Brownian motion Brownian motion

(Brownian motion), the random movement of tiny particles suspended in a liquid or gas under the influence of impacts from environmental molecules; discovered by R. Brown.

BROWNIAN MOTION

BROWNIAN MOTION (Brownian motion), random movement of tiny particles suspended in a liquid or gas, occurring under the influence of impacts from environmental molecules; discovered by R. Brown (cm. BROWN Robert (botanist) in 1827
When observing a suspension of flower pollen in water under a microscope, Brown observed a chaotic movement of particles arising “not from the movement of the liquid or from its evaporation.” Suspended particles 1 µm in size or less, visible only under a microscope, performed disordered independent movements, describing complex zigzag trajectories. Brownian motion does not weaken with time and does not depend on the chemical properties of the medium; its intensity increases with increasing temperature of the medium and with a decrease in its viscosity and particle size. Even a qualitative explanation of the causes of Brownian motion was possible only 50 years later, when the cause of Brownian motion began to be associated with impacts of liquid molecules on the surface of a particle suspended in it.
The first quantitative theory of Brownian motion was given by A. Einstein (cm. EINSTEIN Albert) and M. Smoluchowski (cm. SMOLUCHOWSKI Marian) in 1905-06 based on molecular kinetic theory. It was shown that random walks of Brownian particles are associated with their participation in thermal motion along with the molecules of the medium in which they are suspended. Particles have on average the same kinetic energy, but due to their greater mass they have a lower speed. The theory of Brownian motion explains the random movements of a particle by the action of random forces from molecules and friction forces. According to this theory, the molecules of a liquid or gas are in constant thermal motion, and the impulses of different molecules are not the same in magnitude and direction. If the surface of a particle placed in such a medium is small, as is the case for a Brownian particle, then the impacts experienced by the particle from the molecules surrounding it will not be exactly compensated. Therefore, as a result of “bombardment” by molecules, the Brownian particle comes into random motion, changing the magnitude and direction of its speed approximately 10 14 times per second. From this theory it followed that by measuring the displacement of a particle over a certain time and knowing its radius and the viscosity of the liquid, one can calculate Avogadro’s number (cm. AVOGADRO CONSTANT).
The conclusions of the theory of Brownian motion were confirmed by measurements by J. Perrin (cm. PERRIN Jean Baptiste) and T. Svedberg (cm. Svedberg Theodor) in 1906. Based on these relations, the Boltzmann constant was experimentally determined (cm. BOLZMANN CONSTANT) and Avogadro's constant.
When observing Brownian motion, the position of the particle is recorded at regular intervals. The shorter the time intervals, the more broken the trajectory of the particle will look.
The laws of Brownian motion serve as a clear confirmation of the fundamental principles of molecular kinetic theory. It was finally established that the thermal form of motion of matter is due to the chaotic movement of atoms or molecules that make up macroscopic bodies.
The theory of Brownian motion played an important role in the substantiation of statistical mechanics; the kinetic theory of coagulation of aqueous solutions is based on it. In addition, it also has practical significance in metrology, since Brownian motion is considered as the main factor limiting the accuracy of measuring instruments. For example, the limit of accuracy of the readings of a mirror galvanometer is determined by the vibration of the mirror, like a Brownian particle bombarded by air molecules. The laws of Brownian motion determine the random movement of electrons, which causes noise in electrical circuits. Dielectric losses in dielectrics are explained by random movements of the dipole molecules that make up the dielectric. Random movements of ions in electrolyte solutions increase their electrical resistance.

encyclopedic Dictionary. 2009 .

See what “Brownian motion” is in other dictionaries:

- (Brownian motion), the random movement of small particles suspended in a liquid or gas, occurring under the influence of impacts from environmental molecules. Explored in 1827 by England. scientist R. Brown (Brown; R. Brown), whom he observed through a microscope... ... Physical encyclopedia

BROWNIAN MOTION- (Brown), the movement of tiny particles suspended in a liquid, occurring under the influence of collisions between these particles and the molecules of the liquid. It was first noticed under an English microscope. botanist Brown in 1827. If in sight... ... Great Medical Encyclopedia

- (Brownian motion) random movement of tiny particles suspended in a liquid or gas under the influence of impacts from environmental molecules; discovered by R. Brown... Big Encyclopedic Dictionary

BROWNIAN MOTION, disordered, zigzag movement of particles suspended in a flow (liquid or gas). It is caused by the uneven bombardment of larger particles from different sides by smaller molecules of a moving flow. This… … Scientific and technical encyclopedic dictionary

Brownian motion- – oscillatory, rotational or translational movement of particles of the dispersed phase under the influence of thermal movement of molecules of the dispersion medium. General chemistry: textbook / A. V. Zholnin ... Chemical terms

BROWNIAN MOTION- random movement of tiny particles suspended in a liquid or gas, under the influence of impacts from environmental molecules in thermal motion; plays an important role in some physical chem. processes, limits accuracy... ... Big Polytechnic Encyclopedia

Brownian motion- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN Brownian motion ... Technical Translator's Guide

This article or section needs revision. Please improve the article in accordance with the rules for writing articles... Wikipedia

Continuous chaotic movement of microscopic particles suspended in a gas or liquid, caused by the thermal movement of environmental molecules. This phenomenon was first described in 1827 by the Scottish botanist R. Brown, who studied under... ... Collier's Encyclopedia

More correct is Brownian motion, the random movement of small (several micrometers or less in size) particles suspended in a liquid or gas, occurring under the influence of shocks from the molecules of the environment. Discovered by R. Brown in 1827.… … Great Soviet Encyclopedia

Books

• Brownian motion of a vibrator, Yu.A. Krutkov, Reproduced in the original author's spelling of the 1935 edition (publishing house Izvestia of the USSR Academy of Sciences). IN… Category: Mathematics Publisher:

Brownian motion

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Onishchuk Ekaterina

The concept of Brownian motion

Patterns of Brownian motion and application in science

The concept of Brownian motion from the point of view of Chaos theory

Billiard ball movement

Integration of deterministic fractals and chaos

The concept of Brownian motion

Brownian motion, more correctly Brownian motion, thermal motion of particles of matter (several sizes µm and less) particles suspended in a liquid or gas. The cause of Brownian motion is a series of uncompensated impulses that a Brownian particle receives from the liquid or gas molecules surrounding it. Discovered by R. Brown (1773 - 1858) in 1827. Suspended particles, visible only under a microscope, move independently of each other and describe complex zigzag trajectories. Brownian motion does not weaken over time and does not depend on the chemical properties of the medium. The intensity of Brownian motion increases with increasing temperature of the medium and with a decrease in its viscosity and particle size.

A consistent explanation of Brownian motion was given by A. Einstein and M. Smoluchowski in 1905-06 on the basis of molecular kinetic theory. According to this theory, the molecules of a liquid or gas are in constant thermal motion, and the impulses of different molecules are unequal in magnitude and direction. If the surface of a particle placed in such a medium is small, as is the case for a Brownian particle, then the impacts experienced by the particle from the molecules surrounding it will not be exactly compensated. Therefore, as a result of “bombardment” by molecules, the Brownian particle comes into random motion, changing the magnitude and direction of its speed approximately 10 14 times per second. When observing Brownian motion, it is fixed (see Fig. . 1) the position of the particle at regular intervals. Of course, between observations the particle does not move rectilinearly, but connecting successive positions with straight lines gives a conventional picture of the movement.

Brownian motion of a gum gum particle in water (Fig. 1)

Patterns of Brownian motion

The laws of Brownian motion serve as a clear confirmation of the fundamental principles of molecular kinetic theory. The general picture of Brownian motion is described by Einstein's law for the mean square displacement of a particle

along any x direction. If during the time between two measurements a sufficiently large number of collisions of a particle with molecules occurs, then proportional to this time t: = 2D

Here D- diffusion coefficient, which is determined by the resistance exerted by a viscous medium to a particle moving in it. For spherical particles of radius, and it is equal to:

D = kT/6pha, (2)

where k is the Boltzmann constant, T - absolute temperature, h - dynamic viscosity of the medium. The theory of Brownian motion explains the random movements of a particle by the action of random forces from molecules and frictional forces. The random nature of the force means that its action during the time interval t 1 is completely independent of the action during the interval t 2 if these intervals do not overlap. The average force over a sufficiently long time is zero, and the average displacement of the Brownian particle Dc also turns out to be zero. The conclusions of the theory of Brownian motion are in excellent agreement with experiment; formulas (1) and (2) were confirmed by measurements by J. Perrin and T. Svedberg (1906). Based on these relationships, Boltzmann's constant and Avogadro's number were experimentally determined in accordance with their values ​​obtained by other methods. The theory of Brownian motion played an important role in the foundation of statistical mechanics. In addition, it also has practical significance. First of all, Brownian motion limits the accuracy of measuring instruments. For example, the limit of accuracy of the readings of a mirror galvanometer is determined by the vibration of the mirror, like a Brownian particle bombarded by air molecules. The laws of Brownian motion determine the random movement of electrons, causing noise in electrical circuits. Dielectric losses in dielectrics are explained by random movements of the dipole molecules that make up the dielectric. Random movements of ions in electrolyte solutions increase their electrical resistance.

The concept of Brownian motion from the point of view of Chaos theory

Brownian motion is, for example, the random and chaotic movement of dust particles suspended in water. This type of movement is perhaps the aspect of fractal geometry that has the most practical use. Random Brownian motion produces a frequency pattern that can be used to predict things involving large amounts of data and statistics. A good example is the price of wool, which Mandelbrot predicted using Brownian motion.

Frequency diagrams created by plotting Brownian numbers can also be converted into music. Of course, this type of fractal music is not musical at all and can really bore the listener.

By randomly plotting Brownian numbers on a graph, you can get a Dust Fractal like the one shown here as an example. In addition to using Brownian motion to produce fractals from fractals, it can also be used to create landscapes. Many science fiction films, such as Star Trek, have used the Brownian motion technique to create alien landscapes such as hills and topological patterns of high mountain plateaus.

These techniques are very effective and can be found in Mandelbrot's book The Fractal Geometry of Nature. Mandelbrot used Brownian lines to create fractal coastlines and maps of islands (which were really just randomly drawn dots) from a bird's eye view.

BILLIARD BALL MOVEMENT

Anyone who has ever picked up a pool cue knows that accuracy is the key to the game. The slightest error in the initial impact angle can quickly lead to a huge error in the ball's position after just a few impacts. This sensitivity to initial conditions, called chaos, poses an insurmountable barrier to anyone hoping to predict or control the ball's trajectory after more than six or seven collisions. And don’t think that the problem is dust on the table or an unsteady hand. In fact, if you use your computer to build a model containing a pool table with no friction, no human control over cue positioning accuracy, you still won't be able to predict the ball's trajectory long enough!

How long? This depends partly on the accuracy of your computer, but more on the shape of the table. For a perfectly round table, up to approximately 500 collision positions can be calculated with an error of about 0.1 percent. But if you change the shape of the table so that it becomes at least a little irregular (oval), and the unpredictability of the trajectory can exceed 90 degrees after just 10 collisions! The only way to get a picture of the general behavior of a billiard ball bouncing off a clean table is to depict the angle of bounce or arc length corresponding to each shot. Here are two successive magnifications of such a phase-spatial picture.

Each individual loop or scatter region represents the behavior of the ball resulting from one set of initial conditions. The area of ​​the picture that displays the results of one particular experiment is called the attractor area for a given set of initial conditions. As can be seen, the shape of the table used for these experiments is the main part of the attractor regions, which are repeated sequentially on a decreasing scale. Theoretically, such self-similarity should continue forever and if we enlarge the drawing more and more, we would get all the same shapes. This is called a very popular word today, fractal.

INTEGRATION OF DETERMINISTIC FRACTALS AND CHAOS

From the examples of deterministic fractals discussed above, we can see that they do not exhibit any chaotic behavior and that they are in fact very predictable. As you know, chaos theory uses a fractal to recreate or find patterns in order to predict the behavior of many systems in nature, such as, for example, the problem of bird migration.

Now let's see how this actually happens. Using a fractal called the Pythagorean Tree, not discussed here (which, by the way, was not invented by Pythagoras and has nothing to do with the Pythagorean theorem) and Brownian motion (which is chaotic), let's try to make an imitation of a real tree. The ordering of leaves and branches on a tree is quite complex and random and is probably not something simple enough that a short 12 line program can emulate.

First you need to generate a Pythagorean Tree (left). It is necessary to make the trunk thicker. At this stage, Brownian motion is not used. Instead, each line segment has now become a line of symmetry between the rectangle that becomes the trunk and the branches outside.

The Scottish botanist Robert Brown (sometimes his last name is transcribed as Brown) during his lifetime, as the best plant expert, received the title “Prince of Botanists.” He made many wonderful discoveries. In 1805, after a four-year expedition to Australia, he brought to England about 4,000 species of Australian plants unknown to scientists and spent many years studying them. Described plants brought from Indonesia and Central Africa. He studied plant physiology and for the first time described in detail the nucleus of a plant cell. The St. Petersburg Academy of Sciences made him an honorary member. But the name of the scientist is now widely known not because of these works.

In 1827 Brown conducted research on plant pollen. He was particularly interested in how pollen participates in the process of fertilization. Once he looked under a microscope at pollen cells from a North American plant. Clarkia pulchella(pretty clarkia) elongated cytoplasmic grains suspended in water. Suddenly Brown saw that the smallest solid grains, which could barely be seen in a drop of water, were constantly trembling and moving from place to place. He found that these movements, in his words, “are not associated either with flows in the liquid or with its gradual evaporation, but are inherent in the particles themselves.”

Brown's observation was confirmed by other scientists. The smallest particles behaved as if they were alive, and the “dance” of the particles accelerated with increasing temperature and decreasing particle size and clearly slowed down when replacing water with a more viscous medium. This amazing phenomenon never stopped: it could be observed for as long as desired. At first, Brown even thought that living beings actually fell into the field of the microscope, especially since pollen is the male reproductive cells of plants, but there were also particles from dead plants, even from those dried a hundred years earlier in herbariums. Then Brown thought whether these were “elementary molecules of living beings”, about which the famous French naturalist Georges Buffon (1707–1788), author of a 36-volume book, spoke Natural history. This assumption fell away when Brown began to examine apparently inanimate objects; at first it was very small particles of coal, as well as soot and dust from the London air, then finely ground inorganic substances: glass, many different minerals. “Active molecules” were everywhere: “In every mineral,” wrote Brown, “which I have succeeded in pulverizing to such an extent that it can be suspended in water for some time, I have found, in greater or lesser quantities, these molecules."

It must be said that Brown did not have any of the latest microscopes. In his article, he specifically emphasizes that he had ordinary biconvex lenses, which he used for several years. And he goes on to say: “Throughout the entire study I continued to use the same lenses with which I began the work, in order to give more credibility to my statements and to make them as accessible as possible to ordinary observations.”

Now, to repeat Brown's observation, it is enough to have a not very strong microscope and use it to examine the smoke in a blackened box, illuminated through a side hole with a beam of intense light. In a gas, the phenomenon manifests itself much more clearly than in a liquid: small pieces of ash or soot (depending on the source of the smoke) are visible, scattering light, and continuously jumping back and forth.

As often happens in science, many years later historians discovered that back in 1670, the inventor of the microscope, the Dutchman Antonie Leeuwenhoek, apparently observed a similar phenomenon, but the rarity and imperfection of microscopes, the embryonic state of molecular science at that time did not attract attention to Leeuwenhoek’s observation, therefore the discovery is rightly attributed to Brown, who was the first to study and describe it in detail.

Brownian motion and atomic-molecular theory.

The phenomenon observed by Brown quickly became widely known. He himself showed his experiments to numerous colleagues (Brown lists two dozen names). But neither Brown himself nor many other scientists for many years could explain this mysterious phenomenon, which was called the “Brownian movement”. The movements of the particles were completely random: sketches of their positions made at different points in time (for example, every minute) did not at first glance make it possible to find any pattern in these movements.

An explanation of Brownian motion (as this phenomenon was called) by the movement of invisible molecules was given only in the last quarter of the 19th century, but was not immediately accepted by all scientists. In 1863, a teacher of descriptive geometry from Karlsruhe (Germany), Ludwig Christian Wiener (1826–1896), suggested that the phenomenon was associated with the oscillatory movements of invisible atoms. This was the first, although very far from modern, explanation of Brownian motion by the properties of the atoms and molecules themselves. It is important that Wiener saw the opportunity to use this phenomenon to penetrate the secrets of the structure of matter. He was the first to try to measure the speed of movement of Brownian particles and its dependence on their size. It is curious that in 1921 Reports of the US National Academy of Sciences A work was published on the Brownian motion of another Wiener - Norbert, the famous founder of cybernetics.

The ideas of L.K. Wiener were accepted and developed by a number of scientists - Sigmund Exner in Austria (and 33 years later - his son Felix), Giovanni Cantoni in Italy, Karl Wilhelm Negeli in Germany, Louis Georges Gouy in France, three Belgian priests - Jesuits Carbonelli, Delso and Tirion and others. Among these scientists was the later famous English physicist and chemist William Ramsay. It gradually became clear that the smallest grains of matter were being hit from all sides by even smaller particles, which were no longer visible through a microscope - just as waves rocking a distant boat are not visible from the shore, while the movements of the boat itself are visible quite clearly. As they wrote in one of the articles in 1877, “...the law of large numbers no longer reduces the effect of collisions to average uniform pressure; their resultant will no longer be equal to zero, but will continuously change its direction and its magnitude.”

Qualitatively, the picture was quite plausible and even visual. A small twig or a bug should move in approximately the same way, pushed (or pulled) in different directions by many ants. These smaller particles were actually in the vocabulary of scientists, but no one had ever seen them. They were called molecules; Translated from Latin, this word means “small mass.” Amazingly, this is exactly the explanation given to a similar phenomenon by the Roman philosopher Titus Lucretius Carus (c. 99–55 BC) in his famous poem About the nature of things. In it, he calls the smallest particles invisible to the eye the “primordial principles” of things.

The principles of things first move themselves,
Following them are bodies from their smallest combination,
Close, as it were, in strength to the primary principles,
Hidden from them, receiving shocks, they begin to strive,
Themselves to move, then encouraging larger bodies.
So, starting from the beginning, the movement little by little
It touches our feelings and becomes visible too
To us and in the specks of dust that move in the sunlight,
Even though the tremors from which it occurs are imperceptible...

Subsequently, it turned out that Lucretius was wrong: it is impossible to observe Brownian motion with the naked eye, and dust particles in a sunbeam that penetrated into a dark room “dance” due to vortex movements of the air. But outwardly both phenomena have some similarities. And only in the 19th century. It became obvious to many scientists that the movement of Brownian particles is caused by random impacts of the molecules of the medium. Moving molecules collide with dust particles and other solid particles that are in the water. The higher the temperature, the faster the movement. If a speck of dust is large, for example, has a size of 0.1 mm (the diameter is a million times larger than that of a water molecule), then many simultaneous impacts on it from all sides are mutually balanced and it practically does not “feel” them - approximately the same as a piece of wood the size of a plate will not “feel” the efforts of many ants that will pull or push it in different directions. If the dust particle is relatively small, it will move in one direction or the other under the influence of impacts from surrounding molecules.

Brownian particles have a size of the order of 0.1–1 μm, i.e. from one thousandth to one ten-thousandth of a millimeter, which is why Brown was able to discern their movement because he was looking at tiny cytoplasmic grains, and not the pollen itself (which is often mistakenly written about). The problem is that the pollen cells are too large. Thus, in meadow grass pollen, which is carried by the wind and causes allergic diseases in humans (hay fever), the cell size is usually in the range of 20 - 50 microns, i.e. they are too large to observe Brownian motion. It is also important to note that individual movements of a Brownian particle occur very often and over very short distances, so that it is impossible to see them, but under a microscope, movements that have occurred over a certain period of time are visible.

It would seem that the very fact of the existence of Brownian motion unambiguously proved the molecular structure of matter, but even at the beginning of the 20th century. There were scientists, including physicists and chemists, who did not believe in the existence of molecules. The atomic-molecular theory only slowly and with difficulty gained recognition. Thus, the leading French organic chemist Marcelin Berthelot (1827–1907) wrote: “The concept of a molecule, from the point of view of our knowledge, is uncertain, while another concept - an atom - is purely hypothetical.” The famous French chemist A. Saint-Clair Deville (1818–1881) spoke even more clearly: “I do not accept Avogadro’s law, nor an atom, nor a molecule, for I refuse to believe in what I can neither see nor observe.” And the German physical chemist Wilhelm Ostwald (1853–1932), Nobel Prize laureate, one of the founders of physical chemistry, back in the early 20th century. resolutely denied the existence of atoms. He managed to write a three-volume chemistry textbook in which the word “atom” is never even mentioned. Speaking on April 19, 1904, with a large report at the Royal Institution to members of the English Chemical Society, Ostwald tried to prove that atoms do not exist, and “what we call matter is only a collection of energies collected together in a given place.”

But even those physicists who accepted the molecular theory could not believe that the validity of the atomic-molecular theory was proved in such a simple way, so a variety of alternative reasons were put forward to explain the phenomenon. And this is quite in the spirit of science: until the cause of a phenomenon is unambiguously identified, it is possible (and even necessary) to assume various hypotheses, which should, if possible, be tested experimentally or theoretically. So, back in 1905, a short article by St. Petersburg physics professor N.A. Gezehus, teacher of the famous academician A.F. Ioffe, was published in the Brockhaus and Efron Encyclopedic Dictionary. Gesehus wrote that, according to some scientists, Brownian motion is caused by “light or heat rays passing through a liquid,” and boils down to “simple flows within a liquid that have nothing to do with the movements of molecules,” and these flows can be caused by “evaporation, diffusion and other reasons." After all, it was already known that a very similar movement of dust particles in the air is caused precisely by vortex flows. But the explanation given by Gesehus could easily be refuted experimentally: if you look at two Brownian particles located very close to each other through a strong microscope, their movements will turn out to be completely independent. If these movements were caused by any flows in the liquid, then such neighboring particles would move in concert.

Theory of Brownian motion.

At the beginning of the 20th century. most scientists understood the molecular nature of Brownian motion. But all explanations remained purely qualitative; no quantitative theory could withstand experimental testing. In addition, the experimental results themselves were unclear: the fantastic spectacle of non-stop rushing particles hypnotized the experimenters, and they did not know exactly what characteristics of the phenomenon needed to be measured.

Despite the apparent complete disorder, it was still possible to describe the random movements of Brownian particles by a mathematical relationship. For the first time, a rigorous explanation of Brownian motion was given in 1904 by the Polish physicist Marian Smoluchowski (1872–1917), who in those years worked at Lviv University. At the same time, the theory of this phenomenon was developed by Albert Einstein (1879–1955), a then little-known 2nd class expert at the Patent Office of the Swiss city of Bern. His article, published in May 1905 in the German journal Annalen der Physik, was entitled On the motion of particles suspended in a fluid at rest, required by the molecular kinetic theory of heat. With this name, Einstein wanted to show that the molecular kinetic theory of the structure of matter necessarily implies the existence of random motion of the smallest solid particles in liquids.

It is curious that at the very beginning of this article, Einstein writes that he is familiar with the phenomenon itself, albeit superficially: “It is possible that the movements in question are identical with the so-called Brownian molecular motion, but the data available to me regarding the latter are so inaccurate that I could not formulate a this is a definite opinion.” And decades later, already in his late life, Einstein wrote something different in his memoirs - that he did not know about Brownian motion at all and actually “rediscovered” it purely theoretically: “Not knowing that observations of “Brownian motion” have long been known, I discovered that the atomic theory leads to the existence of observable motion of microscopic suspended particles." Be that as it may, Einstein's theoretical article ended with a direct call to experimenters to test his conclusions experimentally: "If any researcher could soon answer the questions raised here questions!" – he ends his article with such an unusual exclamation.

The answer to Einstein's passionate appeal was not long in coming.

According to the Smoluchowski-Einstein theory, the average value of the squared displacement of a Brownian particle ( s 2) for time t directly proportional to temperature T and inversely proportional to the liquid viscosity h, particle size r and Avogadro's constant

N A: s 2 = 2RTt/6ph rN A,

Where R– gas constant. So, if in 1 minute a particle with a diameter of 1 μm moves by 10 μm, then in 9 minutes - by 10 = 30 μm, in 25 minutes - by 10 = 50 μm, etc. Under similar conditions, a particle with a diameter of 0.25 μm over the same periods of time (1, 9 and 25 min) will move by 20, 60 and 100 μm, respectively, since = 2. It is important that the above formula includes Avogadro’s constant, which thus , can be determined by quantitative measurements of the movement of a Brownian particle, which was done by the French physicist Jean Baptiste Perrin (1870–1942).

In 1908, Perrin began quantitative observations of the motion of Brownian particles under a microscope. He used an ultramicroscope, invented in 1902, which made it possible to detect the smallest particles by scattering light onto them from a powerful side illuminator. Perrin obtained tiny balls of almost spherical shape and approximately the same size from gum, the condensed sap of some tropical trees (it is also used as yellow watercolor paint). These tiny beads were suspended in glycerol containing 12% water; the viscous liquid prevented the appearance of internal flows in it that would blur the picture. Armed with a stopwatch, Perrin noted and then sketched (of course, on a greatly enlarged scale) on a graphed sheet of paper the position of the particles at regular intervals, for example, every half minute. By connecting the resulting points with straight lines, he obtained intricate trajectories, some of them are shown in the figure (they are taken from Perrin’s book Atoms, published in 1920 in Paris). Such a chaotic, disorderly movement of particles leads to the fact that they move in space quite slowly: the sum of the segments is much greater than the displacement of the particle from the first point to the last.

Consecutive positions every 30 seconds of three Brownian particles - gum balls with a size of about 1 micron. One cell corresponds to a distance of 3 µm. If Perrin could determine the position of Brownian particles not after 30, but after 3 seconds, then the straight lines between each neighboring points would turn into the same complex zigzag broken line, only on a smaller scale.

Using the theoretical formula and his results, Perrin obtained a value for Avogadro’s number that was quite accurate for that time: 6.8 . 10 23 . Perrin also used a microscope to study the vertical distribution of Brownian particles ( cm. AVOGADRO'S LAW) and showed that, despite the action of gravity, they remain suspended in solution. Perrin also owns other important works. In 1895, he proved that cathode rays are negative electric charges (electrons), and in 1901 he first proposed a planetary model of the atom. In 1926 he was awarded the Nobel Prize in Physics.

The results obtained by Perrin confirmed Einstein's theoretical conclusions. It made a strong impression. As the American physicist A. Pais wrote many years later, “you never cease to be amazed at this result, obtained in such a simple way: it is enough to prepare a suspension of balls, the size of which is large compared to the size of simple molecules, take a stopwatch and a microscope, and you can determine Avogadro’s constant!” One might also be surprised: descriptions of new experiments on Brownian motion still appear in scientific journals (Nature, Science, Journal of Chemical Education) from time to time! After the publication of Perrin’s results, Ostwald, a former opponent of atomism, admitted that “the coincidence of Brownian motion with the requirements of the kinetic hypothesis... now gives the most cautious scientist the right to talk about experimental proof of the atomic theory of matter. Thus, the atomic theory has been elevated to the rank of a scientific, well-founded theory.” He is echoed by the French mathematician and physicist Henri Poincaré: “The brilliant determination of the number of atoms by Perrin completed the triumph of atomism... The atom of chemists has now become a reality.”

Brownian motion and diffusion.

The movement of Brownian particles is very similar in appearance to the movement of individual molecules as a result of their thermal motion. This movement is called diffusion. Even before the work of Smoluchowski and Einstein, the laws of molecular motion were established in the simplest case of the gaseous state of matter. It turned out that molecules in gases move very quickly - at the speed of a bullet, but they cannot fly far, since they very often collide with other molecules. For example, oxygen and nitrogen molecules in the air, moving at an average speed of approximately 500 m/s, experience more than a billion collisions every second. Therefore, the path of the molecule, if it were possible to follow it, would be a complex broken line. Brownian particles also describe a similar trajectory if their position is recorded at certain time intervals. Both diffusion and Brownian motion are a consequence of the chaotic thermal motion of molecules and are therefore described by similar mathematical relationships. The difference is that molecules in gases move in a straight line until they collide with other molecules, after which they change direction. A Brownian particle, unlike a molecule, does not perform any “free flights”, but experiences very frequent small and irregular “jitters”, as a result of which it chaotically shifts in one direction or the other. Calculations have shown that for a particle 0.1 µm in size, one movement occurs in three billionths of a second over a distance of only 0.5 nm (1 nm = 0.001 µm). As one author aptly puts it, this is reminiscent of moving an empty beer can in a square where a crowd of people has gathered.

Diffusion is much easier to observe than Brownian motion, since it does not require a microscope: movements are observed not of individual particles, but of their huge masses, you just need to ensure that diffusion is not superimposed by convection - mixing of matter as a result of vortex flows (such flows are easy to notice, placing a drop of a colored solution, such as ink, into a glass of hot water).

Diffusion is convenient to observe in thick gels. Such a gel can be prepared, for example, in a penicillin jar by preparing a 4–5% gelatin solution in it. The gelatin must first swell for several hours, and then it is completely dissolved with stirring by lowering the jar into hot water. After cooling, a non-flowing gel is obtained in the form of a transparent, slightly cloudy mass. If, using sharp tweezers, you carefully insert a small crystal of potassium permanganate (“potassium permanganate”) into the center of this mass, the crystal will remain hanging in the place where it was left, since the gel prevents it from falling. Within a few minutes, a violet-colored ball will begin to grow around the crystal; over time, it becomes larger and larger until the walls of the jar distort its shape. The same result can be obtained using a crystal of copper sulfate, only in this case the ball will turn out not purple, but blue.

It’s clear why the ball turned out: MnO 4 – ions formed when the crystal dissolves, go into solution (the gel is mainly water) and, as a result of diffusion, move evenly in all directions, while gravity has virtually no effect on the diffusion rate. Diffusion in liquid is very slow: it will take many hours for the ball to grow several centimeters. In gases, diffusion is much faster, but still, if the air were not mixed, the smell of perfume or ammonia would spread in the room for hours.

Brownian motion theory: random walks.

The Smoluchowski–Einstein theory explains the laws of both diffusion and Brownian motion. We can consider these patterns using the example of diffusion. If the speed of the molecule is u, then, moving in a straight line, in time t will go the distance L = ut, but due to collisions with other molecules, this molecule does not move in a straight line, but continuously changes the direction of its movement. If it were possible to sketch the path of a molecule, it would be fundamentally no different from the drawings obtained by Perrin. From these figures it is clear that due to chaotic movement the molecule is displaced by a distance s, significantly less than L. These quantities are related by the relation s= , where l is the distance that a molecule flies from one collision to another, the mean free path. Measurements have shown that for air molecules at normal atmospheric pressure l ~ 0.1 μm, which means that at a speed of 500 m/s a nitrogen or oxygen molecule will fly the distance in 10,000 seconds (less than three hours) L= 5000 km, and will shift from the original position by only s= 0.7 m (70 cm), which is why substances move so slowly due to diffusion, even in gases.

The path of a molecule as a result of diffusion (or the path of a Brownian particle) is called a random walk. Witty physicists reinterpreted this expression as drunkard's walk - “the path of a drunkard.” Indeed, the movement of a particle from one position to another (or the path of a molecule undergoing many collisions) resembles the movement of a drunk person. Moreover, this analogy also allows one to deduce quite simply the basic equation of such a process is based on the example of one-dimensional motion, which is easy to generalize to three-dimensional.

Suppose a tipsy sailor came out of a tavern late at night and headed along the street. Having walked the path l to the nearest lantern, he rested and went... either further, to the next lantern, or back, to the tavern - after all, he does not remember where he came from. The question is, will he ever leave the zucchini, or will he just wander around it, now moving away, now approaching it? (Another version of the problem states that there are dirty ditches at both ends of the street, where the streetlights end, and asks whether the sailor will be able to avoid falling into one of them.) Intuitively, it seems that the second answer is correct. But it is incorrect: it turns out that the sailor will gradually move further and further away from the zero point, although much more slowly than if he walked only in one direction. Here's how to prove it.

Having passed the first time to the nearest lamp (to the right or to the left), the sailor will be at a distance s 1 = ± l from the starting point. Since we are only interested in its distance from this point, but not its direction, we will get rid of the signs by squaring this expression: s 1 2 = l 2. After some time, the sailor, having already completed N"wandering", will be at a distance

s N= from the beginning. And having walked again (in one direction) to the nearest lantern, at a distance s N+1 = s N± l, or, using the square of the displacement, s 2 N+1 = s 2 N± 2 s N l + l 2. If the sailor repeats this movement many times (from N before N+ 1), then as a result of averaging (it passes with equal probability N th step to the right or left), term ± 2 s N l will cancel, so s 2 N+1 = s2 N+ l 2> (angle brackets indicate the average value). L = 3600 m = 3.6 km, while the displacement from the zero point for the same time will be equal to only s= = 190 m. In three hours it will pass L= 10.8 km, and will shift by s= 330 m, etc.

Work u l in the resulting formula can be compared with the diffusion coefficient, which, as shown by the Irish physicist and mathematician George Gabriel Stokes (1819–1903), depends on the particle size and the viscosity of the medium. Based on similar considerations, Einstein derived his equation.

The theory of Brownian motion in real life.

The theory of random walks has important practical applications. They say that in the absence of landmarks (the sun, stars, the noise of a highway or railroad, etc.), a person wanders in the forest, across a field in a snowstorm or in thick fog in circles, always returning to his original place. In fact, he does not walk in circles, but approximately the same way molecules or Brownian particles move. He can return to his original place, but only by chance. But he crosses his path many times. They also say that people frozen in a snowstorm were found “some kilometer” from the nearest housing or road, but in reality the person had no chance of walking this kilometer, and here’s why.

To calculate how much a person will shift as a result of random walks, you need to know the value of l, i.e. the distance a person can walk in a straight line without any landmarks. This value was measured by Doctor of Geological and Mineralogical Sciences B.S. Gorobets with the help of student volunteers. He, of course, did not leave them in a dense forest or on a snow-covered field, everything was simpler - the student was placed in the center of an empty stadium, blindfolded and asked to walk to the end of the football field in complete silence (to exclude orientation by sounds). It turned out that on average the student walked in a straight line for only about 20 meters (the deviation from the ideal straight line did not exceed 5°), and then began to deviate more and more from the original direction. In the end, he stopped, far from reaching the edge.

Let now a person walk (or rather, wander) in the forest at a speed of 2 kilometers per hour (for a road this is very slow, but for a dense forest it is very fast), then if the value of l is 20 meters, then in an hour he will cover 2 km, but will move only 200 m, in two hours - about 280 m, in three hours - 350 m, in 4 hours - 400 m, etc. And moving in a straight line at such a speed, a person would walk 8 kilometers in 4 hours , therefore, in the safety instructions for field work there is the following rule: if landmarks are lost, you need to stay in place, set up a shelter and wait for the end of bad weather (the sun may come out) or for help. In the forest, landmarks - trees or bushes - will help you move in a straight line, and each time you need to stick to two such landmarks - one in front, the other behind. But, of course, it is best to take a compass with you...

Ilya Leenson

Literature:

Mario Liozzi. History of physics. M., Mir, 1970
Kerker M. Brownian Movements and Molecular Reality Prior to 1900. Journal of Chemical Education, 1974, vol. 51, No. 12
Leenson I.A. Chemical reactions. M., Astrel, 2002



Thermal movement

Any substance consists of tiny particles - molecules. Molecule- this is the smallest particle of a given substance that retains all its chemical properties. Molecules are located discretely in space, i.e. at certain distances from each other, and are in a state of continuous disorderly (chaotic) movement .

Since bodies consist of a large number of molecules and the movement of molecules is random, it is impossible to say exactly how many impacts one or another molecule will experience from others. Therefore, they say that the position of the molecule and its speed at each moment of time are random. However, this does not mean that the movement of molecules does not obey certain laws. In particular, although the speeds of molecules at some point in time are different, most of them have speed values ​​​​close to some specific value. Usually, when speaking about the speed of movement of molecules, they mean average speed (v$cp).

It is impossible to single out any specific direction in which all molecules move. The movement of molecules never stops. We can say that it is continuous. Such continuous chaotic movement of atoms and molecules is called -. This name is determined by the fact that the speed of movement of molecules depends on body temperature. The higher the average speed of movement of the molecules of a body, the higher its temperature. Conversely, the higher the body temperature, the greater the average speed of molecular movement.

The movement of liquid molecules was discovered by observing Brownian motion - the movement of very small particles of solid matter suspended in it. Each particle continuously makes abrupt movements in arbitrary directions, describing trajectories in the form of a broken line. This behavior of particles can be explained by considering that they experience impacts from liquid molecules simultaneously from different sides. The difference in the number of these impacts from opposite directions leads to the movement of the particle, since its mass is commensurate with the masses of the molecules themselves. The movement of such particles was first discovered in 1827 by the English botanist Brown, observing pollen particles in water under a microscope, which is why it was called - Brownian motion.

Brownian motion- random movement of microscopic visible particles of a solid substance suspended in a liquid or gas, caused by the thermal movement of the particles of the liquid or gas. Brownian motion never stops. Brownian motion is related to thermal motion, but these concepts should not be confused. Brownian motion is a consequence and evidence of the existence of thermal motion.

Brownian motion is the most clear experimental confirmation of the concepts of molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation period is large enough for the forces acting on the particle from the molecules of the medium to change their direction many times, then the average square of the projection of its displacement on any axis (in the absence of other external forces) is proportional to time.

When deriving Einstein's law, it is assumed that particle displacements in any direction are equally probable and that the inertia of a Brownian particle can be neglected compared to the influence of friction forces (this is acceptable for sufficiently long times). Formula for coefficient D is based on the application of Stokes' law for hydrodynamic resistance to the motion of a sphere of radius A in a viscous fluid. The relationships for A and D were experimentally confirmed by measurements by J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant was experimentally determined k and Avogadro's constant N A. In addition to translational Brownian motion, there is also rotational Brownian motion - the random rotation of a Brownian particle under the influence of impacts of molecules of the medium. For rotational Brownian motion, the root mean square angular displacement of the particle is proportional to the observation time. These relationships were also confirmed by Perrin's experiments, although this effect is much more difficult to observe than translational Brownian motion.

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  Brownian motion occurs due to the fact that all liquids and gases consist of atoms or molecules - tiny particles that are in constant chaotic thermal motion, and therefore continuously push the Brownian particle from different directions. It was found that large particles with sizes greater than 5 µm practically do not participate in Brownian motion (they are stationary or sediment), smaller particles (less than 3 µm) move forward along very complex trajectories or rotate. When a large body is immersed in a medium, the shocks occurring in huge quantities are averaged and form a constant pressure. If a large body is surrounded by a medium on all sides, then the pressure is practically balanced, only the lifting force of Archimedes remains - such a body smoothly floats up or sinks. If the body is small, like a Brownian particle, then pressure fluctuations become noticeable, which create a noticeable randomly varying force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in the medium.

  Opening

  Brownian motion theory

  Construction of the classical theory

  D = R T 6 N A π a ξ , (\displaystyle D=(\frac (RT)(6N_(A)\pi a\xi )),)

  Where D (\displaystyle D)- diffusion coefficient, R (\displaystyle R)- universal gas constant, T (\displaystyle T)- absolute temperature, N A (\displaystyle N_(A))- Avogadro's constant, a (\displaystyle a)- particle radius, ξ (\displaystyle \xi )- dynamic viscosity.

  Experimental confirmation

  Einstein's formula was confirmed by the experiments of Jean Perrin and his students in 1908-1909. As Brownian particles, they used grains of resin from the mastic tree and gum, the thick milky sap of trees of the genus Garcinia. The validity of the formula was established for various particle sizes - from 0.212 microns to 5.5 microns, for various solutions (sugar solution, glycerin) in which the particles moved.

  Brownian motion as a non-Markov random process

  The theory of Brownian motion, well developed over the last century, is an approximate one. And although in most practically important cases the existing theory gives satisfactory results, in some cases it may require clarification. Thus, experimental work carried out at the beginning of the 21st century at the Polytechnic University of Lausanne, the University of Texas and the European Molecular Biological Laboratory in Heidelberg (under the leadership of S. Jeney) showed the difference in the behavior of the Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable when increasing particle sizes. The studies also touched upon the analysis of the movement of surrounding particles of the medium and showed a significant mutual influence of the movement of the Brownian particle and the movement of the particles of the medium caused by it on each other, that is, the presence of “memory” of the Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory her past behavior. This fact was not taken into account in the Einstein-Smoluchowski theory.

  The process of Brownian motion of a particle in a viscous medium, generally speaking, belongs to the class of non-Markov processes, and for a more accurate description it is necessary to use integral stochastic equations.