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Atomic PhysicsChapter 21
Table of Contents
Section 1 Quantization of Energy
Section 2 Models of the Atom
Section 3 Quantum Mechanics
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Section 1 Quantization of EnergyChapter 21
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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Blackbody Radiation
• Physicists study blackbody radiation by observing a hollow object with a small opening, as shown in the diagram.
Section 1 Quantization of Energy
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.
• A blackbody is a perfect radiator and absorber and emits radiation based only on its temperature.
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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.
Chapter 21 Section 1 Quantization of Energy
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Chapter 21
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.
Section 1 Quantization of Energy
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Chapter 21
Blackbody Radiation and the Ultraviolet Catastrophe
Section 1 Quantization of Energy
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Chapter 21
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.
E = hfenergy of a quantum = Planck’s constant frequency
Planck’s constant (h) ≈ 6.63 10–34 J•s
• The energy of a light quantum, which corresponds to the energy difference between two adjacent levels, is given by the following equation:
Section 1 Quantization of Energy
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Chapter 21
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
Section 1 Quantization of Energy
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Chapter 21
Energy of a Photon
Section 1 Quantization of Energy
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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.
Chapter 21 Section 1 Quantization of Energy
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The Photoelectric Effect
Chapter 21 Section 1 Quantization of Energy
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The Photoelectric Effect
Chapter 21 Section 1 Quantization of Energy
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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.
Chapter 21 Section 1 Quantization of Energy
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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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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.
Chapter 21 Section 1 Quantization of Energy
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Compton Shift
Chapter 21 Section 1 Quantization of Energy
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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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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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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.
Section 2 Models of the Atom
• Most of the alpha particles passed through the foil. Some were deflected through very large angles.
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Rutherford’s Gold Foil Experiment
Section 2 Models of the AtomChapter 21
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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.
Section 2 Models of the AtomChapter 21
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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.
Section 2 Models of the Atom
• 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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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.
Section 2 Models of the Atom
• 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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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.
Section 2 Models of the AtomChapter 21
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Emission and Absorption Spectra of Hydrogen
Section 2 Models of the AtomChapter 21
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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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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.
Section 2 Models of 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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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.
Section 2 Models of the Atom
• 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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The Bohr Model of the Atom
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Sample Problem
Interpreting Energy-Level DiagramsAn 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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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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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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Sample Problem, continued
Tip: Note that electron volts were converted to joules so that the units cancel properly.
Section 2 Models of the Atom
E hf
f E
h
(2.55 eV)(1.60 10–19J/eV)
6.6310–34J•sf 6.15 1014 Hz
2. Use Planck’s equation to find the frequency.
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Sample Problem, continued
Section 2 Models of the Atom
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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Sample Problem, continued
Section 2 Models of the Atom
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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• 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.
Section 2 Models of the Atom
The Bohr Model, continued
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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.
Section 3 Quantum MechanicsChapter 21
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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, particle-like 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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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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The Dual Nature of Light
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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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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:
Section 3 Quantum Mechanics
• As seen by this equation, the larger the momentum of an object, the smaller its wavelength.
h
ph
mv
de Broglie wavelength = Planck's constant
momentum
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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:
Section 3 Quantum Mechanics
• 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).
f E
h
frequency = energy
Planck's constant
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Matter Waves, continued
• He assumed that an electron orbit would be stable only if it contained an integral (whole) number of electron wavelengths.
Section 3 Quantum Mechanics
• De Broglie saw a connection between his theory of matter waves and the stable orbits in the Bohr model.
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De Broglie and the Wave-Particle Nature of Electrons
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The Uncertainty Principle
Section 3 Quantum Mechanics
• 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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The Uncertainty Principle
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The Electron Cloud, continued
Section 3 Quantum Mechanics
• 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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The Electron Cloud
Section 3 Quantum Mechanics
• 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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Multiple Choice
1. What is another word for “quantum of light”?
A. blackbody radiation
B. energy level
C. frequency
D. photon
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Multiple Choice
1. What is another word for “quantum of light”?
A. blackbody radiation
B. energy level
C. frequency
D. photon
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Multiple Choice, continued
2. According to classical physics, when a light illuminates a photosensitive surface, what should determine how long it takes before electrons are ejected from the surface?
F. frequency
G. intensity
H. photon energy
J. wavelength
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Multiple Choice, continued
2. According to classical physics, when a light illuminates a photosensitive surface, what should determine how long it takes before electrons are ejected from the surface?
F. frequency
G. intensity
H. photon energy
J. wavelength
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Multiple Choice, continued
3. According to Einstein’s photon theory of light, what does the intensity of light shining on a metal determine?
A. the number of photons hitting the metal in a given time interval
B. the energy of photons hitting the metal
C. whether or not photoelectrons will be emitted
D. KEmax of emitted photoelectrons
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Multiple Choice, continued
3. According to Einstein’s photon theory of light, what does the intensity of light shining on a metal determine?
A. the number of photons hitting the metal in a given time interval
B. the energy of photons hitting the metal
C. whether or not photoelectrons will be emitted
D. KEmax of emitted photoelectrons
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Multiple Choice, continued
4. An X-ray photon is scattered by a stationary electron. How does the frequency of this scattered photon compare to its frequency before being scattered?
F. The new frequency is higher.
G. The new frequency is lower.
H. The frequency stays the same.
J. The scattered photon has no frequency.
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Multiple Choice, continued
4. An X-ray photon is scattered by a stationary electron. How does the frequency of this scattered photon compare to its frequency before being scattered?
F. The new frequency is higher.
G. The new frequency is lower.
H. The frequency stays the same.
J. The scattered photon has no frequency.
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Multiple Choice, continued
5. Which of the following summarizes Thomson’s model of the atom?
A. Atoms are hard, uniform, indestructible spheres.
B. Electrons are embedded in a sphere of positive charge.
C. Electrons orbit the nucleus in the same way
that planets orbit the sun.
D. Electrons exist only at discrete energy levels.
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Multiple Choice, continued
5. Which of the following summarizes Thomson’s model of the atom?
A. Atoms are hard, uniform, indestructible spheres.
B. Electrons are embedded in a sphere of positive charge.
C. Electrons orbit the nucleus in the same way
that planets orbit the sun.
D. Electrons exist only at discrete energy levels.
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Multiple Choice, continued
6. What happens when an electron moves from a higher energy level to a lower energy level in an atom?
F. Energy is absorbed from a source outside the atom.
G. The energy contained in the electromagnetic field inside the atom increases.
H. Energy is released across a continuous range of values.
J. A photon is emitted with energy equal to the difference in energy between the two
levels.
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Multiple Choice, continued
6. What happens when an electron moves from a higher energy level to a lower energy level in an atom?
F. Energy is absorbed from a source outside the atom.
G. The energy contained in the electromagnetic field inside the atom increases.
H. Energy is released across a continuous range of values.
J. A photon is emitted with energy equal to the difference in energy between the two levels.
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Multiple Choice, continued
Use the energy-level diagram for hydrogen to answer
questions 7–8.
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7. What is the frequency of the photon emitted when an electron jumps from E5 to E2?
A. 2.86 eV
B. 6.15 1014 Hz
C. 6.90 1014 Hz
D. 4.31 1033 Hz
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Multiple Choice, continued
Use the energy-level diagram for hydrogen to answer
questions 7–8.
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7. What is the frequency of the photon emitted when an electron jumps from E5 to E2?
A. 2.86 eV
B. 6.15 1014 Hz
C. 6.90 1014 Hz
D. 4.31 1033 Hz
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Multiple Choice, continued
Use the energy-level diagram for hydrogen to answer
questions 7–8.
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8. What frequency of photon would be absorbed when an electron jumps from E2
to E3?
F. 1.89 eV
G. 4.56 1014 Hz
H. 6.89 1014 Hz
J. 2.85 1033 Hz
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Multiple Choice, continued
Use the energy-level diagram for hydrogen to answer
questions 7–8.
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8. What frequency of photon would be absorbed when an electron jumps from E2
to E3?
F. 1.89 eV
G. 4.56 1014 Hz
H. 6.89 1014 Hz
J. 2.85 1033 Hz
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Multiple Choice, continued
9. What type of spectrum is created by applying a high potential difference to a pure atomic gas?
A. an emission spectrum
B. an absorption spectrum
C. a continuous spectrum
D. a visible spectrum
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Multiple Choice, continued
9. What type of spectrum is created by applying a high potential difference to a pure atomic gas?
A. an emission spectrum
B. an absorption spectrum
C. a continuous spectrum
D. a visible spectrum
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Multiple Choice, continued
10.What type of spectrum is used to identify elements in the atmospheres of stars?
F. an emission spectrum
G. an absorption spectrum
H. a continuous spectrum
J. a visible spectrum
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Multiple Choice, continued
10.What type of spectrum is used to identify elements in the atmospheres of stars?
F. an emission spectrum
G. an absorption spectrum
H. a continuous spectrum
J. a visible spectrum
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Multiple Choice, continued
11.What is the speed of a proton (m = 1.67 10–27 kg) with a de Broglie wavelength of 4.00 10–14 m?
A. 1.59 10–30 m/s
B. 1.01 10–7 m/s
C. 9.93 106 m/s
D. 1.01 107 m/s
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Multiple Choice, continued
11.What is the speed of a proton (m = 1.67 10–27 kg) with a de Broglie wavelength of 4.00 10–14 m?
A. 1.59 10–30 m/s
B. 1.01 10–7 m/s
C. 9.93 106 m/s
D. 1.01 107 m/s
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Multiple Choice, continued
12.What does Heisenberg’s uncertainty principle state?
F. It is impossible to simultaneously measure a particle’s position and momentum with
infinite accuracy.
G. It is impossible to measure both a particle’s position and its momentum.
H. The more accurately we know a particle’s position, the more accurately we know
the particle’s momentum.
J. All measurements are uncertain.
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Multiple Choice, continued
12.What does Heisenberg’s uncertainty principle state?
F. It is impossible to simultaneously measure a particle’s position and momentum with
infinite accuracy.
G. It is impossible to measure both a particle’s position and its momentum.
H. The more accurately we know a particle’s position, the more accurately we know
the particle’s momentum.
J. All measurements are uncertain.
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Short Response
13.What is the energy of a photon of light with frequency f = 2.80 1014 Hz? Give your answer in both J and eV.
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Short Response
13.What is the energy of a photon of light with frequency f = 2.80 1014 Hz? Give your answer in both J and eV.
Answer: 1.86 10–19 J, 1.16 eV
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Short Response, continued
14.Light of wavelength 3.0 10–7 m shines on the metals lithium, iron, and mercury, which have work functions of 2.3 eV, 3.9 eV, and 4.5 eV, respectively. Which of these metals will exhibit the photoelectric effect? For each metal that does exhibit the photoelectric effect, what is the maximum kinetic energy of the photoelectrons?
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Short Response, continued
14.Light of wavelength 3.0 10–7 m shines on the metals lithium, iron, and mercury, which have work functions of 2.3 eV, 3.9 eV, and 4.5 eV, respectively. Which of these metals will exhibit the photoelectric effect? For each metal that does exhibit the photoelectric effect, what is the maximum kinetic energy of the photoelectrons?
Answers: lithium and ironlithium: 1.8 eV, iron: 0.2 eV
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Short Response, continued
15. Identify the behavior of an electron as primarily like a wave or like a particle in each of the following situations:
a. traversing a circular orbit in a magnetic field
b. absorbing a photon and being ejected from the surface of a metal
c. forming an interference pattern
Standardized Test PrepChapter 21
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Short Response, continued
15. Identify the behavior of an electron as primarily like a wave or like a particle in each of the following situations:
a. traversing a circular orbit in a magnetic field
b. absorbing a photon and being ejected from the surface of a metal
c. forming an interference pattern
Answers: a. particle; b. particle; c. wave
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Extended Response
16. Describe Bohr’s model of the atom. Identify the assumptions that Bohr made that were a departure from those of classical physics. Explain how Bohr’s model accounts for atomic spectra.
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Extended Response
16. Describe Bohr’s model of the atom. Identify the assumptions that Bohr made that were a departure from those of classical physics. Explain how Bohr’s model accounts for atomic spectra.
Answer: Answers should describe electrons orbiting a nucleus only in discrete energy levels. The model departs from classical physics in that the electrons are only allowed to have certain energies and they do not lose energy simply by moving in an electromagnetic field. Atomic spectra are the result of photons being emitted or absorbed when electrons jump between energy levels.
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Extended Response, continued
17. Electrons are ejected from a surface with speeds ranging up to 4.6 105 m/s when light with a wavelength of 625 nm is used.
a. What is the work function of this surface?
b. What is the threshold frequency for this surface?
Show all your work.
Standardized Test PrepChapter 21
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Extended Response, continued
17. Electrons are ejected from a surface with speeds ranging up to 4.6 105 m/s when light with a wavelength of 625 nm is used.
a. What is the work function of this surface?
b. What is the threshold frequency for this surface?
Show all your work.
Answers: a. 1.39 eV
b. 3.35 1014 Hz
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Extended Response, continued
18. The wave nature of electrons makes an electron microscope, which uses electrons rather than light, possible. The resolving power of any microscope is approximately equal to the wavelength used. A resolution of approximately 1.0 10–11 m would be required in order to “see” an atom.
a. If electrons were used, what minimum kinetic energy of the electrons (in eV) would be required to obtain this degree of resolution?
b. If photons were used, what minimum photon energy would be required?
Standardized Test PrepChapter 21
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Extended Response, continued
18. The wave nature of electrons makes an electron microscope, which uses electrons rather than light, possible. The resolving power of any microscope is approximately equal to the wavelength used. A resolution of approximately 1.0 10–11 m would be required in order to “see” an atom.
a. If electrons were used, what minimum kinetic energy of the electrons (in eV) would be required to obtain this degree of resolution?
b. If photons were used, what minimum photon energy would be required?
Answers: a. 1.5 104 eV; b. 1.2 105 eV
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Atomic Spectra
Section 2 Models of the AtomChapter 21