Question

On the basis of Bohr's theory, derive an expression for the radius of the n^th orbit of an electron of hydrogen atom.

Answer 1

Let r be the orbit's radius and e, m, and v represent the electron's charge, mass, and velocity. Ze represents the nucleus's positive charge. For an atom of hydrogen, Z = 1. The electrostatic force of attraction provides centripetal force.
Therefore,

`(mv^2)/r = 1/(4piε_0) (Ze xx e)/ r^2`

`mv^2 = (Ze^2)/(4piε_0r)`  ...(i)

By first postulate:

`mvr = (nh)/(2pi)`  ...(ii)

Where n is the quantum number.

Squaring equation (ii) and dividing by equation (i), we get:

`(m^2v^2r^2)/(mv^2) = (n^2h^2)/((4pi^2)/((Ze^2)/(4piε_0r)))`

Then, `r = (n^2h^2ε_0)/(piZe^2m)`