Rydberg’s formula

This relation is called the Bohr frequency condition. If the energies on the right of this expression are each proportional to h ln2, then we have accounted for Rydberg s formula. We still have to explain why the energies have this form, but we have made progress. 

Comparing this expression to the general form of Rydberg s formula we can see that... 

Rydberg s formula (7.1) can now be deduced from the Bohr model. We have... 

Langmuir, whose scheme for the arrangement of the electrons round the atomic nucleus was based on Rydberg s formula, predicted that element 72 would end the rare earth series. UrbainJ had already in 1911 fractionated lutecium residues some 15,000 times in an endeavour to isolate No. 72 and obtained some new lines in the spectrum which he took to indicate its existence but they were really fresh lutecium lines not observable with the less pure specimens. He named the supposed element celtium. 

But it can be established that atoms and molecules in high Rydberg states are similar in many ways (5). Each consists of a core ion and an electron at a large mean distance from the core. For many purposes, the core can be appoximated by a point charge. The coarse energy-level structure is given by Rydberg s formula... 

In the case of covalent interaction, the octet is of primary importance, next to the role of electron pairs, implied by the factor 2 in Rydberg s formula, from which twofold symmetry in the electronic configuration of an atom was inferred. It is important to note that the classification of electrons into s, p, d, f subsets, characterized by four quantum numbers in terms of Pauli s exclusion principle, which states that no two electrons in one atom can have aU four quantum numbers identical, antedates the development of wave mechanics. 

From Rydberg s formula in Eq. (14.4-10), find the wavelength and freqnency of the photons emitted by a hydrogen atom nndergoing the n = 2 n = 1 transition. 

This unit may have come about due to the way some early experiments were done to measure wavelengths and was important in the derivation of Rydberg s formula but it is now established in infrared spectroscopy. In particular, the combined Balmer-Rydberg formula in wave numbers is... 

The first appearance of Rydberg atoms is in the Balmer series of atomic H. Balmer s formula, from 1885, for the wavelengths of the visible series of atomic H, is given by1... 

As shown by Berry [220], the phase shift increases by % at each shape resonance for positive energies above threshold. When the 4/ function drops into the inner well, 5/ moves to occupy the position of 4/ before collapse, 6/ moves to play the role of 5/, etc [201]. As far as QDT is concerned, p relates to the Rydberg energy formula applied in the outer well. Thus, if p = n—v is written with n obtained by node counting, there will be an error of 1, because in fact one should use p = n—l — v when the resonance drops below the bound states. Consequently, Seaton s theorem becomes... 

Balmer s celebrated representation in a single empirical formula of the wavelengths of the principal lines in the visible and near ultra-violet spectrum of hydrogen was not the first attempt in this direction, but was indisputably the most successful. Rydberg [115] records that he heard of Balmer s formula as he was trying out various forms of the function f(n) in his expression... 

This generalized form of Eq. (8.9.1) can be applied only when a single discrete state interacts with one or several continua that do not interact with each other. This situation is typical for a predissociated state. Unfortunately, it does not often occur for an autoionized Rydberg state. When there are several resonances that have large widths relative to their separations, Fano s formula must be replaced by the formula given by Mies (1968) ... 

Bohr s theory was received with a certain amount of scepticism by Rutherford, but it did have the advantage of explaining various features of atomic spectra. There had been numerous attempts to rationalise the lines observed in atomic emission spectra since the invention of the spectroscope by Bunsen and Kirchhoff in 1859 (Chapter 9). Little progress was made until 1885 when Johann Jacob Balmer (1825-1898), a Swiss school teacher, showed that the wavelengths of the four lines then known in the hydrogen spectrum could be expressed in terms of a simple equation. In 1890 Balmer s formula was rearranged by Johannes Robert Rydberg (1854-1919) to the form... 

This formula applies only to atoms/ions with just one electron such as H, He, Li " ", Be, etc., where Z is the number of protons in the nucleus. Although the constant c has been standardized, the value of R is the most accurately measured number in physical science with an relative uncertainty of only 6.6 x 10 in the 90th Edn. of the CRC Handbook. The modem value of the Rydberg constant R is 109737.31568527 cm and early measurements could be made to at least 109737 cm be/ore Bohr derived his formula in 1913. Looking back at Rydberg s work, it is clear that the specific value of his constant is dependent on his choice of (1/X) units, and we will see that this unit is still used in infirared spectroscopy. The use of the reciprocal square of integers is an extension of Balmer s formula. 

Just about the time Rydberg composed his simple mathematical formula, he also discovered Balmer s paper on the spectrum of the hydrogen atom. Rydberg realized immediately that he could... 

Planck s constant and the speed of light are often used to express the Rydberg constant in units of s or length. The formulas below determine the photon frequency or wavelength corresponding to a given electron transition ... 

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