**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 D*E* = *E*(*f*) = *hf*.
In this equation, D*E* 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* = -3*hc*R_{HNA/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 }

D*E* = (-*me*^{4}*N*_{A}/8*h*^{2}e_{0}^{2})(1/1^{2}
- 1/2^{2}) = (-*me*^{4}*N*_{A}/8*h*^{2}e_{0}^{2})(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 D*E*. Light is
emitted or absorbed at only one place in one atom at a time. Since *E*(*f*) = D*E*,
the complete loss from the atom of one electron would have an energy change of

*E*(*f*) = *hc*R_{H}*N*_{A}.

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

D*E* = *me*^{4}*N*_{A}/8*h*^{2}e_{0}^{2}

Equating these two results and solving for the Rydberg constant of hydrogen R_{H}
gives

R_{H} = *me*^{4}/8*h*^{3}e_{0}^{2}*c*

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.) = *me*^{4}/8*h*^{3}e_{0}^{2}*c*

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

R_{H} = (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.

Copyright 1997 James R. Fromm