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In the 1880s, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he 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 (n) 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.

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  • Rydberg formula
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  • In the 1880s, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he 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 (n) 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.
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abstract
  • In the 1880s, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he 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 (n) 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. First he tried the formula: , where n is the line's wavenumber, n0 is the series limit, m is the line's ordinal number in the series, m' is a constant different for different series and C0 is a universal constant. This did not work very well. Rydberg was trying: when he became aware of Balmer's formula for the hydrogen spectrum λ=h m ²/(m ² − 4). In this equation, m is an integer and h is a constant. Rydberg therefore rewrote Balmer's formula in terms of wavenumbers, as n = no − 4no/m ². This suggested that that the Balmer formula for hydrogen might be a special case with m' = 0 and C0 = 4no, where no = 1/h, the reciprocal of Balmer's constant. The term Co was found to be a universal constant common to all elements, equal to 4/h. This constant is now known as the Rydberg constant, and m' is known as the quantum defect. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. The fundamental role of wavenumbers was also emphasized by the Rydberg-Ritz combination principle of 1908. The fundamental reason for this lies in quantum mechanics. Light wavenumber is proportional to frequency (1/λ = frequency/c), and therefore also proportional to light quantum energy E. Thus, 1/λ = E/hc. 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. [Rydberg's 1888 classical expression for the form of the spectral series was not accompanied by a physical explanation. Ritz's pre-quantum 1908 explanation for the mechanism underlying the spectral series was that atomic electrons behaved like magnets and that the magnets could vibrate with respect to the atomic nucleus (at least temporarily) to produce electromagnetic radiation.] 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.
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