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Problem 44 Hard Difficulty

The Paschen series in the hydrogen emission spectrum is formed by electron transitions from $n_{\mathrm{i}}>3$ to $n_{\mathrm{f}}=3.$
(a) What is the longest wavelength in the Paschen series? (b) What is the wavelength of the series limit (the lower bound of the wavelengths in the series)?
(c) In what part or parts of the EM spectrum is the Paschen series found (IR, visible, UV, etc.)?

Answer

a. $$\lambda_{s}=1875 \mathrm{nm}$$
b. $$1875 \mathrm{nm}-820 \mathrm{nm}$$
c. This series lies in the infrared - region of the EM spectrum.

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Top Physics 103 Educators
Elyse G.

Cornell University

Farnaz M.

Simon Fraser University

Zachary M.

Hope College

Meghan M.

McMaster University

Video Transcript

for this exercise, we have to consider the caution Sears of the hydrogen atom that takes and Adam at an original energy level in why larger than three takes it down to the third energy level. So and after the final area level is three. And in the first question, we're asked what is the longest, uh, the longest wavelength from the Maxwell off this Siri's. In order to calculate this, I'm gonna remember that the energy of the energy level of the hydrogen atom is given by minus 13.6 in squared, elegant boats. And that, uh, from the conservation of energy you have that with the energy of the initial energy level is gonna be the energy of the emitted photo, plus the energy of the final energy level. Here, the final energy level is three. So you can calculate the energy of the photon as the energy of the final energy lab, which is minus 13.6 over in. I'm sorry, but the energy of the nature energy level it's minus 13.6 over in I square, plus my mind is the energy of the the third energy level, which is 13.6 over nine. So this is gonna be, uh, 13.6 1/9. Mine is one, and I squared. I noticed that the way the energy of a photon egos age see over London so that the maximum wavelength will happen when the energy is minimum and the minimum energy Ah, that can be achieved from this warm over here would be achieved when and I is the smallest value per possible and the smallest value possible for and Adam to transition from some initial and your final and equals three is an equals four. So we're gonna have N y equals four and we can calculate the the smallest energy. Now, the smallest energy is gonna be 13.6. Let in vote 1/9, minus 1/16. And this is zero point 661 electoral votes. That, and since the wavelength off ah of a photon is getting by age see over its energy and a C is 1240 acting votes, none of meters. We have as the wavelength the longest wavelengths 18 75 millimeters. Okay, and for question be were asked, what is the shortest wavelength of the Siri's so again. I'm gonna right here the energy, which is 13.6 1/9. Mine is one over and I square and as an eye tends to infinity, the energy increases and the wavelength were decreased. So the minimum wavelength will happen when this term here can be disregarded. And it's so big that this term could be in this regard. So the minimum the maximum energy is gonna be 1.51 electoral votes, which it's correspondent to a wavelength. So if we do, we do the same thing as before. We have talkto 40 electing votes centimeters over one point 51 electoral votes and this is the same s 821 centimeters. So this is the the shortest waving and in question see were asked in which region of the electromagnetic spectrum this this fortunes lie and well, notice that the infrared region goes from about 700 nanometers to about one millimeter. So since we have, we have photos that who's shortest wavelength is 821 centimeters and the longest wavelength is 1875 millimeters. We can say that ah are our photon analyzing Fulton's apply in the infrared region of the electromagnetic spectrum

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

Farnaz M.

Simon Fraser University

Zachary M.

Hope College

Meghan M.

McMaster University