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Derive an expression for the potential and kinetic energy of an electron in an orbit of a hydrogen atom according to Bohr’s model.

Answer
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Hint: According to Bohr’s model, in a hydrogen atom, a single electron revolves around a nucleus of charge +e. for an electron moving with a uniform speed in a circular orbit of a given radius, the centripetal force is given by the coulomb’s law of attraction between the electron and nucleus. The gravitational attraction may be neglected as the mass of electrons and protons is very small.

Complete step by step solution:
Since the electron is moving in a fixed orbit, therefore the centripetal force is equal to the attraction between electron and proton.
mv2r=ke2r2
mv2=ke2r , ….equation (1)
Where m is the mass of the electron, r is the radius of the orbit, e is the charge of the electron, and k=14πε0 is a constant
Now express the formula for the angular momentum of the electron in orbit
mvr=nh2π , where h is planck's constant.
v=nh2πmr
We can substitute this expression v in equation 1
m(nh2πmr)2=ke2r
r=n2h24π2kme2 ……equation (2)
From equation (1) we can write
Ee=12mv2=12ke2r
Using equation (2) we can put r
Ee=ke224π2kme2n2h2
Ee=2π2k2me2n2h2
Now for the potential energy we have
Ep=ke2r
Write the expression r from equation 1 in the above equation
Ep=ke24π2kme2n2h2
Ep=4π2k2me4n2h2
Hence Ee and Ep are the kinetic energy and potential energy respectively.

Note: Here the kinetic energy and potential energy for the electron is different from the other objects having more mass. When we talk about electrons we neglect the gravitational attraction of the earth. But we don’t consider its mass zero.
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