Ch.7 - Quantum MechanicsWorksheetSee all chapters
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Ch.1 - Intro to General Chemistry
Ch.2 - Atoms & Elements
Ch.3 - Chemical Reactions
BONUS: Lab Techniques and Procedures
BONUS: Mathematical Operations and Functions
Ch.4 - Chemical Quantities & Aqueous Reactions
Ch.5 - Gases
Ch.6 - Thermochemistry
Ch.7 - Quantum Mechanics
Ch.8 - Periodic Properties of the Elements
Ch.9 - Bonding & Molecular Structure
Ch.10 - Molecular Shapes & Valence Bond Theory
Ch.11 - Liquids, Solids & Intermolecular Forces
Ch.12 - Solutions
Ch.13 - Chemical Kinetics
Ch.14 - Chemical Equilibrium
Ch.15 - Acid and Base Equilibrium
Ch.16 - Aqueous Equilibrium
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Ch.19 - Nuclear Chemistry
Ch.20 - Organic Chemistry
Ch.22 - Chemistry of the Nonmetals
Ch.23 - Transition Metals and Coordination Compounds

The De Broglie Wavelength equation relates wavelength to velocity or speed. 

Wave Nature of Light

Concept #1: The De Broglie Wavelength Equation 

Example #1: Find the wavelength (in nm) of a proton with a speed of 7.33 x 109 . (Mass of a  proton = 1.67 x 10-27 kg)

Practice: What is the speed of an electron that has a wavelength of 895 μm? (Mass of an electron = 9.11 x 10 -31 kg)

Additional Problems
Calculate the wavelength in meters for a bullet weighing 5.0 g and traveling at 400 m/sec. A) 3.3 x 10-34 m B) 5.8 x 10-19 m C) 1.5 x 10-38 m D) 3.0 x 1033 m E) 3.3 x 10-37 m
The smallest atoms can themselves exhibit quantum-mechanical behavior. Calculate the de Broglie wavelength (in pm) of a hydrogen atom traveling at 475 m/s.
Calculate the DeBroglie wavelength of an electron traveling with a velocity of 4.0 x 10  9 cm/sec in an electron microscope. A. 0.18 Å B. 67 Å C. 1.5 Å D. 0.0018 Å E. 1.1 x 10-38 Å
In a scanning electron microscope (SEM), electrons are accelerated to great velocities. Calculate the wavelength of an electron traveling with a velocity of 1.7 x 104 meters per second. The mass of an electron is 9.1 x 10-28 g? a) 4.3 x 10 -11 m b) 4.3 x 10 -8 m c) 1.8 x 10 4 m d) 2.3 x 10 -35 m e) 2 x 10 -33 m
Calculate the wavelength in nanometers for a bullet weighing 5.0 g and traveling at 400 m/sec. A) 3.3 x 10-34 m B) 5.8 x 10-19 m C) 1.5 x 10-38 m D) 3.0 x 1033 m E) 3.3 x 10-37 m
The de Broglie wavelength for a baseball moving at 25.0 m/s is 2 x 10-34 m. If a baseball could travel at the speed of light, what would be its de Broglie wavelength at that speed? A) 1.7 x 10-41 m B) 1.5 x 10-26 m C) 1.7 x 10-43 m D) 1.5 x 10-28 m E) 1.5 x 10-24 m
What is the wavelength of a marble (5 g) traveling at 372,889 m/s.
What is the de Broglie wavelength of a bird being chased by Schrodinger’s cat, Albert? The bird has a mass of 0.5 kg and is flying at 20 m/s. 1. 6.626 × 10−35 m 2. 6.626 × 103 m 3. 6.626 × 10−38 m 4. 6.626 × 10−18 m 5. 6.626 × 10−15 m
Since quantum-mechanical theory is universal, it applies to all objects, regardless of size. Therefore, according to the de Broglie relation, a thrown baseball should also exhibit wave properties. Why don’t we observe such properties at the ballpark?
What is the de Broglie wavelength of an electron?
What determines the value of the de Broglie wavelength for an electron?
Why is the wave nature of matter not important for a baseball?
The first 25 years of the twentieth century were momentous for the rapid pace of change in scientists understanding of the nature of matter.In what ways is de Broglies hypothesis, as it applies to electrons, consistent with J. J. Thomsons conclusion that the electron has mass? In what sense is it consistent with proposals that preceded Thomsons work, that the cathode rays are a wave phenomenon?
Neutron diffraction is an important technique for determining the structures of molecules.Calculate the velocity of a neutron needed to achieve a wavelength of 1.35 Å . (Take the mass of the neutron m=1.674910-27kg).
In August 2011, the Juno spacecraft was launched from Earth with the mission of orbiting Jupiter in 2016. The closest distance between Jupiter and Earth is 391 million miles.Once Juno reaches Jupiter, what is the minimum amount of time it takes for the transmitted signals to travel from the spacecraft to Earth?
The speed of sound in dry air at 20 oC is 343 m/s and the lowest frequency sound wave that the human ear can detect is approximately 20 Hz.What is the wavelength of such a sound wave?
Among the elementary subatomic particles of physics is the muon, which decays within a few nanoseconds after formation. The muon has a rest mass 206.8 times that of an electron.Calculate the de Broglie wavelength associated with a muon traveling at a velocity of 8.60×105 cm/s .
An electron is accelerated through an electric potential to a kinetic energy of 13.6 keV .What is its characteristic wavelength? [Hint: Recall that the kinetic energy of a moving object is large{ ormalsize E = {frac{1}{2}} ormalsize {mv^2}}, where m is the mass of the object and v is the speed of the object.]
The deBroglie wavelength can become significant for an object that is:A. low in mass and low in velocityB. low in mass and high in velocityC. high in mass and low in velocityD. high in mass and high in velocityE. found in degenerate orbitals
The de Broglie wavelength of a 455 kg car is found to be 5.43 × 10 –47 nm. Calculate the speed (m/s) of the car.a) 26.8 m/sb) 37.3 m/sc) 2.68 × 1019 m/sd) 3.73 × 107 m/se) 3.00 × 108 m/s
 An electron (mass = 9.11x10 -31 kg) moves with a velocity of 5.00x10 8 cm s -1. What is its wavelength in angstroms (Å)?a. 1.45b. 9.72x10 -8c. 3.25x10 -9d. 2.91e. 0.970
What is the wavelength of a He atom traveling at 1.96 x10  3 m/sec?(mass of He  = 6.65 x10 -27 kg)
Use the de Broglie relationship to determine the wavelengths of the following objects: (a) an 85-kg person skiing at 50 km/hr
As the velocity of an object doubles, what is expected of its deBroglie wavelength of the object? A. It will increase by a factor of four B. It will increase by a factor of two C. It will remain constant D. It will decrease by a factor of two E. It will decrease by a factor of four
Use the de Broglie relationship to determine the wavelengths of the following objects: (b) a 10.0-g bullet fired at 250 m/s
Use the de Broglie relationship to determine the wavelengths of the following objects: (c) a lithium atom moving at 2.5 x 105 m/s
Use the de Broglie relationship to determine the wavelengths of the following objects: (d) an ozone (O3) molecule in the upper atmosphere moving at 550 m/s.
Among the elementary subatomic particles of physics is the muon, which decays within a few nanoseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at a velocity of 8.85 x 105 cm/s.
Neutron diffraction is an important technique for determining the structures of molecules. Calculate the velocity of a neutron needed to achieve a wavelength of 0.955 Å. (Refer to the inside cover for the mass of the neutron).
The deBroglie wavelengths of a moving electron are given. Which wavelength would correspond to the greatest speed of the moving electron?a. 8.25 x 10 9 mb. 1.35 x 10 -13 mc. 8.25 x 10 12 md. 1.21 x 10 -6 me. 1.21 x 10 -10 m 
The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of 8.95 x 106 m/s. What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?
A certain rifle bullet has a mass of 6.93 g. Calculate the de Broglie wavelength of the bullet traveling at 1025 miles per hour. A= ___m
What is the de Broglie wavelength of a bowling ball rolling down a bowling alley lane? Assume the mass of the ball is 4500 g and it is moving at 4.12 m/s.1. 1.4725 x 10–37 m2. 1.229 x 10–32 m3. 3.57389 x 10–35 m4. 3.574 x 10–38 m
What is the wavelength of a neutron traveling at 4.15 km/day? [1J = 1 kg m  2 /s 2 ; mass of neutron = 1.675×10 −24 g; h = 6.63 x10 -34 J s; 1 km=1000 m]
An electron is traveling at a speed of 3.00 x 105 m/s. What is its de Broglie wavelength?  a) 0.64 nm b) 1.87 nm c) 2.42 nm d) 4.31 nm e) Electrons do not have detectable wavelengths.
The fastest serve in tennis is about 140 miles per hour, or about 63.2 m/s. Calculate the wavelength associated with an electron moving at this same velocity. (Mass of an electron = 9.11 x 10–31 kg)
What is the de Broglie wavelength (in meters) of a pitched baseball with a mass of 0.120 kg and a speed of 44.7 m/s? (1 J = 1 kg•m2/s2)A. 6.24 x 10-34 mB. 1.50 x 10-36 mC 1.24 x 10-34 mD. 0.76 x 10-34 mE. 6.24 x 10-36 m
In an explanation for the quantized energy levels of atoms de Broglie theorized that orbiting electrons might be at a fixed distance from the nucleus and thus only certain wavelength of wavelike motion would be stable. What is the implication of de Broglie's theory on the movement of matter?     (a) Matter does not behave as though it moves in a wave.     (b) All matter behaves as though it moves in a wave.     (c) Only very fast moving  matter behaves as though it moves in a wave.     (d) Only very low mass matter behaves as though it moves in a wave.     (e) Only very high matter behaves as though it moves in a wave.
The mass of an electron is 9.11x10 -31 kg . If the de Broglie wavelength for an electron in a hydrogen atom is 3.31x10-10 m , how fast is the electron moving relative to the speed of light? The speed of light is 3.00 x 108 m/s . Express your answer numerically as a percentage of the speed of light.
The mass of a golf ball is 45.9 g. If it leaves the tee with a speed of 69.0m/s, what is its corresponding wavelength? Express your answer numerically in meters.
The deBroglie wavelengths of a moving electron are given. Which wavelength would correspond to the greatest speed of the moving electron?a. 8.25 x 10 9 mb. 1.35 x 10 -13 mc. 8.25 x 10 12 md. 1.21 x 10 -6 me. 1.21 x 10 -10 m 
What is the wavelength of a neutron traveling at 4.15 km/s? [1J = 1 kg m 2 /s2; mass of neutron = 1.675×10−24 g; h = 6.63 x10 -34 J s; 1 km=1000 m]
The faster an electron is moving, the __________ its kinetic energy, and the __________ its wavelength.(A) higher, shorter(B) higher, longer(C) lower, longer(D) lower, shorter(E) More than one of the answer choices will result in a true statement.
Consider an atom travelling at 1% the speed of light. The de Broglie wavelength is found to be 3.32 x 10-3 pm. Which element is this?A.) HB.) CaC.) BeD.) PE.) F
In a scanning electron microscope the wavelength of the electrons used to image very small objects is 2.74 pm, what is the velocity of the electron in the instrument?  (h =  6.63 x 10 -34 kg•m2/sec; 1.00 pm = 1.00x10 -12 m, me = 9.11x10 -31 kg; 1J = 1Kg•m2/sec2)
Calculate the wavelength of an electron (m = 9.11 x 10-28 g) moving at 3.66 x 106 m/s.a. 5.52 x 10-9 mb. 1.81 x 10-10 mc. 1.99 x 10-10 md. 5.03 x 10-10 me. 2.76 × 10-9 m
The speed of sound in dry air at 20˚C is 343 m/s and the lowest frequency sound wave that the human ear can detect is approximately 20 Hz. What is the wavelength of such a sound wave? What would be the frequency of electromagnetic radiation with the same wavelength?
Does the following equation describe particle-like or wavelike behavior? Does it involve both types of behavior?(e) λ = h/mν
A 0.22-caliber handgun fires a 29-g bullet at a velocity of 770. m/s. Is the wave nature of matter significant for bullets?
A 0.22 caliber handgun fires a 29 g bullet at a velocity of 775 m/s. a. Calculate the de Broglie wavelength of the bullet. b. Is the wave nature of matter significant for bullets?
By using photons of specific wavelengths, chemists can dissociate gaseous HI to produce H atoms with certain speeds. When HI dissociates, the H atoms move away rapidly, whereas the heavier I atoms move more slowly.(a) What is the longest wavelength (in nm) that can dissociate a molecule of HI?(b) If a photon of 254 nm is used, what is the excess energy (in J) over that needed for dissociation?(c) If this excess energy is carried away by the H atom as kinetic energy, what is its speed (in m/s)?
A proton in a linear accelerator has a de Broglie wavelength of 132 pm. What is the speed of the proton?
In certain cases where X-ray diffraction is unsuitable for determining the structure of a crystal, neutron diffraction can be used. Instead of X rays, a beam of neutrons is used to analyze the sample. Calculate the velocity of a beam of neutrons with a wavelength of 2.25 Å .
What is the velocity of an electron that has a de Broglie wavelength approximately the length of a chemical bond? Assume the length of a chemical bond is 1.8×10−10 m . (The mass of an electron is 9.11 x 10-31 kg.)
The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.23 nm in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.23 nm?
Neutron diffraction is used in determining the structures of molecules. Calculate the velocity of a neutron with a wavelength of 75 pm (1 pm = 10-12 m).
Calculate the velocities of electrons with de Broglie wavelengths of 1.0 x 10  2 nm and 1.0 nm.
How fast must a 56.5-g tennis ball travel to have a de Broglie wavelength equal to that of a photon of green light (5400 Å)?
How fast must a 142-g baseball travel to have a de Broglie wavelength equal to that of an x-ray photon with λ = 100. pm?
In order for a thermonuclear fusion reaction of two deuterons (12H+ ) to take place, the deuterons must collide and each must have a velocity of about 1 x 106 m/s. Find the wavelength of such a deuteron.
You may want to reference (Pages 224 - 227)Section 6.4 while completing this problem.Consider the following three moving objects:A golf ball with a mass of 45.9 g moving at a speed of 50.0 m/s.An electron moving at a speed of 3.5 x 105 m/s.A neutron moving at a speed of 2.3 x 102 m/s.List the three objects in order from shortest to longest de Broglie wavelength.
Compare the wavelengths of an electron (mass = 9.11 x 10 −31 kg) and a proton (mass = 1.67 x 10−27 kg), each having a speed of 3.4 x 10 6 m/s.
Compare the wavelengths of an electron (mass = 9.11 x 10 −31 kg) and a proton (mass = 1.67 x 10−27 kg), each having a kinetic energy of 2.7 x 10 −15 J.
Use the de Broglie relationship to determine the wavelengths of the following objects.an 85 kg person skiing at 55.0 km/hr
Use the de Broglie relationship to determine the wavelengths of the following objects.a 10.0-g bullet fired at 250 m/s
Use the de Broglie relationship to determine the wavelengths of the following objects.a lithium atom moving at 2.6×105 m/s
Use the de Broglie relationship to determine the wavelengths of the following objects.an ozone (O3) molecule in the upper atmosphere moving at 560 m/s
For any microscope, the size of the smallest observable object is one-half the wavelength of the radiation used. For example, the smallest object observable with 400-nm light is 2 x 10−7 m. What is the smallest observable object for an electron microscope using electrons moving at 5.5 x 104 m/s?
For any microscope, the size of the smallest observable object is one-half the wavelength of the radiation used. For example, the smallest object observable with 400-nm light is 2 x 10−7 m. What is the smallest observable object for an electron microscope using electrons moving at 3.0 x 107 m/s?
You may want to reference (Pages 225 - 227) Section 6.4 while completing this problem.The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of 9.61 × 106 m/s. What is the characteristic wavelength of this electron? (The mass of the electron is 9.1094 × 10–31 kg)
You may want to reference (Pages 225 - 227) Section 6.4 while completing this problem.The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of 9.61 × 106 m/s. What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms? (The mass of the electron is 9.1094 × 10–31 kg)
Calculate the de Broglie wavelength for an electron with a velocity 10.% of the speed of light.
The smallest atoms can themselves exhibit quantum mechanical behavior. Calculate the de Broglie wavelength (in picometers) of a hydrogen atom traveling at 460 m/s.
What is the de Broglie wavelength of an electron traveling at 1.34×105 m/s?
Calculate the de Broglie wavelength of a 143-g baseball traveling at 90 mph.
Calculate the de Broglie wavelength for a tennis ball (55 g) served at 35 m/s (approx. 80 mi/h)
Technetium (Tc; Z = 43) is a synthetic element used as a radioactive tracer in medical studies. A Tc atom emits a beta particle (electron) with a kinetic energy (Ek = 12 mv 2) of 4.71 x 10−15 J. What is the de Broglie wavelength of this electron?
A 232-lb fullback runs 40 yd at 19.8 ± 0.1 mi/h. What is his de Broglie wavelength (in meters)?
In a technique used for surface analysis called auger electron spectroscopy (AES), electrons are accelerated toward a metal surface. These electrons cause the emissions of secondary electrons-called auger electrons-from the metal surface. The kinetic energy of the auger electrons depends on the composition of the surface. The presence of oxygen atoms on the surface results in auger electrons with a kinetic energy of approximately 510 eV. What is the de Broglie wavelength of this electron? [KE = mv2; 1 electron volt (eV) = 1.602 x 10-19 J]
An alpha particle (mass = 6.6 x 10 −24 g) emitted by a radium isotope travels at 3.4 x 107 ± 0.1 x 10 7 mi/h. What is its de Broglie wavelength (in meters)?
Find the longest wavelength of a wave that can travel around in a circular orbit of radius 5.6 m.
A 0.22-caliber handgun fires a 29-g bullet at a velocity of 770. m/s. Calculate the de Broglie wavelength of the bullet.
A certain rifle bullet has a mass of 8.85 g. Calculate the de Broglie wavelength of the bullet traveling at 1769 miles per hour. Physical constants can be found here.
A particular monochromatic orange light source emits light with a wavelength of 602 nm. What is the energy of a photon emitted from this light source?
A particular monochromatic light source emits light with a frequency of 662 THz. What is the wavelength of the light emitted? What is the color of the light emitted? 
What is the wavelength of a257 Rf nucleus moving at 225 km hr-1? Select one: a. 2.48 x 4-11 m b. 4 13 x 10-35 m c. 1.15 x 10-35 m d. 6.90 x 10-12 m
Some chemical reactions can be initiated by light that carries an energy of 419 kJ/mol. Only light less than a certain wavelength will initiate such reactions. What is the longest wavelength in nanometers that can deliver 419 kJ/mol? Convert the energy m kJ/mol to energy in J/photon. Use Planck's Equation to determine the frequency in hertz. Convert frequency to wavelength in nanometers h = 6.626 x 10^34 J
What is the wavelength of an electron that has a mass of 9.10938188 x 10  -31 kg and a velocity of 2.17 x 106 m/s? Give your answer in angstroms.
The biological effects of a given dose of electromagnetic energy generally become more serious as the energy of the radiation increases: Infrared radiation has a pleasant warming effect: ultraviolet radiation causes tanning and burning: and X rays can cause considerable tissue damage. What energies in kilojoules per mole are associated with the following wavelengths: Part Ainfrared radiation with λ = 1.63 x 10 -6 m? Part Bultraviolet light with λ = 224 nm? Part CX rays with λ = 5.32 nm? 
Calculate the maximum wavelength of light (in nm) required to ionize a single rubidium atom. The first ionization energy of Rb Is 403 kJ/mol. 
What is the wavelength λ of the photon that has been released with ΔE = -2.05 x 10-18 J? Express your answer with the appropriate units.
For 532-nm visible light, calculate its frequency (v, Hz), wavenumber (v, cm-1), and photon energy (J). If a laser were produced at this frequency, what color of light would you observe?
What velocity would an electron (mass = 9.11 x 10-31 kg) need for its de Broglie wavelength to be that of red light (755 nm )?
What is the energy of 1 photon of frequency 3.50 x 10 14 Hz.(a) 5.28 x 1043 J (b) 2.32 x 10-19 J (c) 1.89 x 1019 J (d) 5.28 x 1017 J
A proton in a linear accelerator has a de Broglie wavelength of 131 pm. Part AWhat is the speed of the proton? Express your answer with the appropriate units.
A particular monochromatic cyan light source emits light with a frequency of 631 THz. What is the energy of a photon emitted from this light source?
Part A A heat lamp produces 26.0 watts of power at a wavelength of 7.1 μm.How many photons are emitted per second? (1 watt = 1J/s) Express your answer using two significant figures.
What is the wavelength (in nm) of light having a frequency of 8.8 x 10 13 Hz? What is the frequency (in Hz) of light having a wavelength of 5.82 x 102 nm? 
A proton in a linear accelerator has a de Broglie wavelength of 161 pm. What is the speed of the proton? Express your answer with the appropriate units.
A 0.22-caliber handgun fires a 27-g bullet at a velocity of 775 m/s.Part ACalculate the de Broglie wavelength of the bullet. Express your answer in meters using two significant figures. Part BIs the wave nature of matter significant for bullets?          • yes          • no 
The energy of a particular color of red light is 2.89 x 10 -22 photon. The wavelength of this light is __________ nanometers. (109 nm = 1 m)
A proton in a linear accelerator has a de Broglie wavelength of 154 pm. Part AWhat is the speed of the proton? Express your answer with the appropriate units.
The UV light that is responsible for tanning the skin falls in the 320-to 400-nm region. Calculate the total energy (in joules) absorbed by a person exposed to this radiation for 3.5 h, given that there are 2.0 x 1016 photons hitting Earth's surface per square centimeter per second over a 80-nm (320 to 400 nm) range and that the exposed body area is 0.45 m2. Assume that only half of the radiation is absorbed and the other half is reflected by the body. (Use an average wavelength of 360 nm in calculating the energy of a photon.)
The de Broglie equation predicts that the wavelength (in m) of a proton (1.67 x 10-27 kg) moving at 1000 m/s is 1) 3.96 • 10-10 m 2) 2.52 • 109 m 3) 3.96 • 10-7 m 4) > 1010 m 5) 2.52 • 106 m