Qualitatively, how does the energy required for an electron to transition from n=1 to n=4 in a hydrogen atom compare to transitions between other energy levels?

• A. It is equal to the energy required for any other transition.
• B. No energy is required for this transition.
• C. It is greater than the energy required for transitions between higher principal quantum numbers.
• D. It is less than the energy required for transitions between higher principal quantum numbers.

Answer: (C)

Verified step by step guidance

1. Recall that the energy levels of a hydrogen atom are given by the formula: \(E_n = -\frac{13.6\ \text{eV}}{n^2}\), where \(n\) is the principal quantum number.
2. Understand that the energy required for an electron to transition between two levels \(n_i\) and \(n_f\) is the difference in energy between those levels: \(\Delta E = E_{n_f} - E_{n_i}\).
3. Since the energy levels become closer together as \(n\) increases (because of the \$1/n^2\( dependence), transitions involving lower \)n\( values (like from \)n=1\( to \)n=4\() involve larger energy differences than transitions between higher \)n\( values (like from \)n=4\( to \)n=5$).
4. Therefore, the energy required for the transition from \(n=1\) to \(n=4\) is greater than the energy required for transitions between higher principal quantum numbers.
5. This explains why the transition from \(n=1\) to \(n=4\) requires more energy compared to transitions between levels with higher \(n\) values, due to the larger energy gap at lower quantum numbers.