Mysteries of Atomic Spectra

Mysteries of Atomic Spectra

As noted in Quantization of Energy , the energies of some small systems are quantized. Atomic and molecular emission and absorption spectra have been known for over a century to be discrete (or quantized). (See this figure.) Maxwell and others had realized that there must be a connection between the spectrum of an atom and its structure, something like the resonant frequencies of musical instruments. But, in spite of years of efforts by many great minds, no one had a workable theory. (It was a running joke that any theory of atomic and molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra.) Following Einstein’s proposal of photons with quantized energies directly proportional to their wavelengths, it became even more evident that electrons in atoms can exist only in discrete orbits.

Part (a) shows, from left to right, a discharge tube, slit, and diffraction grating producing a line spectrum. Part (b) shows the emission line spectrum for iron. The discrete lines imply quantized energy states for the atoms that produce them. The line spectrum for each element is unique, providing a powerful and much used analytical tool, and many line spectra were well known for many years before they could be explained with physics. (credit for (b): Yttrium91, Wikimedia Commons)

In some cases, it had been possible to devise formulas that described the emission spectra. As you might expect, the simplest atom—hydrogen, with its single electron—has a relatively simple spectrum. The hydrogen spectrum had been observed in the infrared (IR), visible, and ultraviolet (UV), and several series of spectral lines had been observed. (See this figure.) These series are named after early researchers who studied them in particular depth.

The observed hydrogen-spectrum wavelengths can be calculated using the following formula:

\(\cfrac{1}{\lambda }=R\left(\cfrac{1}{{n}_{\text{f}}^{2}}-\cfrac{1}{{n}_{\text{i}}^{2}}\right),\)

where \(\lambda \) is the wavelength of the emitted EM radiation and \(R\) is the Rydberg constant, determined by the experiment to be

\(R=1\text{.}\text{097}×{\text{10}}^{7}/\text{m}\phantom{\rule{0.25em}{0ex}}({\text{or m}}^{-1}).\)

The constant \({n}_{\text{f}}\) is a positive integer associated with a specific series. For the Lyman series, \({n}_{\text{f}}=1\); for the Balmer series, \({n}_{\text{f}}=2\); for the Paschen series, \({n}_{\text{f}}=3\); and so on. The Lyman series is entirely in the UV, while part of the Balmer series is visible with the remainder UV. The Paschen series and all the rest are entirely IR. There are apparently an unlimited number of series, although they lie progressively farther into the infrared and become difficult to observe as \({n}_{\text{f}}\) increases. The constant \({n}_{\text{i}}\) is a positive integer, but it must be greater than \({n}_{\text{f}}\). Thus, for the Balmer series, \({n}_{\text{f}}=2\) and \({n}_{\text{i}}=3, 4, 5, 6, ...\text{}\). Note that \({n}_{\text{i}}\) can approach infinity. While the formula in the wavelengths equation was just a recipe designed to fit data and was not based on physical principles, it did imply a deeper meaning. Balmer first devised the formula for his series alone, and it was later found to describe all the other series by using different values of \({n}_{\text{f}}\). Bohr was the first to comprehend the deeper meaning. Again, we see the interplay between experiment and theory in physics. Experimentally, the spectra were well established, an equation was found to fit the experimental data, but the theoretical foundation was missing.

A schematic of the hydrogen spectrum shows several series named for those who contributed most to their determination. Part of the Balmer series is in the visible spectrum, while the Lyman series is entirely in the UV, and the Paschen series and others are in the IR. Values of \({n}_{\text{f}}\) and \({n}_{\text{i}}\) are shown for some of the lines.