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Question
Using Rutherford's model of the atom, derive the expression for the total energy of the electron in hydrogen atom. What is the significance of total negative energy possessed by the electron?
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Solution
From Rutherford's model of the atom, the magnitude of this force is
`f_E=1/(4piepsilon_0)((2e)(Ze))/r^2`
For hydrogen atom,
Let,
Fc − Centripetal force required to keep a revolving electron in orbit
Fe − Electrostatic force of attraction between the revolving electron and the nucleus
Then, for a dynamically stable orbit in a hydrogen atom, where Z = 1,
Fc = Fe
`(mv^2)/r=((e)(e))/(4piepsilon_0r^2)`
`r=e^2/(4piepsilon_0mv^2) `
K.E. of electron in the orbit,
`K=1/2mv^2`
From equation (i),
`K=e^2/(8piepsilon_0r)`
Potential energy of electron in orbit,
`U=((e)(-e))/(4piepsilon_0r)=(-e^2)/(4piepsilon_0r)`
∴ Total energy of electron in hydrogen atom
`E=k+U=e^2/(8piepsilon_0r)-e^2/(4piepsilon_0r)`
`E=-e^2/(8piepsilon_0r)`
Here, negative sign indicates that the revolving electron is bound to the positive nucleus.
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