Which electronic transition in a hydrogen atom causes the absorption line at the shortest wavelength?

• A. n = 2 to n = 3
• B. n = 1 to n = 2
• C. n = 1 to n = 3
• D. n = 2 to n = 4

Answer: (C)

Verified step by step guidance

1. Recall that the wavelength of light absorbed or emitted during an electronic transition in a hydrogen atom is related to the energy difference between the initial and final energy levels.
2. Use the Rydberg formula for the wavelength of the photon absorbed or emitted: \\ \( \frac{1}{\lambda} = R \left( \frac{1}{n_{\text{initial}}^2} - \frac{1}{n_{\text{final}}^2} \right) \), where \(R\) is the Rydberg constant, \(n_{\text{initial}}\) is the lower energy level, and \(n_{\text{final}}\) is the higher energy level.
3. Calculate the energy difference for each transition by evaluating the term \( \left( \frac{1}{n_{\text{initial}}^2} - \frac{1}{n_{\text{final}}^2} \right) \) for each given pair of \(n\) values.
4. Recognize that a larger energy difference corresponds to a higher frequency and thus a shorter wavelength, since \( E = h \nu = \frac{hc}{\lambda} \).
5. Compare the calculated values for each transition to determine which has the largest energy difference, and therefore corresponds to the absorption line at the shortest wavelength.