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The Rydberg equation (sometimes called the Balmer equation) is an analytical equation for determining the wavelength of a photon emitted or absorbed when an electron changes energy levels in a hydrogen atom.

In 1884, Johannes Balmer, a Swiss high school teacher, first determined that the prominent emission lines of hydrogen gas had wavelengths which followed the simple equation

1/λ = R × (1/4 - 1/n2)

where λ is the wavelength, R is a constant (now called Rydberg's constant) equal to 1.0967758(±7) × 107 meter-1, and n is an integer greater than two. The optical emission (and absorption) lines of hydrogen have since been known as the Balmer series.

In 1900, the Swede Johannes Rydberg derived this equation independently, but generalized it to cover all transitions in the hydrogen atom, rather than just the Balmer series. It is

1/λ = R × (1/n12 - 1/n22)

where n1 is the lower energy level, and n2 is the higher.

Suppose an electron is in the second electronic level (n1=2). In order to boost that electron to the third electronic level (n2=3), the atom must absorb a photon with a wavelength, λ, of exactly

1/λ = R × (1/22 - 1/32)

or about 6563 Å. If the electron falls from level 3 to level 2, it will emit a photon with that wavelength. You can also use this equation to figure out what photon energy is required to ionize the atom by setting n2 to infinity. Thus a level 1 (ground state) electron can be ejected by a 911 Å photon, a level two electron by a 3645 Å photon, and so on. In this case, however, you do not need a photon with exactly that wavelength, just one that is that wavelength or shorter (shorter wavelengths = higher energy). The extra energy goes into kinetic energy of the atom and electron.

Transition series are given names depending upon what their lowest and highest levels are. The first six are: Lyman (n1=1), Balmer (n1=2), Paschen (n1=3), Brackett (n1=4), Pfund (n1=5), and Humphreys (n1=6). Furthermore, these are each divided into additional groups denoted with Greek letters. For example, Lyman alpha is the transition between levels 1 and 2, Lyman beta is the transition between levels 1 and 3, and so on. (One particularly important line in astronomy is the Balmer alpha line at 6563 Å, often called "H-alpha." It is frequently observed in emission line nebulae.) The Lyman lines are all in the ultraviolet, the Balmer in the optical, and the Paschen, Brackett, Pfund, and Humphreys are all in the infrared regions of the electromagnetic spectrum. Each is named after the scientist who discovered the series.

Rydberg's formula (at the time) supported the Bohr model of the atom, at least for the simple hydrogen atom with its single electron. However, the equation can be modified for other elements ionized to the point that they contain only one electron (singly-ionized helium, doubly-ionized lithium, etc), simply by multiplying the Rydberg constant by the square of the nuclear charge (the atomic number). However, the equation was not perfect. In particular, it could not explain phenomena like the Zeeman and Stark effects. Eventually, the Bohr model was scrapped in favor of the quantum mechanical model of Erwin Schrodinger.

The name Pfund makes me giggle.

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