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Problem 16 Medium Difficulty

Following are four possible transitions for a hydrogen atom
$$\begin{array}{ll}{\text { I. } n_{i}=2 ; n_{f}=5} & {\text { II. } n_{i}=5 ; n_{f}=3} \\ {\text { III. } n_{i}=7 ; n_{f}=4} & {\text { IV. } n_{i}=4 ; n_{f}=7}\end{array}$$
(a) Which transition will emit the shortest - wavelength photon? (b) For which transition will the atom gain the most energy? (c) For which transition(s) does the atom lose energy?

Answer

a) 5$\rightarrow 3$
b) 2$\rightarrow 5$
c) 5$\rightarrow 3$ and 7$\rightarrow 4$

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Top Physics 103 Educators
Andy C.

University of Michigan - Ann Arbor

Farnaz M.

Other Schools

Aspen F.

University of Sheffield

Meghan M.

McMaster University

Video Transcript

In this exercise, we have four transitions of the hydrogen atom where the Adam transitions from an initial energy level and more and I two final energy level and f ah, And in order to calculate to answer all the questions that are asked of us here, the first thing I'm going to do is to calculate Alto E. That is the difference in energy between the final energy GF and the initial energy e I and I have to remember here that the energy of I'm gonna write it in blue here that the energy off the end energy level of the hydrogen atom is given by minus 13.6 electoral votes divided by and square. So this means that Delta E, which is e f minus, equals B F minus E. I is just minor is just 13.6 times one over, and I square mine is one over in F square. Okay, so I can calculate the difference in energy for each of the four transitions you. So let's do it first. The difference in energy for transition one that's gonna be 13.6 times one over an ice square, and I is too So it's one of her four minus one over an F squared, which is one over 25 and this is equal to 2.85 six electoral votes. Then we have the difference in the energy for the transition to, and that's 13.6 times and I, which is and I squared, which is 1/5 square, that is, 1/25 miners one over and naff squared, which is 1/9. And this is minus zero point 967 Noticed that here feel the Adam loses energy, and in this process it emits ah, foot. For the third doubt, E three, we have 13.6 times, 1/49 which is 1/7 square. Mine is one over 16. This is minus zero point 572 electoral votes. Finally, for the fourth, we have 13 point 6/1 times 1/16 minus 1/9. 49 and this is zero point 572 electoral votes. Okay, okay, so now we are in a position to answer the questions in question. A. We asked what is the transition that results in the short, the shortest wavelength off the folding being emitted. So we have only two transitions here that Emmett Fulton's okay, we have the and the transitions that emit Fulton's are all always have a negative difference in energy. Okay, so the elect, the Adam loses energy, and that's why it can emit a phone. So we had of the transition to in the transition three. Okay. And we're looking among them. Which one has the shortest wavelength? Lunda. And we know that the wavelength of the photon is given by HC over the energy of the photo. Okay, And remember that the energy of the Fulton is minus the difference in energy of the ah, of the atom. So the biggest the energy of the election. Okay, the smallest, the the energy. I'm sorry. The smallest, the wavelength of the photon. So we're looking core the difference in energy of the atom that has the the greatest magnitude. Okay, because then the the energy of the fourth movie, Greatest and then the wavelength of the fulton will be the shortest. And that happens for transition to because the magnitude of the surface of the difference in energy into the transition to 0.9 67 electoral votes, and that's greater than the magnitude off 0.572 So the maximum the actually the shortest wavelength Lambda is that happens for transition to the question be we have to say wish transition. In which tradition that the Adam gains the most energy. So we have to see which transitions are positive and that's just one and four. And we have to see among them which one has the greatest energy. And obviously it's the 1st 1 It's 2.80 uh, 856 So the answer is transition to and in C were asked for wish musicians. The Adam loses energy, and that's all we have to do. Still, look at the differences in energy and see which ones are negative. So we have the two and three are negative. Uh, yeah, I said that actually, India had two. It's one right. I pointed to the right one, but I wrote that there are and in seeds two and three. Okay. And this concludes that the exercise

Universidade de Sao Paulo
Top Physics 103 Educators
Andy C.

University of Michigan - Ann Arbor

Farnaz M.

Other Schools

Aspen F.

University of Sheffield

Meghan M.

McMaster University