Question

State Bohr's postulate to explain stable orbits in a hydrogen atom. Prove that the speed with which the electron revolves in n^th orbit is proportional to `(1/"n")`.

Answer 1

The stationary orbits are those orbits for which the angular momentum of the electron is an integral multiple of `"h"/(2π)`, where h is Planck's constant.

Electron in an atom revolves because of the balance of Coulomb force of attraction between the protons and electrons and the centripetal force.

In the n^th orbit,

`("mv"_"n"^2)/"r"_"n" = 1/(4πε_0) "e"^2/"r"_"n"^2`

∴ v_n = `"e"/(sqrt(4πε_0"mr"_"n"))`  ......(i)

Again, r_n = `(ε_0"h"^2"n"^2)/("e"^2π"m")`  .....(ii)

Putting in equation (i)

v_n = `"e"/sqrt(4πε_0"m"(ε_0"h"^2"n"^2)/("e"^2π"m"))`

= `"e"^2/(2ε_0"hn")`

∴ `"v"_"n" ∝ 1/"n"`