The Balmer Formula

Leading up to the Formula: 1869 - 1882


In the years after the work of Kirchhoff and Bunsen, the major goal in spectroscopy was to determine the quantitative relationships between the lines in the spectrum of a given element as well as relationships between lines of different substances.
For example, George Johnstone Stoney in 1869 speculated that spectra arose from the internal motions of molecules. However, his mathematical theory was rejected and in 1881, Arthur Schuster concluded:

"Most probably some law hitherto undiscovered exists . . . . "

One year later, Schuster added:

"It is the ambitious object of spectroscopy to study the vibrations of atoms and molecules in order to obtain what information we can about the nature of forces which bind them together . . . But we must not too soon expect the discovery of any grand or very general law, for the constitution of what we call a molecule is no doubt a very complicated one, and the difficulty of the problem is so great that were it not for the primary importance of the result which we may finally hope to obtain, all but the most sanguine might well be discouraged to engage in an inquiry which, even after many years of work, may turn out to have been fruitless. . . . In the meantime, we must welcome with delight even the smallest step in the desired direction."

The Balmer Formula: 1885
On June 25, 1884, Johann Jacob Balmer took a fairly large step forward when he delivered a lecture to the Naturforschende Gesellschaft in Basel. He first represented the wavelengths of the four visible lines of the hydrogen spectrum in terms of a "basic number" h:

Balmer recognized the numerators as the sequence 3², 4², 5², 6² and the denominators as the sequence 3² - 2², 4² - 2², 5² - 2², 6² - 2².
So he wound up with a simple formula which expressed the known wavelengths (l) of the hydrogen spectrum in terms of two integers m and n:

For hydrogen, n = 2. Now allow m to take on the values 3, 4, 5, . . . . Each calculation in turn will yield a wavelength of the visible hydrogen spectrum. He predicted the existence of a fifth line at 3969.65 x 10¯⁷ mm. He was soon informed that this line, as well as additional lines, had already been discovered.

Here are some calculations using Balmer's formula.

At the time, Balmer was nearly 60 years old and taught mathematics and calligraphy at a high school for girls as well as giving classes at the University of Basle. Balmer was very interested in mathematical and physical ratios and was probably thrilled he could express the wavelengths of the hydrogen spectrum using integers.
Balmer was devoted to numerology and was interested in things like how many sheep were in a flock or the number of steps of a Pyramid. He had reconstructed the design of the Temple given in Chapters 40-43 of the Book of Ezekiel in the Bible. How then, you may ask, did he come to select the hydrogen spectrum as a problem to solve?

One day, as it happened, Balmer complained to a friend he had "run out of things to do." The friend replied: "Well, you are interested in numbers, why don't you see what you can make of this set of numbers that come from the spectrum of hydrogen?" (In 1871 Ångström had measured the wavelengths of the four lines in the visible spectrum of the hydrogen atom.)

Balmer published his work in two papers, both published in 1885. The first, titled 'Notiz über die Spektrallinien des Wasserstoff,' is the source of the equation above. He also gives the value of the constant (3645.6 x 10¯⁷ mm.) and discusses its significance:

"One might call this number the fundamental number of hydrogen; and if one should succeed in finding the corresponding fundamental numbers for other chemical elements as well, then one could speculate that there exist between these fundamental numbers and the atomic weights [of the substances] in question certain relations, which could be expressed as some function."

He goes on to discuss how the constant determined the limiting wavelength of the lines described by the Balmer Formula:

"If the formula for n = 2 is correct for all the main lines of the hydrogen spectrum, then it implies that towards the utraviolet end these spectral lines approach the wavelength 3645.6 in closer and closer sequence, but cannot cross this limit; while at the red end [of the spectrum] the C-line [today called H_a] represents the line of longest possible [wavelength]. Only if in addition lines of higher order existed, would further lines arise in the infrared region."

In this second paper, Balmer shows that his formula applies to all 12 of the known lines in the hydrogen spectrum. Many of the experimentally measured values were very, very close to Balmer's values, within 0.1 Å or less. There was at least one line, however, that was about 4 Å off. Balmer expressed doubt about the experimentally measured value, NOT his formula! He also correctly predicted that no lines longer than the 6562 x 10¯⁷ mm. line would be discovered in this series and that the lines converge at 3645.6 x 10¯⁷ mm.
with m = 2, 3, 4, . . . and n = 1, 2, 3, . . . ; but the two constants change in a particular pattern.
By higher order, he means allow n to take on higher values, such as 3, 4, 5, and so on in this manner:

n	m
3	4, 5, 6, 7, . . .
4	5, 6, 7, 8, . . .
5	6, 7, 8, 9, . . .

There is also this one, but I'm not sure if Balmer discussed it:

n	m
1	2, 3, 4, 5, . . .

Before leaving Balmer, several points:

1) Balmer's Formula is entirely empirical. By this I mean that it is not derived from theory. The equation works, but no one knew why. That is, until a certain person.
2) That certain person was born October 7, 1885 in Copenhagen. His name? Niels Henrik David Bohr.
3) Be careful when you read about Balmer's Formula in other books. Often, a form of the formula using frequency rather than wavelength is used.

At first Balmer's formula produced nothing but puzzlement, since no theoretical explanation seemed possible. In 1890 Johannes Robert Rydberg generalized Balmer's formula and showed that it had a wider applicability. He introduced the concept of the wave number v, the reciprocal of the wavelength l, and wrote his formula as

v = 1/l = R (1/n₁² - 1/n₂²)

where n₁ and n₂ are integers and R is now known as the Rydberg constant (value = 10973731.534 m¯¹). Later many other atomic spectral lines were found to be consistent with this formula.
For the lines in the hydrogen spectrum (today called the Balmer series), n₁ = 2 and n₂ takes on the values 3, 4, 5, 6, . . . . If you try the calculations (I don't mind if you do, I can wait.), remember to do one over the answer, so as to recover the wavelength.

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In 1885, Balmer wrote these prophetic words:

"It appeared to me that hydrogen . . . more than any other substance is destined to open new paths to the knowledge of the structure of matter and its properties. In this respect the numerical relations among the wavelengths of the first four hydrogen spectral lines should attract our attention particularly."

In 1913, Niels Bohr will announce what is now call the Bohr Model of the Atom. He will offer the correct mechanism for the lines in the hydrogen spectrum.