Bohr's atom model

BOHR'S ATOM STRUCTURE MODEL

In the 19th and in the beginning of the 20th centuries the physicists and chemists researched the phenomena of the dispersion of light induced by a prism. The light was emitted by the heated chemical elements. Such light after the dispersion created the spectrum (characteristic for each element). The spectrum, as it occurred, is a different character of each and every element. The phenomena became widely used for chemical analysis. Also the new, unknown before elements were discovered. The more and more precisely systems were made for that phenomena researching. One of them is shown below.

Spectrum of diffrent bodies: the Sun, hydrogen, helium, mercury
Color photograph (1000x1200) - 160k

The system consists of a source of light, a slot through which a parallel beam of light goes, a prism which dispersed them (the waves of the smaller length are more deflected than the waves of the bigger length). The pattern of colourful spectral lines occurs on the screen. Their position is depended on the waves length emitted by the element. Each element during heating emitted a different set of the light waves. The scientists began to mark the systems - the spectrographs so they could define the length of waves emitted by the individual elements. They noticed that many of the spectrum lines occur over the visible range - in the infra-red and the ultraviolet. Hydrogen has the simplest spectrum. Balmer proofed that he hydrogen spectrum near the visible range can bent by the formula:

(1)

where:

(2)

Where

is the length of the light wave, n' is the constant equal two, n is the integer equal 3 or bigger. R is the Rydberg constant found experimentally and equal about 1,09677 * 10⁷ m^-1. As the formula shows when lambda

decreases (x increases), the density of spectrum lines increases. For example for the first Balmer line (n = 3) the length of the light wave is equal 6563 angstroms, for n = 4 it is equal 4861, for n = 5 it is equal 4341 and for n = 6 it is equal 4102.
The system of the lines was called the Balmer spectral series. For hydrogen there are also some other series:

The Lyman spectral series - placed in ultraviolet - is described by the first formula (given above) but n' is equal 1 here, and n is equal 2 or bigger.
The Pachen spectral series - placed in infra-red - is described by the first formula but n' is equal 3 here, and n is equal 4 or bigger.
The Brackett spectral series - placed in infra-red - is described by the first formula but n' is equal 4 here, and n is equal 5 or bigger.

    The spectral lines of the other elements which are heavier than hydrogen are more complicated.
    The lines described above are caused by the emission of the light waves of given length. The light waves are emitted by the atom excited by heating. According to the Rutherford's model of atom the electrons places outside the nucleus must circulate around it on some orbits. Otherwise they would fall on it. If there is some energy transferred to the atom than its electrons would circulate in a farther distance from the nucleus (their potential and kinetic energy increases). They would circulate on the more external orbits. According to the classic laws of the electrodynamics a circulating electron should cause some electromagnetic radiation. The radiation transfers out some of the energy of electron. So the electron loosing its energy should move on the smaller and the smaller orbits and finally fall on the nucleus. But no such phenomena occurs. If it had been so than the electron would has been placed on any orbit which would has been permanently changed by the electron as the energy would has been emitted. According to that all the atoms should be on different energy states and emit the radiation of all waves lengths. The spectrum should be continuous one not the line one. That divergences between theory and practise led the one of the most famous physicists of the beginning of the 20th century - Niels Bohr to a new theory describing the laws governing the atom.
    In 1913 Niels Henrik Bohr published his new theory of the atoms constitution. Just like Rutherford he assumed that electrons circulate around the nucleus. But had the three completely new ideas:

There are some orbits called by him the stationery ones, where the moving electrons don't emit energy.
Each emission or absorption of radiation energy represents the electron transition from the one stationery orbit to the another. The radiation emitted during such transition homogeneous and its frequency is given by the formula hv = E₁-E₂, where h is the Planck constant, E₁ and E₂ are the energies in the two stationary states of the system.
The laws of mechanics describ the dynamic equilibrium of the electrons in stationery states but do not describe the situation of the electron transition from the one stationery orbit to the another.

The electron emits or absorbs the energy changing the orbits

The first idea is in contrary with the classic laws thermodynamics. According to the second one the energy of hydrogen spectral lines can be calculated theoretically.
Bohr noticed also that the stationery orbits are the ones where the angular momentum (the orbital moment) is an integral multiple of the value of h/(2*

).
According to the third idea the electron movement 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:

(3)

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

(4)

For the stationary orbits the both forces counterbalance. So we can equate the formulas (3) and (4) and after the transformation we get:

(5)

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

(6)

The angular momentum of the electron moving on the circular orbit is given by the formula:

(7)

Equating the formulas (6) and (7) we get:

(8)

Calculating v from it we get:

(9)

Placing the (9) into (5) we get:

(10)

Having this formula one can calculated the radius each and every orbit of the Bohr atom - the values of the square brackets are known and n is an integer equal 1 or bigger (for n = 1 one gets the r of the first stationary orbit). The n number was called the main quantum number. After placing the values of

, e, m, h into the formula (10) we get the interdependence between the radius of a given orbit and the quantum number according to the formula:

(11)

Using the experimental methods of measurement the scientists calculated the atom radius with quite a big accuracy. It was equal 0,5* 10^-8 what is approximately equal the first Bohr's model orbit. Bohr calculated also the total energy of 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:

(12)

The kinetic energy is given by the formula:

(13)

But of the formula (3) we get:

(14)

Connecting this two formulas and adding the potential energy calculated from the formula (12) we get:

(15)

In this formula all the values from the right side are known.
The Bohr's theory explained the phenomenon of spectral lines of the hydrogen atom creation. Let's assume that the electron is on the first stationary orbit. After supplying some energy to the atom the electron can "jump" to one of the higher orbits. After that the electron emits the energy and comes back to the first orbit. The emitted energy is in the form of radiation, the frequency ( v ) of which is ecqual:

(16)

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 (15) we get:

v = [(2*pi^2*m*e^4)/(h^3]*[(1/1^2)-(1/n^2]

(17)

where n is the orbit of which the electron comes back. After placing the values into (2*

²*m*e⁴*k²)/h³we see it is equal the Rydberg constant R. So the value of R found experimentally is equal of the value calculated theoretically by the Bohr's theory.
    Identically one can calculate the emitted energy when the electron goes down from the higher orbit to any lower one (i.e. from the fifth to the third one). The frequencies of radiation calculated by Bohr for the successive electron transitions is in agreement with an experimental data (the spectrum lines of hydrogen).
    The Bohr's theory describes well the spectrums of the atoms around which the only one electron circulates. Such atoms are: H, He⁺, Li⁺². Unfortunately the theory doesn't describe the spectrums of the atoms around which the two ore more electrons circulate.
    Soon the new phenomena was discovered - the spectrum lines of atoms are not homogenous but they consist of several convenient lines. For example for n = 2 there are two such lines, just like there were two electron orbits of almost identical energies. The problem was solved in 1916 by Arnold Sommerfeld.