Obtain an expression for the total energy of an electron in the `n^(th)` orbit of hydrogen atom in terms of absolute constants.

Explanation

consider an electron of mass m and charge - e revolving round the nucleus of an atom of atomic number Z in the `n^(th)` orbit of radius 'r'. Let v be the velocity of the electron . The electron also possesses kinetic energy by virtue of its motion.
Potential energy of the electron is given by,
`E_(p)` = (potential at a distance r from the nucleus) (-e)
`=1/(4piepsilon_(0))[(Ze)/(r)] (-e)`
`E_(p)=(Ze^(2))/(4piepsilon_(0)r)" "...(1)`
Kinetic energy of the electron is given by,
`E_(k)=1/2mv^(2)" "...(2)`
From Bohr's postulate,
`(mv^(2))/(r)=(1)/(4piepsilon_(0))[(Ze^(2))/(r^(2))]`
`:. mv^(2)=(Ze^(2))/(4piepsilon_(0)r)`
Substituting this value of `mv^(2)` in equation (2)
`E_(k)=1/2((Ze^(2))/(4piepsilon_(0)r))" "...(3)`
Total energy of the electron revolving in the `n^(th)` orbit is given by `E_(n)=E_(p)+E_(k)`
`E_(n)=(Ze^(2))/(4piepsilon_(0)r)+1/2((Ze^(2))/(4piepsilon_(0)r))` Using (1) and (2)
`=(Ze^(2))/(4piepsilon_(0)r)[(-1)/(2)+(1)/(2)]((Ze^(2))/(4piepsilon_(0)r))[-(1)/(2)\]`
`E_(n)=-(Ze^(2))/(8piepsilon_(0)r)`