Which of the following is NOT a valid set of four quantum numbers (n, l, m_l, m_s) for an electron in an atom?

• A. n = 2, l = 1, m_l = 0, m_s = +1/2
• B. n = 3, l = 2, m_l = -2, m_s = -1/2
• C. n = 4, l = 0, m_l = 0, 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 values 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$.
2. Check the given quantum numbers against these rules. For example, if \(n = 1\), then \(l\) can only be \$0\( because \)l\( ranges from \)0\( to \)n-1$.
3. Look at the set \(n = 1, l = 1, m_l = 0, m_s = +1/2\). Since \(l = 1\) is not allowed when \(n = 1\) (because \(l\) must be less than \(n\)), this set violates the quantum number rules.
4. Verify the other sets to confirm they follow the rules: for \(n = 2\), \(l\) can be \$0\( or \)1\(; for \)n = 3\(, \)l\( can be \)0\(, \)1\(, or \)2\(; for \)n = 4\(, \)l\( can be \)0\(, \)1\(, \)2\(, or \)3\(. Also, \)m_l\( must be between \)-l\( and \)+l\(, and \)m_s\( must be either \)+1/2\( or \)-1/2$.
5. Conclude that the invalid set is the one where \(l\) is not less than \(n\), which is \(n = 1, l = 1, m_l = 0, m_s = +1/2\).