Question

Bohr in 1911 (i.e. about 200 years after Newton), associated Coulomb’s Force F_c=e²/r²reigning between the proton and the electron of mass m, and of charge e, and separated by a distance r, in the Hydrogen atom, with Newton’s law of Motion.

a) () Write this equation, assuming that the orbit is circular.

b) () Write Bohr Quatization rule, knowing that the angular momentum is expressed as mvr, where v is the velocity of the electron in orbit around the proton.

c) () Bohr accordingly was able to obtain, the previously established Rydbergemprical relationship, along with the notation we used in class: 1/ λ=R(1/n²-1/h²). Here λ is the wavelength of the photon emitted by the electron as it steps down from thekth energy level to the nth energy level, both k and n being integer numbers, and R is the Rydberg Constant, i.e. R=2π²e⁴m/ch³ ; h is Planck Constant, c is the speed of light in vacuum. The Rydberg relationship, written above, chiefly pops out when n is fixed, for instance to 2, and k is considered to cover up, all levels above the nth level. Thusly, Rydberg could determine R, empirically to be about 1.1x10⁵ cm^-1. Check this value, using the above Bohr Rydberg Constant R expression, knowing the following data in CGS unit system: e=4.8x10^-10 esu, m=0.9x10^-27 g, c=3x10¹⁰ cm/s, h=6.6x10^-27 ergxs.

Answer 1

(a)

The attractive Coulomb force between proton and electron in the hydrogen atom,

(Here we have taken only the magnitude and is in CGS) ------------------------- (1)

This force provided the neessary centripaetal force for the electron in the circular orit. Thus

---------------------------------------------- (2)

where is the velocity of electron.

---------------------------------------------- (3)

The kinetic energy of the electron in circular motion is then given by

-------------------------------------- (4)

(b).

According to Bohr"s Quatization rule,

---------------------------------------------------------- (5)

Thus the velocity of electron in the nth orbit

--------------------------------------------------------------(6)

Substituting this value of velocity in equation (4), we get

The potential energy of an electron in the hydrogen atom is

THus the total energy of hydrogen atom in the nth state is

  

Substituting the value of ,

  

  

The emission or absorption of radiation takesplace only when an electron makes a transition from one stationary state to the other. The frequency of radiation iis given by

  

where

is the Rydberg constant.

Substituting the values of m, e and h,