The Foundations of Chemistry

5-12 Atomic Spectra and the Bohr Atom 203

stable state). Going away from the nucleus, the allowable orbits are farther apart in distance,
but closer together in energy. Consider the two possible limits of these equations. One limit
is when n1; this describes the electron at the smallest possible distance from the nucleus
and at its lowest (most negative) energy. The other limit is for very large values of n, that
is, as napproaches infinity. As this limit is approached, the electron is very far from the
nucleus, or effectively removed from the atom; the potential energy is as high as possible,
approaching zero.
Each line in the emission spectrum represents the difference in energiesbetween two
allowed energy levels for the electron. When the electron goes from energy level n 2 to
energy level n 1 , the difference in energy is given off as a single photon. The energy of this
photon can be calculated from Bohr’s equation for the energy, as follows.

Eof photonE 2
E 1 




Factoring out the constant 2.180 10
^18 J and rearranging, we get

Eof photon2.180 10
^18 J


The Planck equation, Ehc/, relates the energy of the photon to the wavelength of the
light, so



h



c

2.180 10 
^18 J 
 

Rearranging for 1/, we obtain



2.180

hc

10 
^18 J

 
 

Comparing this to the Balmer-Rydberg equation, Bohr showed that the Rydberg constant
is equivalent to 2.180 10
^18 J/hc. We can use the values for hand cto obtain the same
value, 1.097 107 m
^1 , that was obtained by Rydberg on a solely empirical basis. Further-
more, Bohr showed the physical meaning of the two whole numbers n 1 and n 2 ; they represent
the two energy states between which the transition takes place. Using this approach, Bohr
was able to use fundamental constants to calculate the wavelengths of the observed lines in
the hydrogen emission spectrum. Thus, Bohr’s application of the idea of quantization of
energy to the electron in an atom provided the answer to a half-century-old puzzle
concerning the discrete colors given off in the spectrum.

1



n 22

1



n 12

1





1



n 22

1



n 12

1



n 22

1



n 12

2.180 10 
^18 J



n 12

2.180 10 
^18 J



n 22

We now accept the fact that electrons occupy only certain energy levels in atoms. In
most atoms, some of the energy differences between levels correspond to the energy of
visible light. Thus, colors associated with electronic transitions in such elements can be
observed by the human eye.
Although the Bohr theory satisfactorily explained the spectra of hydrogen and of other
species containing one electron (He, Li^2 , etc.) the wavelengths in the observed spectra
of more complex species could not be calculated. Bohr’s assumption of circular orbits was
modified in 1916 by Arnold Sommerfeld (1868–1951), who assumed elliptical orbits. Even
so, the Bohr approach was doomed to failure, because it modified classical mechanics to
solve a problem that could not be solved by classical mechanics. It was a contrived solu-
tion. This failure of classical mechanics set the stage for the development of a new physics,
quantum mechanics, to deal with small particles. The Bohr theory, however, did intro-
duce the ideas that only certain energy levels are possible, that these energy levels are