Q. 534.0( 3 Votes )

When an electric discharge is passed through hydrogen gas, the hydrogen molecules dissociate to produce excited hydrogen atoms. These excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula :



What points of Bohr’s model of an atom can be used to arrive at this formula? Based on these points derive the above formula giving description of each step and each term.

Answer :

When an electric discharge is passed through hydrogen gas, the hydrogen molecules dissociate to produce excited hydrogen atoms. These excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula :


- this interpretation of the spectrum of atomic hydrogen can be reached by using 2 basic postulates of Bohr’s model of an atom :


1. The criterion of selecting stationary orbits by the electrons follows the principle: “ the angular momentum of the electron should be an integral multiple of h/2π (h= Planck’s constant)


if the mass and velocity of the electron is m and v at an orbit of radius r then its angular momentum


mvr = n. h/2π,n= non zero positive integer, i.e. 1,2,3,..................... etc.


2. When an electron jumps from one orbit to another, the difference of energy between the two energy levels is either emitted or absorbed (in accordance with the quantum theory of radiation).


Thus, when an electron jumps from an orbit with energy E2 to an orbit of energy E1 (E2>E1), the difference in energy (E2-E1 ) is emitted in the form of quantized radiation. if the frequency of this radiation is ϑ then energy is given by :


E2 – E1 = h ϑ


Now using these 2 points we can reach to the Rydberg’s constant –


Let us consider 2 electron orbits with quantum numbers n1 and n2 in a Bohr type (one electron system ) orbit such that n2>n1 and the corresponding energies are En1 and En2 and En1<En2.


Hence the energy emitted while transfer of electrons from n2 to n1 orbit :


E2-E1= h ϑ, ϑ being the frequency of radiation.


According to the Bohr atomic model for hydrogen atom (Z=1):



Therefore : E2-E1 = h ϑ =


Or, ϑ = and , c is the velocity of light.


Substituting the values form, e, ε0 and h


cm-1


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