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Problem 54 Hard Difficulty

Suppose the ionization energy of an atom is 4.100 $\mathrm{eV}$ . In this same atom, we observe emission lines that have wavelengths of $310.0 \mathrm{nm}, 400.0 \mathrm{nm},$ and 1378 $\mathrm{nm} .$ Use this information to construct the energy level diagram with the least number of levels. Assume that the higher energy levels are closer together.

Answer

$-4 \cdot 100 e V$

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Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

Zachary M.

Hope College

Meghan M.

McMaster University

Video Transcript

In this exercise, you have an Adam that has an ionization energy of 4.1 electoral votes. And we have the wavelengths of some of the emission lines of the the Adam. So we have when the one equal shoot 310 centimeters, Linda two equals to 400 a meters in London. Three equals two, 1378 millimeters. Okay. And what we need to do is to draw the diagram of the energy levels of this Adam using the fewest, uh, the fewest lines that we can use. Okay, so the first thing I'm gonna do is to calculate the energy of the elect of the photons that are emitted. Ah, by the Adam whose wavelengths were given to us. So all I'm gonna need is to use the normal age, see over London. HC is 240 centimeters. Electron vote divided by Lunda. And we can apply this formula or these values here. So you have that The energy Girma, one of the first fulton is able to four l a crumb votes the energy. The geometry of the second photo is 3.1 electoral votes and the energy going three of the third. Fulton is 0.9 electoral votes. Okay? And for their more notice that the energy since the ionization energies toward would one election votes, we can assume that the energy of if the ground state of the atom is minus 4.1 electoral votes. Okay, um, and I'm calling the energy other going to state you want. So from these informations from the information that the energy other bones were on state is minus four, but one. And the energy of the Fulton's that are emitted when the Adam that when there is an internal transition in the Adam, we can draw a diagram with 33 lines. Uh, 33 lines of corresponding to the energy levels of the atom. Okay, so I'm gonna do out here to rely on him and gonna explain how we can find their energies. So here we have three lines. The 1st 1 has and equals one. That's the world state. The second has an equals two, and the third has an equals three. The 1st 1 I've read it determined the energy that's the one equal equal to minus 4.1 electoral votes. Okay, and then we have three. Ah, three energies are the Fulton's that are caused by the transition between these, uh, three energy levels of the atom. Okay, so they have some line is caused by the transition from the end equals three energy level to the end equals one notice. You can see this by the figure that this transition that is certainly the one that takes the most. Ah, the most energy. Okay, this is the one that that who's Fulton that are that is emitted. Here is the one is the most energetic fulton among all the transitions possible. So let's go back to our energies of the Fulton and see which one has the the highest energy. And it's certainly this one with four electoral votes. So if the energy of the transition from energy e gamma one of the transition from the third energy level to the first is able to four electoral votes. So this is equal to the difference in energy between the third energy level and the 1st 1 So e three is equal to four electoral votes, plus you want, and this is minus 0.1 electoral votes. So we have A E three is equal to minus 0.1 electoral votes. Okay, Now we confined to notice that the energy in the position from any goes to two and equals one is the second largest. Because we're assuming that ah, levels with with higher values of n are closer together and that that is a good assumption. Okay, Um so the second, the second largest value of the energy of the focus is given by ego MMA to that's three point one electoral votes here. So eager Mattoo is 3.1 electoral votes and this is equal to e tu minus one. So each U is equal to minus one electoral vote. So this is you to equal my It was minus one electoral vote. And in order to check if everything is correct, we should check if the transition from an equals 32 and it was two results in the the only energy of a photon laughed, which is a gamma three. Okay, So if everything is right again, 13 should be e three, man, is it to? So it should be minus 0.1 plus one electoral vote or 0.9 electoral votes. Let's see if this is true here. This clearly is true. So our diagram is correct. OK, this diagram here is the answer to our cat question.

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

Zachary M.

Hope College

Meghan M.

McMaster University