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

• A. 6
• B. 2
• C. 4
• D. 1

Answer: (B)

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 (angular momentum) quantum number indicating the subshell type, and \(m_l\) is the magnetic quantum number indicating the orientation of the orbital within that subshell.
2. Given \(n = 3\), \(l = 2\), and \(m_l = 0\), identify the specific orbital: \(l = 2\) corresponds to the d subshell, and \(m_l = 0\) specifies one particular d orbital within the \(n=3\) energy level.
3. Recall that each orbital (defined by \(n\), \(l\), and \(m_l\)) can hold a maximum of 2 electrons, which must have opposite spins (\(m_s = +\frac{1}{2}\) and \(m_s = -\frac{1}{2}\)).
4. Since the problem fixes \(n\), \(l\), and \(m_l\), the only variable left is the spin quantum number \(m_s\), which can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\), allowing for 2 electrons in this orbital.
5. Therefore, the number of electrons that can have the quantum numbers \(n=3\), \(l=2\), and \(m_l=0\) is 2.