A hydrogen atom in its ground state absorbs a 97-nm ultraviolet photon. It then emits one or mor

A hydrogen atom in its ground state absorbs a 97-nm...

A hydrogen atom in its ground state absorbs a $97-\mathrm{nm}$ ultraviolet photon. It then emits one or more photons to return to the ground state. (a) If the atom is at rest before absorbing the UV photon, what is its recoil speed after absorption? (b) There are several different possible ways for the atom to return to the ground state. How many? (c) For each of the possibilities in (b), give the wavelength of each photon emitted, and classify it as visible, UV, IR, x-ray, etc.

A hydrogen atom in its ground state absorbs a $97-\mathrm{nm}$ ultraviolet photon. It then emits one or more photons to return to the ground state. (a) If the atom is at rest before absorbing the UV photon, what is its recoil speed after absorption? (b) There are several different possible ways for the atom to return to the ground state. How many? (c) For each of the possibilities in
(b), give the wavelength of each photon emitted, and classify it as visible, UV, IR, x-ray, etc.

Key Concepts

Electromagnetic Spectrum and Photon Classification

Photons are classified across different regions of the electromagnetic spectrum—such as ultraviolet, visible, infrared, and x-ray—based on their wavelength (or energy). This classification is essential for understanding the nature of the emitted light from atomic transitions, as different transitions yield photons in different regions of the spectrum depending on the energy differences involved.

Photon Momentum and Recoil

When a photon is absorbed by an atom, its momentum is transferred to that atom, causing the atom to recoil. This concept is grounded in the conservation of momentum, which requires that the momentum of the absorbed photon (calculated using its wavelength) equals the momentum imparted to the atom. This is fundamental to understanding the dynamics of light-matter interaction in quantum mechanics.

Atomic Energy Level Transitions

Atoms possess discrete, quantized energy levels, and transitions between these levels involve the absorption or emission of a photon whose energy matches the difference between the levels. This concept is critical to atomic physics and spectroscopy, as it explains how electrons move from higher to lower energy states (or vice versa) and the specific energies associated with such transitions.

Radiative Cascade Processes

A radiative cascade occurs when an excited atom returns to its ground state through a series of intermediate energy levels rather than a single direct transition. Each step in the cascade emits a photon corresponding to the energy difference between consecutive levels, allowing for multiple pathways of de-excitation. This process underlies the richness of spectral lines observed in atomic emission spectra.

Transcript

00:01 In this exercise, we have a hydrogen atom that absorbs a photon whose wavelength is given by 97 nanometers.

00:10 And we have the information that the hydrogen atom is initially at its ground state and at rest.

00:18 And in the first question, we are asked to calculate what is the speed of the hydrogen atom after the absorption? so this is a question of applying conservation of momentum.

00:31 So the momentum of the photon that's absorbed is going to be equal to the momentum of the hydrogen.

00:40 The momentum of the photon, as we know, is h over lambda, and the momentum of the hydrogen will be the mass of the protein times the speed, so that the speed is going to be h over m lambda.

00:58 And now, h is 6 .63 times 10 to the minus 34 in joules 2nd.

01:08 The mass of the proton is 1 .67 times 10 to the minus 27 kilograms.

01:19 And lambda is 97 nanometer, so it's 97 times 10 to the minus 9 meters.

01:26 So the speed, if we carry out the calculation, this is going to be 4 .1 meters per second.

01:35 And this is the answer to question a.

01:39 In question b, we have that the atom is excited by the photon absorbed, and then from that excited state, goes back to the ground state.

01:53 And the exercise wants us to calculate how many ways are there for the hydrogen atom to jump from this.

02:00 Excited state to the ground state.

02:03 So we know that the photon absorbed had a wavelength of 97 nanometers.

02:11 And the first thing we need to calculate is to what state the hydrogen atom was excited.

02:18 So in order to do that, the first thing we need to do is to calculate the energy that the photon gave the hydrogen atom.

02:26 And the energy of a photon is given by hc over lambda.

02:32 And hc is 1 ,140 electron volts nanometers.

02:40 Lambda is 97 nanometers.

02:43 So the energy of the photon is going to be about 12 .75 electron volts.

02:54 So the energy, the final energy of the hydrogen atom is going to be the initial energy, plus the energy it absorbed from the photon.

03:04 Now the initial energy of the hydrogen atom is minus 13 .6.

03:09 That's just the energy of the ground state in electron volts plus 12 .75.

03:18 This is going to be minus 0 .85.

03:24 And we want to know to what state, to what energy level, this energy corresponds.

03:31 So the energy of the nth energy level in the hydrogen atom is given by the energy level.

03:38 Minus 13 .6 over n squared, this is electron volts, and we're going to make it equal to minus 0 .85, and this corresponds to n equals 4.

03:53 So the hydrogen atom is excited to its fourth energy level, and then it decays from the fourth to the first.

04:06 And we want to know how many ways are there for this transition to happen.

04:11 So the atom can transition directly from the fourth to the first...

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