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

**Concept:** The De Broglie Wavelength Equation

In this new video, we're going to take a look at the wave nature of light. Now, we've talked about light being composed of small packets of electromagnetic energy. We say that each one of these packets of this energy were called quantums way long ago. In modern days now, we call them photons, so we say light is composed of small particles called photons, but we've also talked about the inverse relationships of frequency and wavelength. In those examples, we've talked about light moving as a wave.

In this section, we're going to stick to that type of idea. We're going to see light as moving not as individualized particles but rather as a wave. We're going to say there's an equation that shares this relationship of light moving as a wave. It's called de Broglie wavelength equation. Under this equation, it says light moves as a wave, not as individual photons, individual particles. Now, we're going to say to calculate the wavelength of matter, we simply use this equation.

Now, the equation is for de Broglie wavelength equation is, lambda which is our wavelength equals h which is Planck's constant divided by m times v. We know that lambda is wavelength in meters. H is Planck's constant which we've talked about. Here m represents the mass of the object we're talking about. Usually, this object is subatomic particle. Here it will be the mass in kilograms, not in grams. v does not represent our frequency anymore. V here is actually velocity or speed. Velocity or speed are in meters per second.

What we need to realize here about Planck's constant is Planck's constant is 6.626 x 10-34 joules times seconds. We should realize that joules can be written a different way. Joules also equal kilograms times meters squared over seconds squared. That's what joules equal. If our joules are still multiplying with those seconds, then seconds will be canceled out with one of the seconds here.

In this equation, we're going to say Planck's constant becomes 6.626 x 10-34 kilograms times meters squared over seconds. Just realize how those things canceled out to give me that. Those are the units that you're going to have to remember when it deals with de Broglie wavelength.

**Example:** Find the wavelength (in nm) of a proton with a speed of 7.33 x 10** ^{9}** . (Mass of a proton = 1.67 x 10

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

What is the wavelength of a^{257} 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

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You may want to reference (page) section 7.4 while completing this problem.

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.

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**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.

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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?

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What is the change in energy if the electron from Part A now drops to the ground state?

Express your answer with the appropriate units.

What is the wavelength λ of the photon that has been released in Part B?

Express your answer with the appropriate units.

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What is the energy of 1 photon of frequency 3.50 x 10 ^{14} Hz.

(a) 5.28 x 10^{43} J

(b) 2.32 x 10^{-19} J

(c) 1.89 x 10^{19} J

(d) 5.28 x 10^{17} J

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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 A**

infrared radiation with λ = 1.63 x 10 ^{-6} m?

**Part B**

ultraviolet light with λ = 224 nm?

**Part C**

X rays with λ = 5.32 nm?

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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 10^{16} 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 m^{2}.

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.)

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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 10^{2} nm?

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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.

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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 10^{6} m/s? Give your answer in angstroms.

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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?

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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?

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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?

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The energy of a particular color of red light is 2.89 x 10 ^{-22} photon. The wavelength of this light is __________ nanometers. (10^{9} nm = 1 m)

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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

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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.

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Careful spectral analysis shows that the familiar yellow light of sodium lamps (such as street lamps) is made up of photons of two wavelengths, 589.0 nm and 589.6 nm. What is the difference in energy (in joules) between photons with these wavelengths?

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Consider the following energy levels of a hypothetical atom:

(a) What is the wavelength of the photon needed to excite an electron from E1 to E4? (b) What is the energy (in joules) a photon must have in order to excite an electron from E2 to E3? (c) When an electron drops from the E3 level to the E1 level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

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A particular form of electromagnetic radiation has a frequency of 8.11 × 1014 Hz. (a) What is its wavelength in nanometers? In meters? (b) To what region of the electromagnetic spectrum would you assign it? (c) What is the energy (in joules) of one quantum of this radiation?

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What experimental support did de Broglie’s concept receive?

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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.

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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 10^{8 }m/s . Express your answer numerically as a percentage of the speed of light.

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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

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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} m

b. 1.81 x 10^{-10} m

c. 1.99 x 10^{-10} m

d. 5.03 x 10^{-10} m

e. 2.76 × 10^{-9} m

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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?

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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 10^{4} 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

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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

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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 × 10^{3} m

3. 6.626 × 10^{−38} m

4. 6.626 × 10^{−18} m

5. 6.626 × 10^{−15} m

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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 10^{33} m

E) 3.3 x 10-37 m

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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

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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 10^{33} m

E) 3.3 x 10^{-37} m

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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.

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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]

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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•m^{2}/s^{2})

A. 6.24 x 10^{-34 }m

B. 1.50 x 10^{-36 }m

C 1.24 x 10^{-34 }m

D. 0.76 x 10^{-34 }m

E. 6.24 x 10^{-36 }m

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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} m

2. 1.229 x 10^{–32} m

3. 3.57389 x 10^{–35 }m

4. 3.574 x 10^{–38 }m

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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

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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)

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What is the wavelength of a neutron traveling at 4.15 km/s? [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]

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An electron is traveling at a speed of 3.00 x 10^{5} 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.

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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.

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What is the wavelength of a marble (5 g) traveling at 372,889 m/s.

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What is the wavelength of a He atom traveling at 1.96 x10 ^{3} m/sec?

(mass of He = 6.65 x10 ^{-27} kg)

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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•m^{2}/sec; 1.00 pm = 1.00x10 ^{-12} m, m_{e} = 9.11x10 ^{-31} kg; 1J = 1Kg•m^{2}/sec^{2})

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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.

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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. 1.45

b. 9.72x10 ^{-8}

c. 3.25x10 ^{-9}

d. 2.91

e. 0.970

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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/s

b) 37.3 m/s

c) 2.68 × 10^{19} m/s

d) 3.73 × 10^{7} m/s

e) 3.00 × 10^{8} m/s

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The deBroglie wavelength can become significant for an object that is:

A. low in mass and low in velocity

B. low in mass and high in velocity

C. high in mass and low in velocity

D. high in mass and high in velocity

E. found in degenerate orbitals

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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} m

b. 1.35 x 10 ^{-13} m

c. 8.25 x 10 ^{12} m

d. 1.21 x 10 ^{-6} m

e. 1.21 x 10 ^{-10} m

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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.) H

B.) Ca

C.) Be

D.) P

E.) F

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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} m

b. 1.35 x 10 ^{-13} m

c. 8.25 x 10 ^{12} m

d. 1.21 x 10 ^{-6} m

e. 1.21 x 10 ^{-10} m

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