Multiple Choice
Which of the following is a valid set of quantum numbers (n, l, m_l) for an electron in an atom?
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9. Quantum Mechanics
Quantum Numbers: Principal Quantum Number
Multiple Choice
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
Verified step by step guidance1
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 \(n-1\).
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 \(n-1\).
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.
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 \(-1/2\).
Conclude that the invalid set is the one where \(l\) is not less than \(n\), which is \(n = 1, l = 1, m_l = 0, m_s = +1/2\).
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