Ch.8 - The Quantum-Mechanical Model of the Atom

Problem 52

Chapter 8, Problem 52

A proton in a linear accelerator has a de Broglie wavelength of 122 pm. What is the speed of the proton?

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Identify the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity.

Rearrange the formula to solve for velocity \( v \): \( v = \frac{h}{m\lambda} \).

Substitute the known values into the equation: Planck's constant \( h = 6.626 \times 10^{-34} \) Js, the mass of a proton \( m = 1.673 \times 10^{-27} \) kg, and the wavelength \( \lambda = 122 \times 10^{-12} \) m.

Calculate the velocity \( v \) using the rearranged formula and the substituted values.

Ensure the units are consistent and check the calculation for any possible errors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This concept illustrates the wave-particle duality of matter, indicating that particles like protons exhibit wave-like properties.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In the context of particles, especially in quantum mechanics, momentum plays a crucial role in determining the behavior of particles. For a proton, knowing its mass allows us to calculate its momentum when its speed is known, which is essential for finding its de Broglie wavelength.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. It has a value of approximately 6.626 x 10^-34 Js. In the context of the de Broglie wavelength, Planck's constant is used to connect the wave properties of particles to their momentum, enabling calculations that reveal the wave-like behavior of matter at the quantum level.

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Textbook Question

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 0.20 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 0.20 nm?

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Textbook Question

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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Open Question

What is the de Broglie wavelength of an electron traveling at 1.35 x 10^5 m/s?

Textbook Question

Calculate the de Broglie wavelength of a 143-g baseball traveling at 95 mph. Why is the wave nature of matter not important for a baseball?

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Textbook Question

A 0.22-caliber handgun fires a 1.9-g bullet at a velocity of 765 m/s. Calculate the de Broglie wavelength of the bullet. Is the wave nature of matter significant for bullets?

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Textbook Question

An electron has an uncertainty in its position of 552 pm. What is the uncertainty in its velocity?

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