Which statement best describes the relationship between the principal quantum number n, the energy of an orbital, and its average radius in a hydrogen atom?

• A. As n increases, the energy increases but the average radius decreases.
• B. As n increases, both the energy and the average radius of the orbital increase.
• C. As n increases, both the energy and the average radius of the orbital decrease.
• D. As n increases, the energy decreases but the average radius increases.

Answer: (B)

Verified step by step guidance

1. Understand that the principal quantum number \(n\) determines the main energy level of an electron in a hydrogen atom.
2. Recall that the energy of an electron in a hydrogen atom is given by the formula \(E_n = -\frac{13.6\,\text{eV}}{n^2}\), where \(n\) is a positive integer (1, 2, 3, ...).
3. Note from the formula that as \(n\) increases, the magnitude of the energy (absolute value) decreases, meaning the energy becomes less negative and thus increases toward zero.
4. Recognize that the average radius of the electron's orbital in a hydrogen atom is proportional to \(n^2\), specifically \(\langle r \rangle \propto n^2\), so as \(n\) increases, the average radius increases.
5. Combine these insights to conclude that as \(n\) increases, both the energy of the orbital increases (becomes less negative) and the average radius of the orbital increases.