`text(Bohr's Assumptions)`

1. Electrons move in circular orbits around the proton under the influence of Coulomb force of attraction. Electrons orbit in certain `text("stationary states")` in which the orbiting electrons do not continuously radiate electromagnetic energy. The stationary states have definite total energy. This assumption implies that classical laws of electromagnetic radiation by an accelerated charge do not apply to an electron in stationary orbit.

2. The emission or absorption of electromagnetic radiation (photon) occurs only when there is transition of electrons between two stationary states. `text(The orbits of electrons in atoms are quantized)` . i.e., only cert ain orbits are allowed. Each orbit has a different energy and electrons can move to higher orbit by absorbing energy and drop to a lower orbit by emitting energy. The amount of energy emitted or absorbed is also quantized. The frequency n of absorbed or emitted radiation is proportional to the difference in energy of the stationary states

`E = |E_i - E_f| = h nu......................(1)`

where , h is Planck's constant, `E_i` and `E_f` are the energies of initial and final states.

`3.` `text(The angular momentum of an electron in its orbit is quantized)` The angular momentum of the system in a stationary state is an integral multiple of `h/(2 pi)` (This constant is often written as `barh = h/(2 p)` pronounced "h bar" i.e. `barh` )
Quantization of angular momentum

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

Coulombic attraction provides centripetal acceleration

`(kze^2)/r^2 = (mv^2)/r........................(3)`

`mv^2r = kze^2....................................(4)`

dividing (2 ) by (4)

`1/v = (nh)/(2 pi kz e^2)`

`v = (2pi k z e^2)/(nh)`

`r = (nh)/(2 pi m v) = (nh)/((2pim) ((2pikze^2)/(nh))) , r = (n^2h^2)/(4 pi^2 m k z e^2) = (epsilon_0 n^2h^2)/(pi m z e^2)`

`KE = 1/2 mv^2 = 1/2 m. (4 pi^2 k^2 z^2e^4)/(n^2h^2) , K = KE = (2 pi^2 m k^2z^2e^4)/(n^2h^2)`

`U=PE= -(kze^2)/r= -(kze^2)/{(n^2h^2)/(4pi^2mkze^2)}=-(4pi^2mk^2z^2e^4)/(n^2h^2)`

`U = -2 K`

`TE = U +K = -K = (pi^2mK^2z^2e^2)/(n^2h^2) , BE = -TE =K =- U/2`

`TE = - (2 pi^2 m k^2 z^2 e^4)/(n^2 h^2) , TE = -(2 pi^2 m k^2 z^2 e^4)/(n^2h^2) = - (mz^2e^4)/(8 epsilon_0^2 n^2h^2)`

`TE=(- 13.6 z^2)/n^2`eV

`text(Transition of electron)`

`E_2-E_1 = (2 pi^2mk^2z^2e^4)/h^2[1/n_1^2 - 1/n_2^2]`

`h n = (hc)/lambda = (mz^2 e^4)/(8 epsilon_0^2 h^2)[1/n_1^2 - 1/n_2^2] ` ,

`1/lambda = (2 pi^2mk^2z^2e^4)/(h^3c)[1/n_1^2 - 1/n_2^2] `

`1/lambda = (mz^2 e^4)/(8epsilon^2h^3c)[1/n_1^2 - 1/n_2^2]`

`1/lambda =Rz^2 [1/n_1^2 - 1/n_2^2]`

where `R =(2 pi^2mk^2z^2e^4)/(h^3c) `

`= (me^4)/(8 epsilon_0^2 h^3 c) = 1.09 xx 10^7 m^(-1)`

= Rydberg constant

`TE = - (Rhcz^2)/n^2`

`=(13.6z^2)/n^2` eV

`r = (n^2 h^2)/(4 pi m k z e^2) = (epsilon_0 n^2h^2)/(pi m z e^2) = (0.53 n^2)/z overset@A`

`text(radius of 1st Bohr orbit) = 0.53 overset@A text(for hydrogen atom)`