Multiple Choice
Which of the following correctly describes the allowed values for the four quantum numbers n, l, m_l, and m_s for an electron in an atom?
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9. Quantum Mechanics
Quantum Numbers: Principal Quantum Number
Multiple Choice
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
Verified step by step guidance1
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 \(-\frac{1}{2}\).
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.
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.
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.
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 \(-\frac{1}{2}\).
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