In 1913, Danish physicist Niels Bohr applied Max Planck’s quantum theory to the nuclear atom of Ernest Rutherford, thus formulating the well-known planetary model of the atom, wherein electrons orbit a central nucleus in well-defined levels of energy (Figure 1). Note that Bohr stated that electrons in the atom follow elliptical orbits (not circles as is often pictured). Also note that Bohr viewed the atom as a classical entity when it is not radiating energy, and, importantly, as a quantum entity when it is radiating.

Figure 1. The classic model of the atom.

Quantum jumps

Bohr postulated that electrons in the atom move in what he called energy states. These energy states are “stationary” (of fixed energy, not fixed position) at different distances from the nucleus. He further reasoned that electrons can undergo transitions from one energy state to another. Bohr called these transitions quantum jumps. He agreed with Planck’s assertion that quantum chunks of radiation are emitted when such a quantum jump occurs from a higher to a lower energy state. This is the process of de-excitation (Figure 2).

Figure 2. The processes of atomic excitation and de-excitation.

Energy of emitted radiation

In 1900 Max Planck postulated that the energy of a radiated quantum of energy is proportional to the frequency of radiation: E ~ f. With the constant of proportionality h (Planck’s constant) we have the familiar E = hf. Five years later Einstein proposed not only that material energy is quantized, but that light itself exists as quantum lumps, or “corpuscles,” later named photons.

Interestingly, Bohr didn’t believe in photons when he developed the quantum view of the atom. However, he did recognize that the frequency of emitted radiation is determined by E = hf (actually, f = E/h). Bohr took this a step further, hypothesizing that the energy E is the difference in the atom’s energy when an electron moves from one orbit to another.

Figure 3. Classically, an electron continuously emitting energy should spiral into the nucleus.

The spiraling problem

The classical view of the atom had a glaring difficulty. Accelerated electrons, according to James Clerk Maxwell’s theory, radiate energy in the form of electromagnetic waves. An electron that orbits a nucleus is constantly accelerating. So an orbiting electron should radiate energy continuously, causing the electron to spiral into the nucleus (Figure 3).

Bohr broke with classical physics by stating that the electron doesn’t radiate light while it accelerates around the nucleus; radiation of light occurs only when the electron makes a transition from a higher energy level to a lower energy level. The emitted frequency of radiation is determined by the energy differences in the atom. The amount of energy that boosts an electron to a higher orbit is the same amount of energy carried away when the electron de-excites back to its lower energy state as illustrated in Figure 2. This is energy conservation at the atomic level.

Differences make the difference

So the atom emits a photon whose energy is equal to the difference in energy between the two energy levels, E = hf. The frequency of the emitted photon, its color, depends on the magnitude of the jump. Planck had related the frequency of radiated light to energy change in matter without a model of the atom. Bohr was able to advance to the next step and determine features of individual atoms.

Atomic spectra: Clues to atomic structure

In the nineteenth century, chemists used optical spectroscopes for chemical analysis. When heated, elements emit light. When this light was viewed through a spectroscope, a pattern of spectral lines emerged. Physicists tried to find order in the confusing arrays of spectral lines. The most orderly spectrum was that of hydrogen (Figure 4). The spacing between successive lines becomes smaller and smaller from the first red line to the last ultraviolet one, until the lines become so close that they seem to merge.

Figure 4. A portion of the hydrogen emission spectrum.

A Swiss schoolteacher, Johann Jakob Balmer, first expressed the wavelengths of these spectral lines in a single mathematical formula in 1885. Balmer, however, was unable to provide a reason why his formula worked so successfully. Nonetheless, his guess that his formula could be extended to predict other lines of hydrogen proved to be correct, leading to the prediction of lines that had not yet been observed.

Another regularity in atomic spectra was found by Swedish physicist and mathematician Johannes Rydberg. He noticed that the frequencies of lines of certain series in many elements, not just hydrogen, followed a formula similar to Balmer’s, and that the sum of the frequencies of to lines in such series often equaled the frequency of a third line. This relationship was later advanced as a general principle by Swiss physicist Walter Ritz and is called the Rydberg-Ritz combination principle. It states that the spectral lines of any element include frequencies that are either the sum or the difference of the frequencies of two other lines. Like Balmer, Ritz was unable to offer an explanation for this regularity. These regularities were the clues that Bohr used to understand the structure of the atom itself.

Bohr’s clarification of Ritz’s principle

Bohr’s explanation of the Rydberg-Ritz combination principle is shown in Figure 5, which shows three of many levels in an atom. The three levels produce three spectral lines. The figure depicts an electron jumping from the third level to the second level (red arrow A), and another electron jumping from the second level to the ground state (green arrow B). The sum of the energies (and the frequencies) for these two transitions equals the energy (and the frequency) of the single jump from the third level to the ground state (blue arrow C).

Figure 5. Three of many energy levels in an atom.

Bohr was able to account for x-rays from heavier elements, showing their emissions as electrons jumping from outer to innermost orbits, the innermost orbits being “hydrogen-like.” He predicted x-ray frequencies that were later experimentally confirmed. Bohr was also able to calculate the “ionization energy” of a hydrogen atom—the energy needed to knock the electron out of the atom completely. His calculated value matched what was known experimentally.

Applications

Using measured frequencies of x-rays as well as visible, infrared, and ultraviolet light, investigators could map energy levels of all the elements. Bohr’s model of the atom accounted for the general chemical properties of the elements, even leading to the discovery of a new element—hafnium.

Bohr solved the mystery of atomic spectra while providing an extremely useful model of the atom. He was quick to stress that his model was to be interpreted as a crude beginning, and the picture of electrons whirling about the nucleus like planets about the Sun was not to be taken literally (to which popularizers of science paid no heed). His sharply defined orbits were conceptual representations of an atom whose later description involved waves—quantum mechanics. His ideas of quantum jumps and frequencies being proportional to energy differences remain part of today’s modern theory.

Acknowledgments

I thank my mentor Ken Ford for his many suggestions, more of which can be found at his Web site www.basic-physics.com.

On the web

See complementary student tutorial screencast 115, 116, and 121 on www.HewittDrewIt.com, and on www.ConceptualAcademy.com.

Paul G. Hewitt (pghewitt@aol.com) is the author of Conceptual Physics, new 13th edition; Conceptual Physical Science, 6th edition, coauthored with daughter Leslie Hewitt and nephew John Suchocki; and Conceptual Integrated Science, new 3rd edition, with coauthors Suzanne Lyons, John Suchocki, and Jennifer Yeh.