To determine which electron corresponds to the given quantum numbers [tex]\( n=4, l=2, m_l=-2, m_s=-\frac{1}{2} \)[/tex], let's analyze each quantum number step-by-step:
1. Principal Quantum Number ([tex]\( n \)[/tex]):
- [tex]\( n = 4 \)[/tex]
- This represents the principal energy level or shell. The higher the [tex]\( n \)[/tex], the higher the energy level and the larger the average distance from the nucleus.
2. Azimuthal Quantum Number ([tex]\( l \)[/tex]):
- [tex]\( l = 2 \)[/tex]
- The azimuthal quantum number defines the shape of the orbital and can take on integer values from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex].
- The common representations of [tex]\( l \)[/tex] are:
- [tex]\( l = 0 \)[/tex] -> [tex]\( s \)[/tex] orbital
- [tex]\( l = 1 \)[/tex] -> [tex]\( p \)[/tex] orbital
- [tex]\( l = 2 \)[/tex] -> [tex]\( d \)[/tex] orbital
- [tex]\( l = 3 \)[/tex] -> [tex]\( f \)[/tex] orbital
- Here, since [tex]\( l = 2 \)[/tex], it corresponds to a [tex]\( d \)[/tex] orbital.
3. Magnetic Quantum Number ([tex]\( m_l \)[/tex]):
- [tex]\( m_l = -2 \)[/tex]
- The magnetic quantum number specifies the orientation of the orbital in space and can range from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], including zero.
- For [tex]\( l = 2 \)[/tex], the possible values of [tex]\( m_l \)[/tex] are [tex]\( -2, -1, 0, +1, +2 \)[/tex].
- Therefore, [tex]\( m_l = -2 \)[/tex] is a valid value for a [tex]\( d \)[/tex] orbital.
4. Spin Quantum Number ([tex]\( m_s \)[/tex]):
- [tex]\( m_s = -\frac{1}{2} \)[/tex]
- The spin quantum number indicates the electron's spin and can have values of [tex]\( +\frac{1}{2} \)[/tex] or [tex]\( -\frac{1}{2} \)[/tex].
Since all the quantum numbers are valid and self-consistent, we can conclude that the electron with the given quantum numbers [tex]\( n=4, l=2, m_l=-2, m_s=-\frac{1}{2} \)[/tex] would be located in the [tex]\( 4d \)[/tex] orbital.
Therefore, the correct answer is:
A. A 4d electron.
