Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 32

Without doing detailed calculations, indicate which of the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) \(n=4\) to \(n=3 ;\) (b) \(n=1\) to \(n=2\) (c) \(n=1\) to \(n=6 ;\) (d) \(n=3\) to \(n=2\).

Short Answer

Expert verified
The transition that produces the light of the longest wavelength is \(n=3\) to \(n=2\).

Step by step solution

01

Understand the Relationship between Energy, Frequency, and Wavelength

The energy of a photon is given by the equation \(E=hf\), where \(h\) is Planck's constant and \(f\) is the frequency of the light. We also know that the frequency of light is inversely proportional to the wavelength, and so this can also be written as \(E=hc/\lambda\), where \(c\) is the speed of light and \(\lambda\) is the wavelength.
02

Compare the Energy Changes for Each Transition

The energy levels of the hydrogen atom are given by \(E=-13.6/n^{2} eV\), where \(n\) is the principal quantum number. The energy difference between two levels is given by \(ΔE=E_{f}-E_{i}\), where \(E_{f}\) and \(E_{i}\) represent the final and initial energy states respectively. For (a), the energy difference is \(E_3 - E_4\), for (b) it’s \(E_2 - E_1\), for (c) it’s \(E_6 - E_1\), and for (d) it’s \(E_2 - E_3\). We're looking for the smallest energy difference.
03

Determine the Transition with the Smallest Energy Difference

Without detailed calculations, observe the energy levels involved in each of the transitions. (a) \(n=4\) to \(n=3\), (b) \(n=1\) to \(n=2\), (c) \(n=1\) to \(n=6\), and (d) \(n=3\) to \(n=2\). Notice that the smallest difference in the quantum numbers is for transition (b) \(n=1\) to \(n=2\), and (d) \(n=3\) to \(n=2\). Both have a change of one energy level. However, energy levels in an atom are not equally spaced; they get closer together as the energy level number increases. Therefore, the transition from \(n=3\) to \(n=2\) will have a smaller energy difference than the transition from \(n=1\) to \(n=2\).

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

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

The most intense line in the cerium spectrum is at 418.7 nm. (a) Determine the frequency of the radiation producing this line. (b) In what part of the electromagnetic spectrum does this line occur? (c) Is it visible to the eye? If so, what color is it? If not, is this line at higher or lower energy than visible light?

Without doing detailed calculations, determine which of the following wavelengths represents light of the highest frequency: (a) \(6.7 \times 10^{-4} \mathrm{cm} ;\) (b) \(1.23 \mathrm{mm}\) (c) \(80 \mathrm{nm} ;\) (d) \(6.72 \mu \mathrm{m}\)

What is \(\Delta E\) for the transition of an electron from \(n=5\) to \(n=2\) in a Bohr hydrogen atom? What is the frequency of the spectral line produced?

In the ground state of a hydrogen atom, what is the probability of finding an electron anywhere in a sphere of radius (a) \(a_{0},\) or \(\left(\text { b) } 2 a_{0} ?\right.\)
See all solutions

Recommended explanations on Chemistry Textbooks

Inorganic Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Chemical Reactions

Read Explanation

Making Measurements

Read Explanation

Kinetics

Read Explanation

Physical Chemistry

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