Bohr's atomic model

Bohr's atomic model diagram.

The Bohr model of the atom is a classical model of the atom, but it was the first atomic model in which it was proposed that electrons could only occupy specific orbits, called stable orbits.

Since the quantization of momentum is adequately introduced, the model can be considered transactional in that it falls between classical and quantum mechanics. It was proposed in 1913 by the Danish physicist Niels Bohr, to explain how electrons can have stable orbits around the nucleus and why atoms exhibited characteristic emission spectra (two problems that were ignored in Rutherford's previous model). In addition, Bohr's model incorporated ideas taken from the photoelectric effect, explained by Albert Einstein.

In 1913 Niels Bohr developed a new model of the atom. He proposed that the electrons are arranged in concentric circular orbits around the nucleus. This model is based on the solar system and is known as the planetary model.

In 1926, Erwin Schrödinger, an Austrian physicist, took Bohr's atomic model one step further. This atomic model is known as the quantum mechanical model. Unlike the Bohr model, this model does not define the exact path of an electron, but rather predicts the probabilities of the electron's location. This model can be represented as a nucleus surrounded by a subparticle. Where the cloud is denser, the probability of finding electrons is higher and, conversely, the electron is less likely to be in a less dense area of the cloud.

Until 1932, it was believed that the atom was composed of a positively charged nucleus surrounded by negatively charged electrons. James Chadwick interpreted this radiation as composed of particles with a neutral electrical charge and the approximate mass of a proton. This particle became known as the neutron. With the discovery of this electron cloud, a more adequate model of the atom became available to scientists.

Introduction

Bohr was trying to make an atomic model capable of explaining the stability of matter and the discrete emission and absorption spectra that are observed in gases. He described the hydrogen atom as having a proton in the nucleus, and an electron revolving around it. Bohr's atomic model was conceptually based on Rutherford's atomic model and the emerging ideas about quantization that had emerged a few years earlier with the research of Max Planck and Albert Einstein.

In this model the electrons revolve in circular orbits around the nucleus, occupying the lowest possible energy orbit, or the closest possible orbit to the nucleus. Classical electromagnetism predicted that a charged particle moving in a circular way would emit energy so that the electrons should collapse on the nucleus in short moments of time. To overcome this problem, Bohr assumed that electrons could only move in specific orbits, each one characterized by its energy level. Each orbit can then be identified by an integer n that takes values from 1 onwards. This number "n" It is called the principal quantum number.

Bohr further assumed that the angular momentum of each electron was quantized and could only vary by integer fractions of Planck's constant. According to the principal quantum number he calculated the distances at which each of the orbits allowed in the hydrogen atom was from the nucleus. These levels were originally classified by letters beginning with "K" and ended in the "Q". Later electronic levels ordered by numbers. Each orbit has electrons with different energy levels obtained that later have to be released and for that reason the electron is jumping from one orbit to another until it reaches one that has the appropriate space and level, depending on the energy it possesses, to be released. no problem and return to its original orbit again. However, he did not explain the fine structure spectrum that could be explained a few years later thanks to Sommerfeld's atomic model. Historically, the development of Bohr's atomic model together with the wave-particle duality would allow Erwin Schrödinger to discover the fundamental equation of quantum mechanics.

Bohr's Postulates

In 1913, Niels Bohr developed his famous atomic model according to three fundamental postulates:

First Postulate

Electrons describe circular orbits around the nucleus of the atom without radiating energy.

The reason why the electron does not radiate energy in its orbit is, for the moment, a postulate, since according to classical electrodynamics a charge with an accelerated movement must emit energy in the form of radiation.

To maintain the circular orbit, the force experienced by the electron—the Coulomb force due to the presence of the nucleus—must be equal to the centripetal force. This gives us the following expression:

kZe2r2=mev2r{displaystyle k{Ze^{2} over r^{2}}}={m_{ev^{2} over r}}}

Where the first term is the electrical force or Coulomb, and the second is the centripetal force; k is the constant of the strength of Coulomb, Z is the atomic number of the atom, e is the charge of the electron, me{displaystyle m_{e} is the mass of the electron, v is the speed of the electron in the orbit and r the orbit radius.

In the previous expression we can solve for the radius, obtaining:

r=kZe2mev2{displaystyle r=k{Ze^{2} over m_{e}v^{2}

And now, with this equation, and knowing that the total energy is the sum of the kinetic and potential energies:

E=T+V=12mev2− − kZe2r=− − 12kZe2r{displaystyle E=T+V={1 over 2}m_{e}v^{2}-k{Ze^{2} over r}=-{1 over 2}{kZe^{2} over r}}}}

Where the energy of a circular orbit for the electron is expressed depending on the radius of that orbit.

Note: Sometimes it can be written in terms of the permitivity of the vacuum k=1/4π π ε ε 0{displaystyle k=1/4pi epsilon _{0}} or in electrostatic load units: k=1.

Second Postulate

The only orbits allowed for an electron are those for which the angular moment, L{displaystyle L}the electron is an integer multiple =h2π π {displaystyle hbar ={h over 2pi }}.

where h{displaystyle h} is the constant of Plank, This condition is mathematically written:

L=mevr=n {displaystyle L=m_{e}vr=nhbar }

with n=1,2,3,...... {displaystyle n=1,2,3,dots }

From this condition and the expression for the radio obtained before, we can replace v{displaystyle v} and there is a quantification for the allowed radios:

rn=n2 2kmeZe2{displaystyle r_{n}={n^{2}hbar ^{2} over km_{e}Ze^{2}}}

with n=1,2,3,...... {displaystyle n=1,2,3,dots }; subscript introduced in this expression to emphasize that radio is now a discreet magnitude, unlike what the first postulate said.

Now, giving values to n{displaystyle n}, main quantum number, we get the radios from the allowed orbits. The first of them (with n=1) is called Bohr radio:

a0= 2kmee2=0.529{displaystyle a_{0}={hbar ^{2} over km_{e}e^{2}}}=0.529}

expressing the result in angstroms.

In the same way we can now replace the allowed radios rn{displaystyle r_{n}} in the expression for the energy of the orbit and thus obtain the corresponding energy at each permitted level:

En=− − 12k2meZ2e4n2 2{displaystyle E_{n}=-{1 over 2}{k^{2}m_{eZ}{2}{2}{2}{2}{2}{2}}}{2}}}}}}}{2}}}

As before, for the hydrogen atom (Z=1) and the first allowed level (n=1), we obtain:

E0=− − 12k2mee4 2=− − 13.6eV{displaystyle E_{0}=-{1 over 2}{k^{2}m_{e}^{4} over hbar ^{2}}}=-13.6{text{ eV}}}}}}}

which is the so-called energy of the ground state of the Hydrogen atom. And we can express the rest of the energies for any Z and n as:

En=Z2n2E0{displaystyle E_{n}={Z^{2} over n^{2}}

This postulate, however, is incompatible with modern quantum mechanics because (1) it presupposes that v and r (and angular momentum) take on well-defined values, in contradiction with the uncertainty principle, and (2) attributes to the first level a non-zero value of the angular momentum.

Third Postulate

The electron only emits or absorbs energy in the jumps from one allowed orbit to another. In this change it emits or absorbs a photon whose energy is the difference of energy between both levels. This photon, according to Planck's law, has an energy:

Eγ γ =h.. =Enf− − Eni{displaystyle E_{gamma }=hnu =E_{n_{f}}-E_{n_{i}}}}

where ni{displaystyle n_{i}} identify the initial orbit and nf{displaystyle n_{f}} the end, and .. {displaystyle nu } It's the frequency.

Then the frequencies of the photons emitted or absorbed in the transition will be:

.. =k2meZ2e42h 2(1nf2− − 1ni2){displaystyle nu ={k^{2}m_{e}Z^{2}e^{4 over 2hhbar ^{2}}left({1 over n_{f}{2}}{2}}}-{1 over n_{i}{2}}{2}}{2}}}{1right)}}

Sometimes, instead of the frequency, the inverse of the wavelength is usually given:

.. ! ! =1λ λ =k2meZ2e42hc 2(1nf2− − 1ni2){displaystyle {overline {nu }}={1 over lambda }={k^{2}m_{eZ}{2}e^{4} over 2hchbar ^{2}}}{1 left({1 over n_{f}{2}}}{1 over n_{i^}{2}{2}{2}}}}}}}}}{1 over

This last expression was very well received because it theoretically explained the phenomenological formula found earlier by Balmer to describe the spectral lines observed since the end of the 19th century in the de-excitation of Hydrogen, which were given by: Bohr described the fundamental atom of hydrogen as an electron moving in circular orbits around a proton, the latter representing the nucleus of the atom, which Bohr locates in its central part and giving a robust explanation regarding the stability of the electron orbit and of the atom as a whole...

.. ! ! =1λ λ =RH(122− − 1n2){displaystyle {overline {nu }}={1 over lambda }=R_{H}left({1 over 2^{2}}}-{1 over n^{2}}}{right)}}

with n=3,4,5,...... {displaystyle n=3,4,5,dots }and where RH{displaystyle R_{H}} is Rydberg's constant for hydrogen. And as we see, the theoretical expression for the case nf=2{displaystyle n_{f}=2}, is the expression predicted by Balmer, and the experimentally measured value of Rydberg's constant (1.097↓ ↓ 107m− − 1{displaystyle 1.097*10^{7}m^{-1}), coincides with the value of the theoretical formula.

It can be shown that this set of hypothesis corresponds to the hypothesis that stable electrons orbiting an atom are described by stationary wave functions. An atomic model is a representation that describes the parts that have an atom and how they are willing to form a whole. Based on Planck's constant E=h.. {displaystyle E=hnu ,} He managed to quantify orbits by observing the lines of the spectrum.