Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 38: Problem 27

An excited hydrogen atom emits a photon with an energy of \(1.133 \mathrm{eV}\). What were the initial and final states of the hydrogen atom before and after emitting the photon?

Short Answer

Expert verified
Answer: The initial state is 3, and the final state is 2.

Step by step solution

01

Recall the formula for energy levels in the hydrogen atom

The energy levels of the hydrogen atom can be determined using the following formula: \(E_n = -\dfrac{13.6 \mathrm{eV}}{n^2}\), where \(n\) is the principal quantum number (state) and \(E_n\) denotes the energy of the nth state.
02

Calculate the energy difference

The energy of the emitted photon is the difference in energy between the initial and final states of the atom. Thus, we can write the energy difference as: \(\Delta E = E_{initial} - E_{final} = 1.133 \mathrm{eV}\)
03

Use the formula for energy levels to find the possible state transitions

Since the energy difference is caused by a transition between two energy states, we can set up an equation to find the possible initial and final states: \(E_{initial} - E_{final} = \dfrac{13.6}{n^2_{final}} - \dfrac{13.6}{n^2_{initial}} = 1.133 \mathrm{eV}\)
04

Find the ratio of initial and final states

To simplify our calculation, let's call this ratio \(k\): \(k = \dfrac{1}{n^2_{initial}} - \dfrac{1}{n^2_{final}}\). Now, we can rewrite our equation as: \(13.6 \times k = 1.133 \mathrm{eV}\) Solving for \(k\), we get: \(k =\dfrac{1.133}{13.6} = 0.0833\)
05

Find the initial and final states that satisfy this ratio

Since \(n_{initial}\) and \(n_{final}\) are integers and \(n_{initial}>n_{final}\), we can find pairs of values \((n_{initial}, n_{final})\) that satisfy the condition \(k = \dfrac{1}{n^2_{initial}} - \dfrac{1}{n^2_{final}}\). After checking possible energy state transitions, we find that only the following pair of states satisfies the obtained ratio: \((n_{initial}, n_{final}) = (3, 2)\)
06

Conclude the initial and final states

Based on our calculations, the initial and final states of the hydrogen atom before and after emitting the photon are: Initial state: \(n_{initial} = 3\) Final state: \(n_{final} = 2\)

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 ionization energy of a hydrogen atom excited to the \(n=2\) state?

What is the energy of a transition capable of producing light of wavelength \(10.6 \mu \mathrm{m}\) ? (This is the wavelength of light associated with a commonly available infrared laser.)

A ruby laser consists mostly of alumina \(\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)\) and a small amount of chromium ions, responsible for its red color. One such laser of power \(3.00 \mathrm{~kW}\) emits light pulse of duration \(10.0 \mathrm{~ns}\) and of wavelength \(685 \mathrm{nm}\). a) What is the energy of the photons in the pulse? b) Determine the number of chromium atoms undergoing stimulated emission to produce this pulse.

In a hydrogen atom, the electron is in the \(n=5\) state. Which of the following sets could correspond to the \(\ell, m\) states of the electron? a) 5,-3 b) 4,-5 c) 3,-2 d) 4,-6

A collection of hydrogen atoms have all been placed into the \(n=4\) excited state. What wavelengths of photons will be emitted by the hydrogen atoms as they transition back to the ground state?
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Oscillations

Read Explanation

Medical Physics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Electromagnetism

Read Explanation

Quantum Physics

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