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

• A. 3
• B. 1
• C. 7
• D. 5

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, and \(l\) is the azimuthal (angular momentum) quantum number indicating the subshell or shape of the orbital.
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. The magnetic quantum number \(m_l\) determines the orientation of the orbital and can take integer values from \(-l\) to \(+l\), including zero.
4. Since \(l = 2\), \(m_l\) can be \(-2, -1, 0, +1, +2\), which means there are 5 possible values of \(m_l\).
5. Each unique set of quantum numbers \((n, l, m_l)\) corresponds to one orbital, so the number of orbitals with \(n=3\) and \(l=2\) is 5.