To determine the possible value of the third quantum number, [tex]\( m_l \)[/tex], for an electron in the 2p orbital of phosphorus, let's analyze the quantum numbers associated with this electron.
1. Principal Quantum Number (n):
- The electron is in the 2p orbital, so the principal quantum number [tex]\( n \)[/tex] is 2.
2. Azimuthal Quantum Number (l):
- For a p-orbital, which is where the electron is located, the azimuthal quantum number [tex]\( l \)[/tex] is 1 (since for s, [tex]\( l=0 \)[/tex]; for p, [tex]\( l=1 \)[/tex]; for d, [tex]\( l=2 \)[/tex], and so on).
3. Magnetic Quantum Number (m_l):
- The magnetic quantum number [tex]\( m_l \)[/tex] depends on the azimuthal quantum number [tex]\( l \)[/tex].
- For [tex]\( l=1 \)[/tex], the possible values of [tex]\( m_l \)[/tex] are [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex] in integer steps, which means [tex]\( m_l \)[/tex] can be [tex]\( -1, 0, \)[/tex] or [tex]\( 1 \)[/tex].
Now, let's examine the given options one by one to see which matches the possible values of [tex]\( m_l \)[/tex]:
- Option A: [tex]\( m_l = -2 \)[/tex]:
This is not a possible value because [tex]\( m_l \)[/tex] must be within the range [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex].
- Option B: [tex]\( m_l = 2 \)[/tex]:
This is not a possible value because [tex]\( m_l \)[/tex] must be within the range [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex].
- Option C: [tex]\( m_l = -1 \)[/tex]:
This is a possible value because it falls within the range [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex].
- Option D: [tex]\( m_l = 3 \)[/tex]:
This is not a possible value because [tex]\( m_l \)[/tex] must be within the range [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex].
Based on the analysis above, the correct answer is Option C: [tex]\( m_l = -1 \)[/tex].
