Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 36: Problem 4

Which of the following has the smallest de Broglie wavelength? a) an electron traveling at \(80 \%\) the speed of light b) a proton traveling at \(20 \%\) the speed of light c) a carbon nucleus traveling at \(70 \%\) the speed of light d) a helium nucleus traveling at \(80 \%\) the speed of light e) a lithium nucleus traveling at \(50 \%\) the speed of light

Short Answer

Expert verified
a) an electron at \(80\%\) the speed of light b) a proton at \(20\%\) the speed of light c) a carbon nucleus at \(70\%\) the speed of light d) a helium nucleus at \(80\%\) the speed of light e) a lithium nucleus at \(50\%\) the speed of light Answer: c) a carbon nucleus at \(70\%\) the speed of light

Step by step solution

01

Write down de Broglie wavelength formula

\(\lambda = \frac{h}{mv}\)
02

Find the mass of each particle

Mass of an electron: \(m_e = 9.11 \times 10^{-31} kg\) (approx.) Mass of a proton: \(m_p = 1.67 \times 10^{-27} kg\) (approx.) Mass of a carbon nucleus: \(m_C = 12 \times m_p\) Mass of a helium nucleus: \(m_{He} = 4 \times m_p\) Mass of a lithium nucleus: \(m_{Li} = 7 \times m_p\)
03

Calculate the velocity of each particle

Velocity of an electron: \(v_e = 0.8c\) Velocity of a proton: \(v_p = 0.2c\) Velocity of a carbon nucleus: \(v_C = 0.7c\) Velocity of a helium nucleus: \(v_{He} = 0.8c\) Velocity of a lithium nucleus: \(v_{Li} = 0.5c\) Here, \(c\) represents the speed of light.
04

Calculate the de Broglie wavelength for each particle

Wavelength of an electron: \(\lambda_e = \frac{h}{m_e v_e} = \frac{h}{(9.11 \times 10^{-31} kg)(0.8c)}\) Wavelength of a proton: \(\lambda_p = \frac{h}{m_p v_p} = \frac{h}{(1.67 \times 10^{-27} kg)(0.2c)}\) Wavelength of a carbon nucleus: \(\lambda_C = \frac{h}{m_C v_C} = \frac{h}{(12 \times 1.67 \times 10^{-27} kg)(0.7c)}\) Wavelength of a helium nucleus: \(\lambda_{He} = \frac{h}{m_{He} v_{He}} = \frac{h}{(4 \times 1.67 \times 10^{-27} kg)(0.8c)}\) Wavelength of a lithium nucleus: \(\lambda_{Li} = \frac{h}{m_{Li} v_{Li}} = \frac{h}{(7 \times 1.67 \times 10^{-27} kg)(0.5c)}\)
05

Compare the de Broglie wavelengths

We can see that the denominators in each of the wavelengths determine their relative sizes. Since all velocities are fractional values of the speed of light, the smallest de Broglie wavelength will correspond to the particle with the highest mass and the highest velocity. Because a carbon nucleus has the highest mass and a velocity of \(70\%\) the speed of light, we expect that the de Broglie wavelength will be smaller than the other particle velocities. Therefore, the smallest de Broglie wavelength is from the carbon nucleus with a velocity of \(70\%\) the speed of light (option c).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In classical mechanics, for a particle with no net force on it, what information is needed in order to predict where the particle will be some later time? Why is this prediction not possible in quantum mechanics?

Given that the work function of tungsten is \(4.55 \mathrm{eV}\) what is the stopping potential in an experiment using tungsten cathodes at \(360 \mathrm{nm} ?\)

What would a classical physicist expect would be the result of shining a brighter UV lamp on a metal surface, in terms of the energy of emitted electrons? How does this differ from what the theory of the photoelectric effect predicts?

A photovoltaic device uses monochromatic light of wavelength 700 . \(\mathrm{nm}\) that is incident normally on a surface of area \(10.0 \mathrm{~cm}^{2}\). Calculate the photon flux rate if the light intensity is \(0.300 \mathrm{~W} / \mathrm{cm}^{2}\).

Which of the following has the higher temperature? a) a white-hot object c) a blue-hot object b) a red-hot object
See all solutions

Recommended explanations on Physics Textbooks

Translational Dynamics

Read Explanation

Space Physics

Read Explanation

Work Energy and Power

Read Explanation

Electromagnetism

Read Explanation

Physics of Motion

Read Explanation

Thermodynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free