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

• A. 1
• B. 6
• C. 3
• 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 or shell, \(l\) is the azimuthal (angular momentum) quantum number indicating the subshell or orbital shape, and \(m_l\) is the magnetic quantum number indicating the orientation of the orbital within the subshell.
2. Given \(n = 3\), this means we are looking at the third energy level. For \(l = 2\), this corresponds to the d subshell (since \(l = 0\) is s, \(l = 1\) is p, \(l = 2\) is d).
3. The magnetic quantum number \(m_l = 0\) specifies one particular orbital within the d subshell. The d subshell has five orbitals with \(m_l\) values of \(-2, -1, 0, +1, +2\), so \(m_l = 0\) refers to exactly one of these orbitals.
4. Each orbital can hold a maximum of 2 electrons, which must have opposite spins (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 the maximum number of electrons in one orbital, which is 2.