객관식
How many orbitals are contained in the third principal energy level (n = 3) of a given atom?
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9. Quantum Mechanics
Quantum Numbers: Principal Quantum Number
객관식
Which of the following sets of quantum numbers is valid for an electron in an atom?
A
n = 1, l = 1, m_l = 0, m_s = +1/2
B
n = 2, l = 0, m_l = 2, 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
검증된 단계별 안내1
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 be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
Check the first set: \(n = 1\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\). Since \(l\) must be less than \(n\), and here \(l = 1\) is not less than \(n = 1\), this set is invalid.
Check the second set: \(n = 2\), \(l = 0\), \(m_l = 2\), \(m_s = +\frac{1}{2}\). For \(l = 0\), \(m_l\) must be between \(-0\) and \(0\), so \(m_l\) can only be \(0\). Here, \(m_l = 2\) is outside this range, so this set is invalid.
Check the third set: \(n = 3\), \(l = 3\), \(m_l = 2\), \(m_s = -\frac{1}{2}\). Since \(l\) must be less than \(n\), and here \(l = 3\) is not less than \(n = 3\), this set is invalid.
Therefore, the only valid set must have \(l\) less than \(n\), \(m_l\) within \(-l\) to \(+l\), and \(m_s\) equal to \(+\frac{1}{2}\) or \(-\frac{1}{2}\). The correct valid set is \(n = 2\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\).
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