Suppose that you have a glass tube filled with atomic...

Suppose that you have a glass tube filled with atomic hydrogen gas (H, not $\mathrm{H}_{2}$ ). Assume that the atoms start out in their ground states. You illuminate the gas with monochromatic light of various wavelengths, ranging through the entire IR, visible, and UV parts of the spectrum. At some wavelengths, visible light is emitted from the $\mathrm{H}$ atoms. (a) If there are two and only two visible wavelengths in the emitted light, what is the wavelength of the incident radiation? (b) What is the largest wavelength of incident radiation that causes the $\mathrm{H}$ atoms to emit visible light? What wavelength(s) is/are emitted for incident radiation at that wavelength? (c) For what wavelengths of incident light are hydrogen ions $\left(\mathrm{H}^{+}\right)$ formed?

Key Concepts

Photoionization Threshold

Photoionization occurs when a photon has sufficient energy to completely remove an electron from an atom, converting it into an ion. In the case of hydrogen, this threshold energy corresponds to a photon wavelength calculated from the energy required to ionize the ground state electron. Understanding the concept of a photoionization threshold helps to determine which incident wavelengths will not only excite electrons to higher levels but also completely ionize the atom.

Balmer Series

The Balmer series is the set of spectral line emissions of the hydrogen atom that result from electron transitions from higher energy levels down to the n = 2 level. Since these transitions fall within the visible region of the electromagnetic spectrum, they are fundamental to explaining why hydrogen emits visible light under certain conditions. They illustrate how specific incident photon energies can lead to visible emission lines.

Photon Absorption and Emission

When an atom absorbs a photon, the electron is promoted from a lower energy level to a higher energy level if the photon’s energy matches the energy difference between these levels. The subsequent emission of light occurs when the electron returns to lower energy levels, emitting photons whose wavelengths correspond to the differences between the energy levels. This process is responsible for the discrete spectra observed in atomic hydrogen.

Quantized Energy Levels

In atomic physics, electrons in an atom reside in discrete, quantized energy levels rather than a continuous spectrum of energies. The energy difference between these levels is fixed, and transitions between levels occur only when the atom absorbs or emits photons with energies that exactly match these differences. This concept is essential for explaining both the absorption of incident light and the emission of light at specific wavelengths in hydrogen.

Transcript

00:01 In this exercise, we have a hydrogen atom being shined upon by different electromagnetic radiation.

00:08 And we also have the information that the hydrogen atom itself emits photons in specific wavelengths.

00:17 So in question a, we are given the information that the atom only emits photons in two visible wavelengths.

00:27 And then we are asked to calculate based on this information, what is the wavelength of the incident, the incident radiation.

00:40 So let's do it.

00:41 Let's remind ourselves that the wavelength of the radiation emitted by a hydrogen atom due to electronic transitions is given by 1 over the rate bar constant times 1 minus an f minus 1 over an f squared minus 1 over n i squared here nf is the is the energy level of the the final energy level of the hydrogen atom and i is the initial energy level of the hydrogen atom so that transition goes from n i to an f r the rate bear constant is given by 1 .1 times 10 to the minus i'm sorry 10 to the 7 meters to the minus 1.

01:35 So i have to calculate what are the possible ni and f such that lambda is in the visible range.

01:44 So i'm going to also write here that the visible range, visible range goes from 400 nanometers to 700 nanometers.

01:59 Okay.

02:03 So let's let's check for nf equals 1, which is called the linemen series.

02:10 What is maximum wavelength well the maximum wavelength for nf equals 1 so lambda max is going to happen when n i equals 2 and that's going to be 121 .6 nanometers notice that the maximum wavelength emitted by linens series that is when nf equals 1 is shorter than the minimum wavelength in the visible range so the linens series is out of question let's also check for for the poshin series, which means that nf equals 3, what is the minimum wavelength? and if you calculate it, all you have to do is put ni equals infinite.

02:56 I'm sorry, not infinite here.

02:58 The minimum wavelength that you're going to have is 820 .62 nanometers, which is larger than the maximum wavelength in the visible range.

03:10 So both the lyman and the poshin series are out of question we are not going to consider them here because they do not produce visible wavelengths at all.

03:21 What we're going to do is to look at the balmer series, which has an f equals 2, balmer, because it produces visible radiation.

03:35 And we want to know what are the two lowest energy levels and i that will produce a visible wavelength.

03:43 So we can calculate it.

03:44 You can calculate the wavelengths for the transition 3 to 2.

03:54 So that's simply a matter of putting it in the equation up here.

03:58 And f is going to be 2, and i is going to be 3, and r is going to be 1 .1 times 10 to the 7th meters to the minus 1.

04:06 So the wavelength is going to be 656 nanometers, so that's visible, that's good.

04:14 And the transition from the fourth to the second is going to have a wavelength of 486 centimeters.

04:27 So these two transitions give rise to wavelengths that are within the visible range, which is great.

04:37 So the two transitions here that are responsible for the two visible wavelengths that the photon emit can only be from the fourth to the second level and from the third to the second level.

04:53 Okay? so what we have to calculate is lambda incident.

05:03 Okay? that's the wavelength of the incident photon...

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