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Question

Using Bohr's postulates, obtain the expression for the total energy of the electron in the stationary states of the hydrogen atom. Hence draw the energy level diagram showing how the line spectra corresponding to Balmer series occur due to transition energy levels .

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Solution

According Bohr's postulates, in a hydrogen atom, a single electron revolves around a nucleus of charge +e. Then the centripetal force is provided by Coulomb force of gravitational attraction. So,
mv2r=ke2r2
mv2=ke2r..................(1)
Where, m= mass of electron
r= radius of electron orbit
v= velocity of electron
Again,
mvr=nh2π
v=nh2πmr

Substituting v in (1) we get,
m(nh2πmr)2=ke2r

n2h24π2mre2=r...........(2)

Using equation (2) we get:
Ek=ke24π2kme22n2h2

=2π2k2me4n2h2

(ii) Potential energy
Ep=k(e)×(e)r
=ke2r
=ke2×4π2kme2n2h2
=4π2k2me4n2h2

Hence, total energy of the element in the nth orbit
E=EP+EK

=4π2kme4n2h2+2π2k2me4n2h2

=2π2k2me4n2h2=13.6n2cV

In H-atom, when an electron jumps from the orbit ni to orbit nf the wavelength of the emitted radiation is given by:
1λ=R1n2f1n2i
Where, R Rydberg's constant =1.09678×1010m1

For Balmer series, nf=2andni=3,4,5,....(i)
1λ=R(1221n2i)
Where, ni=3,4,5,....
These spectral lines lie in the visible region.

560767_506206_ans_c280f873614443d4aee3977e4053b6bb.png

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