Bohr's atomic model

The Bohr model is the first widely accepted model of the atom , the elements of quantum mechanics contains. It was developed by Niels Bohr in 1913 . In this model, atoms consist of a heavy, positively charged atomic nucleus and light, negatively charged electrons , which circle the atomic nucleus in closed orbits. Using three postulates , Bohr partially overrode classical physics within the model . Unlike older atomic models, Bohr's atomic model shows many of the properties observed on the hydrogen atom . On the other hand, many details become more spectroscopicMeasurements not recorded by him. Chemical bonds cannot explain it. The concept of electrons moving in narrow paths around the nucleus contradicts the uncertainty principle .

According to Bohr's atomic model, electrons move on circular paths with a certain energy. Here a single electron changes from the 3rd to the 2nd circular path; a photon of the appropriate frequency is emitted.

Bohr's model of the atom paved the way for understanding the structure of the atomic shell . The vivid idea of ​​electrons orbiting the atomic nucleus like planets the sun has shaped the popular image of atoms ever since. However, the image of the orbits of electrons has been replaced in all quantum mechanical atomic models since around 1925 by the fact that the electrons are only assigned certain probabilities of location , see atomic orbital .

overview

Bohr took as a starting point the Rutherford model of the atom, which had only been postulated two years earlier and was still little widespread at the time . An atom consists of a positively charged nucleus and negatively charged electrons, which are located in a shell around the core, whereby nothing more precise is given about the movement of the electrons in the shell. With the assumption of the applicability of classical mechanics and Coulomb's law, it could be concluded that electrons move around the nucleus like planets around the sun. This model is inconsistent, however, because according to classical electrodynamics , a circular charge generates electromagnetic waves with which energy is radiated. As a result, every circling electron would lose energy and would have to fall into the nucleus on a spiral path. There could be no stable atoms. But since there are atoms of stable size, the model has been refuted in this form.

In order to be able to describe atoms that are stable despite circling electrons, Bohr partly abandoned the validity of classical mechanics and electrodynamics in 1913. He assumed that there are certain orbits for electrons in the atom, on which they orbit the nucleus in a stable form without generating electromagnetic waves, and that all other orbits that would also be possible according to classical mechanics do not occur in nature . The atom only emits radiation when an electron passes from one of the permitted orbits into another, whereby - again in contradiction to electrodynamics - the frequency of the generated wave has nothing to do with the frequencies with which the electron is before or afterwards Circle moves. Only in the borderline case of very large orbits, in which the orbital frequencies differ only very little, does the radiation have approximately the same frequency (see principle of correspondence ). However, no further statements can be made about the exact course of this quantum leap . In doing so, Bohr broke with the previously valid principle of natura non facit saltus (“nature does not make jumps”). In establishing these principles, with which Bohr wanted to achieve a useful explanation of the observed properties of the hydrogen atom, he was largely guided by his intuition.

Bohr set three postulates for the selection of stable paths. His aim was to consider Planck's quantum of action as the characteristic natural constant suitable for microscopic processes in the equations so that the results of his model fit as well as possible with the facts observed on atoms. His model successfully showed that some exceptions from classical physics can be combined with some new but seemingly simple conditions in such a way that many properties of the atoms can be derived from them. In particular, the results reflect the data of the hydrogen atom within the scope of the measurement accuracy possible at that time: its size, the fact that the atom only emits radiation with certain wavelengths and the exact size of these wavelengths ( Rydberg formula ), as well as the ionization energy . This agreement with experimental results legitimized the z. T. revolutionary postulates. The model therefore played an outstanding role in the further development of atomic physics . Because of its clarity, Bohr's atomic model is still often used today as a basis for the qualitative description of atomic processes. However, it was precisely its clarity that could not be maintained in the much better models of quantum mechanics from 1925.

Predecessor and successor models

Arthur Erich Haas' (1910) atomic model can be described as a forerunner of Bohr's model . Haas pursued the idea of understanding Planck's quantum of action in terms of the properties of electrons and atoms. He started from Thomson's atomic model, which was widely accepted at the time, and assumed for the simplest atom that an electron circles within a positively charged sphere the size of the atom. Then he equated the energy difference between the rest position of the electron in the center and its largest possible circular path - arbitrarily - with the energy of the light quantum, which according to the equation corresponds to the limit frequency of the spectral lines of the Balmer series . From this he obtained an equation for the radius of the orbit, which agrees well with the known radius of the H atom. Conversely, this equation allows the limit frequency of the spectral lines and thus the ( Rydberg constant ) to be calculated from the radius . Bohr derived the same equation in his 1913 model, but now for the smallest possible orbit of the electron (while the Balmer series in his model includes a four times larger orbit). After all, Haas' atomic model showed that Planck's quantum of action is a natural constant that can be used to calculate atomic quantities if it is inserted into a physical atomic model - even if it appears arbitrarily. ${\displaystyle h}$ ${\displaystyle E=hf_{\infty }}$ ${\displaystyle f_{\infty }}$

With regard to quantization , John William Nicholson proposed in 1912 that the quantization of angular momentum in steps should be preferred over quantization of energy in steps . His reason was that the angular momentum of circular orbits as in Haas' model is on the one hand the quotient of energy and angular frequency and on the other hand a multiple of . This meaning of Planck's constant turned out to be fundamental (see Planck's quantum of action # angular momentum ). However, due to the lack of any further explanatory value, neither Haas nor Nicholson's ideas were accepted as useful atomic models. ${\displaystyle \hbar =h/2\pi }$ ${\displaystyle E=hf}$ ${\displaystyle \hbar =h/2\pi }$

The direct successor to Bohr's model was the Bohr-Sommerfeld atomic model from 1916 . According to Arnold Sommerfeld's suggestion, elliptical orbits were included in this in order to obtain more and more precise results after improved experimental methods had produced increasingly small deviations from the predictions of the Bohr model.

The model

Bohr's postulates

According to Rutherford's 1911 atomic model, an atom consists of a positively charged, very small and heavy atomic nucleus surrounded by a number of electrons. Bohr took up this idea. He investigated the periodic orbital motion of a single electron, as it results from the formulas of classical mechanics when the force between the nucleus and the electrons comes from electrostatic attraction . In order to adapt this model to the observed properties of the hydrogen atom, he expanded it to include three postulates:

1. Not all classically possible orbits are available to the electron, but only certain selected ones. It does not generate electromagnetic radiation on these orbits, but retains its energy. These are the stationary states of the atom.
2. The electron can jump from one stationary state to another. This process, known as a quantum leap, lies outside the scope of classical mechanics and electrodynamics. During the quantum leap between stationary states with different energies, the energy levels , electromagnetic radiation is emitted or absorbed . The frequency of  the radiation is not determined by the orbital frequency of the electron, but exclusively by the energy difference between  the two states according to the formula discovered by Max Planck for thermal radiation ${\displaystyle f}$ ${\displaystyle \Delta E}$ ${\displaystyle f=\Delta E/h.}$
3. The frequency of the generated or absorbed radiation approaches the orbital frequency of the electron if the electron moves only slowly in the initial state and jumps into the energetically closest state.

In the first two postulates, Bohr formulates that the laws of classical mechanics and electrodynamics only apply to a limited extent at the atomic level. In contrast to classical mechanics, there is no continuous transition between two states, but a quantum leap. In the detailed calculation he implements the first postulate in such a way that he assumes for certain circular orbits that, in direct contradiction to the theory of electrodynamics, the electrons do not lose any energy in the form of electromagnetic radiation during rotation. The second postulate also contradicts electrodynamics, because in its model the frequency of the emitted radiation does not have to match the orbital frequency of the particle generating the wave. As a result (and with the help of a further, but absurd and incorrect additional assumption) he succeeds in deriving completely new formulas for the relationship between the electron movement (with the parameters orbital radius, energy, orbital frequency) and the emitted radiation (parameter frequency), which the Look similar to Rydberg's formula .

In order to select the right one from these still too general formulas, in his third postulate he uses for the first time the principle of correspondence between classical and quantum physics, which he discovered (but only later referred to as) : despite the stark contrasts as stated in the first two postulates there must be a smooth transition from the familiar and proven classical physics to the new quantum physics. After some calculations it follows from the third postulate that the stable electron orbits are characterized by the orbital angular momentum of the electron being an integral multiple of Planck's reduced quantum : ${\displaystyle L}$ ${\displaystyle \hbar ={\tfrac {h}{2\pi }}}$

${\displaystyle L=n\hbar \quad (n=1,2,\dotsc )}$

This, too, is sometimes referred to as Bohr's third postulate, because it enables a strict derivation of the formulas of Bohr's atomic model without using the correspondence principle or the mentioned false additional assumption (see below mathematical formulation ). Bohr himself later only referred to the first two assumptions as his postulates.

Experimental review

The Bohr model of the atom could explain a number of physical measurement results of the nascent atomic physics. In subsequent experiments carried out with greater accuracy, however, there were also significant deviations between the model and reality.

Size of atoms

The diameter of atoms calculated with the few basic assumptions of the model is of the correct order of magnitude for many elements . In particular, they roughly agreed with the experiments on X-ray diffraction carried out for the first time by Max von Laue and William H. Bragg at the same time . The small but finite size of atoms was one of their key properties in the then still vague ideas about the atomic structure of matter. Hence, the Bohr model's ability to infer size from general assumptions was considered a success.

Spectral transitions

In the first half of the 19th century, spectral lines were discovered in the hydrogen atom. For the position of the lines within the respective series, Johann Jakob Balmer and Johannes Rydberg were able to provide numerical formulas based on measured line spectra as early as 1885 and 1888 ( Balmer series , Rydberg formulas ). However, the physical background to these formulas remained a mystery for nearly thirty years. The spectral transitions of electrons from one shell to the other introduced by Bohr made it possible to derive the Balmer and Rydberg formula from general principles. They also gave an intuitively illuminating picture of the processes in the atom. A spectral series corresponds to the transitions from electrons of different higher levels to the same basic level. The higher the energy of the initial level, the greater the energy difference during the quantum jump, i.e. the energy of the photons generated and thus their frequency.

States of discrete energy

The existence of the excited stationary states of Bohr's atomic model was proven in 1913/1914 with the Franck-Hertz experiment . In the experiment was in the mercury atoms ground state be shown that during impact by a free electron, a certain amount of energy must be transferred to reach the first excited state. This confirmed the first postulate of Bohr's atomic model in an independent manner.

Weaknesses and contradictions

Some weaknesses and contradictions of the model were already clear when it was published in 1913. Others later became apparent with improved experiments and more elaborated theory of quantum mechanics.

• The postulates are justified by no fundamental principle, but solely by their success. They contradict classical electrodynamics.
• Bohr's model describes the behavior of hydrogen atoms and of ions with only one electron . Multi-electron systems are not recorded.
• The theory of relativity is not taken into account, although almost 1% of the speed of light is ascribed to the electron in the hydrogen ground state .
• The hydrogen atom in Bohr's model should be a flat disk.
• Chemical bonds cannot be understood with Bohr's model.
• In all stationary states the orbital angular momentum of the electron comes out too large. In particular, it should be in the basic state according to Bohr , but in fact it is 0. ${\displaystyle 1\hbar }$ ${\displaystyle 1\hbar }$
• The even-numbered splitting of many spectral lines under the influence of magnetic fields ( anomalous Zeeman effect ) cannot be explained.
• Certain spectral lines of hydrogen turn out to be double lines on more precise measurements. The Bohr model cannot explain this separation, called the Lamb Shift after its discoverer .
• The 21 cm line of hydrogen, which is important in radio astronomy, cannot be derived from the Bohr model.
• The idea of ​​a defined orbit of the electron around the atomic nucleus violates the uncertainty principle discovered by Werner Heisenberg in 1927 .

Quantum physics, whose statements agree in every detail with the experimental findings to this day, paints a somewhat different picture of the atom with the orbital model . Contrary to what the Bohr model assumes, the electrons in the atom have a finite probability of being in large areas, sometimes right into the nucleus. They don't move on orbits. The idea of ​​a cloud is more appropriate. However, three basic statements of Bohr's model have been adopted in all atomic models to this day:

• Electrons only stay in defined areas in the atom, while Bohr's orbits have evolved into today's orbitals.
• Electrons can only take on certain energy values ​​in the atom; the areas where they are located include discrete energy levels.
• Energy exchange with an electron of an atom is only possible if this electron changes its energy level and absorbs or releases the corresponding energy difference.

Mathematical formulation

As much as Bohr's model of the atom ignores reality, it is clearly superior to the previous atomic models. It allows a series of numerical results to be compared with experimental results, above all the position of the lines in the hydrogen spectrum. In contrast to more modern atomic models, the mathematics required for this can be done using formulas and simple transformations of equations:

Bohr's atomic model regards the electron as a point-like particle of mass that is attracted by the opposite electrical charge of the nucleus. With this Coulomb force as the centripetal force , ${\displaystyle m}$ ${\displaystyle e}$

${\displaystyle {\frac {e^{2}}{4\pi \varepsilon _{0}}}{\frac {1}{r^{2}}}={\frac {mv^{2}}{r}},}$

the electron can move on a circular path with suitable combinations of radius and speed . ${\displaystyle r}$ ${\displaystyle v}$

If the above equation is multiplied by, the virial theorem is there ${\displaystyle r}$

${\displaystyle -E_{\mathrm {pot} }=2E_{\mathrm {kin} },}$

see. Coulomb potential and kinetic energy .

If it is multiplied by again , the additionally postulated angular momentum quantization can be used to eliminate: ${\displaystyle mr^{2}}$ ${\displaystyle L=mvr=n\hbar }$ ${\displaystyle v}$

${\displaystyle {\frac {e^{2}}{4\pi \varepsilon _{0}}}mr=n^{2}\hbar ^{2}}$

This results in the orbit radius in the state with the principal quantum number to ${\displaystyle n}$

${\displaystyle r_{n}=n^{2}{\frac {4\pi \varepsilon _{0}\hbar ^{2}}{me^{2}}}.}$

The smallest radius is about 52.9  picometers and is known as the classic or Bohr atomic radius . ${\displaystyle r_{1}}$

The following applies to the total energy:

${\displaystyle E_{n}=E_{\mathrm {pot} }+E_{\mathrm {kin} }={\frac {1}{2}}E_{\mathrm {pot} }=-{\frac {1}{2}}{\frac {e^{2}}{4\pi \varepsilon _{0}}}{\frac {1}{r}}=-\left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{2}{\frac {m}{2\hbar ^{2}}}{\frac {1}{n^{2}}}=-{\frac {1}{n^{2}}}E_{\mathrm {R} }}$

with the Rydberg energy . ${\displaystyle E_{\mathrm {R} }=\left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{2}{\frac {m}{2\hbar ^{2}}}}$

For energy differences between states and we get ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$

${\displaystyle \Delta E=E_{n_{2}}-E_{n_{1}}=\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)E_{\mathrm {R} }.}$

This is (apart from a correction of for the co-movement of the nucleus) precisely the Rydberg formula for the hydrogen spectrum , which Johannes Rydberg had already read from the observed line spectra in 1888 - without knowledge of an atomic model and without the interpretation of the Rydberg energy as a combination of Constants of nature. ${\displaystyle 0{,}05\%}$ ${\displaystyle E_{\mathrm {R} }}$

If you let go towards infinity in the Rydberg formula , you get the ionization energy for ionization from the state . ${\displaystyle n_{2}}$ ${\displaystyle n_{1}}$

outlook

Bohr's atomic model from 1913 was expanded in various ways in Bohr-Sommerfeld's atomic model from 1916. Among other things, a second and third quantum number were added to explain the intensities and fine structure splits of the spectral lines. The Stern-Gerlach experiment of 1922 again added spin to the model .

Both models were replaced with quantum mechanics, but at the same time Bohr's postulates were fully justified. It became clear why Bohr's model and its extensions were successful in many areas, that is, led to correct predictions.

Logo of Ulm University (excerpt)

Culture

See also : atomic symbol

⚛️

Emoji

Bohr's atomic model can be found as an iconic representation of the atom in logos , in coats of arms (see atomic symbol in heraldry ), in caricatures and in popular scientific illustrations. Such a representation is standardized in Unicode as the character U + 269B ⚛ atom symbol and as an emoji .

swell

• Niels Bohr: On the Constitution of Atoms and Molecules, Part I . In: Philosophical Magazine . 26, 1913, pp. 1-25.
• Niels Bohr: On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus . In: Philosophical Magazine . 26, 1913, pp. 476-502.
• Niels Bohr: On the Constitution of Atoms and Molecules, Part III Systems containing several nuclei . In: Philosophical Magazine . 26, 1913, pp. 857-875.
• Niels Bohr: The spectra of helium and hydrogen . In: Nature . 92, 1914, pp. 231-232. doi : 10.1038 / 092231d0 .
• Niels Bohr: Atomic Structure . In: Nature . 107, 1921, pp. 104-107. doi : 10.1038 / 107104a0 .
• Michael Eckert: The Birth of Modern Atomic Theory . In: Physics in Our Time . 44 (4), 2013, p. 168.

Web links

Commons : Bohr's atomic model  - collection of images, videos and audio files

• Netchemie Periodic Table Animated representation of all elements according to the Bohr model
• ChemGlobe - Bohr's atomic model
• Presentation of Bohr's atomic model in the video alteso.de explains Bohr's atomic model.
• Bohr's model of the atom - an introduction to school (Ulm University). ( Memento from February 10, 2010 in the Internet Archive ).
• scinexx.de: 100 years of Bohr's atomic model June 21, 2013

Individual evidence

1. ↑ ^{a b} Jagdish Mehra, Helmut Rechenberg: The Historical Development of Quantum Theory. Vol. 1, chap. II.2, Springer-Verlag 1987.
2. ↑ Abraham Pais: Inward Bound: Of Matter and Forces in the Physical World. Clarendon Press, Oxford, 1986.
3. ↑ N. Bohr: On the Constitution of Atoms and Molecules . In: Philosophical Magazine . tape 26 , 1913, pp. 4 . Bohr arbitrarily assumes that when a free electron is captured in an orbit with a rotational frequency, the binding energy is given off in the form of light quanta of the energy . This number , which later turns out to be the (main) quantum number , is originally actually a number of quanta. ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle {\tfrac {1}{2}}hf}$ ${\displaystyle n}$
4. ↑ N. Bohr: On the application of the quantum theory to the atomic structure . In: Zeitschr. f. Physics . tape 13 , 1923, pp. 117-165 .
5. ↑ Example: IAEA logo
6. ↑ Example: Iran-Winter-Games.htm ( memento from October 10, 2012 in the Internet Archive ), a 2010 cartoon on Iran's nuclear policy .
7. ↑ Example: First direct observation of atoms in gas. scinexx .de, September 19, 2016, accessed on September 19, 2016 .  The cover illustration was made available to the press by MIT .