, let's check each of the provided quantum numbers:
1. [tex]\( I = 3 \)[/tex]
- This is not valid because it does not lie within the range [tex]\( 0 \leq I \leq 2 \)[/tex].
2. [tex]\( m = 3 \)[/tex]
- For any [tex]\( I \)[/tex], [tex]\( m \)[/tex] values range from [tex]\( -I \)[/tex] to [tex]\( I \)[/tex]. Since the highest [tex]\( I \)[/tex] can be [tex]\( 2 \)[/tex], [tex]\( m = 3 \)[/tex] is not valid.
3. [tex]\( I = 0 \)[/tex]
- This is valid because [tex]\( 0 \leq I \leq 2 \)[/tex].
4. [tex]\( m = -2 \)[/tex]
- For [tex]\( I = 2 \)[/tex], [tex]\( m = -2 \)[/tex] is valid as it lies within [tex]\( -2 \leq m \leq 2 \)[/tex].
5. [tex]\( I = -1 \)[/tex]
- This is not valid because [tex]\( I \)[/tex] must be non-negative and [tex]\( -1 \)[/tex] is less than [tex]\( 0 \)[/tex].
6. [tex]\( m = 2 \)[/tex]
- For [tex]\( I = 2 \)[/tex], [tex]\( m = 2 \)[/tex] is valid as it lies within [tex]\( -2 \leq m \leq 2 \)[/tex].
Summarizing our findings, the valid quantum numbers from the given options are:
[tex]\[ I = 0 \][/tex]
[tex]\[ m = -2 \][/tex]
[tex]\[ m = 2 \][/tex]
Thus, the valid quantum numbers for the electron shell [tex]\( n = 3 \)[/tex] from the given list are:
[tex]\[ ([0], [-2, 2]). \][/tex]
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