Bohr's atomic model

Illustration of the atomic model of Borh

For the year 1910, it was certain that the atoms had electrons, this is manifested in the experiments carried out with photoelectric effect, the dispersion of X-rays and others, such as the e/m ratio of the electron explained in the previous post, the results of these works establish a number Z, which indicates the number of electrons in the atom, and since the atoms are basically neutral, the numbers of negatively charged electrons must be equal to those of positive charge, it was also known that the mass of the electrons was very small with respect to the total mass of the atom, so it was assumed that the positive charge contained the greatest amount of mass. All these data and considerations raised the problem of the distribution of electrons in the atom so I take to the development of the famous atomic models, built by Thomson, Rutherford and Bohr, in order to know the behavior of these particles in the atom

In a very summarized way in the atomic model of Thomson the electrons were considered as small particles located in "fixed" points within a continuous distribution of positive charge and spherical shape, in which given to the repulsions between the charges, these must be located uniformly in the sphere. This model is known as "plum pudding model" illustrated in figure 1.

Figure 1

Later in 1911 Rutherford by dispersing the alpha particles by atoms, I can show that the positive charge was not distributed throughout the atom, but rather in a small region in the nucleus where almost all the mass of the element is concentrated (figure 2), leaving the electrons orbiting around this. Even when it was possible to advance in a more exact perception of the atoms, there were strong doubts about how the electrons are distributed around the nucleus,and this because the Coulomb Force that exists between the particles does not allow to conceive an arrangement where the electrons do not fall to the nucleus,and if there is such a way as the way the planets orbit around the sun, which works in classical mechanics, there was no explanation for the case when the electrons were excited and consequently accelerated causing an electromagnetic radiation that would weaken the mechanical energy of the electron which would force them to fall spirally towards the nucleus. Then, as you can see, there was no description of the problem of the stability of the atom, until in 1913 Niels Bohr proposed his model that successfully managed to predict the atomic spectra of some elements[1]

Figure 2

Atomic model of Bohr

The atomic model of Bohr, gives a quantitative and qualitative explanation of the behavior of atomic spectra, this is defined by the following postulates:

1. An electron in an atom moves in a circular orbit around the nucleus under the influence of the Coulomb attraction between the electron and the nucleus, subjecting itself to the laws of classical mechanics.
2. Instead of an infinity of orbits that would be possible in classical mechanics, for an electron it is only possible to move in an orbit for which its angular impulse L is an integral multiple of , the Planck constant divided between 2.
3. Even though the electron constantly accelerates when it moves in one of these allowed orbits, it does not radiate electromagnetic energy. Then, his total energy E remains constant.
4. Electromagnetic radiation is emitted if an electron, which initially moves in a total energy orbit , changes its movement discontinuously to move in a total energy orbit . The frequency of the radiation emitted is equal to the amount divided between the constant .
   [2]

The above corresponds to the quantitative explanation that establishes Bohr for the structure and behavior of electrons in the atom, one can notice a curious mixture of the laws of classical mechanics with the innovation of the quantization of the energy of Planck at that time, although seeing it well it does not sound so crazy to apply such concepts since we are talking about quantum kingdom, very studied today. Now with respect to the quantitative form, it is easy to see this by developing the following example:

Consider an atom with a single electron orbiting its nucleus, that is to say, we will study the hydrogen atom the most abundant and simple element of the universe that consists of Z = 1. So by applying Newton's second law and the Coulomb Force in the in the circular path of the electron, we have:

(1)

Coulomb force and circular motion

Where:
:the mass of the electron.
: the speed of the electron in its circular path.
: the radius of the circular path.

It is important to mention that (1) is an idealization of the phenomenon, like all modeling, but with considerations of physical sense, since the electron is under the influence of a centripetal force apart from the Coulomb Force, it is clear then that the acceleration experienced by the electron is given by the expression that accompanies the mass in the right member of the equation. Now applying the second postulate:

(2)

(2)represents the angular impulse of the electron according to Bohr where it is possible to take only integer values as follows n = 1, 2, 3 ... And since L it is constant with a force acting in the radial direction we can write:

(3)

Clearing of (3) and replacing in(1).

(4)

With n= 1, 2, 3, 4...

We take and replace in(1), then we clear and we have:

(5)

For n=1, Z=1 (hydrogen atom) and the constants already known, We can get m, and , this information represents the Bohr radius for the hydrogen atom when the electron are in their ground state of energy, this is appreciable in (5) where r are the circular orbits whose values are given by n which are in this case the allowed orbits. Now calculating the energy in these trajectories we apply the definition of potential, with the Coulomb Force shown in (1)

And for the kinetic energy also taken out of (1)

Therefore the energy of the electron is given by the sum of the 2 previous expressions:

Taking into account (5) we have left:

(6)

(6)indicates the energy that each possible level or orbit that can occupy the electro in the hydrogen atom whose values are represented in the following figure.

Figure 3

These values agree very well with the data obtained experimentally, as well as the frequency of the electromagnetic radiation given by:

(7)

Figure 4

obeying the fourth postulate of Bohr, where the relationship is considered , rewriting (7) as follows:

(8)

with:

(8) represents the reciprocal of the wavelength of the electromagnetic radiation of all possible transitions of the electron in the energy levels of the hydrogen atom, with the condition of >. In fact, this last expression contains the series of Balmer, Lyman, Paschen, Brackett y Pfund establishing the corresponding value of , for each interval, that is to say, the experimental data of the atomic hydrogen spectrum that are the basis of the mentioned series,are deductible through the Bohr atomic model, which give the values of the spectral lines of hydrogen in all ranges, as shown in Figure 5.

Figure 5

The above shows the correctness of this theory, although only limited to certain elements that meet the conditions proposed by Borh, represents a remarkable scientific advance in the area of quantum physics, for me it is great and appropriate how simple and elegant this method, that at that time was supported by the results of the research carried out by Planck and Einstein, which clearly is a fundamental help, I am impressed by the mere fact of having the vision in order to put together the pieces necessary to give shape to this successful theory.

Enjoy a lot doing this little post about the most important and interesting advances of science in the area of Physics, I hope it will be useful or pleasant for you and I appreciate your attention and support. Regards

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References:
[2] Física cuántica. Eisberg- Resnick. Limusa Wiley. Modelo atomico de Bohr. pp.129-132.
Física Moderna. Virgilio Acosta. Clyde L. Covan. Graham. El modelo de Borh pp. 129-133.

Equations made in word 2007 and pasted as image