Consequences of the Bohr Model of Hydrogen

James Richard Fromm

The Bohr model of the hydrogen atom and the Planck equation E(f) = hf, where h is the Planck constant and f is the frequency of the light, link together the spectrum and the energy structure of the hydrogen atom. In equation form this link becomes DE = E(f) = hf. In this equation, DE is the energy difference between any two of the Bohr orbits of hydrogen. Consider the lowest-energy, and lowest-frequency, spectral line of hydrogen found in the Lyman series. According to the Rydberg equation, the energy for any orbit is given by E = -3hcRHNA/n2.   For the lowest Bohr orbit n = 1 and E = -3hcRHNA. For the next lowest orbit n = 2 and E(f) = 3hcRHNA/4. The Bohr model considers the energy difference between the lowest energy level or orbit, that for which n = 1, and the next highest energy level or orbit, that for which n = 2, to be

DE = (-me4NA/8h2e02)(1/12 - 1/22) = (-me4NA/8h2e02)(3/4)

Energy is conserved, and emission of a particular frequency of light as a photon or quantum of energy E(f) = hf means that a single electron making the transition between orbitals must provide the necessary energy change DE. Light is emitted or absorbed at only one place in one atom at a time. Since E(f) = DE, the complete loss from the atom of one electron would have an energy change of

E(f) = hcRHNA.

This would be the energy required if the electron were lost from the innermost Bohr orbit, since (1/1 - 1/infinity) is one. This would be the energy on a molar basis. For the same reason and on the same basis, the Bohr model gives this same energy as

DE = me4NA/8h2e02

Equating these two results and solving for the Rydberg constant of hydrogen RH gives

RH = me4/8h3e02c

All of the terms on the right-hand side of this equation are fundamental physical constants which can be accurately measured in other ways. As a consequence, we now see that the Bohr model and the Planck equation taken together quantitatively give a precise value to the otherwise empirical Rydberg constant.

Experimentally, the Rydberg constant for hydrogen does not have exactly this value but one very close to it. It is instructive for us to ask why this is so. The best modern value for the Rydberg constant of hydrogen is, as stated in a previous section, 10967578.1 m-1 while the combination of fundamental constants above gives 10973731.534(13) m-1 as indicated in the table of fundamental constants. There is, in fact, a good reason for this difference. In any system in which two bodies move, one orbiting about the other, the rotation is not actually of one about the other but of both rotating about their common center of mass. The planets of the solar system orbit about its center of mass, which is close to but not exactly at the center of mass of the sun; the difference is due primarily to the large mass of Jupiter. Likewise, the Moon orbits around the center of mass of the Earth-Moon pair rather than about the Earth itself. The value of 10973735.534(13) m-1 corresponds to R(inf.), the Rydberg constant for a nucleus of "infinite" mass, that is, a nucleus whose mass is so large that the mass of the electron is negligible. The equation which is correct is

R(inf.) = me4/8h3e02c

For atoms which are much heavier than hydrogen this difference is extremely small, but for hydrogen it is significant enough to be measurable:

RH = (1672.6485)/(1672.6845 + 0.910954))R(inf.)

This correction quantitatively accounts for the slightly lower values of the Rydberg constants of all of the lighter elements.

The Bohr model of the hydrogen atom explains the empirical law embodied in the Rydberg equation. Bohr's success in calculating the exact value of one of the most accurately known physical constants in terms of others was a major advance in the physics of the atom.