Which of the following sets of three quantum numbers does NOT specify a valid orbital in the hydrogen atom?

• A. n = 3, l = 2, m_l = -1
• B. n = 4, l = 3, m_l = 2
• C. n = 2, l = 1, m_l = 2
• D. n = 1, l = 0, m_l = 0

Answer: (C)

Verified step by step guidance

1. Recall the rules for quantum numbers in the hydrogen atom: the principal quantum number \(n\) must be a positive integer (\(n = 1, 2, 3, \ldots\)).
2. The azimuthal quantum number \(l\) can take integer values from \$0\( up to \)n-1\(, so \)l\( must satisfy \)0 \leq l \leq n-1$.
3. The magnetic quantum number \(m_l\) can take integer values from \(-l\) to \(+l\), so \(m_l\) must satisfy \(-l \leq m_l \leq l\).
4. Check each set of quantum numbers against these rules. For example, for \(n=2\), \(l=1\), the allowed \(m_l\) values are \(-1, 0, 1\). If \(m_l=2\), this is outside the allowed range, so this set is invalid.
5. Confirm that the other sets satisfy all the conditions for \(n\), \(l\), and \(m_l\) to specify valid orbitals.