Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
