Concept: The wavelength of an emitted photon is inversely proportional to the energy transition. The shortest possible wavelength across all series corresponds to the maximum possible energy transition in the hydrogen atom.
Formula:$$ \frac{1}{\lambda} = R_H \left( \frac{1}{p^2} - \frac{1}{n^2} \right) $$
Solution:- To minimize \( \lambda \), the expression \( \left( \frac{1}{p^2} - \frac{1}{n^2} \right) \) must be maximized.
- This happens when the electron falls to the lowest possible ground state (\( p = 1 \), the Lyman series) from the highest possible initial state (\( n = \infty \)).
- This maximum energy gap (13.6 eV) translates to the absolute shortest wavelength in the entire hydrogen spectrum.
Why other options are incorrect:Balmer, Paschen, and Brackett series end at higher energy levels (\( p=2, 3, 4 \)), meaning their largest possible energy drops are always smaller than the Lyman series, resulting in strictly longer wavelengths.
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