Physical Chemistry , 1st ed.

Bohr stated that this Emust equal the energy of the photon:

Eh (9.37)

Now that Bohr had derived an equation for the total energies of the hydrogen

atom, he could substitute into equations 9.36 and 9.37:

EhEfEi

8 

m

(^20) n
ee
f^2
4
h^2

8 
m
(^20) n
ee
i^2
4
h^2




8

m



e

(^20)
e
h
4
 (^2) n

1

i^2



n

1

f^2

 (9.38)

For emission,Eis negative (that is, energy is given off ), and for absorption

Eis positive (energy is absorbed). In terms of wavenumber ̃, equation 9.38

becomes

 ̃ 

8

m

^20

e

h

e^4

 (^3) cn

1

i^2



n

1

f^2

 (9.39)

Compare this with Rydberg’s equation, 9.17. It is the same expression! Bohr

therefore derived an equation that predicts the spectrum of the hydrogen

atom. Also, Bohr is predicting that the Rydberg constant RHis

RH

8

m

^20

e

h

e^4

 (^3) c (9.40)
Substituting for the values of the constants as they were known at that time,
Bohr calculated from his assumptions a value for RHthat differed less than
7% from the experimentally determined value. Current accepted values for
the constants in equation 9.40 yield a theoretical value for RHthat differs by
less than 0.1% from the experimental value. (This can be made even closer
to experimental values by using the reduced mass of the H atom, rather
than the mass of the electron. We will consider reduced masses in the next
chapter.)
The importance of this conclusion cannot be overemphasized. By using
some simple classical mechanics, ignoring the problem with Maxwell’s elec-
tromagnetic theory, and making one single new assumption—the quantization
of angular momentum of the electron—Bohr was able to deduce the spectrum
of the hydrogen atom, a feat unattained by classical mechanics. By deducing
the value of the Rydberg constant, an experimentally determined parameter,
Bohr was showing the scientific community that new ideas about nature were
crucial to the understanding of atoms and molecules. Scientists of his time
were unable to shrug off the fact that Bohr had come up with a way to under-
stand the spectrum of an atom, whatever the source of the derivation. This
crucial step, regarding other measurable quantities like angular momentum as
quantized, was what made the Bohr theory of the hydrogen atom one of the
most important steps in the modern understanding of atoms and molecules.
The limitations of Bohr’s conclusion, however, also cannot be forgotten. It
applies to the hydrogen atom, and onlythe hydrogen atom. Therefore it is lim-
ited, and it is not applicable to any other element that has more than one elec-
tron. The Bohr theory is, however, applicable to an atomic system that has only
a single electron (which means that the systems involved were highly charged
cations), and the ultimate equation for the energy of the system is revised to
Etot
8

Z



2

(^20)
m
n^2
ee
h
4
 2 (9.41)
266 CHAPTER 9 Pre-Quantum Mechanics