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Show that the speed of the electron in the nth Bohr orbit in hydrogen is given by

$$v_{n}=\frac{k_{e} e^{2}}{n \hbar}$$

$v_{n}=\frac{k e^{2}}{n \hbar}$

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University of Michigan - Ann Arbor

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University of Sheffield

this exercise. We have to find the speed the orbital speed of an electron in a hydrogen atom that is, at the end energy level. Eso. In order to do that, the first thing I'm gonna write down here is the formula for the the angular momentum for the hydrogen atom according to force theory. So having the angular momentum, l is able to end times H bar. And we also know that the angular momentum can be redness, linear momentum, witches and V times the radius r of the orbit. So if we isolate be here, we have a V is able to end age divided by M. R. So let me just hi like this for a moment because we're gonna come back to this. Now, remember that the radius of the end energy the orbital level of the hydrogen atom is given by and square times Age bar squared over. He k e e square time Zim. Okay, so I'm gonna just substitute this back into the equation for the speed. So having the speed is able to end H bar or M times are, which is, and square H bar squared over k e. He squared em. So the M skin so out one of the end age bars cancel out as well. And we have V equals K e. Yeah, over an H bar. Just writing down here to be a little tighter, So v equals K e e over in age more as we wanted to show.

Universidade de Sao Paulo