Brownian motion explained. Brownian motion: definition

Table 13

Comparison of Brownian motion intensity and silver particle settling rate (Burton's calculation)

If the dispersed phase settles to the bottom of the vessel or floats to the surface in a relatively short time, the system is called kinetically unstable. An example is a suspension of sand in water.

If the particles are small enough and Brownian motion prevents them from settling completely, the system is said to be kinetically stable.

Due to random Brownian motion in a kinetically stable disperse system, an unequal distribution of particles in height along the action of gravity is established. The nature of the distribution is described by the equation:

where with 1 h 1 ;since 2- concentration of particles at height h2; t- mass of particles; d- their density; D 0 - density of the dispersion medium. With the help of this equation, the most important constant of the molecular kinetic theory was determined for the first time -. Avogadro's number N 0 . Having counted under a microscope the number of particles of gummigut suspended in water at various levels, J. Perrin obtained the numerical value of the constant N 0 , which varied in different experiments from 6.5 10 23 to 7.2 10 23 . According to modern data, Avogadro's number is 6.02 10 23 .

Currently, when the constant N 0 known to be very accurate, counting particles at various levels is used to find their size and mass.

Article on Brownian motion

Today we will consider an important topic in detail - we will define the Brownian motion of small pieces of matter in a liquid or gas.

Map and coordinates

Some schoolchildren, tormented by boring lessons, do not understand why they should study physics. Meanwhile, it was this science that once made it possible to discover America!

Let's start from afar. In a sense, the ancient civilizations of the Mediterranean were lucky: they developed on the shores of a closed inland reservoir. The Mediterranean Sea is called so because it is surrounded on all sides by land. And ancient travelers could advance quite far with their expedition without losing sight of the shores. The outlines of the land helped to navigate. And the first maps were drawn more descriptively than geographically. Thanks to these relatively short voyages, the Greeks, Phoenicians and Egyptians learned how to build ships well. And where the best equipment is, there is the desire to push the boundaries of your world.

Therefore, one fine day, the European powers decided to go out into the ocean. While sailing through the vast expanses between the continents, sailors saw only water for many months, and they had to somehow navigate. The invention of an accurate watch and a high-quality compass helped determine their coordinates.

Clock and compass

The invention of small hand-held chronometers helped navigators a lot. To determine exactly where they were, they needed to have a simple instrument that measured the height of the sun above the horizon, and know exactly when it was noon. And thanks to the compass, the captains of the ships knew where they were going. Both the clock and the properties of the magnetic needle were studied and created by physicists. Thanks to this, the whole world was opened to Europeans.

The new continents were terra incognita, uncharted lands. Strange plants grew on them and incomprehensible animals were found.

Plants and physics

All the natural scientists of the civilized world rushed to study these strange new ecological systems. And of course, they wanted to take advantage of them.

Robert Brown was an English botanist. He made trips to Australia and Tasmania, collecting plant collections there. Already at home, in England, he worked hard on the description and classification of the brought material. And this scientist was very meticulous. Once, while observing the movement of pollen in plant sap, he noticed that small particles constantly make chaotic zigzag movements. This is the definition of the Brownian motion of small elements in gases and liquids. Thanks to the discovery, the amazing botanist wrote his name into the history of physics!

Brown and Gooey

In European science, it is customary to name an effect or phenomenon by the name of the one who discovered it. But often it happens by accident. But a person who describes, discovers the importance, or explores a physical law in more detail, finds himself in the shadows. So it happened with the Frenchman Louis Georges Gui. It was he who gave the definition of Brownian motion (Grade 7 definitely does not hear about him when he studies this topic in physics).

Gouy's research and properties of Brownian motion

The French experimenter Louis Georges Gouy observed the movement of various types of particles in several liquids, including solutions. The science of that time already knew how to accurately determine the size of pieces of matter up to tenths of a micrometer. Exploring what Brownian motion is (it was Gouy who gave the definition in physics to this phenomenon), the scientist realized that the intensity of the movement of particles increases if they are placed in a less viscous medium. Being a broad-spectrum experimenter, he exposed the suspension to the action of light and electromagnetic fields of various powers. The scientist found that these factors do not affect the chaotic zigzag jumps of particles. Gouy unequivocally showed what Brownian motion proves: the thermal movement of the molecules of a liquid or gas.

Collective and mass

And now we will describe in more detail the mechanism of zigzag jumps of small pieces of matter in a liquid.

Any substance is made up of atoms or molecules. These elements of the world are very small, not a single optical microscope is able to see them. In a liquid, they vibrate and move all the time. When any visible particle enters the solution, its mass is thousands of times greater than one atom. The Brownian motion of liquid molecules occurs randomly. But nevertheless, all atoms or molecules are a collective, they are connected to each other, like people who join hands. Therefore, sometimes it happens that the atoms of the liquid on one side of the particle move in such a way that they "press" on it, while on the other side of the particle a less dense medium is created. Therefore, the dust particle moves in the space of the solution. Elsewhere, the collective motion of fluid molecules randomly acts on the other side of the more massive component. This is exactly how the Brownian motion of particles takes place.

Time and Einstein

If a substance has a non-zero temperature, its atoms perform thermal vibrations. Therefore, even in a very cold or supercooled liquid, Brownian motion exists. These chaotic jumps of small suspended particles never stop.

Albert Einstein is perhaps the most famous scientist of the twentieth century. Everyone who is at least somewhat interested in physics knows the formula E = mc 2 . Also, many may recall the photoelectric effect, for which he was given the Nobel Prize, and the special theory of relativity. But few people know that Einstein developed the formula for Brownian motion.

Based on the molecular kinetic theory, the scientist derived the diffusion coefficient of suspended particles in a liquid. And it happened in 1905. The formula looks like this:

D = (R * T) / (6 * N A * a * π * ξ),

where D is the desired coefficient, R is the universal gas constant, T is the absolute temperature (expressed in Kelvin), N A is the Avogadro constant (corresponding to one mole of a substance, or about 10 23 molecules), a is the approximate average particle radius, ξ is the dynamic viscosity of a liquid or solution.

And already in 1908, the French physicist Jean Perrin and his students experimentally proved the correctness of Einstein's calculations.

One particle in the warrior field

Above, we described the collective action of the medium on many particles. But even one foreign element in a liquid can give some regularities and dependencies. For example, if you observe a Brownian particle for a long time, then you can fix all its movements. And out of this chaos, a coherent system will emerge. The average advance of a Brownian particle along any one direction is proportional to time.

During experiments on a particle in a liquid, the following quantities were refined:

• Boltzmann's constant;
• Avogadro's number.

In addition to linear motion, chaotic rotation is also characteristic. And the average angular displacement is also proportional to the observation time.

Sizes and shapes

After such reasoning, a logical question may arise: why is this effect not observed for large bodies? Because when the length of an object immersed in a liquid is greater than a certain value, then all these random collective “shocks” of molecules turn into constant pressure, as they are averaged. And the general Archimedes is already acting on the body. Thus, a large piece of iron sinks, and metal dust floats in the water.

The particle size, on the example of which the fluctuation of liquid molecules is revealed, should not exceed 5 micrometers. As for objects with large sizes, this effect will not be noticeable here.

BROWNIAN MOTION(Brownian motion) - chaotic movement of small particles suspended in a liquid or gas, occurring under the influence of impacts of environmental molecules. Investigated in 1827 by P. Brown (Brown; R. Brown), to-ry observed in the microscope the movement of pollen suspended in water. Observed particles (Brownian) with a size of ~1 μm and less make disordered independent movements, describing complex zigzag trajectories. The intensity of B. d. does not depend on time, but increases with an increase in the temperature of the medium, a decrease in its viscosity and particle size (regardless of their chemical nature). The complete theory of B. d. was given by A. Einstein and M. Smoluchowski in 1905-06.

The causes of B. D. are the thermal motion of the molecules of the medium and the absence of exact compensation for the impacts experienced by the particle from the molecules surrounding it, i.e., B. D. is due to fluctuations pressure. Impacts of the molecules of the medium lead the particle into random motion: its speed rapidly changes in magnitude and direction. If the position of the particles is fixed at small equal time intervals, then the trajectory constructed by this method turns out to be extremely complex and confusing (Fig.).

B. d. - Naib. visual experiment. confirmation of representations molecular-kinetic. theories about chaos. thermal motion of atoms and molecules. If the observation interval t is large enough so that the forces acting on the particle from the molecules of the medium change their direction many times, then cf. the square of the projection of its displacement on to-l. axis (in the absence of other external forces) is proportional to time t (Einstein's law):

where D- coefficient diffusion of a Brownian particle. For spherical particle radius a: (T- abs. temp-ra, - dynamic. medium viscosity). 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 effect of friction forces (this is acceptable for sufficiently large ones). F-la for the coefficient. D based on the application stokes law for hydrodynamic resistance to the movement of a sphere with a radius a in a viscous liquid. Relations for and D were experimentally confirmed by the measurements of J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant is experimentally determined k and Avogadro constant N A.

In addition to translational B. D., there is also rotational B. D. - Random rotation of a Brownian particle under the influence of impacts of the molecules of the medium. For rotating B. d. cf. quadratic angular displacement of the particle is proportional to the observation time

where D vp - coefficient. diffusion rotate. B. d., equal to spherical. particles: . These ratios were also confirmed by the experiments of Perrin, although this effect is much more difficult to observe than the progressive B. d.

The theory of B. D. proceeds from the concept of the motion of a particle under the influence of a "random" generalized force f(<), к-рая описывает влияние ударов молекул и в среднем равна нулю, систематич. внеш. силы X, which may depend on time, and the friction force - arising when a particle moves in a medium with a speed of . Equation of random motion of a Brownian particle - Langevin equation- looks like:

where t is the mass of the particle (or, if X- angle, its moment of inertia), h- coefficient friction during the motion of a particle in a medium. For sufficiently large time intervals, the inertia of the particle (i.e., the term) can be neglected and, by integrating the Langevin equation, provided that cf. the product of impulses of a random force for non-overlapping time intervals is equal to zero, find cf. fluctuation squared, i.e., derive Einstein's relation. In a more general problem of the theory of particle dynamics, the sequence of values ​​of the coordinates and momenta of particles at regular intervals is considered as Markov random process, which is another formulation of the assumption about the independence of shocks experienced by particles in different non-overlapping time intervals. In this case, the probability of the state X in the moment t completely determined by the probability of the state x0 in the moment t0 and you can introduce a function - the probability density of the transition from the state x0 in a state for which X lies within x, x+dx at the time t. The probability density satisfies the integral equation of Smoluchovsky, which expresses the absence of "memory" of the beginning. state for a random Markov process. This equation for many problems in the theory of B. d. can be reduced to a dif. Fokker - Planck equation in partial derivatives - to the generalized equation of diffusion in phase space. Therefore, the solution of problems in the theory of b. border and early conditions. Mat. model B. d. is Wiener random process.

Brownian motion of three particles of gum in water (according to Perrin). The dots mark the positions of the particles every 30 s. The particle radius is 0.52 µm, the distance between grid divisions is 3.4 µm.

Brownian motion - Random movement of microscopic particles of a solid substance, visible, suspended in a liquid or gas, caused by the thermal movement of particles of a 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 obvious experimental confirmation of the ideas of the molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation interval is large enough so that the forces acting on the particle from the molecules of the medium 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). The formula for the coefficient D is based on the application of Stokes' law for the hydrodynamic resistance to the motion of a sphere of radius a in a viscous fluid. The relationships for and D were experimentally confirmed by the measurements of J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant k and the Avogadro constant NA are experimentally determined. In addition to the translational Brownian motion, there is also a rotational Brownian motion - random rotation of a Brownian particle under the influence of impacts of the molecules of the medium. For rotational Brownian motion, the rms angular displacement of a 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.

The essence of the phenomenon

Brownian motion occurs due to the fact that all liquids and gases consist of atoms or molecules - the smallest particles that are in constant chaotic thermal motion, and therefore continuously push the Brownian particle from different sides. It was found that large particles larger than 5 µm practically do not participate in Brownian motion (they are immobile or sediment), smaller particles (less than 3 µm) move progressively along very complex trajectories or rotate. When a large body is immersed in the medium, the shocks that occur in large numbers 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 changing force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in a medium.

Brownian motion theory

In 1905, Albert Einstein created a molecular kinetic theory for a quantitative description of Brownian motion. In particular, he derived a formula for the diffusion coefficient of spherical Brownian particles:

where D- diffusion coefficient, R is the universal gas constant, T is the absolute temperature, N A is the Avogadro constant, a- particle radius, ξ - dynamic viscosity.

Brownian motion as non-Markovian
random process

The theory of Brownian motion, well developed over the last century, is approximate. And although in most cases of practical importance 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 Biology Laboratory in Heidelberg (under the direction of S. Dzheney) showed the difference in the behavior of a Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable when increase in particle size. The studies also touched upon the analysis of the movement of the 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 a "memory" in the Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory her behavior in the past. 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 its more accurate description it is necessary to use integral stochastic equations.

The Scottish botanist Robert Brown (sometimes his surname is transcribed as Brown) during his lifetime, as the best connoisseur of plants, received the title of "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. Studied plant physiology, first described in detail the nucleus of a plant cell. 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, in particular, was interested in how pollen is involved in the process of fertilization. Once he looked under a microscope isolated from pollen cells of a North American plant Clarkia pulchella(Pretty Clarkia) elongated cytoplasmic grains suspended in water. Suddenly, Brown saw that the smallest hard grains, which could hardly 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 water was replaced by a more viscous medium. This amazing phenomenon never stopped: it could be observed for an arbitrarily long time. At first, Brown even thought that living creatures really got into the field of the microscope, especially since pollen is the male germ cells of plants, but particles from dead plants, even from those dried a hundred years earlier in herbariums, also led. Then Brown thought if these were the “elementary molecules of living beings”, which the famous French naturalist Georges Buffon (1707–1788), the author of the 36-volume natural history. This assumption fell away when Brown began to explore 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,” Brown wrote, “which I managed to grind into dust to such an extent that it could be suspended in water for some time, I found, in greater or lesser quantities, these molecules.

I must say 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 further writes: "Throughout the study, I continued to use the same lenses with which I began work, in order to give more persuasiveness to my statements and to make them as accessible as possible to ordinary observations."

Now, in order 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 vividly than in a liquid: small patches of ash or soot (depending on the source of the smoke) are visible scattering light, which continuously jump back and forth.

As is often the case in science, many years later, historians discovered that back in 1670, the inventor of the microscope, the Dutchman Anthony 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 first studied and described 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 nor many other scientists could explain this mysterious phenomenon, which was called "Brownian motion", for many years. The movements of the particles were completely random: sketches of their positions made at different points in time (for example, every minute) did not give at first glance any possibility of finding any regularity in these movements.

The explanation of Brownian motion (as this phenomenon was called) by the motion of invisible molecules was given only in the last quarter of the 19th century, but was not immediately accepted by all scientists. In 1863, Ludwig Christian Wiener (1826–1896), a teacher of descriptive geometry from Karlsruhe (Germany), suggested that the phenomenon is 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 an opportunity to penetrate the secrets of the structure of matter with the help of this phenomenon. He first tried to measure the speed of movement of Brownian particles and its dependence on their size. Curiously, in 1921 Reports of the US National Academy of Sciences The work on the Brownian motion of another Wiener, Norbert, the famous founder of cybernetics, was published.

The ideas of L.K. Wiener were accepted and developed by a number of scientists - Sigmund Exner in Austria (and 33 years later - and his son Felix), Giovanni Cantoni in Italy, Carl Wilhelm Negeli in Germany, Louis Georges Gui in France, three Belgian priests - the Jesuits Carbonelli, Delso and Tirion and others. Among these scientists was the later famous English physicist and chemist William Ramsay. Gradually it became clear that the smallest grains of matter are hit from all sides by even smaller particles, which are no longer visible in the microscope - just as the waves rocking a distant boat are not visible from the shore, while the movements of the boat itself are quite clearly visible. As they wrote in one of the articles in 1877, "... the law of large numbers now does not reduce the effect of collisions to an 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 bug should move in approximately the same way, which are pushed (or pulled) in different directions by many ants. These smaller particles were actually in the lexicon of scientists, only no one had ever seen them. They called them 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 Car (c. 99–55 BC) in his famous poem On the nature of things. In it, he calls the smallest particles invisible to the eye the “primordial principles” of things.

The origin of things first move themselves,
Behind them are bodies from their smallest combination,
Close, how to say, in strength to the beginnings of the primary,
Hidden from them, receiving pushes, they begin to strive,
Themselves to the movement then prompting the larger body.
So, starting from the beginning, the movement little by little
Our feelings touches, and becomes visible as well
To us and in the dust particles it is that move in the sunlight,
Though imperceptible shocks from which it occurs ...

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 a dark room “dance” due to the vortex movements of 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 grain of dust is large, for example, has a size of 0.1 mm (a million times larger than a water molecule), then many simultaneous impacts on it from all sides are mutually balanced and it practically does not “feel” them - about 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, on the other hand, a grain of dust is relatively small, it will move first in one direction, then in the other, under the influence of the impacts of the 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, that he examined tiny cytoplasmic grains, and not the pollen itself (which is often mistakenly reported). The fact 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 small 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 gained recognition only slowly and with difficulty. So, the largest French organic chemist Marcelin Berthelot (1827-1907) wrote: "The concept of a molecule, from the point of view of our knowledge, is indefinite, while another concept - an atom - is purely hypothetical." The well-known French chemist A. Saint-Clair Deville (1818–1881) spoke even more clearly: “I do not allow either Avogadro’s law, or an atom, or a molecule, because I refuse to believe in what I can neither see nor observe.” And the German physical chemist Wilhelm Ostwald (1853–1932), Nobel Prize winner, one of the founders of physical chemistry, back in the early 20th century. strongly denied the existence of atoms. He managed to write a three-volume chemistry textbook in which the word "atom" is never even mentioned. Speaking April 19, 1904 with a big report at the Royal Institute 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 gathered together in a given place."

But even those physicists who accepted the molecular theory could not believe that the truth of the atomic-molecular doctrine 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 verified experimentally or theoretically. So, back in 1905, a small article was published in the Encyclopedic Dictionary of Brockhaus and Efron by the St. Petersburg professor of physics N.A. Gezehus, teacher of the famous academician A.F. Ioffe. Gezehus wrote that, according to some scientists, Brownian motion is caused by "light or heat rays passing through the liquid", is reduced to "simple flows within the liquid, which 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 Gezehus could easily be refuted experimentally: if two Brownian particles that are very close to each other are examined through a strong microscope, then 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 verification. In addition, the experimental results themselves were indistinct: the fantastic spectacle of non-stop rushing particles hypnotized the experimenters, and they did not know exactly what characteristics of the phenomenon should be measured.

Despite the apparent complete disorder, it was still possible to describe the random movements of Brownian particles by mathematical dependence. The first 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 little-known expert of the 2nd class at the Patent Office of the Swiss city of Bern. His article, published in May 1905 in the German journal Annalen der Physik, was titled On the motion of particles suspended in a fluid at rest, required by the molecular-kinetic theory of heat. By this name, Einstein wanted to show that the existence of a random motion of the smallest solid particles in liquids necessarily follows from the molecular-kinetic theory of the structure of matter.

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 motions 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 this particular opinion." And decades later, already on the slope of his 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 on“ Brownian motion ”have long been known, I discovered that the atomistic theory leads to the existence of an observable motion of microscopic suspended particles." Be that as it may, Einstein's theoretical article ended with a direct appeal to experimenters to test his conclusions in practice: "If any researcher could soon answer the questions raised here questions!" - he ends his article with such an unusual exclamation.

Einstein's impassioned 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 fluid viscosity h, particle size r and the Avogadro constant

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

where R is the gas constant. So, if in 1 min a particle with a diameter of 1 µm is displaced by 10 µm, then in 9 min – by 10 = 30 µm, in 25 min – by 10 = 50 µm, etc. Under similar conditions, a particle with a diameter of 0.25 µm will shift by 20, 60, and 100 µm, respectively, in the same time intervals (1, 9, and 25 min), since = 2. It is important that the above formula includes the Avogadro constant, which is thus , can be determined by quantitative measurements of the movement of a Brownian particle, which is what the French physicist Jean Baptiste Perrin (1870–1942) did.

In 1908, Perrin began quantitative observations of the movement of Brownian particles under a microscope. He used the ultramicroscope, invented in 1902, which made it possible to detect the smallest particles due to the scattering of light from a powerful side illuminator on them. Tiny balls of almost spherical shape and approximately the same size Perrin obtained from gummigut - the condensed juice of some tropical trees (it is also used as a yellow watercolor paint). These tiny balls were weighed in glycerin containing 12% water; the viscous liquid prevented the appearance of internal flows in it, which would have smeared 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 a minute. By connecting the obtained points with straight lines, he obtained intricate trajectories, some of which are shown in the figure (they are taken from Perrin's book atoms published in 1920 in Paris). Such a chaotic, chaotic movement of particles leads to the fact that they move in space rather slowly: the sum of the segments is much greater than the displacement of the particle from the first point to the last.

Sequential positions every 30 seconds of three Brownian particles - gummigut balls about 1 micron in size. 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 the value of Avogadro's number, which was quite accurate for that time: 6.8 . 10 23 . Perrin also investigated using a microscope the distribution of Brownian particles along the vertical ( cm. AVOGADRO LAW) and showed that, despite the action of terrestrial gravity, they remain in solution in a suspended state. Perrin also owns other important works. In 1895 he proved that cathode rays are negative electric charges (electrons), 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. This 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 the Avogadro constant!” One can be surprised in another way: until now, in scientific journals (Nature, Science, Journal of Chemical Education), descriptions of new experiments on Brownian motion appear from time to time! After the publication of Perrin's results, the former opponent of atomism, Ostwald, admitted that “the coincidence of Brownian motion with the requirements of the kinetic hypothesis ... now gives the right to the most cautious scientist to speak about the experimental proof of the atomistic theory of matter. Thus, the atomistic theory is elevated to the rank of a scientific, firmly established theory. He is echoed by the French mathematician and physicist Henri Poincaré: "Perrin's brilliant determination of the number of atoms completed the triumph of atomism ... The atom of chemists has now become a reality."

Brownian motion and diffusion.

The movement of Brownian particles looks very much like 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 motion of molecules were established in the simplest case of the gaseous state of matter. It turned out that the molecules in gases move very quickly - at the speed of a bullet, but they cannot “fly away” far, as they very often collide with other molecules. For example, oxygen and nitrogen molecules in the air, moving at an average speed of about 500 m/s, experience more than a billion collisions every second. Therefore, the path of the molecule, if it could be traced, would be a complex broken line. A similar trajectory is described by Brownian particles if their position is fixed at certain time intervals. Both diffusion and Brownian motion are a consequence of the chaotic thermal motion of molecules and therefore are 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 randomly shifts to one side or the other. Calculations have shown that for a 0.1 µm particle, one movement occurs in three billionths of a second over a distance of only 0.5 nm (1 nm = 0.001 µm). According to the apt expression of one author, this is reminiscent of the movement of 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: it is not the movements of individual particles, but their huge masses that are observed, it is only necessary to ensure that convection is not superimposed on diffusion - the mixing of matter as a result of vortex flows (such flows are easy to notice, by dropping a drop of a colored solution, such as ink, into a glass of hot water).

Diffusion is conveniently observed in thick gels. Such a gel can be prepared, for example, in a penicillin jar by preparing a 4–5% gelatin solution in it. Gelatin must first swell for several hours, and then it is completely dissolved with stirring, lowering the jar into hot water. After cooling, a non-flowing gel is obtained in the form of a transparent, slightly cloudy mass. If, with the help of sharp tweezers, a small crystal of potassium permanganate (“potassium permanganate”) is carefully introduced into the center of this mass, then the crystal will remain hanging in the place where it was left, since the gel does not allow it to fall. Within a few minutes, a purple-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 with the help of a crystal of copper sulphate, only in this case the ball will turn out not purple, but blue.

Why the ball turned out is clear: the MnO 4 - ions, formed during the dissolution of the crystal, go into solution (gel is mainly water) and, as a result of diffusion, move uniformly in all directions, while gravity has practically no effect on the diffusion rate. Diffusion in a liquid is very slow: it takes many hours for the ball to grow a few centimeters. In gases, diffusion goes much faster, but still, if the air did not mix, then the smell of perfume or ammonia would spread in the room for hours.

Brownian motion theory: random walks.

The Smoluchowski-Einstein theory explains the patterns of both diffusion and Brownian motion. We can consider these regularities on the example of diffusion. If the speed of the molecule is u, then, moving in a straight line, it takes time t will pass 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 not fundamentally differ from the drawings obtained by Perrin. It can be seen from such figures that, due to the chaotic motion, the molecule is displaced by a distance s, much less than L. These quantities are related by the relation s= , where l is the distance that the molecule flies from one collision to another, the mean free path. Measurements showed that for air molecules at normal atmospheric pressure l ~ 0.1 μm, which means that at a speed of 500 m / s a ​​molecule of nitrogen or oxygen will fly in 10,000 seconds (less than three hours) L= 5000 km, and will shift from the original position by only s\u003d 0.7 m (70 cm), therefore substances due to diffusion move so slowly 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 (in English random walk). Witty physicists reinterpreted this expression into drunkard's walk - “the path of a drunkard.” Indeed, moving 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 makes it quite easy to derive the basic equation of such a process - on the example of one-dimensional motion, which is easy to generalize to three-dimensional.

Let the tipsy sailor leave the tavern late in the evening and head 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 tavern, or will he just wander around it, now moving away, now approaching it? (In another version of the problem, it is said that at both ends of the street where the lanterns end, there are dirty ditches, and the question is whether the sailor will be able to avoid falling into one of them). Intuitively, the second answer seems to be correct. But he is wrong: 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 the direction, we get rid of the signs by squaring this expression: s 1 2 \u003d l 2. After some time, the sailor, having already N"wandering", will be at a distance

s N= from start. And having passed once again (to one side) to the nearest lantern, - at a distance s N+1 = s N± l, or, using the square of the offset, 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 right or left), term ± 2 s N l cancels out so that s 2 N+1 = s2 N+ l 2> (angle brackets indicate the average value). L \u003d 3600 m \u003d 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 he will pass L= 10.8 km, and will shift to 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 such considerations, Einstein derived his equation.

The theory of Brownian motion in real life.

The theory of random walks has an important practical application. It is said that in the absence of landmarks (the sun, stars, the noise of a highway or railway, etc.), a person wanders in a 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 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 who were frozen in a blizzard were found “some kilometer” from the nearest housing or road, but in fact a person had no chance to walk this kilometer, and here's why.

To calculate how much a person will move as a result of random walks, you need to know the value of l, i.e. the distance that a person can walk in a straight line without any reference points. This value was measured by the doctor of geological and mineralogical sciences B.S. Gorobets with the help of student volunteers. Of course, he did not leave them in a dense forest or on a snowy field, everything was simpler - they put the student in the center of an empty stadium, blindfolded him and asked him to go in complete silence (to exclude orientation by sounds) to the end of the football field. 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.

Now let 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 go 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 such a rule: if the landmarks are lost, you must stay in place, equip the shelter and wait for the end of the bad weather (the sun may come out) or help. In the forest, landmarks - trees or bushes - will help you move in a straight line, and each time you need to keep two such landmarks - one in front, the other behind. But, of course, it's best to take a compass with you...

Ilya Leenson

Literature:

Mario Lozzi. 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