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Problem 29 Hard Difficulty

A general expression for the energy levels of one - electron atoms and ions is
$$E_{n}=-\frac{\mu k_{e}^{2} q_{1}^{2} q_{2}^{2}}{2 \hbar^{2} n^{2}}$$
Here $\mu$ is the reduced mass of the atom, given by $\mu=m_{1} m_{2} /$ $\left(m_{1}+m_{2}\right),$ where $m_{1}$ is the mass of the electron and $m_{2}$ is the mass of the nucleus; $k_{e}$ is the Coulomb constant; and $q_{1}$ and $q_{2}$ are the charges of the electron and the nucleus, respectively. The wavelength for the $n=3$ to $n=2$ transition of the hydrogen atom is 656.3 $\mathrm{nm}$ (visible red light). What are the wavelengths for this same transition in (a) positronium, which consists of an electron and a positron, and (b) singly ionized helium? Note: A positron is a positively charged electron.

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Top Physics 103 Educators
Elyse G.

Cornell University

LB
Liev B.

Numerade Educator

Farnaz M.

Other Schools

Jared E.

University of Winnipeg

Video Transcript

in this exercise were given the information that the energy level z an off one elector in Adams and Nyhan's is given by minors. Mu que square he walked. Q. One squared times Q two square, divided by two h bar, square and square. Okay, I'll wear mu is the reduced mass that's equal to the mass of the electorate and one times the mess off the nucleus and to divided by him wimp. Listen to Kyu Wan is the charge of the electron, and Q two is the chart of the nuclear's, and we're also giving the information that when the hydrogen atom transitions from their state and equals three to the state n equals two it a maid's a Fulton whose wave ling Linda A. Is equal to 656.2 nanometers and based on these informations were asked in question eight to calculate what is the wavelength emitted by the positive tone in which I'm gonna call Linda P. Remember that the Pope positron Ian is consists of an electron that orbits Ah, positive and in question be we have to calculate the wavelength Lunda that will be produced by an ionized helium atom. Okay. And in both situations, we have to consider that the transition. Ah, that the Adam like object, uh, undergoes is from the third energy level to the 2nd 1 So it's from an equals 32 and equals two. Okay, so let's solve question a first of all, uh, let's calculate what is the reduced mass of the positive Tonia. So we know that the mess, the richest mess is a mess of the electric em times the mass of the positive, which is the same as the mess of the election. So it's m square divided by the mass of the electron, um, plus the massive deposit training, which is also him. So where they reduced mass is m over to. That's the reduced mass of the posit. Tonia. Okay, notice that the energy of the hydrogen atom I'm gonna leave it. Leave the age here too explicit that it's the hydrogen atom in the 10th energy level is given by minus the mass of the electron. Because there were just massively hydrogen atom is just the mass of the electorate Times K square que que on square, which is just the charge of the electric times the charge of the protein, which is also e and that square. So it's it to the fourth about it. Divided by two and squared H bar squared. That's the energy off the hydrogen atom. No, the energy of the positron Ian is given by minus the reduced mass, which is am over to times case. Where times the charge off the electron square times the charge of oppose it turns out squared, which is, and the charge of the post it turn is equal to the charge of the election. So this is it to the fourth as well divided by two and square divided by at times H Bar square. So notice that this is the same energy as a hydrogen atom, but divided by two. Okay, so have that. The energy of the post. Joining E Pierre is equal to the energy of a hydrogen atom. Ian E h m divided by two. Okay. And we have the information that the hydrogen atom emits Fulton that has a labeling 656.2 centimeters. Now, remember that the energy of a Fulton is given by HC over London. Okay, So if the energy and the energy off the fulton, this proportional to the energy of the energy levels of the hydrogen era. Okay, So if the energy of the energy levels of the post budget earn in our half the energy of the energy levels off the hydrogen, then the wavelength London will be two times the weevil England, London of the party training will be two times the wavelength off the hydrogen ETA. Okay, that that's created by the hydrogen atom because the the energy is proportional to one over Lambda. So from here, we can get that the wavelength of the posit, joining his two times 656.2 centimeters. So the wavelength of the positive China is 1312.6 centimeters. Okay. And in question, be, um we have to calculate the wavelength off the lights that's emitted by the helium. Ah, the iron eyes. He in mind. And what we have to do there is to notice that the energy of the helium is even by minus. Uh, I'm square in because the reduced mass of the heat him is just the mass of the electorate because it's, uh, the mass of the heat him. The mass of the electron is much smaller than the mass of the problem. Okay, so this is just the mass of the electorate. Times case square times the charge of the electrons square, which is e I'm squared times the charge of the nucleus squared and the charge of the nuclear is off. The helium is to e. So the charge of the new place square will be four e square divided by to age, square and square. I noticed that this is four times the energy of the hydrogen. Okay, so if the energy of the hydrogen atom is four eyes, if the energy of the heated madam is four times the energy of the hydrogen atom, then the wavelength produced by the heat him. Adam is 1/4 the wavelength produced by the hajj. In Adam, it's so this is 1/4 off 656.2, which is 164.1 millimeters. And this is the wavelength produced by the helium out

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

LB
Liev B.

Numerade Educator

Farnaz M.

Other Schools

Jared E.

University of Winnipeg