In a set of experiments on a hypothetical one-electron atom,...

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground state $(n=1),$ as shown in the energy-level diagram in Fig. E39.31. You also observe that it takes 17.50 eV to ionize this atom. (a) What is the energy of the atom in each of the levels $(n=1, n=2,$ etc. $)$ shown in the figure? (b) If an electron made a transition from the $n=4$ to the $n=2$ level, what wavelength of light would it emit?

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the
photons emitted from transitions ending in the ground state $(n=1),$ as shown in
the energy-level diagram in Fig. E39.31. You also observe that it takes 17.50 eV to ionize this atom.
(a) What is the energy of the atom in each of the levels $(n=1, n=2,$ etc. $)$ shown in the figure? (b) If an electron made a transition from the $n=4$ to the $n=2$ level, what
wavelength of light would it emit?

Key Concepts

Electron Transitions and Photon Emission

When an electron moves between quantized energy levels, the change in energy is emitted or absorbed as a photon. The energy of the photon corresponds to the difference between the initial and final energy states of the electron. This process is fundamental to spectroscopy, where the wavelengths of the emitted or absorbed photons provide direct information about the energy spacing between levels in the atom.

Energy-Wavelength Relationship in Photon Emission

The energy of a photon is directly related to its wavelength by the equation E = hc/?, where h is Planck’s constant and c is the speed of light. This relationship allows us to calculate the wavelength of light emitted when an electron transitions between energy levels by first finding the energy difference of the transition. This principle is key in understanding how spectral lines are formed and how they provide insights into the energy structure of atoms.

Ionization Energy

Ionization energy is the minimum amount of energy needed to completely remove an electron from an atom, taking it from a bound state to a free state. In the context of one?electron atoms, the ionization energy provides a benchmark for determining the energy scale of the bound states. This concept is crucial because it allows the absolute energy levels to be defined relative to the point of complete electron removal, which is typically set as the zero potential energy level.

Quantum Energy Levels

In quantum mechanics, electrons in atoms occupy discrete energy levels (or orbitals) rather than a continuum of energies. In one?electron (hydrogen-like) atoms, these energy levels are quantized and typically depend on the principal quantum number n. The energy associated with each level can be calculated using formulas that incorporate fundamental constants, indicating that the electron can only exist in these well-defined states and must absorb or emit specific amounts of energy when transitioning between them.

Transcript

00:01 In this question, we are told that we performed a set of experiments on a one electron atom, and that we have measured the wavelengths of photons emitted from transitions ending in the ground state, and we've been given an energy level diagram.

00:18 We're also supposed to assume that the ionization energy of this atom was found to be 17 .5 ev.

00:26 So in this question in part a, we are asked to calculate the energy of the atom in, in each of the levels and equals one and equals two, et cetera.

00:36 So in order to do this, we're going to take advantage of that ionization energy.

00:42 So i'm just going to write that as ei.

00:47 And the ionization energy is basically the amount of energy we need to give to the ground state atom in order to liberate the electron from the nucleus of the atom.

01:00 So this is actually.

01:02 The energy that is associated with the atom in the ground state.

01:09 So the energy of the atom in the n equals one state is just going to be negative 17 .5 ev.

01:19 And then if we give it 17 .5 ev then the electron will have the ability to escape the atom.

01:32 Now for the rest of the energy, levels, what we do is we take that ionization energy and we divide by the energy level squared.

01:48 So if we're in the second state, then we're dividing by four squared and that will give us the energy level of that particular state.

02:01 So this comes from the bore model of the atom.

02:05 So we're just taking that directly from there.

02:08 So let's go ahead and put this into practice.

02:10 So the energy of the second state will be 17 .5 divided by 4.

02:21 And that will give us about negative .438 ev, and then so on and so forth.

02:29 So e3 will be negative 17 .5 ev over 9.

02:35 So that will give 1 .94.

02:46 And then we've got e4 will be divided by 16, 4 squared as 16.

02:57 And so we get 1 .0, yeah, about 1 .09 ev, negative as well.

03:09 And then e5 will be negative 17 .5 ev over 25...

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