Explanation 1) Concept: Consider a balloon filled with helium gas at room temperature and atmospheric pressure. The average wavelength of helium atoms can be found using de Broglie hypothesis. Also kinetic energy of atoms can be calculated using 3/2 kT. 2) Given: i) Room Temperature, T = 27 ° = 300 K ii) Atomic weight of He, A = 4 a m u iii) Avogadro’s number , n = 6.023 × 1 0 23 iv) Boltzmann constant, k = 1.38 × 1 0 - 23 J / m o l / K v) Planck’s constant, h = 6.63 × 1 0 - 34 J . s vi) Atmospheric pressure, P = 1 a t m = 1.01 × 1 0 5 P a 3) Formula: i) Kinetic energy of neutron, E = 3 2 k T ii) De Broglie wavelength, λ = h / 2 m E iii) Ideal gas equation, P V = R T 4) Calculation: a) To calculate the de Broglie wavelength of He atoms, we have λ a v g = h p a v g We can write the average momentum as p a v g = 2 m K a v g Substituting this, we get λ a v g = h 2 m K a v g We have average kinetic energy written as K a v g = 3 2 k T Substituting this and simplifying, we get λ a v g = h c 3 m c 2 k T Substituting the values, we get λ a v g = 1240 e V

10th Edition

ISBN: 9781118230725

Author: David Halliday, Robert Resnick, Jearl Walker

Publisher: Wiley, John & Sons, Incorporated

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Chapter 38, Problem 84P

To determine

To find:

a) Average de Broglie wavelength of He

b) Average distance between atoms

c) Whether the atoms consider as particles

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Chapter 38, Problem 83P

Chapter 38, Problem 85P

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Consider a balloon filled with helium gas at room temperature and atmospheric pressure. Calculate (a) the average de Broglie wavelength of the helium atoms and (b) the average distance between atoms under these conditions. The average kinetic energy of an atom is equal to (3/2)kT, where k is the Boltzmann constant. (c) Can the atoms be treated as particles under these conditions? Explain.

The root mean square speed of the hydrogen molecules at temperature t °C is given by 3x8.31 x (t+273) m 2 x 10-3 Calculate the de Broglie wavelength (in nanometers) of the hydrogen molecules at temperature 24 °C. The mass of the hydrogen molecule is 2 x 1.66 x 10-27 kg. Use two decimals in your answer.

a. Conceptually, discuss the particle-wave duality of light. Discuss the implications of this in combination with the de Broglie (pronounced “de Broy”) equation. b. The electron of a hydrogen atom is usually no further than 1.0 Å from the proton. We can therefore say the upper limit of the radius of an isolated hydrogen atom is roughly 1.0 Å. How does the de Broglie wavelength of the electron compare to this radius? (The velocity of an electron in the first principal energy level is about 2.2 x 106 m/s). Explain why wave-particle duality is so important for quantum mechanics, yet not required in macroscopic systems that are well described by classical mechanics. c. Comment as to whether neutrons with velocity 4.14 x 103 m/s may be used to determine structures of molecules in a diffraction-based experiment. You may consider the relevant distance between atoms in molecules to be on the order of 1 Å.

Chapter 38 Solutions

Fundamentals of Physics Extended

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a) Average de Broglie wavelength of He b) Average distance between atoms c) Whether the atoms consider as particles

Solution Summary: The author explains how the average wavelength of helium atoms can be found using de Broglie hypothesis and kinetic energy.

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To find: a) Average de Broglie wavelength of He b) Average distance between atoms c) Whether the atoms consider as particles

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Chapter 38 Solutions

To find:

a) Average de Broglie wavelength of He

b) Average distance between atoms

c) Whether the atoms consider as particles