Bohr’s atomic model: characteristics, postulates, limitations

The Bohr model is the project of the Danish physicist Niels Bohr (1885-1962) concerning the structure of the atom, published in 1913. In the Bohr atom, the electrons around the nucleus occupy only certain permitted orbits thanks to a restriction called quantization .

For Bohr, the image of the atom as a miniature solar system, with electrons orbiting around the nucleus, was not entirely consistent with the fact that electrical charges, when accelerated, radiate energy.

This atom would not be stable, as it would end up collapsing sooner or later because the electrons would spiral into the nucleus. And so, the characteristic patterns of light emitted by hydrogen and other gases when heated were known 50 years ago.

The pattern or spectrum consists of a series of bright lines of certain very specific wavelengths. And the hydrogen atom doesn’t collapse when emitting light.

To explain why the atom is stable despite being able to radiate electromagnetic energy, Bohr proposed that angular momentum could only have certain values ​​and therefore energy as well. This is what is meant by quantization.

Accepting that the energy was quantized, the electron would have the necessary stability not to rush towards the nucleus, destroying the atom.

And the atom only radiates light energy when the electron moves from one orbit to another, always in discrete quantities. This explains the presence of emission patterns in hydrogen.

In this way, Bohr composed a vision of the atom by integrating concepts known from classical mechanics with the newly discovered ones, such as the Planck constant, the photon, the electron, the atomic nucleus (Rutherford had been Bohr’s mentor) and the mentioned spectra. question.

Main features of the Bohr model

Bohr’s atomic model assumes that the electron moves in a circular orbit around the nucleus by the action of the Coulomb electrostatic attraction force and proposes that the angular momentum of the electron be quantized.

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Let’s see how to integrate the two concepts in mathematical form:

Let L be the magnitude of the angular momentum, m the electron’s mass, v the electron’s velocity, and the radius of the orbit. To calculate L, we have:

L = m⋅r⋅v

Bohr proposed that L is equal to integer multiples of the constant h / 2π, where h is Planck’s constant , recently introduced by physicist Max Planck (1858-1947) in solving the problem of energy emitted by a blackbody. , a theoretical object that absorbs all incident light.

Its value is h = 6,626 × 10 -34 J · s, while ah / 2π is denoted as H, which is read   “h bar”.

Therefore, the angular momentum L remains:

m⋅r⋅v = n ħ , with n = 1,2, 3 …

And from this condition the radii of the allowed orbits for the electron are deduced, as we will see below.

Calculation of the radius of the electronic orbit

Next, we’ll assume the simplest atom: hydrogen, which consists of a single proton and an electron, both with a charge of magnitude e.

The centripetal force that holds the electron in its circular orbit is provided by the electrostatic attraction, whose magnitude F is:

F = ke 2 / r 2

Where k is the electrostatic constant of Coulomb’s law was the electron-proton distance. Knowing that in a circular motion the centripetal acceleration a c is given by the ratio between the square of the velocity and the distance r:

c = v 2 / r

By Newton’s second law, the net force is the product of the mass m and the acceleration:

mv 2 / r = ke 2 / r 2

Simplifying the radius r we obtain:

m⋅v 2 r = ke 2

Combining this expression with that of angular momentum, we have a system of equations, given by:

1) mv 2 r = ke 2

2) r = n ħ / mv

The idea is to solve the system and determine r, the radius of the allowed orbit. A little elementary algebra leads to the answer:

r = (n ħ ) 2 / k⋅m⋅e 2

With n = 1, 2, 3, 4, 5 …

For n = 1, we have the smallest of the radii, called  Bohr’s radius a or  with a value of 0 , 5 2 9 × 1 0 −10  m. The radii of the other orbits are expressed in terms of  or .

In this way, Bohr introduces the principal quantum number n , noting that the allowed radii are a function of Planck’s constant, the electrostatic constant, and the electron’s mass and charge.

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Bohr’s atomic model postulates

Bohr deftly combines Newtonian mechanics with the new discoveries that were continually being made during the second half of the 19th century and the beginning of the 20th century. Among them, the revolutionary concept of “quantum”, of which Planck himself said he was not very convinced.

Through his theory, Bohr was able to successfully explain the series of hydrogen spectra and predict energy emissions in the ultraviolet and infrared range that had not yet been observed.

We can summarize their postulates as follows:

Electrons describe circular paths

The electron rotates around the nucleus in a stable circular orbit, with uniform circular motion. The movement is due to the electrostatic attraction that the nucleus exerts on it.

Angular momentum is quantized

The angular momentum of the electron is quantized according to the expression:

L = mvr = n ħ

Where n is an integer: n = 1, 2, 3, 4 …, which leads to the fact that the electron can only be in certain defined orbits, whose radii are:

r = (n ħ ) 2 / kme 2

Electrons emit or absorb photons as they move from one energy state to another

As angular momentum is quantized, so is E energy. It can be demonstrated that E is given by:

The electronic volt, or eV, is another unit of energy, widely used in atomic physics. The negative sign of energy guarantees the stability of the orbit, indicating that work would have to be done to separate the electron from that position.

While the electron is in its orbit, it neither absorbs nor emits light. But when it jumps from a higher energy orbit to a lower one, it does.

The frequency f of the emitted light depends on the difference between the energy levels of the orbits:

E = hf = E initial – E final

Limitations

Bohr’s model has certain limitations:

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-It is only successfully applied to the hydrogen atom. Attempts to apply it to more complex atoms have not worked.

-Does not answer why some orbits are stable and others are not. The fact that the energy in the atom had been quantized worked very well, but the model didn’t provide a reason, and that was something that made scientists uncomfortable.

-Another important limitation is that it did not explain the additional lines emitted by atoms in the presence of electromagnetic fields (Zeeman effect and Stark effect). Or why some lines in the spectrum were more intense than others.

-Bohr’s model also does not consider relativistic effects, which must be taken into account, since it was experimentally determined that electrons are capable of reaching speeds very close to the speed of light in a vacuum.

It assumes that it is possible to know precisely the position and velocity of the electron, but what is actually calculated is the probability of the electron occupying a certain position.

Despite its limitations, the model was very successful at the time, not only because it integrated new discoveries with already-known elements, but because it revealed new questions, making it clear that the path to a satisfactory explanation of the atom lay in quantum mechanics. .

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