Question

Prove that, in Bohr model of hydrogen atom, the time period of revolution of an electron in n^th orbit is proportional to n³.

Answer 1

Using Bohr’s theory:

Centripetal force = Electrostatic force

`(mv^2)/r = (kZe^2)/r^2`

⇒ r = `(kZe^2)/(mv^2)`    ...(i)

Angular momentum is quantised, i.e.,

mvr = `(nh)/(2pi)`

⇒ r = `(nh)/(2pimv)`    ...(ii)

Equating (i) and (ii),

`(nh)/(2pimv) = (kZe^2)/(mv^2)`

v = `(2pikZe^2)/(nh)`    ...(iii)

Putting Eq. (iii) in (ii),

r = `(n^2h^2)/(2pim xx 2pikZe^2)`

= `(n^2h^2)/(4pi^2mkZe^2)`    ...(iv)

ω = `v/r`

T = `(2pi)/omega`

= `(2pir)/v`

= `(n^3h^3)/(4pi^2mk^2Z^2e^4)`    ...(Putting the values of v and r from (iii) and (iv))