Which of the following best explains whether the quantum state n=3, l=3, m_l=-2, m_s=1/2 is an allowable state for an electron in an atom?

• A. No; the value of l cannot be equal to n. For n=3, l must be 0, 1, or 2.
• B. Yes; all quantum numbers are within their allowed ranges.
• C. No; m_s must be -1/2 for this state to be allowed.
• D. No; m_l must be positive when m_s=1/2.

Answer: (A)

Verified step by step guidance

1. Recall the definitions and allowed ranges of the four quantum numbers for an electron in an atom: the principal quantum number \(n\) (positive integers: 1, 2, 3, ...), the azimuthal quantum number \(l\) (integers from 0 to \(n-1\)), the magnetic quantum number \(m_l\) (integers from \(-l\) to \(+l\)), and the spin quantum number \(m_s\) (either \(+\frac{1}{2}\) or \(-\frac{1}{2}\)).
2. Check the given quantum numbers: \(n=3\), \(l=3\), \(m_l=-2\), and \(m_s=+\frac{1}{2}\). Verify if each quantum number falls within its allowed range based on the rules.
3. Focus on the azimuthal quantum number \(l\). Since \(l\) must satisfy \$0 \leq l \leq n-1\(, for \)n=3\(, \)l\( can only be 0, 1, or 2. Here, \)l=3$ which is outside the allowed range.
4. Since \(l=3\) is not allowed for \(n=3\), the quantum state is not valid regardless of the values of \(m_l\) and \(m_s\). There is no need to check the other quantum numbers further for validity in this case.
5. Conclude that the quantum state is not allowable because the azimuthal quantum number \(l\) cannot be equal to \(n\); it must be less than \(n\).