To determine the possible third quantum number [tex]\( m_l \)[/tex] (magnetic quantum number) for a [tex]\( 2p \)[/tex] electron in phosphorus with electron configuration [tex]\( 1s^2 2s^2 2p^6 3s^2 3p^3 \)[/tex], we need to understand the quantum numbers associated with the [tex]\( 2p \)[/tex] orbital.
1. Principal Quantum Number ([tex]\( n \)[/tex]): This is the energy level, which in this case is [tex]\( n = 2 \)[/tex].
2. Azimuthal Quantum Number ([tex]\( l \)[/tex]): This corresponds to the subshell. For a [tex]\( p \)[/tex]-orbital, [tex]\( l = 1 \)[/tex].
3. Magnetic Quantum Number ([tex]\( m_l \)[/tex]): This describes the orientation of the orbital and can take on any integer value between [tex]\(-l\)[/tex] and [tex]\( +l \)[/tex], inclusive. For [tex]\( l = 1 \)[/tex], the possible values of [tex]\( m_l \)[/tex] are:
[tex]\[
m_l = -1, 0, +1
\][/tex]
Given the possible values of [tex]\( m_l \)[/tex]:
Now we check each given option:
A. [tex]\( m_l = 3 \)[/tex] - This value is not one of the allowed values for [tex]\( m_l \)[/tex] (i.e., -1, 0, 1), so it is incorrect.
B. [tex]\( m_l = -1 \)[/tex] - This value is within the allowed range, so it is correct.
C. [tex]\( m_l = 2 \)[/tex] - This value is not one of the allowed values for [tex]\( m_l \)[/tex] (i.e., -1, 0, 1), so it is incorrect.
D. [tex]\( m_l = -2 \)[/tex] - This value is not one of the allowed values for [tex]\( m_l \)[/tex] (i.e., -1, 0, 1), so it is incorrect.
Therefore, the correct value from the given options for the third quantum number [tex]\( m_l \)[/tex] of a [tex]\( 2p \)[/tex] electron in phosphorus is:
B. [tex]\( m_l = -1 \)[/tex]
