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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?
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9. Quantum Mechanics
Quantum Numbers: Principal Quantum Number
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Which type of orbital has the lowest energy for a given principal quantum number n?
A
s orbital
B
f orbital
C
d orbital
D
p orbital
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Recall that for a given principal quantum number \(n\), the energy of an orbital depends primarily on the azimuthal quantum number \(l\), which defines the type of orbital: \(s\) (\(l=0\)), \(p\) (\(l=1\)), \(d\) (\(l=2\)), and \(f\) (\(l=3\)).
Understand that within the same principal energy level \(n\), orbitals with lower \(l\) values have lower energy due to their shape and penetration effect closer to the nucleus.
Recognize that \(s\) orbitals (\(l=0\)) are spherical and penetrate closer to the nucleus than \(p\), \(d\), or \(f\) orbitals, resulting in lower energy because electrons in \(s\) orbitals experience less shielding and more nuclear attraction.
Compare the energy ordering for orbitals within the same shell: \(s < p < d < f\) in terms of increasing energy.
Conclude that for a given principal quantum number \(n\), the \(s\) orbital has the lowest energy among the options provided.
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