Question

(a) For an He+ ion, do the \(2 s\) and \(2 p\) orbitals have the same energy? If not, which orbital has a lower energy? (b) If we add one electron to form the He atom, would your answer to part (a) change?

Short answer

(a) In an He+ ion, the 2s and 2p orbitals have the same energy, as there is no electron shielding effect involved. (b) When an electron is added to form a neutral He atom, the 2s orbital will have lower energy than the 2p orbital due to electron shielding effects.

Step-by-step solution

Analyze the He+ ion

In an He+ ion, there is only one electron present. As a result, there will be no electron-electron interactions and no shielding effect, which often cause energy differences between different orbitals (such as 2s and 2p).

Compare the energy of 2s and 2p orbitals in He+ ion

Since there is no electron shielding effect in an He+ ion, the 2s and 2p orbitals will have the same energy. This is because they have the same principal quantum number (n=2), and the electron shielding effect is the primary factor that causes energy differences between orbitals with the same principal quantum number.

Analyze the He atom

When we add an electron to form a neutral helium atom (He), there are now 2 electrons present. The shielding effect will come into play, as electrons in different orbitals will experience different amounts of shielding from each other.

Compare the energy of 2s and 2p orbitals in He atom

In a neutral helium atom, the more the electron in an orbital is shielded, the less the net attractive force from the nucleus it experiences, and thus the higher the energy of the orbital. In general, s orbitals experience less effective shielding than p orbitals do. Therefore, in a neutral helium atom, the 2s orbital would have lower energy than the 2p orbital.

Final Answer (a)

In an He+ ion, the 2s and 2p orbitals have the same energy.

Final Answer (b)

When an electron is added to form a neutral He atom, the energies of the 2s and 2p orbitals change due to electron shielding effects. The 2s orbital will have lower energy than the 2p orbital in the neutral helium atom.

Key concepts

Electron Orbitals

Electron orbitals are regions around the nucleus where electrons are likely to be found. They are defined by quantum numbers that describe the size, shape, and orientation in space of an electron's probability density. Each orbital can host a maximum of two electrons.

• Principal Quantum Number ( ): Indicates the main energy level occupied by the electron.
• Angular Momentum Quantum Number ( ): Defines the shape of the orbital (s, p, d, f).

In the case of the helium ion ( he^{+} ), the 2s and 2p orbitals refer to the second energy level. Generally, these would have different energies due to differences in electron shielding. However, since he^{+} has only one electron, the shielding effect is negligible, making their energies equivalent.

Ionization

Ionization refers to the process by which an atom or molecule gains or loses electrons to form ions. This can significantly affect the energy levels of electron orbitals. When an atom loses an electron, it becomes a positively charged ion, like He^{+} . In this state, the complexity of interactions, such as electron shielding, is reduced.
In the context of He^{+} , the absence of an additional electron means that the other electron's energy levels are solely determined by its interaction with the nucleus, simplifying the energy landscape. When the helium atom gains back an electron to become a neutral helium atom, internal interactions, including electron shielding, considerably affect the energy distribution among orbitals.

Electron Shielding

Electron shielding describes the phenomenon where inner-shell electrons can shield outer-shell electrons from the full attraction force of the nucleus. This influences the energy of electron orbitals, particularly for multi-electron atoms and ions.

• Shielding Effect: Refers to the reduction in effective nuclear charge on the electron cloud due to repulsive forces among electrons.
  
• Effect on Orbitals: Generally, electron shielding results in outer orbital electrons experiencing a weaker net nuclear charge, increasing their energy in comparison to more tightly bound inner electrons.

In a helium atom, the presence of two electrons causes significant shielding. The 2s orbital experiences less shielding compared to 2p , making it lower in energy when the atom is neutral. Conversely, in He^{+} with a single electron, shielding is not a factor in determining orbital energies.

Quantum Mechanics

Quantum mechanics is the fundamental theory in physics that describes the behavior of particles on an atomic and subatomic level. It explains the physical properties of atoms, molecules, and solids. Several principles in quantum mechanics underpin our understanding of atomic structures and behaviors:

• Wave-Particle Duality: Electrons exhibit properties of both waves and particles. This allows them to exist in 'orbitals' rather than fixed paths.
• Uncertainty Principle: Introduced by Heisenberg, it states that the exact position and momentum of an electron cannot be simultaneously known. This principle explains why we use probability densities (like orbitals) to describe electron positions.
• Quantum State: Defined by quantum numbers, it represents the state of an electron in an atom. Changes in quantum states can result in electron transition between orbitals.
  

In our context, quantum mechanics helps explain why He^{+} , with its single electron, sees no energy difference between the 2s and 2p orbitals, and why in the neutral helium atom, these orbitals differ in energy due to electron interactions predicted by quantum mechanics.