In 1885 Johann Jacob Balmer (1825-1898) produced an empirical relationship between the four prominent lines in the hydrogen spectrum. Utilizing the wavelengths (lambda) published earlier by the Swedish physicist and astronomer Anders Jonas Ångstrom, he showed that these agreed with the wavelengths calculated by the formula
where n is a whole number, 3 for the red line, 4 for the green, 5 for the blue and 6 for the violet. Using other values of n in addition to these, the equation produces a series of lines which crowd together as n gets successively larger. If n is taken as infinity, the series converges at a wavelength limit of 3645.6 Å. Without having seen such a line, Balmer suggested that there should be one at the edge of the visible violet. A colleague informed him that this line and several more in the ultraviolet region had been detected. These corresponded almost exactly with the frequencies calculated by Balmer's formula. In his paper, Balmer suggested that there might be other series of lines in the hydrogen spectrum corresponding to wavelengths calculated by replacing 22 in his formula by 12, 32, 42, etc.
In 1890, Balmer's formula was revised by Johannes Robert Rydberg (1854-1919) to be expressed in wave numbers (1/lambda). Rydberg's formula for hydrogen is written
Rydberg's value for R is 109,720 cm-1. If this equation is generalized following Balmer's speculation it may be written
It was not until after 1908 that improvements in spectroscopy allowed other series of lines to be actually found in the infrared and ultraviolet region of the electromagnetic spectrum.
Example Calculation: Using Balmer's original formula, calculate the wavelength of the wavelength of the line Balmer predicted at the edge of the violet. The next line in the visible spectrum should be the one with n=7. , the wavelength would then have a value of
Exercise 1: Practice using the Balmer Equation by calculating the wavelength in units for the first 4 line in the hydrogen spectrum.
Exercise 2: Calculate the value wave numbers (in cm-1) for the first five lines in the hydrogen spectrum using the Rydberg equation (Equation 2) above.