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Problem 23 Medium Difficulty

A photon with energy 2.28 $\mathrm{eV}$ is absorbed by a hydrogen atom. Find (a) the minimum $n$ for a hydrogen atom that can be ionized by such a photon and (b) the speed of the electron released from the state in part (a) when it is far from the nucleus.

Answer

a) 3
b) $5.2 \times 10^{5} \mathrm{m} / \mathrm{s}$

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Top Physics 103 Educators
Elyse G.

Cornell University

Christina K.

Rutgers, The State University of New Jersey

Marshall S.

University of Washington

Meghan M.

McMaster University

Video Transcript

In this exercise, we have a fulton off Energy e Gomer equals to 2.28 electoral votes that's absorbed by, ah, hydrogen atom. And we want to know what is the smallest energy level end that the electrons in the hydrogen atom is in so that the hydrogen atom is ionized. After absorbing this photo, we know that the the the Adam will be ionized if its initial energy Yeah, plus the energy of the Fulton Gamma is greater or equal than zero. Okay, so we have to check for different levels of energy in if the energy correspondent to the end level plus the energy of a photon is greater equal than zero. Remember that the energy yen for each atomic level, the hydrogen atom is 13.6. You had a bias where electoral votes. So from this we have that when you write it here in blue, you want is equal to minus 13.6 electoral votes. He too is minus 13.6 divided life for So this is minus 3.4. Electoral votes in three is minus 13.4, 13.6 divided by four electoral votes. And this is equal to minus 1.51 electoral votes. Okay. Ah, Now notice that the first energy level and for reach, e n plus gamma is greater or equal than zero is n equals three. Because in that case, E three less you go is equal to 0.77. So this is where than zero. So the answer to question a is n equals three. Okay. And in question be we have to calculate the speed of the immediate actor if the Adam is originally and at the third energy level and when the electron is very far away from the nucleus. Well, here we're going apply conservation of energy. So the initial energy of the system is the energy of the hydrogen atom in the third energy level, plus the energy of the photon gamma. And this is equal to the final energy of the system, which is the kinetic energy of the election. And when the electorate is very far away, there's no binding energy. So it's only the kinetic energy that we have to take into account. And the Connecticut image of the election is M V squared over two. Okay. And we've already calculated e three pleasant gamma that 0.77 electoral votes. So you have that okay is equal to 0.77. And like votes, this is equal to M V squared over two. So we have The V is equal to the square root off two times 0.77 electoral votes divided by the mass of the election, which is 9.1 time Stine to the minus 31 kilograms. What I'm gonna do here, issue it, Multiply the energy. The kinetic energy is your 0.77 electoral votes by 1.6 times 10 to the minus 19. You'll spare election vote. I'm just transforming the the energy the unit of energy from electoral votes to Jules. So have that speed is equal to five points to times 10 to the fifth meters per second. And this is the answer. The question

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

Christina K.

Rutgers, The State University of New Jersey

Marshall S.

University of Washington

Meghan M.

McMaster University