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.

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,mv²/r=ke²/r²
mv²=ke²/r. (1)
Where,m=mass of electron
r=radius of electron orbit
v=velocity of electron
Again,mvr=nh²/2π
v=nh²/2πmr
Substituting v in (1) we get,
m(nh²/2πmr)²=ke²/r
n²h²/4π²mr²=ke²/r
n²h²/4π²m=ke²r
r=n²h²/4π²mke².. (2)
Using equation (2) we get:
Ek=ke²/2r=ke²/2(n²h²/4π²mke²)=2π²k²me⁴/n²h² (ii)
Potential energy
Ep=−k(e)×(e)/r=ke²/r=−ke²×4π²mke²/n²h²=−2π²k²me⁴/n²h²
Hence, total energy of the element in the nth orbit
E=EP+EK=−2π²k²me⁴/n²h²+2π²k²me⁴/n²h²=−2π²k²me⁴/n²h²=-13.6/n²eV
In H-atom, when an electron jumps from the orbit nᵢ to orbit nf the wavelength of the emitted radiation is given by:
1/λ=R[(1/nf²)-(1/nᵢ²)]
Where,R→Rydberg's constant=1.09678×10⁷m⁻¹
For Balmer series,nf=2 and ni=3,4,5,
(i)1/λ=R(1/2²−1/nᵢ²)
Where,ni=3,4,5,
These spectral lines lie in the visible region.