# Balmer series

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## spectrum

**spectrum,** arrangement or display of light or other form of radiation separated according to wavelength, frequency, energy, or some other property. Beams of charged particles can be separated into a spectrum according to mass in a mass spectrometer (see mass spectrograph). Physicists often find it useful to separate a beam of particles into a spectrum according to their energy.

### Continuous and Line Spectra

Dispersion, the separation of visible light into a spectrum, may be accomplished by means of a prism or a diffraction grating. Each different wavelength or frequency of visible light corresponds to a different color, so that the spectrum appears as a band of colors ranging from violet at the short-wavelength (high-frequency) end of the spectrum through indigo, blue, green, yellow, and orange, to red at the long-wavelength (low-frequency) end of the spectrum. In addition to visible light, other types of electromagnetic radiation may be spread into a spectrum according to frequency or wavelength.

The spectrum formed from white light contains all colors, or frequencies, and is known as a continuous spectrum. Continuous spectra are produced by all incandescent solids and liquids and by gases under high pressure. A gas under low pressure does not produce a continuous spectrum but instead produces a line spectrum, i.e., one composed of individual lines at specific frequencies characteristic of the gas, rather than a continuous band of all frequencies. If the gas is made incandescent by heat or an electric discharge, the resulting spectrum is a bright-line, or emission, spectrum, consisting of a series of bright lines against a dark background. A dark-line, or absorption, spectrum is the reverse of a bright-line spectrum; it is produced when white light containing all frequencies passes through a gas not hot enough to be incandescent. It consists of a series of dark lines superimposed on a continuous spectrum, each line corresponding to a frequency where a bright line would appear if the gas were incandescent. The Fraunhofer lines appearing in the spectrum of the sun are an example of a dark-line spectrum; they are caused by the absorption of certain frequencies of light by the cooler, outer layers of the solar atmosphere. Line spectra of either type are useful in chemical analysis, since they reveal the presence of particular elements. The instrument used for studying line spectra is the spectroscope.

### The Quantum Explanation of Spectral Lines

The explanation for exact spectral lines for each substance was provided by the quantum theory. In his 1913 model of the hydrogen atom Niels Bohr showed that the observed series of lines could be explained by assuming that electrons are restricted to atomic orbits in which their orbital angular momentum is an integral multiple of the quantity *h*/2π, where *h* is Planck's constant. The integer multiple (e.g., 1, 2, 3 …) of *h*/2π is usually called the quantum number and represented by the symbol *n.*

When an electron changes from an orbit of higher energy (higher angular momentum) to one of lower energy, a photon of light energy is emitted whose frequency ν is related to the energy difference Δ*E* by the equation ν=Δ*E*/*h.* For hydrogen, the frequencies of the spectral lines are given by ν=*cR* (1/*n*_{f}^{2}−1/*n*_{i}^{2}) where *c* is the speed of light, *R* is the Rydberg constant, and *n*_{f} and *n*_{i} are the final and initial quantum numbers of the electron orbits (*n*_{i} is always greater than *n*_{f}). The series of spectral lines for which *n*_{f}=1 is known as the Lyman series; that for *n*_{f}=2 is the Balmer series; that for *n*_{f}=3 is the Paschen series; that for *n*_{f}=4 is the Brackett series; and that for *n*_{f}=5 is the Pfund series. The Bohr theory was not as successful in explaining the spectra of other substances, but later developments of the quantum theory showed that all aspects of atomic and molecular spectra can be explained quantitatively in terms of energy transitions between different allowed quantum states.