Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 38: Problem 24

Hydrogen atoms are bombarded with 13.1-eV electrons. Determine the shortest wavelength line the atom will emit.

Short Answer

Expert verified
Answer: The shortest wavelength line is 121.6 nm.

Step by step solution

01

Calculate the Energy of an Electron

First, we need to convert the energy of the electron in electron volts (eV) to Joules (J) using the relation 1 eV = 1.6 x 10^-19 J. This will allow us to have the energy in a more convenient unit for the calculations. Energy of electron in eV = 13.1 eV Energy of electron in Joules (J) = 13.1 eV * (1.6 * 10^-19 J/eV) = 2.096 x 10^-18 J.
02

Calculate the Maximum Change in Energy Levels

Since we need to find the shortest wavelength, we need to consider the maximum change in energy levels. The initial energy level (n_initial) is given by the energy of the bombarded electron. The hydrogen atom energy levels are given by the Rydberg formula: E_n = -13.6 eV/n^2, where E_n is the energy of the level n. Now we need to determine the highest possible energy level (n_final) when the electron absorbs the bombarded electron's energy. E_final = E_initial + 13.1 eV, where E_initial = -13.6 eV/n_initial^2, and E_final = -13.6 eV/n_final^2.
03

Solve for the Initial and Final Energy Levels

Solve the equation for n_final in terms of n_initial: -13.6 eV/n_final^2 = -13.6 eV/n_initial^2 + 13.1 eV. Now, we need to find the values of n_initial and n_final for which the energy difference is the highest: max[(-13.6 eV/n_initial^2) - (-13.6 eV/n_final^2)] = 13.1 eV, by using trial and error, we find n_initial = 1 and n_final = 3.
04

Calculate the Wavelength using the Rydberg Formula

Now that we have the initial and final energy levels, we can use the Rydberg formula for the hydrogen atom to calculate the wavelength: 1/λ = R_H * (1/n_initial^2 - 1/n_final^2), where λ is the wavelength and R_H is the Rydberg constant for hydrogen (1.097 x 10^7 m^-1). Plug in the values for n_initial = 1 and n_final = 3: 1/λ = 1.097 x 10^7 m^-1 * (1/1^2 - 1/3^2) = 1.097 x 10^7 m^-1 * (1 - 1/9) = 1.097 x 10^7 m^-1 * (8/9).
05

Calculate the Shortest Wavelength

Finally, calculate the shortest wavelength: λ = 1 / [1.097 x 10^7 m^-1 * (8/9)] = 1.216 x 10^-7 m. Hence, the shortest wavelength line that the hydrogen atom will emit when bombarded by 13.1-eV electrons is 1.216 x 10^-7 m or 121.6 nm.

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

The wavelength of the fourth line in the Lyman series is a) \(80.0 \mathrm{nm}\). b) \(85.0 \mathrm{nm}\). c) \(90.2 \mathrm{nm}\). d) \(94.9 \mathrm{nm}\).

A hydrogen atom is in its fifth excited state, with principal quantum number \(n=6 .\) The atom emits a photon with a wavelength of \(410 \mathrm{nm}\). Determine the maximum possible orbital angular momentum of the electron after emission.

Given that the hydrogen atom has an infinite number of energy levels, why can't a hydrogen atom in the ground state absorb all possible wavelengths of light?

Which of the following can be used to explain why you can't walk through walls? a) Coulomb repulsion d) the Pauli exclusion b) the strong nuclear force \(\quad\) principle c) gravity e) none of the above

The hydrogen atom wave function \(\psi_{200}\) is zero when \(r=2 a_{0} .\) Does this mean that the electron in that state can never be observed at a distance of \(2 a_{0}\) from the nucleus or that the electron can never be observed passing through the spherical surface defined by \(r=2 a_{0}\) ? Is there a difference between those two descriptions?
See all solutions

Recommended explanations on Physics Textbooks

Engineering Physics

Read Explanation

Famous Physicists

Read Explanation

Kinematics Physics

Read Explanation

Space Physics

Read Explanation

Electricity

Read Explanation

Atoms and Radioactivity

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