To determine what kind of electron corresponds to the given quantum numbers [tex]\( n=4 \)[/tex], [tex]\( l=2 \)[/tex], [tex]\( m_l=-2 \)[/tex], and [tex]\( m_s=-\frac{1}{2} \)[/tex], we need to understand the role of each quantum number and how they map to specific electron orbitals.
1. Principal Quantum Number ([tex]\( n \)[/tex]):
This indicates the main energy level of the electron. In this case, [tex]\( n = 4 \)[/tex], so the electron is in the fourth energy level.
2. Azimuthal Quantum Number ([tex]\( l \)[/tex]):
This determines the shape of the orbital and is sometimes referred to as the angular momentum quantum number. The value of [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex] and corresponds to different types of orbitals:
- [tex]\( l = 0 \)[/tex] corresponds to an 's' orbital,
- [tex]\( l = 1 \)[/tex] corresponds to a 'p' orbital,
- [tex]\( l = 2 \)[/tex] corresponds to a 'd' orbital,
- [tex]\( l = 3 \)[/tex] corresponds to an 'f' orbital.
Given [tex]\( l = 2 \)[/tex], the electron is in a 'd' orbital.
3. Magnetic Quantum Number ([tex]\( m_l \)[/tex]):
This number specifies the orientation of the orbital in space. The values of [tex]\( m_l \)[/tex] 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 -2, -1, 0, 1, and 2. Since [tex]\( m_l \)[/tex] is given as -2, it is valid for a 'd' orbital.
4. Spin Quantum Number ([tex]\( m_s \)[/tex]):
This specifies the direction of the electron's spin and can be either [tex]\( +\frac{1}{2} \)[/tex] or [tex]\( -\frac{1}{2} \)[/tex]. In this case, [tex]\( m_s = -\frac{1}{2} \)[/tex], indicating the electron has a specific spin orientation.
Bringing all of these together, the electron with quantum numbers [tex]\( n=4 \)[/tex], [tex]\( l=2 \)[/tex], [tex]\( m_l=-2 \)[/tex], and [tex]\( m_s=-\frac{1}{2} \)[/tex] is in the fourth energy level, in a 'd' orbital (because [tex]\( l=2 \)[/tex]). Hence, it is a 4d electron.
Thus, the correct answer is:
A. A 4d electron
