(a) Find the energies of the first four levels of doubly...

(a) Find the energies of the first four levels of doubly ionized lithium $\left(\mathrm{Li}^{2+}\right),$ starting with $n=1 .$ (b) What are the energies of the photons emitted or absorbed when the electron makes a transition between these levels? (c) Are any of the photons in the visible part of the EM spectrum?

(a) Find the energies of the first four levels of doubly ionized lithium $\left(\mathrm{Li}^{2+}\right),$ starting with $n=1 .$
(b) What are the energies of the photons emitted or absorbed when the electron makes a transition between these levels?
(c) Are any of the photons in the visible part of the EM spectrum?

Key Concepts

Electron Transitions and Photon Emission/Absorption

When an electron transitions between quantized energy levels, the change in energy corresponds to the energy of a photon that is either emitted or absorbed. The photon’s energy is given by the difference between the initial and final energy states, and analyzing these differences allows one to predict the wavelengths of the spectral lines produced.

Electromagnetic Spectrum and Visible Light

The electromagnetic spectrum encompasses all possible wavelengths of electromagnetic radiation, with the visible spectrum specifically covering wavelengths from about 400 to 700 nanometers. Understanding the location of the transition energies within this spectrum helps determine whether the resulting photons from electron transitions fall in the visible range or in other parts of the spectrum, such as ultraviolet or infrared.

Quantized Energy Levels

In quantum mechanics, electrons in atoms occupy discrete energy states. The energy levels are typically given by formulas that depend on quantum numbers, and transitions between these levels involve specific amounts of energy. This quantization explains why only certain energies are emitted or absorbed, and it underpins the analysis of spectral lines.

Hydrogen-like Atoms

Hydrogen-like atoms or ions are those with only one electron orbiting the nucleus. Their energy levels can be described by formulas similar to that of hydrogen, modified by the nuclear charge. This concept is crucial for understanding systems like doubly ionized lithium because the energy levels depend on the square of the nuclear charge, leading to different quantization compared to hydrogen.

Transcript

00:01 For the first part of this exercise, we have to find the first four energy levels of the doubly ionized lithium atom.

00:10 So let's do it.

00:12 Remember that the nth energy level of a hydrogen -like atom is given by minus z squared.

00:20 Z is the atomic number over n squared times 13 .6 electron volts.

00:29 And it's legitimate to consider the doubly ionized lithium as hydrogen -like atom, because since it's been doubly ionized, it has only one electron orbiting the nucleus.

00:45 And so let's also remember that the lithium has an atomic number of three, so that the energy levels of the lithium can be written as minus three squared, which is 9 times 13 .6 electron volts over n squared and 9 times 13 .6 is 122 .4 over n squared.

01:16 So now we are in a position to calculate the first four levels.

01:21 Okay.

01:23 The first level and equals one.

01:25 So the first level is just minus 122 .4 electron volts.

01:31 The second level you're going to have minus 122 .4 over 4, and that is minus 30 .6 electron volts.

01:48 For the third level, we're going to have minus 122 .4 over 9, and that is minus 13 .6, and final, electon volts, of course.

02:04 And finally, for the fourth, for fourth, for fourth, energy level we're going to have minus 122 .4 over 16, and this is minus 7 .165 electron volts.

02:21 So this is the question to the answer to question a.

02:25 Question b asks us to calculate the energy of the photons emitted in each transition between these levels.

02:36 So let's start by the transition from the second level to the first one.

02:41 The photon, the emitted photon, it's going to have an energy that is the difference between the energy of the second level and the energy of the first one.

02:53 And this is going to be minus 30 .6 plus 122 .4, and this is 91 .8, electron volts.

03:06 From the third to the second, we're going to have e3 minus e2, and this is 17, electron volts...

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