In 1913 Niels Henrik Bohr published his new theory of the atoms constitution. Just like Rutherford he assumed that electrons rotate around the nucleus. But had the three completely new ideas:

1. There are some orbits called by him the stationery ones, where the moving electrons don't emit energy.

2. Each emission or absorption of radiation energy represents the electron transition from one stationery orbit to another. The radiation emitted during such transition is homogeneous and its frequency is given by the formula: 

       

where h is the Planck constant, E_n and E_l are the energies in the two stationary states.

3. The laws of mechanics describe the dynamic equilibrium of electrons in stationery states but do not describe the situation of the electron transition from one stationery orbit to another.

Let's now think what each postulate means.

The first one says that electrons can't move on unlimited orbits around the nucleus. Only some orbits are permissible. Electrons moving on them don't loose energy for radiation. The postulate was in complete disagreement with other theories, and especially with the Maxwell theory of electromagnetism. Bohr formulated the postulate ad hoc. He didn't know what it might come from. But he was of the opinion that to properly understand the nature of the atom one has to accept his idea.

The second postulate says that in an atom an electron can change orbits. On each orbit the electron has some defined energy. The energy of the electron is different on different orbits. The bigger the orbit is, the bigger the energy is. If the electron change a higher orbit into a lower one then it emits a quant of energy that is the same as the difference of energy of the higher and lower orbit. To change a lower orbit into a higher one the electron has to absorb an adequate quant of energy. The quant of energy is proportional to the frequency of the emitted radiation. The second postulate explains why does the atom emit radiation of strictly defined wavelengths.

The third postulate is in complete disagreement with the classic theory. According to that postulate the laws of mechanics can only describe electrons moving on stationary orbits and not while changing their orbits.

Well, all right, but can these assumptions make us calculate wavelengths of the electromagnetic waves representing the respective hydrogen spectral lines? Bohr gave the positive answer to that question, but under the condition that the stationery orbits are the ones where the angular momentum (the orbital moment) is an integral multiple of h/(2*p).

Let's see how to achieve Balmer's equation from that assumption.
According to the third postulate the movement of the electron on the orbit can be described by the classical physical formulas. According to the Newton's law the centrifugal force influencing the electron can be given by the formula:

          (1)

where v is the electron velocity, r is the radius of an orbit, m is the mass of the electron. According to Coulomb's law the force of the electrostatic attraction influencing the electron (the charge of the hydrogen nucleus is equal to e the elementary charge) is equal to:

          (2)

For the stationary orbits the both forces counterbalance. So we can equate the formulas (1) and (2) and after the transformation we get:

          (3)

In this formula the values of r and v. are both unknown. According to the Bohr's idea describing the angular momentum M there is: :

          (4)

where n is a natural number. The angular momentum of the electron moving on the circular orbit is given by the formula:

          (5)

          (6)

          (7)

          (8)

Having this formula one can calculate the radius of the respective orbits in the Bohr's hydrogen atom. All the values in the square brackets are known and n is a natural number equal 1 or more (for n = 1 one gets the r of the first stationary orbit, for n = 2 of the second and so on). The n number was called the main quantum number. After placing the values of p, e, m, h, into the formula (8) we get the interdependence between the radius of a given orbit and the quantum number:

          (9)

Using experimental methods of measurement scientists calculated the approximate radius of the hydrogen atom with quite a big accuracy. It was equal to 0,5* 10^-8 what is approximately equal to the first orbit of the Bohr's model. Bohr calculated also the total energy of the hydrogen electron for any stationary orbit. The total energy is a sum of the potential and kinetic energies of the electron. The potential energy can be calculated from the formula:

          (10)

          (11)

          (12)

Connecting these two formulas and adding the potential energy calculated from the formula (10) we get:

          (13)

In this formula all the values from the right side are known except n, which is a variable.

As it was said before, when the electron jumps to a lower orbit, it emits a photon having energy, which according to the second Bohr's postulate equals:

          (14)

where E_n is the energy of the electron on the orbit, of which it comes back, E₁ is the energy of the electron on the first orbit. Using the formula (13) we get:

          (15)

where n is the orbit of which the electron comes back. After placing the values into (2*p²*m*e⁴*k²)/h³ we see it is equal to Rydberg's constant R. So the value of R found experimentally is equal yo the value calculated theoretically by Bohr. We saw then that Bohr gave the description of the Lyman series, which was discovered experimentally. And the Balmer series corresponds with the electron's jump-down to the second orbit in Bohr's model.

Bohr's theory describes well the spectra of the atoms around the nuclei of which only one electron rotates. Such atoms are: H, He⁺, Li²⁺. Unfortunately the theory doesn't describe the spectra of the atoms around the nuclei of which two ore more electrons rotate.