Calculate the shortest wavelength of visible light (in...

Calculate the shortest wavelength of visible light (in nanometers) seen in the spectrum of the hydrogen atom. What are the principal quantum numbers for the levels in this transition? Does Figure $7.11$ include all visible lines?

Key Concepts

Rydberg Formula and Series Limit

The Rydberg formula relates the wavelength of emitted or absorbed light to the initial and final principal quantum numbers of the electron transition. In calculating the shortest wavelength of visible light (the series limit) for the Balmer series, one considers the transition from n = ? to n = 2. This limiting value represents the highest energy (shortest wavelength) photon that can be emitted within that series.

Hydrogen Emission Spectrum

The hydrogen emission spectrum arises from electrons transitioning between discrete energy levels, releasing photons that have specific wavelengths. This quantized nature of energy levels in the hydrogen atom results in distinct spectral lines that are the hallmark of its emission pattern.

Principal Quantum Numbers

Principal quantum numbers are integers (n = 1, 2, 3, …) that label the energy levels of electrons in an atom. In the context of electron transitions, the difference between the lower and higher principal quantum numbers determines the energy of the emitted or absorbed photon, and thus its corresponding wavelength.

Balmer Series

The Balmer series is a set of spectral lines in the hydrogen emission spectrum that occur when electrons fall from higher energy levels (n ? 3) to the n = 2 energy level. These transitions produce wavelengths in the visible range, making the Balmer series particularly significant in both laboratory spectroscopy and astrophysical observations.

Visible Spectrum Range and Graphical Representation

The visible spectrum spans a finite range of wavelengths, roughly from 400 nm to 700 nm. In diagrams or figures that display spectral lines, such as a typical Balmer series plot, not all theoretically possible transitions may be shown or fall within the visible range. Hence, figures might only depict the strongest or most relevant lines, and may omit those transitions that fall outside the observable visible spectrum.

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