How many electrons in an atom can have the quantum numbers n = 3, l = 2, m_l = 0?

• A. 4
• B. 6
• C. 1
• D. 2

Answer: (D)

Verified step by step guidance

1. Understand the meaning of each quantum number: \(n\) is the principal quantum number indicating the energy level, \(l\) is the azimuthal quantum number indicating the subshell or shape of the orbital, and \(m_l\) is the magnetic quantum number indicating the orientation of the orbital within the subshell.
2. Given \(n = 3\) and \(l = 2\), identify the type of orbital: \(l = 2\) corresponds to a d-subshell, which contains 5 orbitals with \(m_l\) values of \(-2, -1, 0, +1, +2\).
3. Since \(m_l = 0\) is specified, we are focusing on exactly one orbital within the d-subshell (the one with magnetic quantum number zero).
4. Recall that each orbital can hold a maximum of 2 electrons, distinguished by their spin quantum number \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\).
5. Therefore, the number of electrons that can have the quantum numbers \(n=3\), \(l=2\), and \(m_l=0\) is 2, corresponding to the two possible spin states in that single orbital.