Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 104

The angular momentum of an electron in the Bohr hydrogen atom is mur , where \(m\) is the mass of the electron, \(u,\) its velocity, and \(r,\) the radius of the Bohr orbit. The angular momentum can have only the values nh/2 \(\pi\), where \(n\) is an integer (the number of the Bohr orbit). Show that the circum frences of the various Bohr orbits are integral multiples of the de Broglie wavelengths of the electron treated as a matter wave.

Short Answer

Expert verified
The circumference for each Bohr orbit is an integral multiple of an electron's de Broglie wavelength.

Step by step solution

01

Write down the known facts

Angluar Momentum L = \(mur = \frac{nh}{2\pi}\), where m is mass of electron, u its velocity, r the radius, n is an integer denoting orbit number, and h is Planck's constant. According to de Broglie hypothesis, \(\lambda = \frac{h}{mu}\), where \(\lambda\) is the wavelength.
02

Substituting for velocity

From the angular momentum equation, the velocity can be written as \(u = \frac{nh}{2\pi m r}\). Substitute this into the wavelength formula, and we find \(\lambda = \frac{2\pi r}{n}\)
03

Relate the circumference to the wavelength.

The circumference of an orbit is given by \(2\pi r\). Thus, each orbit's circumference can be written as \(n\lambda\), meaning that each orbit's circumference is an integral multiple of the electron's de Broglie wavelength.

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

Concerning the concept of subshells and orbitals, (a) How many subshells are found in the \(n=3\) level? (b) What are the names of the subshells in the \(n=3\) level? (c) How many orbitals have the values \(n=4\) and \(\ell=3 ?\) (d) How many orbitals have the values \(n=3, \ell=2\) and \(m_{\ell}=-2 ?\) (e) What is the total number of orbitals in the \(n=4\) level?

What is the length of a string that has a standing wave with four nodes (including those at the ends) and \(\lambda=17 \mathrm{cm} ?\)

Which must possess a greater velocity to produce matter waves of the same wavelength (such as \(1 \mathrm{nm}\) ), protons or electrons? Explain your reasoning.

High-pressure sodium vapor lamps are used in street lighting. The two brightest lines in the sodium spectrum are at 589.00 and \(589.59 \mathrm{nm}\). What is the difference in energy per photon of the radiations corresponding to these two lines?

Construct a concept map representing the ideas of modern quantum mechanics.
See all solutions

Recommended explanations on Chemistry Textbooks

Inorganic Chemistry

Read Explanation

Chemical Analysis

Read Explanation

Making Measurements

Read Explanation

Nuclear Chemistry

Read Explanation

Ionic and Molecular Compounds

Read Explanation

The Earths Atmosphere

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