Bohr's Theory of Hydrogen atom -5

 Spectral feature in terms of matter wavelength of orbital electron 

     The electrons in atomic orbits behave like matter waves .It provides an acceptable explanation for the discreetness of the electronic orbits in atoms. When an electron jumps from outer orbit to any one the inner orbits, its radius and potential energy  are  decreased ,orbital velocity and binding energy are increased and these changes result with a decrease in the matter waves associated with the orbital electron.  During this jumping the excess energy is radiated out as electromagnetic radiation of wavelength  λwave,.  As they are all the after-effects of the above electronic transition, one may believe that they are linked indirectly. Based on this assumption, let us determine the wavelength of em radiation emitted in a transition from orbit with quantum number n2  to orbit with quantum number n1.

 

. Two electronic orbits with matter waves of orbital electron

   The energy of transition  (ΔE) from n2 to n1   = E2 - E1,  ,where E1,E2 denote the total energy of the orbital electron in the orbit 1 and 2 respectively.

                                       E2- E1 = ΔE = h ν = hc/ λwave  or λwave  =  hc/ΔE2→1

The wavelengths of matter wave associated with the electron in the orbit with quantum number n2 and n1  are h/mv2 and h/mv1 respectively. It gives  mv1 = h/λmatter1 and mv2   = h/λmatter2.  With this knowledge one can estimate the kinetic and potential energies of the orbital electron in terms of  its wavelength of matter wave. 

Kinetic energy of the electron in orbit n2 = (1/2) m v2^2 = (1/2m)  h^2/λ^2matter2.

Potential energy of the electron in orbit n2 = - e^2 /Kr2 = - m v2^2  = - (1/m) h^2/λ^2matter2.

Total energy of the electron in the orbit n2 = Sum of kinetic and potential energies E2 = - (1/2m)  h^2/λ^2matter2

Total energy of the electron in the orbit n1 = E1 =  - (1/2m) h2/λ^2matter1

 ΔE2→1 = (h^2/2m) [ 1/λ^2matter1 - 1/λ^2matter2 ]

The radius of the electronic orbit with quantum number n in hydrogen atom is shown as rn = n^2 ao , where ao is Bohr's radius, the radius of the innermost orbit of the electron in the hydrogen atom. From this one can derive mv^2 = e^2/ Kn^2ao and λ^2matter = h^2 Kn^2 ao /me^2 = (2πnao)^2 .  This can be verified by substituting the values for matter wavelength in  ΔE2→1

      ΔE2→1  = (h^2/2m) [1/(2πao)^2][1/n2^2 - 1/n1^2] = [h^2 /8π^2m ao^2][1/n2^2 - 1/n1^2]  = [e^4 m/8 h^2 εo^2] [1/n2^2 - 1/n1^2]

and  λ2→1 = 8 c εo^2 h^3 / (1/nf^2 - 1/ni^2)m e^4