To determine the validity of the given quantum numbers for the electron shell where [tex]\( n = 3 \)[/tex], we need to understand the constraints for each type of quantum number.
1. Azimuthal Quantum Number [tex]\( I \)[/tex]:
- The azimuthal quantum number [tex]\( I \)[/tex] can take integer values ranging from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex].
- For [tex]\( n = 3 \)[/tex], [tex]\( I \)[/tex] can be 0, 1, or 2.
Let's check each provided [tex]\( I \)[/tex] value:
- [tex]\( I = 3 \)[/tex]: This is not valid because it exceeds the maximum value of [tex]\( n - 1 \)[/tex], which is 2.
- [tex]\( I = 0 \)[/tex]: This is valid because it is within the range 0 to 2.
- [tex]\( I = -1 \)[/tex]: This is not valid because it is less than 0.
2. Magnetic Quantum Number [tex]\( m \)[/tex]:
- The magnetic quantum number [tex]\( m \)[/tex] can take integer values from [tex]\( -I \)[/tex] to [tex]\( +I \)[/tex] for a given [tex]\( I \)[/tex].
- For the valid [tex]\( I \)[/tex] values (0, 1, 2), [tex]\( m \)[/tex] values range from:
- When [tex]\( I = 0 \)[/tex]: [tex]\( m \)[/tex] can only be 0.
- When [tex]\( I = 1 \)[/tex]: [tex]\( m \)[/tex] can be -1, 0, or +1.
- When [tex]\( I = 2 \)[/tex]: [tex]\( m \)[/tex] can be -2, -1, 0, +1, or +2.
Let's check each provided [tex]\( m \)[/tex] value:
- [tex]\( m = 3 \)[/tex]: This is not valid for any [tex]\( I \)[/tex] since [tex]\( m \)[/tex] cannot exceed [tex]\( I \)[/tex], and the maximum [tex]\( I \)[/tex] here is 2.
- [tex]\( m = -2 \)[/tex]: This could be valid if [tex]\( I \)[/tex] were 2, but we also need to know that [tex]\( I = 2 \)[/tex] is a valid azimuthal quantum number (in our valid [tex]\( I \)[/tex] list, we only have [tex]\( I = 0 \)[/tex]).
- [tex]\( m = 2 \)[/tex]: Like [tex]\( m = -2 \)[/tex], this would be valid for [tex]\( I = 2 \)[/tex], but given our valid [tex]\( I \)[/tex] list, there is no such corresponding [tex]\( I \)[/tex].
Thus, after analyzing the allowed ranges and checking each given value:
- Valid [tex]\( I \)[/tex] values: 0
- Valid [tex]\( m \)[/tex] values: None, because none of the provided [tex]\( m \)[/tex] values match the magnetic quantum number requirement for the valid [tex]\( I = 0 \)[/tex].
Therefore, the final result is:
- Valid [tex]\( I \)[/tex] numbers: [0]
- Valid [tex]\( m \)[/tex] numbers: []
