Rydberg Formula - Rydberg Formula For Any Hydrogen-like Element

Rydberg Formula For Any Hydrogen-like Element

The formula above can be extended for use with any hydrogen-like chemical elements with

where

is the wavelength of the light emitted in vacuum;

is the Rydberg constant for this element;

is the atomic number, i.e. the number of protons in the atomic nucleus of this element;

and are integers such that .

It's important to notice that this formula can be directly applied only to hydrogen-like, also called hydrogenic atoms of chemical elements, i.e. atoms with only one electron being affected by an effective nuclear charge (which is easily estimated). Examples would include He+, Li2+, Be3+ etc., where no other electrons exist in the atom.

But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1.

Finally, with certain modifications (replacement of Z by Z−1, and use of the integers 1 and 2 for the ns to give a numerical value of 3⁄₄ for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. This is analogous to the Lyman-alpha line transition for hydrogen, and has the same frequency factor. Because the 2p electron is not screened by any other electrons in the atom from the nucleus, the nuclear charge is diminished only by the single remaining 1s electron, causing the system to be effectively a hydrogenic atom, but with a diminished nuclear charge Z−1. Its frequency is thus the Lyman-alpha hydrogen frequency, increased by a factor of (Z−1)2. This formula of f = c/λ = (Lyman-alpha frequency)⋅(Z−1)2 is historically known as Moseley's law (having added a factor c to convert wavelength to frequency), and can be used to predict wavelengths of the K_α (K-alpha) X-ray spectral emission lines of chemical elements from aluminum to gold. See the biography of Henry Moseley for the historical importance of this law, which was derived empirically at about the same time it was explained by the Bohr model of the atom.

For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides incorrect results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above.