For atomic hydrogen, the Paschen series of lines occurs when nf=3, whereas the Brackett series o

step 1 So, a spectral line of hydrogen is given by the redberg formula and it is given as 1 by lambda i

For atomic hydrogen, the Paschen series of lines occurs when...

For atomic hydrogen, the Paschen series of lines occurs when $n_{\mathrm{f}}=3,$ whereas the Brackett series occurs when $n_{\mathrm{f}}=4$ in Equation 30.14 Using this equation, show that the ranges of wavelengths in these two series overlap.

Key Concepts

Rydberg Formula

The Rydberg formula is a fundamental equation in atomic physics that relates the wavelengths of the spectral lines of hydrogen to the transitions between energy levels. It expresses the inverse wavelength (or wavenumber) as a difference in the inverses of the squares of the principal quantum numbers, thereby allowing calculation of the wavelengths emitted when an electron transitions from a higher energy level to a lower one.

Spectral Series

Spectral series are groups of spectral lines that arise when electrons fall to a specific lower energy level in an atom. In hydrogen, each series—such as Lyman (n_f=1), Balmer (n_f=2), Paschen (n_f=3), and Brackett (n_f=4)—is characterized by the same final quantum number (n_f) and encompasses transitions from higher quantum levels. These series illustrate how different ranges of wavelengths are produced as electrons cascade into fixed energy states.

Quantum Transitions

Quantum transitions refer to the changes in an electron’s energy state within an atom, where the electron absorbs or emits energy in the form of a photon. The energy, and therefore the wavelength of the emitted photon, depends on the difference between the initial and final quantum energy levels. In the context of the spectral series, each transition adds to the overall pattern of emission or absorption lines observed in the spectrum.

Wavelength Range Overlap

Wavelength range overlap occurs when the intervals of wavelengths corresponding to different spectral series intersect. In the case of hydrogen, despite the fact that the Paschen series (with n_f=3) and the Brackett series (with n_f=4) are defined by different final quantum numbers, certain transitions from higher energy levels (n_i) in each series can result in emitted photons of similar wavelengths. This overlapping is due to the mathematical relationship in the Rydberg formula, where the differences in 1/n_f^2 and 1/n_i^2 may yield common wavelength values across different series.

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