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Problem 22 Hard Difficulty

Consider a large number of hydrogen atoms, with electrons all initially in the $n=4$ .(a) How many different wavelengths would be observed in the emission spectrum of these atoms? (b) What is the longest wavelength that could be observed? (c) To which series does the wavelength found in (b) belong?

Answer

a) 6
b) 1879 $\mathrm{nm}$

(c) Paschen series

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Top Physics 103 Educators
Andy C.

University of Michigan - Ann Arbor

Marshall S.

University of Washington

Aspen F.

University of Sheffield

Jared E.

University of Winnipeg

Video Transcript

in this exercise. We have several hydrogen atoms initially in the fourth interview level and in question eight, we want to calculate how many wavelengths can be emitted by these Adams. So no notice that an emission occurs when the Adam transitions from a certain energy level to a lower one. So from the fourth energy level, we can transition to the third or to the second or to the first. OK, but eight if you can tradition to the third, then from the 30 can traditions of the second, or to the first into the SEC from the second week in transition to the first. So in total, there are six ways. ST six wavelengths that can be immediate by Miss Adam. Um in question be we want to calculate what's the longest wavelengths that can be omitted. So notice that the energy of a photon is a C over London. So if you're looking for the longest wavelength lumber, that can be immediate. We're actually looking for the smallest full time energy. And remember that the energy of the emitted Fulton is given by the energy of the initial energy level. E I minus the image of the final energy level. And that's just by conservation of energy. And I'm going to define this quantity here by Delta E. Okay, um, so what we're looking for actually stored out to eat to be the smallest possible among these six transitions that I wrote. You're in the answer to questioning, but actually, since we're looking for, they'll tell you to be the smallest possible. I can restrict myself to look for, to look in three off the other transitions okay from the fourth this third from the third to the second and from the second to the first. And that's because all other threes in three transitions will surely be greater. Have greater energies than these three. Here's that I laid out because, for example, the transition from the fourth, the second energy, the second energy level well, surely release more energy then the transition from the fourth to the third. OK, so let us look, uh, and the energies of these three positions that I marked in blue. Okay, So the first thing to remember before writing the guilty is that the energy of the level of the hydrogen atom is minus 13.6 and I can vote over and swear so dot org is 13.6 times one over N F Square. That's the N. F is the final interview level. Mine is one over and I square and I is the initial energy level. This is in electoral votes, so Don t. For the transition from the fourth to the third energy level is 13.6 one over 91 is 1/16 and this is equal to zero point 66. Said 66 electoral votes Don't say for the transition from the third, the second is 13.6, that was 1/4 man is 19 and this is equal to 1.89 electoral votes. They'll tell you for the transition from the second to the first is 13.6 one miners went forth and this is equal to 10.2 electoral votes. So if we're looking for the smallest energy here we have it. This is It's in the transition between the fourth and the third and every level and the energy is 0.66 electoral votes. The wavelength doubt London will just be, uh, h c over the energy. So this is when 1204 electoral votes and meters divided by the energy, which is 0.66 electoral votes. So Lunda is actually 1879. No, no meters. Okay, this is the longest wavelength possible. And in question, See were asked, What is the Siri's that this transition that leads to the wavelength that we calculated in question be ah to what syriza disposition belongs? And since the final energy level is and F equals three, we know that for four transitions that end in the third and energy level for the hydrogen atom, the name of the Siri's is the passion Siri's. So this is the answer.

Universidade de Sao Paulo
Top Physics 103 Educators
Andy C.

University of Michigan - Ann Arbor

Marshall S.

University of Washington

Aspen F.

University of Sheffield

Jared E.

University of Winnipeg