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

• A. n = 2, l = 0, m_l = 2, m_s = -1/2
• B. n = 1, l = 1, m_l = 0, m_s = +1/2
• C. n = 3, l = 3, m_l = 2, m_s = -1/2
• D. n = 2, 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\( to \)n-1\(, the magnetic quantum number \)m_l\( ranges from \)-l\( to \)+l\(, and the spin quantum number \)m_s\( can be either \)+\frac{1}{2}\( or \)-\frac{1}{2}$.
2. Check the first set: \(n = 2\), \(l = 0\), \(m_l = 2\), \(m_s = -\frac{1}{2}\). Since \(l = 0\), \(m_l\) must be between \(-0\) and \$0\(, so \)m_l = 2$ is not allowed. Therefore, this set is invalid.
3. Check the second set: \(n = 1\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\). For \(n = 1\), \(l\) must be between \$0\( and \)n-1 = 0\(, so \)l = 1$ is not allowed. This set is invalid.
4. Check the third set: \(n = 3\), \(l = 3\), \(m_l = 2\), \(m_s = -\frac{1}{2}\). For \(n = 3\), \(l\) must be between \$0\( and \)2\(, so \)l = 3$ is not allowed. This set is invalid.
5. The correct set must satisfy all quantum number rules. For example, \(n = 2\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\) is valid because \(l\) is between \$0\( and \)1\(, \)m_l\( is between \)-1\( and \)1\(, and \)m_s\( is either \)+\frac{1}{2}\( or \)-\frac{1}{2}$.