Which of the following set of quantum numbers is possible?

• A. n = 8, l = 3 , ml_l = 0, m_s = 0
• B. n = 7, l = 2, m_l = 1, m_s = –1/2
• C. n = 9, l = 1, m_l = –2, m_s = +1/2
• D. n = 3, l = 0, m_l = +3, m_s = +1/2
• E. n = 4, l = 2, m_l = –2, m_s = +1

Answer: (B)

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 only be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
2. Check each set of quantum numbers against these rules. For example, verify if \(l\) is less than \(n\), if \(m_l\) lies between \(-l\) and \(+l\), and if \(m_s\) is either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
3. For the first set: \(n=8\), \(l=3\), \(m_l=0\), \(m_s=0\). Since \(m_s\) must be \(\pm \frac{1}{2}\), this set is invalid.
4. For the second set: \(n=7\), \(l=2\), \(m_l=1\), \(m_s=-\frac{1}{2}\). Check if \(l < n\) (2 < 7), \(m_l\) in \([-2,2]\), and \(m_s\) valid. This set meets all criteria.
5. For the remaining sets, similarly verify each quantum number against the allowed ranges to determine if they are possible or not.