Question
How are the first three quantum numbers (n, l, and mL) related for two spin-paired electrons in the same orbital?
Answers
GPT-4o mini
In quantum mechanics, the behavior of electrons in atoms is described by quantum numbers. For two spin-paired electrons in the same orbital, the first three quantum numbers (n, l, and m_l) relate in the following ways:
1. **Principal Quantum Number (n)**: This quantum number indicates the energy level or shell of an electron. For both electrons in the same orbital, \( n \) must be identical, meaning both electrons occupy the same energy level.
2. **Azimuthal Quantum Number (l)**: This quantum number defines the subshell and is associated with the shape of the orbital. Again, for two spin-paired electrons occupying the same orbital, \( l \) must be the same for both electrons.
3. **Magnetic Quantum Number (m_l)**: This quantum number represents the orientation of the orbital in space and can take on values from \(-l\) to \(+l\), including zero. For two spin-paired electrons in the same orbital, they must have the same value of \( m_l \) because they occupy the same spatial orbital.
In summary, when two electrons are spin-paired in the same orbital:
- \( n \) (principal quantum number) is the same for both.
- \( l \) (azimuthal quantum number) is the same for both.
- \( m_l \) (magnetic quantum number) is the same for both.
The only differing property for the two electrons in the same orbital is their **spin quantum number (m_s)**, where one electron has \( m_s = +\frac{1}{2} \) and the other has \( m_s = -\frac{1}{2} \).
1. **Principal Quantum Number (n)**: This quantum number indicates the energy level or shell of an electron. For both electrons in the same orbital, \( n \) must be identical, meaning both electrons occupy the same energy level.
2. **Azimuthal Quantum Number (l)**: This quantum number defines the subshell and is associated with the shape of the orbital. Again, for two spin-paired electrons occupying the same orbital, \( l \) must be the same for both electrons.
3. **Magnetic Quantum Number (m_l)**: This quantum number represents the orientation of the orbital in space and can take on values from \(-l\) to \(+l\), including zero. For two spin-paired electrons in the same orbital, they must have the same value of \( m_l \) because they occupy the same spatial orbital.
In summary, when two electrons are spin-paired in the same orbital:
- \( n \) (principal quantum number) is the same for both.
- \( l \) (azimuthal quantum number) is the same for both.
- \( m_l \) (magnetic quantum number) is the same for both.
The only differing property for the two electrons in the same orbital is their **spin quantum number (m_s)**, where one electron has \( m_s = +\frac{1}{2} \) and the other has \( m_s = -\frac{1}{2} \).