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

• A. 2
• B. 6
• C. 14
• D. 10

Answer: (D)

Verified step by step guidance

1. Understand the meaning of the quantum numbers: \(n\) is the principal quantum number indicating the energy level or shell, and \(l\) is the azimuthal (angular momentum) quantum number indicating the subshell or orbital type within that shell.
2. For \(n = 3\), the possible values of \(l\) range from 0 to \(n-1\), so \(l\) can be 0, 1, or 2. Here, \(l = 2\) corresponds to the d subshell.
3. Determine the number of orbitals in the \(l = 2\) subshell. The magnetic quantum number \(m_l\) can take values from \(-l\) to \(+l\), so the number of orbitals is \$2l + 1\(. For \)l = 2\(, the number of orbitals is \)2 \times 2 + 1 = 5$.
4. Each orbital can hold 2 electrons with opposite spins (spin quantum number \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\)). Therefore, the total number of electrons in the \(n=3\), \(l=2\) subshell is \$5 \text{ orbitals} \times 2 \text{ electrons/orbital} = 10$ electrons.
5. Thus, the number of electrons that can have the quantum numbers \(n=3\) and \(l=2\) is 10.