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Chapter 6: Problem 67

(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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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