§ 3 Bohr's postulates

By 1913, there were three experimental facts that can not be explained in terms of classical physics:

1) The empirical regularities of the line spectrum of the hydrogen atom, expressed in Balmer - Rydberg.

2) The nuclear model of the atom Rutherford.

3) The quantum nature of the emission and absorption of light (heat radiation and the photoelectric effect).

To be able to resolve the difficulty, Niels Bohr (Danish scientist) formulated three postulates for hydrogen and hydrogen-like atoms - a nucleus of charge Z_e and a single electron moving around the nucleus.

I - first postulate - the postulate of stationary states:

In the system, there are some steady states that do not change over time without external influences. In these states, the atom emits no light.

II – nd postulate - the rule of quantization of orbits:

In the stationary state of the atom the electron moving in a circular orbit with the acceleration does not radiate light, must be discrete (quantized) values ??of angular momentum

III - rd postulate - the rule orbits:
Radiation emitted or absorbed in the form of a light quantum energy in the transition of an electron from one steady state to another.
The value of the light quantum is equal to the energy difference of the stationary states between which the transition of the electron

,

n > m – emission of a photon,

n < m – absorption of a photon.
The set of possible discrete frequencies

quantum transitions and determine the line spectrum of the atom.

§ 4 The experiments of Franck and Hertz

The first and third postulates Bohr were experimentally confirmed in experiments of Franck and Hertz (German scientists) in 1913. Vacuum tube filled with mercury vapor (pressure p ~ 13 Pa) contains a cathode (K), two grids (G₁ and G₂) and the anode (A). The electrons emitted by the cathode are accelerated by a potential difference is applied between the K and G₁. Between the grid G₂ and A applied a small retarding potential 0.5 V. The electrons are accelerated in region 1, fall in the region between the two grids, which collide with mercury atoms. The electrons that have enough energy after the collision to overcome the retarding potential in the 3 (on the image), reach the anode. In inelastic collisions of electrons with mercury atoms can be excited by the latest. According to Bohr's theory, each of the mercury atoms may receive only a definite energy, going with one of the excited states. (Ground state n = 1, excited - n = 2, 3, 4, ...) So, if the atoms do exist stationary states, the electrons collide with mercury atoms must lose energy discretely defined portions equal to the energy difference between the corresponding stationary states.

From experience, it follows that increasing the acceleration potential to 4.86 in the anode current increases monotonically. Passing at U = 4.86 ??V through maximum anode current drops sharply. Then increasing again when the U = 4.86 ??÷ 2 • 4.86 ??V. If U = 2 • 4.86 ??to falls and then rises again, etc. Closest to the ground state of the mercury atom is excited state, separated from the main 4.86 eV. As long as the potential difference U_G1-K <4.86 electrons are in elastic collisions, and under the influence of the field fly to A. When U_G1-K = 4.86 in the electron energy is absorbed by mercury vapor and the electron energy is not enough to overcome the retarding potential and etc.

Mercury atoms to the ground state, emits light at λ = 255 nm (UV), which was found in the experiment. Thus, the experience of Franck and Hertz confirmed I and III - rd Bohr's postulates.

§ 5 The spectrum of the hydrogen atom in Bohr's model

Motion of an electron in orbit Coulomb force is centripetal. Then

By II - th postulate of Bohr

therefore,

The radius of the first Bohr orbit is

r₀= 0,529 Å

r ~ n².

The internal energy of the atom is the sum of kinetic and potential energy

of

Follows

Then

Substitute in the expression for r, we obtain the allowed values ??of the energy:

                       (1)

where the minus sign means that the electron is in a bound state. From (1) follows,  that the energy states of the atoms form a sequence of energy levels, which vary depending on the value of n. Integer n in (1) determining the energy levels of the atom, called the principal quantum number. Energy state with n = 1 is the ground state. The state with n > 1 is called excited. The energy level to the ground state, called the ground, all others - excited.

Ionization of atoms - electron detachment from the atom. The ionization energy of the hydrogen atom is 13.6 eV.

According to II - th postulate of Bohr hydrogen atom in the transition from steady state n to the steady state m (n> m) emitted photon with energy

- Balmer - Rydberg formula,

where   - the Rydberg constant.