Balmer series equation

Comparison with the empirical Equation (1.4) shows that = /re /S/z eg and that n" = 2 for the Balmer series. Similarly n" = 1, 3, 4, and 5 for the Lyman, Paschen, Brackett and Pfimd series, although it is important to realize that there is an infinite number of series. Many series with high n" have been observed, by techniques of radioastronomy, in the interstellar medium, where there is a large amount of atomic hydrogen. For example, the (n = 167) — ( " = 166) transition has been observed with V = 1.425 GFIz (1 = 21.04 cm). 

Question. Using Equations (1.11) and (1.12) calculate, to six significant figures, the wavenumbers, in cm of the first two (lowest n") members of the Balmer series of the hydrogen atom. Then convert these to wavelengths, in nm. 

Substituting the value for the population ratio 2/ i = 5.15 x 1CT9 derived from the intensity of the transitions in the Balmer series into Equation 4.4 allows the Balmer temperature to be calculated ... 

This equation was discovered by Balmer in 1885.7 These speotral lines constitute the Balmer series. Other series of lines for hydrogen correspond to transitions from upper states to the state with n = 1 (the Lyman series), to the state with n = 3 (the Paschen series), and sp on. 

The Lyman series is given by the Balmer-Rydberg equation with m = 1 and n > 1. The wavelength A is greatest when n is smallest that is, when n = 2 and n = 3. 

PROBLEM 5.4 The Balmer equation can be extended beyond the visible portion of the electromagnetic spectrum to include lines in the ultraviolet. What is the wavelength (in nanometers) of ultraviolet light in the Balmer series corresponding to a value of n = 7 ... 

Lines in the Brackett series of the hydrogen spectrum are caused by emission of energy accompanying the fall of an electron from outer shells to the fourth shell. The lines can be calculated using the Balmer-Rydberg equation ... 

Balmer s equation was subsequently refined to give an equation that predicts the frequency, v, of any of the lines in any part of the hydrogen spectrum rather than just for his series. It turns out that his was not the most fundamental series, just the first to be discovered. 

Answer Since Hf = 2, this transition gives rise to a spectral line in the Balmer series (see Figure 7.10). From Equation (7.5) we write... 

For Z = 1 the spectrum of the hydrogen atom is obtained from equation (6), and, for raa=2, in particular, the long-familiar Balmer series ... 

The equations of Bohr s theory are in agreement with the observed frequencies in the hydrogen spectrum, as are the observed spectral series— Lyman series (when electrons excited to higher levels relax to the n 1 state) and Balmer series (when electrons excited to higher levels relax to the n 2 state, and so on). Working backward, the observations can also be used to determine the value of Planck s constant. The value obtained in this way was found to be in agreement with the result deduced from the blackbody radiation and photoelectric effect. ... 

For the Balmer series, nf is simply 2 and n, takes the values 3, 4, 5, or 6. In 1908 the German physicist Friedrich Paschen (1865-1947) discovered new spectral lines fitting the above equation if nf = 3 and n = 4 and n, = 5. In 1906, Harvard physicist Theodore Lyman (1874-1954) discovered an ultraviolet series of spectral lines from hydrogen corresponding to nf = 1 and some 16 years later infrared spectral lines were discovered corresponding to nf = 4 and nf = 5. 

Other series of spectral lines occur in the ultraviolet (Lyman series) and infrared (Paschen, Brackett and Pfund series). All lines in all the series obey the general expression given in equation 1.5 where n > n. For the Lyman series, n=, for the Balmer series, n = 2, and for the Paschen, Brackett and Pfund series, k = 3, 4 and 5 respectively. Figure 1.3 shows some of the allowed transitions of the Lyman and Balmer series in the emission spectrum of atomic H. Note the use of the word allowed, the transitions must obey selection... 

Theoretical considerations of emission spectra were slow to develop, although they started in the later 1800 s and extended into the twentieth century. Balmer s equation for the Balmer series of lines of hydrogen started the search for an explanation for the origin of atomic spectra. Later Ritz (1908) noted that lines of hydrogen observed in the ultraviolet by Lyman (1904) fit the Balmer equation if the constant was changed. This work was extended by Rydberg, Kayser, Runge, and Paschen. It was the work of Bohr, with his concept of the astronomical atom and certain postulates... 

Analysis of equation (2-2) for the Balmer series of hydrogen lines indicates that the spectral emission lines are given by the difference between R /4 and R /n. The ratio Ryi/4 is called a fixed term, while Ru/n is called a running term. Similar treatment of the Lyman series produces similar results if the fixed term is Rh/, a fact suggested by Ritz. Thus, the wave-numbers of lines of any series are the results of differences between two terms, one of them being of fixed value. 

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