Atomic Models – The Balmer Formula

The four visible lines of the hydrogen spectrum correspond to electromagnetic oscillations with frequencies in the order of magnitude of one million times one billion oscillations per second, namely, 731·10¹²Hz, 691·10¹²Hz, 617·10¹²Hz and 457·10¹²Hz. Anders Jonas Angstrom measured this as early as in 1862. He could not, however, explain it theoretically.

Can we describe this spectrum using a mathematical formula? To do this, we need a specific concept. Johann Jakob Balmer, a Swiss teacher, proposed using natural numbers m in his formula. He did so perhaps by analogy with the spectrum of a guitar string, because it involves multiples, that is, m times a fundamental frequency. We could assign natural numbers to each of these frequencies, and check whether this will allow us to create a useful formula. We do not know how Balmer came up with his formula. He discovered that the combination of m²-4/m² correctly describes all frequencies for m equal to 3, 4, 5, and 6.

Rydberg rewrote Balmer’s formula as a difference between 1/2²–1/m². This means that Rydberg assigned two natural numbers to each frequency. In this case, 2 and 3, 2 and 4, 2 and 5, and 2 and 6. Using the Rydberg constant, we can obtain the following numbers: 457·10¹², 617·10¹², 691·10¹², and 731·10¹².

Johannes Rydberg extended his formula for use with any m and n. It can thus be used to predict frequencies in the hydrogen emission spectrum that are outside the visible region. With this prediction, he hit the mark.