(a) For an He+ ion, do the 2s and 2p orbitals have the same energy? If not, which orbital has a lower energy?

검증된 단계별 안내

Understand that for hydrogen-like ions (such as He+), the energy levels depend only on the principal quantum number (n) and not on the type of orbital (s, p, d, etc.).

Recognize that in multi-electron atoms, the energy levels of orbitals with the same principal quantum number can differ due to electron-electron interactions and shielding effects.

For the He+ ion, which is a hydrogen-like ion with only one electron, the 2s and 2p orbitals have the same energy because there are no other electrons to cause splitting of energy levels.

Conclude that in the case of He+, the 2s and 2p orbitals are degenerate, meaning they have the same energy.

Note that this degeneracy is a characteristic of hydrogen-like ions and does not apply to multi-electron atoms where electron-electron interactions cause energy differences between orbitals with the same principal quantum number.

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주요 개념

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Hydrogen-like Ions

Hydrogen-like ions, such as He+, have only one electron and exhibit similar energy level structures to hydrogen. In these ions, the energy levels depend primarily on the principal quantum number (n) and the effective nuclear charge (Z). For He+, the electron experiences a stronger attraction to the nucleus compared to hydrogen, leading to different energy considerations for orbitals.

Orbital Energy Levels

In multi-electron atoms, orbitals of the same principal quantum number (n) can have different energy levels due to electron-electron interactions and shielding effects. However, in hydrogen-like ions, the 2s and 2p orbitals are influenced by the same effective nuclear charge, but the 2s orbital is lower in energy than the 2p orbital due to its spherical shape and closer proximity to the nucleus.

Penetration and Shielding

Penetration refers to how close an electron can get to the nucleus, while shielding describes how inner electrons can reduce the effective nuclear charge felt by outer electrons. The 2s orbital penetrates closer to the nucleus than the 2p orbital, resulting in lower energy for the 2s orbital in He+. This difference in penetration leads to the 2s orbital being more stable and lower in energy compared to the 2p orbital.