Contemporary theory of atomic structure

The Ministry of Science and Higher Education of the Russian Federation Kazan National Research Technological University






L. Safina, A. Kuznetsov

CONTEMPORARY THEORY OF ATOMIC STRUCTURE

Tutorial









Kazan
KNRTU Press
2021

         UDC 54:546(075)


Published by the decision of the Editorial Review Board of the Kazan National Research Technological University


Reviewers:
Prof. R. Amirov Prof. V. Furer










         Safina L.
         Contemporary theory of atomic structure : tutorial / L. Safina, A. Kuznetsov; The Ministry of Education and Science of the Russian Federation, Kazan National Research Technological University. - Kazan : KNRTU Press, 2021. - 84 p.

         ISBN 978-5-7882-3103-7

       The tutorial is dedicated to the 95th anniversary of the birth of Professor N. Akhmetov. It contains well-known facts from the history of the theory of atomic structure, the fundamentals of this theory from the viewpoint of quantum mechanics as well as the periodic properties of chemical elements in their relation with the Periodic Law.
       The tutorial intended for the students studying in the educational program in the field “Chemical Technology”, the discipline “General and Inorganic Chemistry”, as well as for international students majoring in the framework of international educational programs, and for university graduates and teachers of chemistry.
       Prepared by the department of Inorganic Chemistry named after Professor N. S. Akhmetov.


UDC 54:546(075)


ISBN 978-5-7882-3103-7    © L. Safina, A. Kuznetsov, 2021
                           © Kazan National Research Technological University, 2021

                PREFACE





Nail Sibgatovich Akhmetov (1926-2006)

This tutorial is dedicated to the 95th anniversary of the birth of Doctor of Chemical Sciences, Professor, full member of the Academy of Sciences of the Republic of Tatarstan, Honored Scientist of the Republic of Tatarstan and the Russian Federation, Akhmetov Nail Sib-gatovich. From 1974 to 1996, Professor Akhmetov was the head of the Department of Inorganic Chemistry, which now bears his name. Due to the activities of Professor Akhmetov, the Department of Inorganic Chemistry of Kazan Chemical Technological Institute (now KNRTU) has become one of the recognized centers of scientific methods for teaching general and inorganic chemistry. Professor Akhmetov was the

author of school chemistry textbooks for students of grades 9-11, and his textbook "General and Inorganic Chemistry" for higher education was widely recognized both in our country and abroad. In particular, this textbook, as well as the manual “Problems and Laboratory Experiments in Inorganic Chemistry” (N. Akhmetov, M. Azizova, and L. Badygina) were published in 1982 in English by the publishing company MIR Publishers (Moscow).
         In recent years, many international students have been studying at KNRTU, so there is an urgent need to teach various disciplines not only in Russian, but also in English. So, in a number of faculties of the university in certain student groups, teaching is conducted in English, in particular, at the Department of Inorganic Chemistry, considerable experience has been already accumulated in lecturing and conducting laboratory experiments and seminars in English. Active work is held to introduce and improve the teaching of chemistry in English: methodological materials and tutorials already published in Russian and well proven in educational practice are translated into English, as well as new ones are being written in English. The authors consider the proposed tutorial as a certain contribution to the development of the English-language teaching of chemical disciplines.
         The tutorial intended for the students studying in the educational program in the field “Chemical Technology”, the discipline “General and Inorganic Chemistry”, as well as for international students majoring in the framework of international educational programs, and for university graduates and teachers of chemistry.

                                        Liliya R. Safina and Andrey M. Kuznetsov May 2021



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                1. FUNDAMENTALS OF THE THEORY
                OF ATOMIC STRUCTURE




1.1.    Historical background of the theory of atomic structure


      The fundamental particle electron, which is the carrier of an elementary electric charge, is the basis of the structure of the atom and its properties. Assumptions about the existence of such a charge were expressed back in the 18th century, i. e. long before the discovery of the electron. In 1830, the English physicist and chemist Michael Faraday confirmed these assumptions by discovery of the laws of electrolysis, according to which, to obtain 1 gram-equivalent of any substance, an equal amount of electricity is required, and the charge of each ion is equal to an integer multiple of the elementary charge.
      In 1891, the Irish physicist and mathematician George Johnston Stoney proposed the term electron. He wrote: “In each chemical atom there can be several elementary charges. These charges, which are conveniently called electrons, cannot be separated from atoms, but they are detected when atoms form chemical compounds“.
      In 1897, the English physicist Joseph John Thomson conducted a series of experiments to study the deviation of cathode rays in a magnetic and electric field. Cathode rays are a stream of electrons, which occurs under glow discharge conditions on a negatively charged conductive surface (cathode). The results of the measurements did not depend on the type of gas in which the discharge was carried out. This meant that the particles that make up the cathode rays (electrons), are contained in the atoms of any gas. Thomson calculated the ratio of the electron charge to its mass: e / me = 1.759-1011 C/kg. In 1906, he was awarded the Nobel Prize in Physics for the discovery of the electron.
      In 1909, the American physicist Robert Millikan measured the electron charge. The experiment was to study the behavior of charged drops of oil in the electric field of a capacitor. Millikan repeated the experiment, changing the voltage, and came to the conclusion that

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the dimensions of the negative charge always take on the same values. The smallest of them is the charge of one electron. According to Millikan's calculation, the magnitude of the negative electron charge turned out to be 1.592-10-19 C, which is somewhat less than the value further specified, and accepted at the present time - 1.602-10—19 C. From the ratio of the electron charge to its mass e/m , obtained by Thomson, and the magnitude of the elementary charge of the electron, it became possible to calculate the electron mass: m = 9.1095-10⁻³¹ kg. In 1923, for outstanding achievements in the field of physics, Millikan was awarded the Nobel Prize.
        By the beginning of the 20th century, the development of the basic sections of classical physics: Newton's mechanics, theory of elasticity, thermodynamics and electrodynamics, was fully completed. The creation of vacuum devices, the birth of radio engineering and the improvement of devices for the study of physical phenomena eventually led to the discovery of the electron, X-rays and radioactivity. It became known, that molecules consist of atoms, solids have a crystal lattice structure, and a fundamental difference between metals and dielectrics was established. However, one of the unsolved problems remained the study of the nature of the interaction of electromagnetic radiation with matter and radiation of matter itself. The application of the laws of classical physics to describe this process led the English physicists D. Rayleigh and D. Jeans to a paradoxical result. The formula, they obtained (Rayleigh-Jeans formula, 1905) for the intensity of thermal radiation of a heated body, depending on the frequency, led to a good agreement with the experiment in the long-wave radiation region, i. e. at small frequency values. However, this formula did not show the maximum on the radiation curve, which is observed in the experiment. Moreover, according to the Rayleigh-Jeans formula, the spectral intensity of radiation increases as the square of frequency and becomes infinite in the limit of very high frequencies. Very high frequencies correspond to ultraviolet, X-ray and g-radiation. Therefore, such a paradox of classical physics was called ultraviolet catastrophe. This and many other problems were solved with the creation of M. Planck's quantum theory of radiation.

        The quantum theory of radiation

        In 1900, the German physicist Max Planck introduced new ideas into physics, which had a decisive influence on its development since the beginning of the twentieth century. To explain the dependence of the radiation


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intensity of heated bodies on the wavelength, he, based on the representation of the discreteness of matter, proposed that the energy is absorbed or emitted by the substance in indivisible portions - quanta. There is a direct proportionality between the quantum energy and the frequency of electromagnetic radiation:
E=hn,                             (1.1)
where h is the proportionality coefficient, called the Planck constant (h = = 6.626-10—34 J-s), the frequency v is measured in Hertz (1 Hz = 1 s⁻¹).
        The frequency of electromagnetic radiation can be expressed in terms of its wavelength by the ratio n=c/l, where c - the speed of light in vacuum, equal to 2.998-10⁸ m/s. Then, E=hc/l. Instead of h, the so-called reduced Planck constant h = h/2p is often used.
        The idea of discreteness was originally applied to matter, not electromagnetic radiation. The assumption about the discreteness of the electromagnetic radiation was made in 1905 by Albert Einstein. Einstein wrote: “...The energy of a beam of light, emitted from a point, is not distributed continuously in an ever-increasing volume, but is made up of a finite number of indivisible energy quanta localized in space, which are absorbed or arise only as a whole.” The idea was born that light is a stream of corpuscles (particles) of its own form, or photons of light, which are also called photons. The corpuscular nature of radiation has been experimentally confirmed in the photoelectric effect and the Compton effect. The photoelectric effect (discovered in 1887 by the German physicist G. Hertz and subsequently experimentally studied by A. G. Stoletov) is a process of electrons detaching from a substance under the influence of electromagnetic radiation. A theoretical explanation of the photoelectric effect was given by Einstein in 1905. When the surface of a metal plate is illuminated, light quanta interact with the electrons of the metal. Light is absorbed, and each quantum transfers its energy to one of the electrons with which it interacts. The kinetic energy, acquired by photoelectrons, can be determined from the Einstein equation:
eₖ = hn - A,                        (1.2)
where A - the electron work function for a given metal.

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        It follows from this equation that there must be some minimum frequency of the quantum at which the photoelectric effect can be observed. In this case, the quantum energy must be equal to, or greater than the electron output. The minimum (boundary) frequency can be found from the condition e = 0, then hn = A. The frequency hn = A, as well as the correspond-k               min                       min
ing wavelength lᵣₑd, is called the red border of the photoelectric effect for a given substance. Quantum radiation theory was able to explain all the regularities of the photoelectric effect that did not fit into the classical physics of the interaction of light with matter.
        Thus, the light propagating in space as an electromagnetic wave, exhibits properties of photons in the photoelectric effect. Not less brightly, than at a photoelectric effect, corpuscular properties of electromagnetic radiation was detected by the so-called Compton scattering of X-rays. The Compton effect (discovered in 1922 by the American physicist A. Compton) is the change in the wavelength of electromagnetic radiation during scattering. In the study of X-ray scattering by electrons it was found that the change in wavelength is associated with the angle of propagation g by the relationship:
Dl= 2plC (1 - cosg)                       (1.3)
                                                 ,
where the constant lC is called the Compton wavelength. The Compton effect has been explained only in the framework of quantum interpretations, i. e. if we assume that the light is not only wave propagation, but also the flow of light particles - photons. According to quantum theory, the Compton wavelength is determined by the ratio lC = й /mec = 3.86 • 10⁻¹³m.

        The first models of atomic structure

        One of the first model of the atomic structure was proposed in 1903 by J. J. Thomson:

     •    The atom is a ball, over the entire volume of which a positive charge is evenly distributed.
     •    There are electrons inside the ball.
     •    The positive charge of the ball is compensated by the total negative charge of the electrons, therefore the atom, as a whole, is neutral.

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        In 1904, Thomson also suggested, that the electrons in the atom form different configurations that determine the periodic properties of the elements. Thus, he tried to establish a connection between the electronic structure of the atom and its chemical properties.
        Thomson's model needed experimental verification: it was necessary to find out whether the positive charge is distributed throughout the volume of an atom with a constant density?
        In 1911, the English physicist Ernest Rutherford published the article “Scattering of a- and b-particles by matter and the structure of the atom”, in which he described the experiments, he conducted together with his colleagues, on the composition and structure of atoms.

    •  a-particles are the nuclei of the helium atom, consisting of two protons and two neutrons. They are formed during the a -decay ofatomic nuclei.
    •  bparticles consist of electrons (e~) and positrons (e+) emitted during the b-decay of atomic nuclei.


        The essence of Rutherford's experiment was as follows. When passing the flow of a-particles through a thin gold foil, it was found that the bulk of the particles penetrated through the foil, and only a negligible fraction deviated from the straight path at an angle of more than 120o. Based on the theory of scattering, Rutherford came to the following conclusions:

   •   The bulk of the atom is concentrated in a positively charged nucleus of extremely small volume (linear size of the order of 10~¹⁴-10~¹⁵ m).

   •   Negatively charged particles (electrons) revolve around the nucleus in closed circular orbits.

   •   The negative charge of all electrons is distributed over the entire volume of the atom and is compensated by the positive charge of the nucleus.


        In the atomic model, proposed by Rutherford, electrons in an atom move in closed circular orbits around the nucleus similar to the rotation of planets around the Sun, and therefore it was called the planetary model of the atom.


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        Of course, the creation of the planetary model of the atom was a significant achievement in the development of ideas about the structure of the atom, but it could not explain a number of experimental facts.
        First, according to the laws of classical electrodynamics, all charged particles, moving with acceleration, must emit electromagnetic energy into space. Consequently, an electron, moving with centripetal acceleration in a circular orbit, should emit energy and completely lose it and fall into the nucleus, which should lead to the disappearance of the atom itself. In fact, atoms are stable and can exist indefinitely long without any changes.
        Second, the Rutherford model failed to explain the nature of atomic spectra. Indeed, if we assume that when approaching the nucleus, the electron changes its energy continuously, then its spectrum should be continuous. However, the optical atomic spectra have a linear character, which indicates a discrete change in energy during electron transitions from one state to another.
        Thus, the planetary model of the atom failed to explain either the stability of atoms or the linear nature of the spectra of individual atoms.
        Rutherford's planetary model of the atom was developed further in the works of the Danish theoretical physicist Niels Bohr.
        In 1913, in his work “On the construction of atoms and molecules”, Bohr wrote: “For reasons we do not yet understand, the atom is characterized not by classical orbits, but by a number of states, that are essentially stationary, and do not change with time.” The planetary model was supplemented by Bohr’s provisions, which he formulated in the form of two postulates:
        Postulate I. Atom can exist only in certain stationary states with constant energy of the electron. In stationary states, the atom does not emit energy. In each state, the electron rotates around the nucleus in a circular orbit, and its angular momentum L = m ur can take strictly defined discrete values according to the relationship:
meur = n h, n = 1,2,3...,                   (1.4)

where ₘ - the mass of the electron; u - the linear velocity of its rotation in orbit; r - the radius of the orbit; h = h /2p, h - Planck’s constant, appearing in the Planck equation (1.1).
        In this case, it is customary to say that the angular momentum of the electron is quantized, and its quantized values are determined by the quantum number n.


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         When the electron moves uniformly in a circular orbit, the centrifugal force meu² / r acting on it is balanced by the centripetal force, which in this case is the Coulomb force of attraction of the electron to the nucleus e² / r² . From the equality of these forces

m u² e
r

e_ r²

(1.5)

taking into account the ratio (1.4) one can obtain an expression for the radius of the orbits:


n ²h²
m e² e


= n²a₀.

(1.6)

        In this formula a0 = h² / me² is the smallest radius of the orbit, which corresponds to the quantum number n = 1. In this case, it is customary to say that the angular momentum of the electron is quantized, and its quantized values are determined by the quantum number n. The numerical value of the radius is 0.529 A (1 A = 10⁻¹⁰ m). This radius is called the Bohr radius or the Bohr radius of the first (closest to the nucleus) electron orbit in the hydrogen atom. For different values of the quantum number n from the formula (1.6) it is possible to obtain a number of values of the radius of orbits of the circular motion of the electron: a0, 4a0, 9a0, 16a0 etc.
        The total energy of an electron is summed up of its kinetic energy and the potential energy of the Coulomb attraction to the nucleus:


22 mₑu e
—
rr


(1.7)

          Using the above ratios, it is easy to obtain the total energy formula


E n

m e⁴
___e___
                                         2n ²h².


(1.8)

         From this ratio it follows that the energy of the electron in the hydrogen atom, as well as its angular momentum, is quantized by a quantum number n . This quantum number is called the principal quantum number, since in the hydrogen atom only it determines the permissible values of the electron energy - it's main characteristic. To emphasize the dependence of E on n in the left part of the formula (1.8), it is customary to indicate the index n for energy.


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