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

In a He^+ ion, the 2s and 2p orbitals have the same energy since energy levels depend only on the principal quantum number (n). However, in a neutral helium atom (He), the 2s orbital has a lower energy than the 2p orbital because the energy levels of orbitals in multi-electron atoms depend on both n and l, the angular momentum quantum number.

Step-by-step solution

Part (a): Energy comparison of 2s and 2p orbitals in He^+

In the He^+ ion, there is only one electron present. According to the hydrogen-like atomic model, the energy levels of orbitals depend only on the principal quantum number (n) and are independent of the angular momentum quantum number (l). In this case, the 2s (n = 2, l = 0) and 2p (n = 2, l = 1) orbitals have the same principal quantum number, n = 2. Therefore, the 2s and 2p orbitals have the same energy in a He^+ ion.

Part (a): Identifying the lower energy orbital

Since the 2s and 2p orbitals have the same energy in a He^+ ion, there is no lower energy orbital between them in this case.

Part (b): Effect of adding an electron on energy levels

When we add an electron to the He^+ ion to form a neutral helium atom, the situation changes. In a multi-electron atom, such as neutral helium (He), the energy levels of the orbitals depend on both the principal quantum number (n) and the angular momentum quantum number (l). The energy of an orbital increases with increasing values of n and l.

Part (b): Energy comparison of 2s and 2p orbitals in He

In a neutral helium atom (He), the 2s orbital (n = 2, l = 0) has a lower energy than the 2p orbital (n = 2, l = 1) due to the different values of their angular momentum quantum number (l). The energy levels of orbitals in multi-electron atoms depend on both n and l. In conclusion, the 2s and 2p orbitals have the same energy in a He^+ ion, but when an electron is added to form a neutral helium atom, the 2s orbital has lower energy than the 2p orbital.

Key concepts

He+ ion

The He⁺ ion is created when a helium atom loses one of its electrons. This makes He⁺ a hydrogen-like ion, with only a single electron orbiting the nucleus. Because it has only one electron, this ion is simpler to analyze compared to atoms with more electrons.

In a hydrogen-like ion such as He⁺, the energy of the orbitals depends solely on the principal quantum number, denoted as n. This means that in the He⁺ ion, even though there are different types of orbitals like the 2s and 2p, their energies are identical if they have the same n value.

This property makes it straightforward to understand the energies of orbitals in the He⁺ ion, as their energy levels are not influenced by other factors like electron-electron interactions.

principal quantum number

The principal quantum number, often symbolized as n, is one of the four quantum numbers that describe the unique quantum state of an electron. It fundamentally determines the size and energy level of an orbital within an atom.

• Values for n are positive integers: 1, 2, 3, and so on.
• The larger the value of n, the higher the energy level and the further the orbital is from the nucleus.

In the context of the He⁺ ion, the principal quantum number is crucial because it alone dictates the energy of the orbitals. For instance, both the 2s and 2p orbitals of He⁺ have an n value of 2, giving them equal energy. This changes in more complex atoms, where other quantum numbers also play a role.

angular momentum quantum number

The angular momentum quantum number, designated as l, further describes the shape and energy of orbitals. While the principal quantum number determines the orbital's size and energy, the angular momentum quantum number specifies its shape and substantially influences energy in atoms with more than one electron.

• The value of l ranges from 0 to n-1 for each principal quantum number n.
• When l = 0, it's an "s" orbital; l = 1 corresponds to a "p" orbital; l = 2 is a "d" orbital; and l = 3 is an "f" orbital, etc.

For the He⁺ ion, even though the 2s (where l = 0) and 2p (where l = 1) orbitals have the same energy due to their shared principal quantum number, the angular momentum quantum number will play a significant role in multi-electron atoms, like neutral helium.

multi-electron atom

A multi-electron atom, such as neutral helium ( He), contains more than one electron. This additional complexity introduces electron-electron interactions that significantly affect the energy levels of the atom's orbitals. These interactions make energy levels in multi-electron atoms dependent on both the principal quantum number n and the angular momentum quantum number l.

Unlike the case for hydrogen-like ions, orbitals in multi-electron atoms no longer have the same energy just because they share the same n. Instead, the energy also depends on the l. Thus, in the neutral helium atom, the 2s orbital (lower l value) is found to be at a lower energy level than the 2p orbital.

These differences stem from the fact that "s" orbitals are closer to the nucleus and experience less electron shielding than "p" orbitals. This results in lower energy for "s" orbitals, illustrating how the structure becomes more complex with additional electrons.