# Question #0c91a

Apr 1, 2017

In order to explain the discrete nature of radiation caused by an excited atom Bohr postulated quantisation of electron orbits in an atom.
According to his hypothesis electrons are allowed only to rotate around the nucleus in those circular orbits where the angular momentum of rotation of electron is integral multiple of $\frac{h}{2 \pi}$

I.e. $m v r = \frac{n h}{2 \pi} \ldots \ldots . \left[1\right]$

where

$m \to \text{mass of electron}$

$v \to \text{velocityof electron}$

$r \to \text{radius of the circular orbit}$

$h \to \text{Planck's constant}$

$n \to \text{principal quantum number taking values 1,2,3,... }$

An interpretation of Bohr's equation is obtained later by the theory of matter wave put forwarded by de Broglie.

According to de Broglie the wave length ($\lambda$) of matter wave of a particle of mass $m$, moving with velocity $v$ is as follows

$\lambda = \frac{h}{m v} \ldots \ldots \left[2\right]$

The movement of an electron in a circular orbit of radius $r$ may be assumed to be in the the form of an standing wave when a whole number of wavelengths must fit along the circumference of the electron's orbit as shown in the figure below and then

$2 \pi r = n \lambda \ldots \ldots . \left[3\right]$

Combining [2] and [3] we get

Bohr's equation[1} $m v r = \frac{n h}{2 \pi}$

So according to equation [3} the number of electron waves in an orbit is the principal quantum number (n) of the orbit which is related to the azimuthal quantum number (l)as follows

$l = n - 1$

Again the magnetic quantum number ${m}_{l}$ varies from ($- l \to + l$)

Here the maximum value of ${m}_{l} = 3 = + l$

So $n = l + 1 = 3 + 1 = 4$

Hence the number of waves should be $= 4$