Rydberg formula

The Rydberg formula (Rydberg-Ritz formula) is used in atomic physics for determining the full spectrum of light emission from hydrogen, later extended to be useful with any element by use of the Rydberg-Ritz combination principle.

A piece of the original document detailing the Rydberg formula in 1888.

The spectrum is the set of wavelengths of photons emitted when electrons jump between discrete energy levels, "shells" around the atom of a certain chemical element. This discovery was later to provide motivation for the creation of quantum physics.

The formula was invented by the Swedish physicist Johannes Rydberg and presented on November 5, 1888.

Rydberg formula for hydrogen

${\frac {1}{\lambda _{\mathrm {vac} }}}=R_{\mathrm {H} }Z^{2}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)$

Where

$\lambda _{\mathrm {vac} }$ is the wavelength of the light emitted in vacuum,

$R_{\mathrm {H} }$ is the Rydberg constant for hydrogen,

$n_{1}$ and $n_{2}$ are integers such that $n_{1}<n_{2}$,

Z is the atomic number, which is 1 for hydrogen.

By setting $n_{1}$ to 1 and letting $n_{2}$ run from 2 to infinity, the spectral lines known as the Lyman series converging to 91nm are obtained, in the same manner:

$n_{1}$	$n_{2}$	Name	Converge toward
1	$2\rightarrow \infty$	Lyman series	91nm
2	$3\rightarrow \infty$	Balmer series	365nm
3	$4\rightarrow \infty$	Paschen series	821nm
4	$5\rightarrow \infty$	Brackett series	1459nm
5	$6\rightarrow \infty$	Pfund series	2280nm
6	$7\rightarrow \infty$	Humphreys series	3283nm

The Lyman series is in the ultraviolet while the Balmer series is in the visible and the Paschen, Brackett, Pfund, and Humphreys series are in the infrared.

Rydberg formula for any hydrogen-like element

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

${\frac {1}{\lambda _{\mathrm {vac} }}}=RZ^{2}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)$

where

$\lambda _{\mathrm {vac} }$ is the wavelength of the light emitted in vacuum;

$R$ is the Rydberg constant for this element;

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

$n_{1}$ and $n_{2}$ are integers such that $n_{1}<n_{2}$.

It's important to notice that this formula can be applied only to hydrogen-like, also called hydrogenic atoms chemical elements, i.e. atoms with only one electron on external system of orbitals. Examples would include He⁺, Li²⁺, Be³⁺ etc.

History

By 1890, Rydberg had discovered a formula describing the relation between the wavelengths in lines of alkali metals and found that the Balmer equation was a special case. Although the Rydberg formula was later found to be imprecise with heavier atoms, it is still considered accurate for all the hydrogen series and for alkali metal atoms with a single valency electron orbiting well clear of the inner electron core. By 1906, Lyman had begun to analyze the hydrogen series of wavelengths in the ultraviolet spectrum named for him that were already known to fit the Rydberg formula.

Rydberg simplified his calculations by using the ‘wavenumber’ (the number of waves occupying a set unit of length) as his unit of measurement. He plotted the wavenumbers of successive lines in each series against consecutive integers which represented the order of the lines in that particular series. Finding that the resulting curves were similarly shaped, he sought a single function which could generate all of them when appropriate constants were inserted.

References

Mike Sutton, “Getting the numbers right – the lonely struggle of Rydberg” Chemistry World, Vol. 1, No. 7, July 2004.

See also

Rydberg-Ritz combination principle