Question

How did DeBroglie's hypothesis account for the fact that the energy in a hydrogen atom is quantised?

Answer 1

Bohr assumed that electrons move in an orbit around the central nucleus and only certain orbits are allowed.

The electron can be considered as a standing wave. This means that only an integral number of wavelengths can fit into a circular orbit. So we can write:

`nlambda =2pir`

`n` is an integer

`lambda` = wavelength of electron

`r` = radius of orbit.

The wavelength of the electron is given by the de Broglie expression:

`lambda =(h)/(mv)`

Where:

`h` = the Planck Constant

`m` = mass of electron

`v` = velocity of electron

Substituting for `lambda` into the 1st equation we get:

`(nh)/(mv)=2pir`

The angular momentum of the electron = `mvr` so rearranging we get:

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

This is an important result in that it tells us that the angular momentum of the electron can only take integral values of `(h)/(2pi)` I.e it is quantised.