Question

what are the four quantum numbers for the electron circled in the diagram below?

Explanation:

Step1: Determine the principal quantum number (\(n\))

The electron is in the 3d orbital. The principal quantum number \(n\) corresponds to the energy level. For the 3d orbital, \(n = 3\).

Step2: Determine the azimuthal quantum number (\(l\))

The azimuthal quantum number \(l\) depends on the orbital type. For d - orbitals, \(l=2\) (s: \(l = 0\), p: \(l = 1\), d: \(l = 2\), f: \(l=3\)).

Step3: Determine the magnetic quantum number (\(m_l\))

The magnetic quantum number \(m_l\) ranges from \(-l\) to \(+l\). For a d - orbital (\(l = 2\)), the orbitals are filled in order. The first d - orbital (when considering the filling order) has \(m_l=- 2\) (the possible values of \(m_l\) for \(l = 2\) are \(-2,-1,0, + 1,+2\)).

Step4: Determine the spin quantum number (\(m_s\))

The electron is shown with a downward arrow, which means the spin quantum number \(m_s=-\frac{1}{2}\) (upward arrow would be \(m_s =+\frac{1}{2}\)).

Answer:

The four quantum numbers are \(n = 3\), \(l=2\), \(m_l=-2\), \(m_s =-\frac{1}{2}\)