The wavelengths of the Paschen series for hydrogen are given...

The wavelengths of the Paschen series for hydrogen are given by $$ \frac{1}{\lambda}=R_{\mathrm{H}}\left(\frac{1}{3^{2}}-\frac{1}{n^{2}}\right) \quad n=4,5,6, \ldots $$ (a) Calculate the wavelengths of the first three lines in this series. (b) Identity the region of the electromagnetic spectrum in which these lines appear.

The wavelengths of the Paschen series for hydrogen are given by
$$
\frac{1}{\lambda}=R_{\mathrm{H}}\left(\frac{1}{3^{2}}-\frac{1}{n^{2}}\right) \quad n=4,5,6, \ldots
$$
(a) Calculate the wavelengths of the first three lines in this series. (b) Identity the region of the electromagnetic spectrum in which these lines appear.

Key Concepts

Rydberg Formula

The Rydberg formula is a fundamental equation in atomic physics that relates the wavelengths of spectral lines to the transitions between energy levels in a hydrogen atom. It is expressed in terms of the Rydberg constant, which encapsulates the physical parameters governing these transitions. The formula is crucial for predicting the wavelengths of light emitted or absorbed when an electron moves between different quantum states.

Electronic Transitions

Electronic transitions refer to the movement of an electron between discrete energy levels or orbitals within an atom. When an electron transitions from a higher energy level to a lower one, it emits a photon, and the energy of this photon corresponds to the difference between the two energy states. This concept is central to understanding how atomic spectra are produced and analyzed.

Spectral Series and the Paschen Series

Spectral series are sets of wavelengths corresponding to groups of transitions between specific energy levels in the hydrogen atom. The Paschen series, in particular, involves transitions where the electron falls to the third energy level (n = 3) from higher energy levels. This series produces spectral lines primarily in the infrared portion of the electromagnetic spectrum.

Electromagnetic Spectrum (Infrared Region)

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from gamma rays to radio waves. The infrared region, in which the Paschen series lines typically appear, is characterized by wavelengths longer than those of visible light. This region is important in various applications, from thermal imaging to astronomical observations, and its identification is key when studying the emission spectra of atomic transitions.

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