Question

Using Bohr's postulates derive the expression for the radius of n^th orbit of the electron.

Answer 1

Let e, m, v be the charge, mass and velocity of the electron and r be the radius of the orbit. Positive charge on the nucleus is Ze. In the case of hydrogen atom, Z = 1. Centripetal force is provided by electrostatic force of attraction. Therefore,

`(mv)^2/"r" = 1/(4piepsilon_0) ("Ze" xx "e")/"r"^2`

`mv^2 = ("Ze"^2)/(4 pi epsilon_0 "r")`     ...(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^2 v^2 "r"^2)/(mv^2) = (((n^2h^2)/(4pi^2))/("Ze"^2/(4piepsilon_0"r")))`

Then, r = `("n"^2"h"^2epsilon_0)/(pi"Ze"^2""m")`