Which of the following sets of quantum numbers is invalid for an electron in an atom?

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

Answer: (D)

Verified step by step guidance

1. Recall the allowed ranges for each quantum number: the principal quantum number \(n\) must be a positive integer (\(n = 1, 2, 3, \ldots\)), the azimuthal quantum number \(l\) can take integer values from \$0\( up to \)n-1\(, the magnetic quantum number \)m_l\( ranges from \)-l\( to \)+l\( in integer steps, and the spin quantum number \)m_s\( can be either \)+\frac{1}{2}\( or \)-\frac{1}{2}$.
2. Check the given quantum numbers for each set against these rules. For example, for \(n=4\), \(l=0\) is valid because \(l\) can be from \$0\( to \)3\( when \)n=4\(; similarly, \)m_l=0\( is valid since \)m_l\( ranges from \)-l\( to \)+l$.
3. Focus on the set \(n=1\), \(l=1\), \(m_l=0\), \(m_s=+\frac{1}{2}\). Since \(l\) must be less than \(n\), and here \(l=1\) is not less than \(n=1\), this violates the rule for \(l\).
4. Confirm that the other quantum numbers in this set (\(m_l=0\) and \(m_s=+\frac{1}{2}\)) are within their allowed ranges, but the invalidity arises solely from the \(l\) value relative to \(n\).
5. Conclude that the set with \(n=1\), \(l=1\), \(m_l=0\), \(m_s=+\frac{1}{2}\) is invalid because the azimuthal quantum number \(l\) cannot be equal to or greater than the principal quantum number \(n\).