Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 40

Consider a transition of the electron in the hydrogen atom from \(n=8\) to \(n=3\). (a) Is \(\Delta E\) for this process positive or negative? (b) Determine the wavelength of light that is associated with this transition. Will the light be absorbed or emitted? (c) In which portion of the electromagnetic spectrum is the light in part (b)?

Short Answer

Expert verified
(a) ΔE is negative. (b) The wavelength of light is 957.3 nm and is emitted. (c) The light is in the infrared portion of the electromagnetic spectrum.

Step by step solution

01

Calculate the energy of the initial and final levels

We use the formula for the energy of an electron in the nth energy level of a hydrogen atom: \[E_n = - \frac{13.6 eV}{n^2}\] For the initial level (n=8): \[E_{initial} = - \frac{13.6 eV}{8^2} = -\frac{13.6 eV}{64} = -0.2125 eV\] For the final level (n=3): \[E_{final} = - \frac{13.6 eV}{3^2} = -\frac{13.6 eV}{9} = -1.511 eV\]
02

Calculate the change in energy, ΔE

The change in energy during the transition is given as: \[\Delta E = E_{final} - E_{initial}\] \[\Delta E = -1.511 eV - (-0.2125 eV) = -1.511 eV + 0.2125 eV = -1.2985 eV\] Since ΔE is negative, that means energy is released during the transition.
03

Calculate the wavelength associated with the change in energy

To calculate the wavelength of light associated with the transition, we use the energy-wavelength formula: \[\Delta E = \frac{hc}{\lambda}\] Where h is Planck's constant (h = 4.1357 × 10⁻¹⁵ eV·s), c is the speed of light (c = 2.998 × 10⁸ m/s), and λ is the wavelength. Solving for the wavelength: \[\lambda = \frac{hc}{\Delta E}\] Now we plug in the numbers: \[\lambda = \frac{(4.1357 × 10^{-15} eV·s)(2.998 × 10^8 m/s)}{-1.2985 eV} = 9.573 × 10^{-8} m = 957.3 nm\]
04

Determine if the light is absorbed or emitted

Since ΔE is negative and energy was released during the transition, the light associated with this transition must be emitted.
05

Identify the portion of the electromagnetic spectrum

The wavelength of the light emitted (957.3 nm) falls in the infrared (IR) portion of the electromagnetic spectrum, which typically has wavelengths between 700 nm and 1 mm. Final answers: (a) The change in energy, ΔE, is negative. (b) The wavelength of light associated with this transition is 957.3 nm, and the light is emitted. (c) The light is in the infrared (IR) portion of the electromagnetic spectrum.

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

As discussed in the A Closer Look box on "Measurement and the Uncertainty Principle," the essence of the uncertainty principle is that we can't make a measurement without disturbing the system that we are measuring. (a) Why can't we measure the position of a subatomic particle without disturbing it? (b) How is this concept related to the paradox discussed in the Closer Look box on "Thought Experiments and Schrödinger's Cat"?

Place the following transitions of the hydrogen atom in order from shortest to longest wavelength of the photon emitted: \(n=5\) to \(n=2, n=4\) to \(n=3, n=8\) to \(n=4,\) and \(n=4\) to \(n=2\).

The familiar phenomenon of a rainbow results from the diffraction of sunlight through raindrops. (a) Does the wavelength of light increase or decrease as we proceed outward from the innermost band of the rainbow? (b) Does the frequency of light increase or decrease as we proceed outward? [Section 6.3]

Sketch the shape and orientation of the following types of orbitals: \((\mathbf{a}) s,(\mathbf{b}) p_{z},(\mathbf{c}) d_{x y}\)

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) \(n=3, l=3, m_{l}=2, m_{5}=+\frac{1}{2}\) (b) \(n=4, l=3, m_{l}=-3, m_{\mathrm{s}}=+\frac{1}{2}\) (c) \(n=3, l=1, m_{l}=2, m_{s}=+\frac{1}{2}\) (d) \(n=5, l=0, m_{l}=0, m_{\mathrm{s}}=0\) (e) \(n=2, l=1, m_{l}=1, m_{s}=-\frac{1}{2}\)
See all solutions

Recommended explanations on Chemistry Textbooks

Kinetics

Read Explanation

Making Measurements

Read Explanation

Chemical Reactions

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Organic Chemistry

Read Explanation

Inorganic 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