Rydberg formula

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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.

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.

1 History
2 Rydberg formula for hydrogen
3 Rydberg formula for any hydrogen-like element
4 References
5 See also

[edit] History

By 1890, Rydberg had discovered a formula describing the relation between the wavelengths in spectral lines of alkali metals, and also found that the 1885 Balmer's formula for the visible lines of hydrogen was a special case of a more general law.

Specifically, Rydberg found that he could simplify his calculations by using the wavenumber (the number of waves occupying a set unit of length, equal to 1/λ, the inverse of the wavelength) 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.

Although the Rydberg formula was later found to be imprecise for most spectral lines of atoms of higher atomic number than hydrogen, it was found to be quite precise for hydrogen, generating not only the Balmer series in the visible spectrum, but also other series of lines in the ultraviolet and infrared. By 1906, Theodore Lyman had begun to analyze the hydrogen Lyman series of wavelengths in the ultraviolet spectrum named for him, that were already known to fit the Rydberg formula. Other hydrogen line-series in the infrared, named for other discoverers, are described below. All are described by the Rydberg equation.

Besides describing the hydrogen series spectral lines, the Rydberg equation also describes alkali metal atoms with a single valence electron orbiting well clear of the inner electron core. It also generates good values for K-alpha (K_α) spectral lines in most elements, with modification as described by Moseley's law.

Expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery, but the fundamental reason for this was not discovered for another 15 years. Light wavenumber is proportional to frequency (1/λ = frequency/c), and therefore also proportional to light quantum energy E. Thus, 1/λ = (h/c)E. Modern understanding is that Rydberg's plots were simplified because of the underlying simplicity of the behavior of spectral lines, in terms of fixed (quantized) energy differences between electron orbitals in atoms. This phenomenon was first understood by Niels Bohr in 1913, as incorporated in the Bohr model of the atom.

In Bohr's conception of the atom, the integer Rydberg (and Balmer) n numbers represent electron orbitals at different integral distances from the atom. A frequency (or spectral energy) emitted in a transition from n₁ to n₂ therefore represents the photon energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2.

[edit] 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

λ vac

is the wavelength of the light emitted in vacuum,

R 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.

[edit] 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 of chemical elements, i.e. atoms with only one electron being affected by an easy-to-estimate effective nuclear charge. Examples would include He⁺, Li²⁺, Be³⁺ etc., where no other electrons exist in the atom.

The Rydberg formula provides correct wavelengths for extremely 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 n's 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.

[edit] References

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

Categories: Atomic physics | Foundational quantum physics | Hydrogen physics | History of physics

Rydberg formula

From Wikipedia, the free encyclopedia

Contents

[edit] History

[edit] Rydberg formula for hydrogen

[edit] Rydberg formula for any hydrogen-like element

[edit] References

[edit] See also

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