To determine the third quantum number [tex]\( m_l \)[/tex] for one of the electrons in the [tex]\( 4p \)[/tex] energy sublevel of bromine, we need to understand the quantum numbers associated with electron orbitals.
1. Principal Quantum Number (n): For the [tex]\( 4p \)[/tex] sublevel, the principal quantum number [tex]\( n \)[/tex] is 4.
2. Azimuthal Quantum Number (l): The azimuthal quantum number, [tex]\( l \)[/tex], determines the shape of the orbital and is associated with the sublevel. For the [tex]\( p \)[/tex] sublevel, [tex]\( l = 1 \)[/tex].
3. Magnetic Quantum Number (m_l): The magnetic quantum number [tex]\( m_l \)[/tex] describes the orientation of the orbital in space and can take on integer values ranging from [tex]\(-l\)[/tex] to [tex]\(+l\)[/tex] (inclusive). Thus, for [tex]\( l = 1 \)[/tex]:
[tex]\[
m_l = -1, 0, +1
\][/tex]
Next, we consider the given options:
- A. [tex]\( m_l = -1 \)[/tex]: This value is valid because it falls within the range of possible [tex]\( m_l \)[/tex] values for the [tex]\( p \)[/tex] sublevel.
- B. [tex]\( m_l = +\frac{1}{2} \)[/tex]: This is not a valid [tex]\( m_l \)[/tex] value because [tex]\( m_l \)[/tex] must be an integer.
- C. [tex]\( m_l = 4 \)[/tex]: This value is not valid because it is outside the range of possible [tex]\( m_l \)[/tex] values for the [tex]\( p \)[/tex] sublevel.
- D. [tex]\( m_l = 5 \)[/tex]: This value is not valid because it is outside the range of possible [tex]\( m_l \)[/tex] values for the [tex]\( p \)[/tex] sublevel.
Therefore, the valid third quantum number for one of the electrons in the [tex]\( 4p \)[/tex] energy sublevel of bromine is:
[tex]\[
\boxed{m_l = -1}
\][/tex]
