
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.
, ….equation (1)
Where is the mass of the electron, r is the radius of the orbit, is the charge of the electron, and is a constant
Now express the formula for the angular momentum of the electron in orbit
, where is planck's constant.
We can substitute this expression in equation 1
……equation (2)
From equation (1) we can write
Using equation (2) we can put
Now for the potential energy we have
Write the expression from equation 1 in the above equation
Hence and 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.
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.
Where
Now express the formula for the angular momentum of the electron in orbit
We can substitute this expression
From equation (1) we can write
Using equation (2) we can put
Now for the potential energy we have
Write the expression
Hence
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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