Balmer equation

The frequencies of hydrogen emission lines in the infrared region of the spectrum other than the visible region would be predicted by replacing the constant 2 in the Balmer equation by the variable m, where m is an integer smaller than n m = 3,4,... 

First we determine the frequency of the radiation, and then match it with the Balmer equation. 

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 ... 

Use the Balmer equation to calculate the wavelength (in nanometers) of the spectral line for hydrogen when n = 6. What is the energy (in kilojoules per mole) of the radiation corresponding to this line ... 

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... 

The Balmer equation, written in terms of wavenumbers, becomes... 

Soon after the development of the Balmer equation another series of spectral lines of hydrogen in the ultraviolet region were reported by Lyman. These lines were found to fit the Balmer equation (2-2) if the first term in the parentheses is 1/1 rather than 1/4. It appeared therefore that a more general form of the Balmer equation would be... 

In this relationship, m is an integer greater than 2, with each value of m representing a different spectral line. Balmer was able to predict the wavelength of some spectral lines that were in the near ultraviolet range. The success of Balmer s equation was strengthened when other spectral series of emission lines were discovered in the ultraviolet (Lyman series) and in the infrared (Paschen series). The lines in their series could be determined by modified Balmer equations ... 

This has the same type denominator as the Bahner formula and when the other numbers are compared, it is found that the Bohr equation is essentially the same as the Balmer equation. There is only a slight difference due to the fact that the nucleus in the Bohr model is fixed at die center of the atom while the real spectra include the fact that the electron and proton both orbit around the center-of-mass (the see-saw balance point) of the two particles. That is really very close to the position of the proton because it is much more massive than the electron. When this correction is made to the Bohr formula, the agreement with the experimental spectra is essentially exact. 

How would the Balmer equation (8.4) have to be modified to predict lines in the infrared spectrum of hydrogen [Hint Compare equations (8.4) and (8.6).]... 

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. 

The constants in Bohr s equation and Balmer s equation are related through E = hv. 

The study of the hydrogen atom also played an important role in the development of quantum theory. The Lyman, Balmer, and Paschen series of spectral lines observed in incandescent atomic hydrogen were found to obey the empirical equation... 

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. 

Thinking about the hydrogen spectrum and trying by trial-and-error to organize the data in various ways, Balmer discovered that the wavelengths of the four lines in the hydrogen spectrum can be expressed by the equation... 

Similarly, a value of n = 4 gives the blue-green line at 486.1 nm, a value of n = 5 gives the blue line at 434.0 nm, and so on. Solve Balmer s equation yourself to make sure. 

By adapting Balmer s equation, the Swedish physicist Johannes Rydberg was able to show that every line in the entire spectrum of hydrogen can be fit by a generalized Balmer-Rydberg equation ... 

We ll look further at the Balmer-Rydberg equation and see what the integers m and n represent in Section 5.9. 

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. 

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