Bohr proposed that the “quantum” of angular momentum, L, possessed by the electron in a hydrogen atom is h/(2).

Therefore, the “angular momentum levels” are given by

where n is an integer. So, in the lowest level L = h/2, in the next level L = 2h/2, etc

Angular momentum,  L = mvr

The force acting on the electron is given by

If the electron orbits in a circular path, the force is also equal to

so, the speed, v, of the electron is given by

Substituting this into equation 1 allows us to calculate the radii of the allowed orbits

The K.E. of the electron  ½mv² which gives

The potential, V, at a point a distance r away from a proton is given by

Therefore the potential energy possessed by an electron at a distance r away from a proton is given by

The total energy (K.E. + P.E.) is therefore

Substituting for r from equation 2, we have

Where E_n is the total energy (K.E.+P.E.) possessed by the electron when it is in level n.

We see that E_n is proportional to 1/n² where the constant of proportionality is

If an electron falls from one energy level, n_initial to a lower level, n_final, then the energy of the quantum of electro-magnetic radiation emitted is equal to the difference between the two levels and is therefore given by

The energy of a quantum of e.m. radiation, E = hc/
So, we can write (ignoring the minus sign)

which has the same form as the Balmer series if we put n_final = 2

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