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 stationary orbit of radius .r.. Let v be the velocity of the electron. The electron posses potential energy because it is in the electrostatic field of the nucleus it also possess kinetic energy by virtue of its motion.
For stationary orbit, total energy E=KE+PE ---(1)
From Rutherford.s atom model,
For stationary orbit, centripetal force=Electrostatic force.
`(mv^2)/(r)=(1)/(4pi epsilon_0)(Ze)/(r^2)(-e)`
(`:.` Electrostatic force, according to Coulomb.s law)
On dividing by 2 on both sides, we get
`(mv^2)/(2)=(Ze^2)/(8pi epsilon_0 r)`
`KE=(Ze^2)/(8pi epsilon_0 r)` -----(2) `[:. KE=1/2mv^2]`
We have PE = Electric potential at a distance are due to `+ Ze xx (-e)`
`PE=(1)/(4pi epsilon_0)(Ze)/(r)(-e)" "[:. V=(1)/(4pi epsilon_0)q/r,"Hence" q=+Ze]`
`PE= -(Ze^2)/(4pi epsilon_0 r) ------rarr(3)`
Equation (2) and (3) in (1), we get
`E=(Ze^2)/(4piepsilon_0 r)[1/2-1]`
`E= -(Ze^2)/(8pi epsilon_0 r)`
The radius of `n^("th")` permitted orbit of electron is given by
`r=(epsilon_0 n^2h^2)/(pi mZ_e^2)`
Substituting the value of .r. in equation (4)
`E=(-Ze^2)/(8pi epsilon_0)xx(pi mZe^2)/(epsilon_0 n^2h^2)`
i.e., `E=(-Z^2 me^4)/(8epsilon_0^2 n^2h^2)`
For hydrogen Z=1
`E=(-me^4)/(8 epsilon_0^2 n^2h^2)`