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matter, the existence of matter waves. According to equation 8.5,
8.3 QUANTUM MECHANICS
any moving object should exhibit wave properties. However, an
The Bohr model of the atom successfully accounted for the line ordinary-sized object would have wavelengths so small that they
spectrum of hydrogen and provided an understandable mecha- could not be observed. This is different for electrons because
nism for the emission of photons by atoms. However, the model they have such a tiny mass.
did not predict the spectra of any atom larger than hydrogen,
and there were other limitations. A new, better theory was EXAMPLE 8.5
needed. The roots of a new theory would again come from
What is the wavelength associated with a 0.150 kg baseball with a
experiments with light. Experiments with light had established
velocity of 50.0 m/s?
that sometimes light behaves as a stream of particles, and at
other times it behaves as a wave (see chapter 7). Eventually, sci-
entists began to accept that light has both wave properties and SOLUTION
particle properties, an idea now referred to as the wave-particle
m = 0.150 kg
duality of light. This dual nature of light was recognized in 1905,
v = 50.0 m/s
when Einstein applied Planck’s quantum concept to the energy –34
of a photon with the relationship found in equation 8.1, E = hf, h = 6.63 × 10 J⋅s
where E is the energy of a photon particle, f is the frequency λ = ?
of the associated wave, and h is Planck’s constant. h _

λ =

mv
–34
6.63 × 10 J⋅s
__

MATTER WAVES = m _
(0.150 kg) (50.0 )
s
In 1923, Louis de Broglie, a French physicist, reasoned that
J⋅s
–34
6.63 × 10
symmetry is usually found in nature, so if a particle of light = __ _

m _

has a dual nature, then particles such as electrons should too. (0.150)(50.0) kg ×
s
De Broglie reasoned further that if this is true, an electron in its _ 2
kg⋅ m

circular path around the nucleus would have to have a particu- –34 2 ⋅s

6.63 × 10
s

lar wavelength that would fit into the circumference of the orbit = __ _

7.50 _
kg⋅m
(Figure 8.13). De Broglie derived a relationship from equations s

concerning light and energy, which was
–35
= 8.84 × 10 m
h _
λ = What is the wavelength associated with an electron with a velocity of

mv 6
6.00 × 10 m/s?
equation 8.5
where λ is the wavelength, m is mass, v is velocity, and h is again SOLUTION
Planck’s constant. This equation means that any moving particle
–31
m = 9.11 × 10 kg
has a wavelength that is associated with its mass and velocity. In
6
v = 6.00 × 10 m/s
other words, de Broglie was proposing a wave-particle duality of
–34
h = 6.63 × 10 J⋅s
Orbit λ = ?
h _
λ =
mv
–34
6.63 × 10 J⋅s
___

= 6 m _
–31
(9.11 × 10 kg) (6.00 × 10 )

s
–34
J⋅s
___
6.63 × 10
=

m _
5.47 × 10 –24 kg ×

s
kg⋅m
_ 2

⋅ s
2
–10 _
s

= 1.21 × 10
kg⋅m
_

Standing s
A electron B Misfit = 1.21 × 10 m
–10
wave
–35
The baseball wavelength of 8.84 × 10 m is much too small to be
FIGURE 8.13 (A) Schematic of de Broglie wave, where the –10
detected or measured. The electron wavelength of 1.21 × 10 m, on the
standing wave pattern will just fit in the circumference of an orbit.
This is an allowed orbit. (B) This orbit does not have a circumfer- other hand, is comparable to the distances between atoms in a crystal, so
a beam of electrons through a crystal should produce diffraction.
ence that will match a whole number of wavelengths; it is not an
allowed orbit.
212 CHAPTER 8 Atoms and Periodic Properties 8-10