Answer :
To determine the validity of the given quantum numbers for the [tex]\( n = 3 \)[/tex] electron shell, we need to understand the rules governing quantum numbers.
1. Principal Quantum Number ([tex]\( n \)[/tex]):
- Defines the electron shell and is given as [tex]\( n = 3 \)[/tex].
2. Azimuthal Quantum Number ([tex]\( l \)[/tex]):
- Defines the subshell and can range from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex].
- For [tex]\( n = 3 \)[/tex], [tex]\( l \)[/tex] can be [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].
3. Magnetic Quantum Number ([tex]\( m_l \)[/tex]):
- Defines the orientation of the orbital and can range from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex].
- For each [tex]\( l \)[/tex]:
- If [tex]\( l = 0 \)[/tex], [tex]\( m_l = 0 \)[/tex].
- If [tex]\( l = 1 \)[/tex], [tex]\( m_l = -1, 0, 1 \)[/tex].
- If [tex]\( l = 2 \)[/tex], [tex]\( m_l = -2, -1, 0, 1, 2 \)[/tex].
Let's analyze each quantum number given:
1. [tex]\( I = 3 \)[/tex]:
- The azimuthal quantum number [tex]\( l = 3 \)[/tex] is invalid because [tex]\( l \)[/tex] must be between [tex]\( 0 \)[/tex] and [tex]\( n-1 = 2 \)[/tex].
2. [tex]\( m = 3 \)[/tex]:
- The magnetic quantum number [tex]\( m_l = 3 \)[/tex] would be valid only if [tex]\( l \geq 3 \)[/tex]. However, since [tex]\( l \)[/tex] for [tex]\( n = 3 \)[/tex] can only be [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex], [tex]\( m_l = 3 \)[/tex] is invalid.
3. [tex]\( I = 0 \)[/tex]:
- The azimuthal quantum number [tex]\( l = 0 \)[/tex] is valid because it is within the range [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex].
4. [tex]\( m = -2 \)[/tex]:
- The magnetic quantum number [tex]\( m_l = -2 \)[/tex] is valid for [tex]\( l \geq 2 \)[/tex]. Since [tex]\( n = 3 \)[/tex] allows [tex]\( l = 2 \)[/tex], [tex]\( m_l = -2 \)[/tex] is valid.
5. [tex]\( I = -1 \)[/tex]:
- The azimuthal quantum number [tex]\( l = -1 \)[/tex] is invalid because [tex]\( l \)[/tex] cannot be negative.
6. [tex]\( m = 2 \)[/tex]:
- The magnetic quantum number [tex]\( m_l = 2 \)[/tex] is valid for [tex]\( l \geq 2 \)[/tex]. Since [tex]\( n = 3 \)[/tex] allows [tex]\( l = 2 \)[/tex], [tex]\( m_l = 2 \)[/tex] is valid.
Based on this analysis, the validity of the quantum numbers is:
- [tex]\( I = 3 \)[/tex]: False
- [tex]\( m = 3 \)[/tex]: False
- [tex]\( I = 0 \)[/tex]: True
- [tex]\( m = -2 \)[/tex]: True
- [tex]\( I = -1 \)[/tex]: False
- [tex]\( m = 2 \)[/tex]: True
So, the result for the given [tex]\( n = 3 \)[/tex] electron shell is:
[tex]\( (False, False, True, True, False, True) \)[/tex]
1. Principal Quantum Number ([tex]\( n \)[/tex]):
- Defines the electron shell and is given as [tex]\( n = 3 \)[/tex].
2. Azimuthal Quantum Number ([tex]\( l \)[/tex]):
- Defines the subshell and can range from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex].
- For [tex]\( n = 3 \)[/tex], [tex]\( l \)[/tex] can be [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].
3. Magnetic Quantum Number ([tex]\( m_l \)[/tex]):
- Defines the orientation of the orbital and can range from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex].
- For each [tex]\( l \)[/tex]:
- If [tex]\( l = 0 \)[/tex], [tex]\( m_l = 0 \)[/tex].
- If [tex]\( l = 1 \)[/tex], [tex]\( m_l = -1, 0, 1 \)[/tex].
- If [tex]\( l = 2 \)[/tex], [tex]\( m_l = -2, -1, 0, 1, 2 \)[/tex].
Let's analyze each quantum number given:
1. [tex]\( I = 3 \)[/tex]:
- The azimuthal quantum number [tex]\( l = 3 \)[/tex] is invalid because [tex]\( l \)[/tex] must be between [tex]\( 0 \)[/tex] and [tex]\( n-1 = 2 \)[/tex].
2. [tex]\( m = 3 \)[/tex]:
- The magnetic quantum number [tex]\( m_l = 3 \)[/tex] would be valid only if [tex]\( l \geq 3 \)[/tex]. However, since [tex]\( l \)[/tex] for [tex]\( n = 3 \)[/tex] can only be [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex], [tex]\( m_l = 3 \)[/tex] is invalid.
3. [tex]\( I = 0 \)[/tex]:
- The azimuthal quantum number [tex]\( l = 0 \)[/tex] is valid because it is within the range [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex].
4. [tex]\( m = -2 \)[/tex]:
- The magnetic quantum number [tex]\( m_l = -2 \)[/tex] is valid for [tex]\( l \geq 2 \)[/tex]. Since [tex]\( n = 3 \)[/tex] allows [tex]\( l = 2 \)[/tex], [tex]\( m_l = -2 \)[/tex] is valid.
5. [tex]\( I = -1 \)[/tex]:
- The azimuthal quantum number [tex]\( l = -1 \)[/tex] is invalid because [tex]\( l \)[/tex] cannot be negative.
6. [tex]\( m = 2 \)[/tex]:
- The magnetic quantum number [tex]\( m_l = 2 \)[/tex] is valid for [tex]\( l \geq 2 \)[/tex]. Since [tex]\( n = 3 \)[/tex] allows [tex]\( l = 2 \)[/tex], [tex]\( m_l = 2 \)[/tex] is valid.
Based on this analysis, the validity of the quantum numbers is:
- [tex]\( I = 3 \)[/tex]: False
- [tex]\( m = 3 \)[/tex]: False
- [tex]\( I = 0 \)[/tex]: True
- [tex]\( m = -2 \)[/tex]: True
- [tex]\( I = -1 \)[/tex]: False
- [tex]\( m = 2 \)[/tex]: True
So, the result for the given [tex]\( n = 3 \)[/tex] electron shell is:
[tex]\( (False, False, True, True, False, True) \)[/tex]
