Physical Chemistry , 1st ed.

where Zis the charge on the nucleus. So Bohr’s theory is applicable to U^91 ,
which has all but one of its electrons stripped from its nucleus. (However, rel-
ativistic effects will be present, so applicability of Bohr’s equation is even more
limited.) Unfortunately, most matter of interest to chemists is not composed
of single-electron atoms, and the Bohr theory is inherently limited.
But it opened the eyes of contemporary scientists to new ideas: ideas that
some measurable quantities, called observables,are not continuous in their
possible values, like positions in a number line. Rather, they are discrete or
quantized,and can have only certain values. This idea became one of the cen-
tral tenets of the new quantum mechanics.

9.10 The de Broglie Equation

Between the introduction of Bohr’s theory and the development of quantum
mechanics, there was very little in the way of new contributions to the under-
standing of matter—except for an important idea put forth by Louis de Broglie
in 1924. De Broglie, a scientist whose family was part of the French aristocracy,
hypothesized that if a wave like light can have particle properties, why can’t
particles like electrons, protons, and so on have waveproperties?
We can understand de Broglie’s hypothesis by equating the expression for
energy from special relativity and from quantum theory:

Emc^2

Eh

Therefore

mc^2 h

Since c (that is, the speed of light equals its frequency times its wave-
length; this is a standard conversion), we can substitute for the frequency :

mc^2 h

c

Canceling cout of both sides and realizing that cis a velocity and that mass
times velocity is momentum p, we can rearrange:



m

h

c



h

p



De Broglie suggested that this relationship applied to particles, for which the
momentum equals mass times velocity (pmv). The de Broglie equationis
written for particles as



m

h

v

h

p

 (9.42)

This equation states that the wavelength of a particle is inversely proportional
to its momentum,mv, and the proportionality constant is h, Planck’s constant.
That is, de Broglie’s equation implies that a particle of mass macts as a wave.
Only a wave, remember, can have a wavelength.
That photons have momentum was hinted at experimentally only one year
before when Compton announced the change of energy of X rays upon de-
flection by graphite. This Compton effect involves a simultaneous transfer of
energy and momentum when a photon collides with an electron. An under-
standing of the conservation of energy as well as the conservation of momen-
tum allowed one to correctly predict not only the new energies of the photons

9.10 The de Broglie Equation 267