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Chapter 21
Section 1 Quantization of Energy
Preview
• Objectives
• Blackbody Radiation
• Quantum Energy
• The Photoelectric Effect
• Compton Shift
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Chapter 21
Section 1 Quantization of Energy
Objectives
• Explain how Planck resolved the ultraviolet
catastrophe in blackbody radiation.
• Calculate energy of quanta using Planck’s equation.
• Solve problems involving maximum kinetic energy,
work function, and threshold frequency in the
photoelectric effect.
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Chapter 21
Section 1 Quantization of Energy
Blackbody Radiation
• Physicists study blackbody radiation by
observing a hollow object with a small opening, as
shown in the diagram.
• A blackbody is a perfect radiator and absorber
and emits radiation based only on its temperature.
•
Light enters this hollow object through
the small opening and strikes the interior
wall. Some of the energy is absorbed, but
some is reflected at a random angle. After
many reflections, essentially all of the
incoming energy is absorbed by the cavity
wall.
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Chapter 21
Section 1 Quantization of Energy
Blackbody Radiation, continued
• The ultraviolet catastrophe is the failed
prediction of classical physics that the energy
radiated by a blackbody at extremely short
wavelengths is extremely large and that the total
energy radiated is infinite.
• Max Planck (1858–1947) developed a formula for
blackbody radiation that was in complete
agreement with experimental data at all
wavelengths by assuming that energy comes in
discrete units, or is quantized.
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Chapter 21
Section 1 Quantization of Energy
Blackbody Radiation
The graph on the left shows the intensity of blackbody radiation at
three different temperatures. Classical theory’s prediction for
blackbody radiation (the blue curve) did not correspond to the
experimental data (the red data points) at all wavelengths, whereas
Planck’s theory (the red curve) did.
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Chapter 21
Section 1 Quantization of Energy
Blackbody Radiation and the Ultraviolet
Catastrophe
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Chapter 21
Section 1 Quantization of Energy
Quantum Energy
• Einstein later applied the concept of quantized
energy to light. The units of light energy called
quanta (now called photons) are absorbed or
given off as a result of electrons “jumping” from
one quantum state to another.
• The energy of a light quantum, which
corresponds to the energy difference between two
adjacent levels, is given by the following equation:
E = hf
energy of a quantum = Planck’s constant  frequency
Planck’s constant (h) ≈ 6.63  10–34 J•s
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Chapter 21
Section 1 Quantization of Energy
Quantum Energy
• If Planck’s constant is expressed in units of J•s,
the equation E = hf gives the energy in joules.
• However, in atomic physics, energy is often
expressed in units of the electron volt, eV.
• An electron volt is defined as the energy that an
electron or proton gains when it is accelerated
through a potential difference of 1 V.
• The relation between the electron volt and the
joule is as follows:
1 eV = 1.60  10–19 J
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Chapter 21
Section 1 Quantization of Energy
Energy of a Photon
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Chapter 21
Section 1 Quantization of Energy
The Photoelectric Effect
• The photoelectric effect is the emission of
electrons from a material surface that occurs when
light of certain frequencies shines on the surface of
the material.
• Classical physics cannot explain the photoelectric
effect.
• Einstein assumed that an electromagnetic wave
can be viewed as a stream of particles called
photons. Photon theory accounts for observations
of the photoelectric effect.
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Chapter 21
Section 1 Quantization of Energy
The Photoelectric Effect
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Chapter 21
Section 1 Quantization of Energy
The Photoelectric Effect
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Chapter 21
Section 1 Quantization of Energy
The Photoelectric Effect, continued
• No electrons are emitted if the frequency of the
incoming light falls below a certain frequency,
called the threshold frequency (ft).
• The smallest amount of energy the electron must
have to escape the surface of a metal is the work
function of the metal.
• The work function is equal to hft.
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Chapter 21
Section 1 Quantization of Energy
The Photoelectric Effect, continued
Because energy must be conserved, the maximum
kinetic energy (of photoelectrons ejected from the
surface) is the difference between the photon
energy and the work function of the metal.
maximum kinetic energy of a photoelectron
KEmax = hf – hft
maximum kinetic energy = (Planck’s constant 
frequency of incoming photon) – work function
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Chapter 21
Section 1 Quantization of Energy
Compton Shift
• If light behaves like a particle, then photons should
have momentum as well as energy; both quantities
should be conserved in elastic collisions.
• The American physicist Arthur Compton directed X
rays toward a block of graphite to test this theory.
• He found that the scattered waves had less energy
and longer wavelengths than the incoming waves,
just as he had predicted.
• This change in wavelength, known as the Compton
shift, supports Einstein’s photon theory of light.
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Chapter 21
Section 1 Quantization of Energy
Compton Shift
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Chapter 21
Section 2 Models of the Atom
Preview
• Objectives
• Early Models of the Atom
• Atomic Spectra
• The Bohr Model of the Hydrogen Atom
• Sample Problem
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Chapter 21
Section 2 Models of the Atom
Objectives
• Explain the strengths and weaknesses of
Rutherford’s model of the atom.
• Recognize that each element has a unique emission
and absorption spectrum.
• Explain atomic spectra using Bohr’s model of the
atom.
• Interpret energy-level diagrams.
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Chapter 21
Section 2 Models of the Atom
Early Models of the Atom
• The model of the atom in the
days of Newton was that of a tiny,
hard, indestructible sphere.
• The discovery of the electron in
1897 prompted J. J. Thomson
(1856–1940) to suggest a new
model of the atom.
• In Thomson’s model, electrons
are embedded in a spherical
volume of positive charge like
seeds in a watermelon.
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Chapter 21
Section 2 Models of the Atom
Early Models of the Atom, continued
• Ernest Rutherford (1871–1937) later proved that
Thomson’s model could not be correct.
• In his experiment, a beam of positively charged alpha
particles was projected against a thin metal foil.
• Most of the
alpha particles
passed through
the foil. Some
were deflected
through very
large angles.
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Chapter 21
Section 2 Models of the Atom
Rutherford’s Gold Foil Experiment
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Chapter 21
Section 2 Models of the Atom
Early Models of the Atom, continued
• Rutherford concluded that all of the positive
charge in an atom and most of the atom’s mass
are found in a region that is small compared to the
size of the atom.
• He called this region the the nucleus of the atom.
• Any electrons in the atom were assumed to be in
the relatively large volume outside the nucleus.
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Chapter 21
Section 2 Models of the Atom
Early Models of the Atom, continued
• To explain why electrons were not pulled into the
nucleus, Rutherford viewed the electrons as moving
in orbits about the nucleus.
• However, accelerated charges
should radiate electromagnetic
waves, losing energy. This would
lead to a rapid collapse of the atom.
• This difficulty led scientists to
continue searching for a new model
of the atom.
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Chapter 21
Section 2 Models of the Atom
Atomic Spectra
When the light given off
by an atomic gas is
passed through a prism, a
series of distinct bright
lines is seen. Each line
corresponds to a different
wavelength, or color.
• A diagram or graph that indicates the wavelengths of
radiant energy that a substance emits is called an
emission spectrum.
• Every element has a distinct emission spectrum.
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Chapter 21
Section 2 Models of the Atom
Atomic Spectra, continued
• An element can also absorb light at specific
wavelengths.
• The spectral lines corresponding to this process form
what is known as an absorption spectrum.
• An absorption spectrum can be seen by passing light
containing all wavelengths through a vapor of the
element being analyzed.
• Each line in the absorption spectrum of a given
element coincides with a line in the emission
spectrum of that element.
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Chapter 21
Section 2 Models of the Atom
Emission and Absorption Spectra of
Hydrogen
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Chapter 21
Section 2 Models of the Atom
The Bohr Model of the Hydrogen Atom
• In 1913, the Danish physicist Niels Bohr (1885–
1962) proposed a new model of the hydrogen atom
that explained atomic spectra.
• In Bohr’s model, only certain orbits are allowed.
The electron is never found between these orbits;
instead, it is said to “jump” instantly from one orbit to
another.
• In Bohr’s model, transitions between stable orbits
with different energy levels account for the
discrete spectral lines.
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Chapter 21
Section 2 Models of the Atom
The Bohr Model, continued
• When light of a continuous
spectrum shines on the atom, only
the photons whose energy (hf )
matches the energy separation
between two levels can be
absorbed by the atom.
• When this occurs, an electron jumps from a lower
energy state to a higher energy state, which
corresponds to an orbit farther from the nucleus.
• This is called an excited state. The absorbed photons
account for the dark lines in the absorption spectrum.
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Chapter 21
Section 2 Models of the Atom
The Bohr Model, continued
• Once an electron is in an excited
state, there is a certain
probability that it will jump back
to a lower energy level by
emitting a photon.
• This process is called
spontaneous emission.
• The emitted photons are responsible for the bright lines
in the emission spectrum.
• In both cases, there is a correlation between the “size”
of an electron’s jump and the energy of the photon.
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Chapter 21
Section 2 Models of the Atom
The Bohr Model of the Atom
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Chapter 21
Section 2 Models of the Atom
Sample Problem
Interpreting Energy-Level Diagrams
An electron in a hydrogen atom drops from energy level E4 to
energy level E2. What is the frequency of the emitted photon,
and which line in the emission spectrum corresponds to this
event?
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Chapter 21
Section 2 Models of the Atom
Sample Problem, continued
1. Find the energy of the photon.
The energy of the photon is equal to the change in
the energy of the electron. The electron’s initial
energy level was E4, and the electron’s final energy
level was E2. Using the values from the energy-level
diagram gives the following:
E = Einitial – Efinal = E4 – E2
E = (–0.850 eV) – (–3.40 eV) = 2.55 eV
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Chapter 21
Section 2 Models of the Atom
Sample Problem, continued
Tip: Note that the energies for each energy level are
negative. The reason is that the energy of an electron in
an atom is defined with respect to the amount of work
required to remove the electron from the atom. In some
energy-level diagrams, the energy of E1 is defined as
zero, and the higher energy levels are positive.
In either case, the difference between a higher energy
level and a lower one is always positive, indicating that
the electron loses energy when it drops to a lower level.
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Chapter 21
Section 2 Models of the Atom
Sample Problem, continued
2. Use Planck’s equation to find the frequency.
E  hf
E (2.55 eV)(1.60  10 –19 J/eV)
f  
h
6.63  10 –34 J•s
f  6.15  1014 Hz
Tip: Note that electron volts were converted to joules
so that the units cancel properly.
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Chapter 21
Section 2 Models of the Atom
Sample Problem, continued
3. Find the corresponding line in the emission
spectrum.
Examination of the
diagram shows that
the electron’s jump
from energy level E4
to energy level E2
corresponds to Line
3 in the emission
spectrum.
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Chapter 21
Section 2 Models of the Atom
Sample Problem, continued
4. Evaluate your answer.
Line 3 is in the visible part of the electromagnetic
spectrum and appears to be blue. The frequency
f = 6.15  1014 Hz lies within the range of the
visible spectrum and is toward the violet end, so
it is reasonable that light of this frequency would
be visible blue light.
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Chapter 21
Section 2 Models of the Atom
The Bohr Model, continued
• Bohr’s model was not considered to be a
complete picture of the structure of the atom.
– Bohr assumed that electrons do not radiate
energy when they are in a stable orbit, but his
model offered no explanation for this.
– Another problem with Bohr’s model was that it
could not explain why electrons always have
certain stable orbits
• For these reasons, scientists continued to search
for a new model of the atom.
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Chapter 21
Section 3 Quantum Mechanics
Preview
• Objectives
• The Dual Nature of Light
• Matter Waves
• The Uncertainty Principle
• The Electron Cloud
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Chapter 21
Section 3 Quantum Mechanics
Objectives
• Recognize the dual nature of light and matter.
• Calculate the de Broglie wavelength of matter
waves.
• Distinguish between classical ideas of measurement
and Heisenberg’s uncertainty principle.
• Describe the quantum-mechanical picture of the
atom, including the electron cloud and probability
waves.
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Chapter 21
Section 3 Quantum Mechanics
The Dual Nature of Light
• As seen earlier, there is considerable evidence for
the photon theory of light. In this theory, all
electromagnetic waves consist of photons, particlelike pulses that have energy and momentum.
• On the other hand, light and other electromagnetic
waves exhibit interference and diffraction effects that
are considered to be wave behaviors.
• So, which model is correct?
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Chapter 21
Section 3 Quantum Mechanics
The Dual Nature of Light, continued
• Some experiments can be better explained or only
explained by the photon concept, whereas others
require a wave model.
• Most physicists accept both models and believe
that the true nature of light is not describable in
terms of a single classical picture.
– At one extreme, the electromagnetic wave description
suits the overall interference pattern formed by a large
number of photons.
– At the other extreme, the particle description is more
suitable for dealing with highly energetic photons of very
short wavelengths.
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Chapter 21
Section 3 Quantum Mechanics
The Dual Nature of Light
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Chapter 21
Section 3 Quantum Mechanics
Matter Waves
• In 1924, the French physicist Louis de Broglie
(1892–1987) extended the wave-particle duality.
De Broglie proposed that all forms of matter may
have both wave properties and particle properties.
• Three years after de Broglie’s proposal, C. J.
Davisson and L. Germer, of the United States,
discovered that electrons can be diffracted by a
single crystal of nickel. This important discovery
provided the first experimental confirmation of
de Broglie’s theory.
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Chapter 21
Section 3 Quantum Mechanics
Matter Waves, continued
• The wavelength of a photon is equal to Planck’s
constant (h) divided by the photon’s momentum (p).
De Broglie speculated that this relationship might
also hold for matter waves, as follows:
h
h
 
p mv
Planck's constant
de Broglie wavelength =
momentum
• As seen by this equation, the larger the momentum
of an object, the smaller its wavelength.
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Chapter 21
Section 3 Quantum Mechanics
Matter Waves, continued
• In an analogy with photons, de Broglie postulated
that the frequency of a matter wave can be found
with Planck’s equation, as illustrated below:
E
f 
h
energy
frequency =
Planck's constant
• The dual nature of matter suggested by de Broglie is
quite apparent in the wavelength and frequency
equations, both of which contain particle concepts
(E and mv) and wave concepts ( and f).
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Chapter 21
Section 3 Quantum Mechanics
Matter Waves, continued
• De Broglie saw a connection between his theory of
matter waves and the stable orbits in the Bohr model.
• He assumed that an
electron orbit would
be stable only if it
contained an
integral (whole)
number of electron
wavelengths.
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Chapter 21
Section 3 Quantum Mechanics
De Broglie and the Wave-Particle Nature of
Electrons
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Chapter 21
Section 3 Quantum Mechanics
The Uncertainty Principle
• In 1927, Werner Heisenberg argued that it is
fundamentally impossible to make simultaneous
measurements of a particle’s position and momentum
with infinite accuracy.
• In fact, the more we learn about a particle’s
momentum, the less we know of its position, and the
reverse is also true.
• This principle is known as Heisenberg’s uncertainty
principle.
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Chapter 21
Section 3 Quantum Mechanics
The Uncertainty Principle
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Chapter 21
Section 3 Quantum Mechanics
The Electron Cloud
• Quantum mechanics also predicts that the electron
can be found in a spherical region surrounding
the nucleus.
• This result is often interpreted by viewing the
electron as a cloud surrounding the nucleus.
• Analysis of each of the energy levels of hydrogen
reveals that the most probable electron location
in each case is in agreement with each of the radii
predicted by the Bohr theory.
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Chapter 21
Section 3 Quantum Mechanics
The Electron Cloud, continued
• Because the electron’s location
cannot be precisely determined,
it is useful to discuss the
probability of finding the
electron at different locations.
• The diagram shows the
probability per unit distance of
finding the electron at various
distances from the nucleus in the
ground state of hydrogen.
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