A hydrogen atom in the ground state absorbs a 12.75 eV...

Name: Problem 61, Chapter 29: College Physics: A Strategic Approach
Uploaded: 2020-12-16T19:56:51-08:00
Duration: 5 min 20 s
Channel: Maria Gabriela Cota Moreira
Description: A hydrogen atom in the ground state absorbs a 12.75 eV photon. Immediately after the absorption, the atom undergoes a quantum jump to the next-lowest energy level. What is the wavelength of the photon emitted in this quantum jump?

A hydrogen atom in the ground state absorbs a 12.75 eV photon. Immediately after the absorption, the atom undergoes a quantum jump to the next-lowest energy level. What is the wavelength of the photon emitted in this quantum jump?

Key Concepts

Hydrogen Atom Energy Levels

Hydrogen’s energy levels are quantized and defined by the equation E? = -13.6 eV/n², where n is the principal quantum number. This concept underlies the structure of the hydrogen atom and dictates that electrons can only exist in specific energy states, which sets the stage for understanding absorption and emission processes in the atom.

Quantum Transitions

Quantum transitions, or quantum jumps, describe the process by which an electron transitions between discrete energy levels. When an electron absorbs energy, it moves to a higher energy state, and when it drops to a lower energy state, it emits a photon. These transitions conserve energy, with the energy of the emitted or absorbed photon equal to the difference between the initial and final energy levels.

Photon Energy-Wavelength Relationship

The energy of a photon is inversely related to its wavelength, as given by the equation E = hc/?, where h is Planck’s constant and c is the speed of light. This relationship is fundamental in linking the energy change during a quantum transition to the wavelength of the photon that is either absorbed or emitted.

Transcript

00:01 Today we're going to talk about the hydrogen spectrum.

00:03 So let's remember that the energies of different quantum levels of the hydrogen atom are given by this formula.

00:12 So it's minus 13 .6 electron volts divided by n squared, on which n is the principal quantum number.

00:25 In this exercise, we're going to suppose that we have a hydrogen atom.

00:29 So here i'm going to draw the energy levels, some levels.

00:35 And this hydrogen atom is initially at the ground state.

00:39 And this hydrogen atom is going to absorb the energy from a photon that has an energy of 12 .75 electron volts.

00:50 After absorbing this energy, the hydrogen atom is going to, from the ground state to a higher state.

00:58 The exercise tells us that after absorbing the photon's energy, the hydrogen will almost immediately after decay to one level lower.

01:13 So here.

01:16 And the question asks us what is going to be the wavelength of the photon emitted during this decay.

01:31 So the first thing we can do is to find to which a level does the hydrogen atom goes after absorbing this energy.

01:45 So one thing we can do is to use this formula to the energy levels of the hydrogen and simply calculate the energies of the higher states.

02:00 So let's do this.

02:02 So the ground state, let me raise this, the ground state, if we substitute n by 1, has an energy of minus 13 .6 electron volts.

02:16 Then the first excited state has an energy of minus 3 .4 electron volts...

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