Question

Show that the radius of the orbit in hydrogen atom varies as n², where n is the principal quantum number of the atom.

Answer 1

According to the Bohr's theory of hydrogen atom, the angular momentum of a revolving electron is given by

 

`mvr=(nh)/(2pi)" ............(1)"`

Here,
m = Mass of the electron
v = Velocity the electron
r = Radius of the orbit
h = Planck's constant
n = Principal quantum number of the atom

If an electron of mass m and velocity v is moving in a circular orbit of radius r, then the centripetal force  required is given by `F=(mv^2)/r`

Also, if the charge on the nucleus is Ze, then the force of electrostatic attraction between the nucleus and the electron will provide the necessary centripetal force.

`F=1/(4piepsilon_0)((Ze)(e))/r^2=(kZe^2)/r^2`

 Where

`k=1/(4piepsilon_0)`

`:.(mv^2)/r=(kZe^2)/r^2 "  ......(ii)"`

From (i), we get

`v=(nh)/(2pimr)`

Putting this value in (ii), we get

`m/r (n^2h^2)/(4pi^2m^2r^2)=(kZe^2)/r^2`

`=>r = (n^2h^2)/(4pi^2mkZe^2)`

⇒r ∝ n²