Balmers Integer Formula

Our story regarding atomic and molecular spectra really starts with the work of a Swiss mathematician Johann Bahner (1825-1898), in 1885, when he was successful in fitting a formula to the available wavelengths of the H spectmm [2]. The main point of Balmer s formula is that it involves 

A significant extension of Balmer s work occurred in 1888 when the Swedish physicist Johannes Rydberg (1854—1919) developed a similar formula using the reciprocal of the wavelength  

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. 

---= this balances the centripetal force of the electron with the electrostatic attrac- 

The key step occurs right here, in that Bohr solved for the velocity in terms of the velocity instead of taking the square root to find v. Thus, he used an unusual algebra step so that he could insert the quantization of the angular momentum  

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The validity of Balmer s formula itself, however, was called in question by Curtis [27], who undertook precise measurements of the wavelengths of the first six members of the series. Using Balmer s formula, he calculated the value of R (equation 3.2) from the measured wavelengths. The results showed a definite trend in R with the running integer n. Although it was... 

Further support for Bohr s theory came from the discovery of line series in the hydrogen spectrum for which the other integer m took values other than 2. The far ultra-violet series for which m — l has already been mentioned. Lyman announced the discovery of the first two members a year after Bohr s paper, recognizing the close connection between the wavelengths of these lines and Balmer s formula. Balmer himself had asked whether ther- might not exist a series for which m — 3, and Paschen had observed its first two members in the infra-red in 1908 [101], The series with m = 4 was found by Brackett in 1922 [18], with ra — 5 by Pfund in 1924 [108], and with m = 6 by Humphreys in 1953 [66]. It is to be understood that in all these series, the running integer n takes (m +1) as its first value. 

Bohr felt instinctively that Planck s quantized energies were related to the discrete lines of elemental spectra— and to the planetary model of the atom— but he could not find the connection. Thirty years earlier Johann Jakob Balmer, a teacher at a girls secondary school, part-time lecturer at the University of Basel (where, we may note, Paracelsus burned the works of Galen), and mathematics hobbyist had found a numerical relationship between frequencies of the lines in the hydrogen spectrum. The relationship was not obvious because it depended on the reciprocal squares of integers, and this was the very feature that caught Bohr s attention. He later said, As soon as I saw Balmer s formula, the whole thing was immediately clear to me. ... 

The calculated frequency of oscillation, expressed as a function of a row of integers, shows a close qualitative resemblance with observed line spectra and with the digital formulae of Balmer and others. 

This is known as the Balmer Formula — why is n set equal to integers from 3 to infinity We ll see shortly This equation was later generalized by Rydberg to what we now call the Rydberg Equation... 

Aware of only these four hues, Balmer calculated 1 for a fifth hue lyn = 7). A hue with a wavelength very close to the predicted value was observed experimentally. Balmer suggested that his formula might also predict wavelengths of other series of spectral fines by using integer values for n other than 2 and rw n L 1. Other series of hydrogen lines were not known then, but were subsequently discovered (the Lyman, Paschen, Brackett, and Pfund series of fines). 

The quantum theory of matter developed from an innovative suggestion by Johann Balmer to account for the optical spectrum of atomic hydrogen, observed as a series of emission lines at narrowly defined wavelengths. He related the measured wavelengths to a series of integers by the formula... 

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