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Problem 52 Hard Difficulty

An electron has a de Broglie wavelength equal to the diameter of a hydrogen atom in its ground state. (a) What is the kinetic energy of the electron? (b) How does this energy compare with the magnitude of the ground - state energy of the hydrogen atom?

Answer

a) 135 $\mathrm{eV}$
b) 10

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Top Physics 103 Educators
LB
Liev B.

Numerade Educator

Marshall S.

University of Washington

Zachary M.

Hope College

Jared E.

University of Winnipeg

Video Transcript

in the sector. Science. We have an electric whose wavelength is equal to the diameter off the hydrogen atom in its ground state. And in question eight, we have to find what is the kinetic energy ke off the electric. So noticed that the wavelength of the electric is two times the radius of the hajj and Adam in the ground state. And you know, from boars theory that the Raiders of the hydrogen atom in the ground state is people's equal to the Bohr radius a zero which is five points 29 times 10. The mine is 11 meters. So Lunda is equal to two times are one which is 5.29 times sent to the minors. 11 meters. So this is 1.5 age time. Sense of the minus 10 meters here, 0.1058 nano meters. Okay, but But we also know from the liberalise formula that the momentum p is equal to H over the wavelength London and we know that the kinetic energy ke is able to p squared over to em. So from here we have that P is equal to the square root of two and K. And so we have that the square root of two. In case instituting this into the release formula, have that the square root of two and K is equal to H over Lunder. So I'm gonna square both sides and have to em okay is equal to age square overlooking the square. So K is equal to age square over to M Linda's where now we can subset of the numbers. So age is 6.63 times since the minus 34 Jews second square divided by two times the mass of the electorate. And the Mass is 9.1 time Stand to the minus 31 kilograms. And Lunda, as we found, is 10 at 1.858 times since the Manus Stan. So this is one point 058 times 10 to the minus 10 meters square. So if if we carry out the calculation we have the K, it's equal to two 0.16 times 10 to the minus 17 jewels, and I'm gonna divide it by 1.6 times. Tend to that minus 19. You spare election votes, so I'm basically he is transforming units here from modules to electoral votes, and we get that this is equal to 135 electoral votes. Okay, that's the kinetic energy of the electron and question Be asked us to compare this with the energy of the ground state of the hydrogen atoms. So we know that the energy overbought ST E one is equal to minus, uh 13.6. So I'm going. I'm just gonna take the magnitude of the interview. That's 13.6 from votes. And we have the kinetic energy of the electron divided by the magnitude of the energy of the the hydrogen atom is of the order of 10. Okay, and this is the answer to aggression be

Universidade de Sao Paulo
Top Physics 103 Educators
LB
Liev B.

Numerade Educator

Marshall S.

University of Washington

Zachary M.

Hope College

Jared E.

University of Winnipeg