Brownian motion. Brownian motion - the history of the development of colloidal chemistry

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 was in its infancy molecular science Leeuwenhoek's observation was not noticed at the time, so 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 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 of 1877, “...the law large numbers does not now 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. Approximately the same way a small twig or bug should move when pushed (or pulled) into different sides lots of 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 sunbeam who entered a dark room, “dance” due to the 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, the pollen of meadow grasses, 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 clearly proved molecular structure matter, however, even at the beginning of the 20th century. There were scientists, including physicists and chemists, who did not believe in the existence of molecules. Atomic-molecular theory only slowly and with difficulty won 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), laureate Nobel Prize, 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 such in a simple way The validity of the atomic-molecular theory was proven, 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 Encyclopedic Dictionary Brockhaus and Efron published a short article by St. Petersburg professor of physics N.A. Gezehus, teacher of the famous academician A.F. Ioffe. 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 quantity theory did not 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).

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 may be surprised by another thing: still 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, 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 could be followed, 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 empty cans from under beer in the square where a crowd of people had 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 colored crystal will begin to grow around the crystal. purple ball, 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 goes much faster, but still, if the air were not mixed, then the smell of perfume or ammonia 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, highway noise or railway etc.) a person wanders in the forest, across a field in a snowstorm or in thick fog in circles, all the time returning to old 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

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 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, pushed (or pulled) in different directions by many ants, should move in approximately the same way. 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.

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.

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

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

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 could be followed, 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 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 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.

Ilya Leenson

Literature:

Robert Brown, a famous British botanist, was born on December 21, 1773 in the Scottish city of Montrose, studied in Aberdeen, and studied medicine and botany at the University of Edinburgh in 1789-1795.

Thanks to hard work natural sciences and friendship with the botanist Joseph Banks, he was appointed botanist on an expedition sent in 1801 to explore the coast of Australia. In 1805, Brown returned to England, bringing with him about 4,000 species of Australian plants, many birds and minerals, and subsequently published his works on the plant world.

Member of the Royal Society of London (since 1810). From 1810 to 1820, Robert Brown was in charge of the Linnean Library. In 1820 he became librarian and curator of the botanical department of the British Museum, where, after the death of Banks, the latter's collections were transferred. Thanks to these collections and the library and the mass of plants from various countries with which he was always surrounded, Brown was the best expert on plants.

Brownian motion

This phenomenon, discovered by R. Brown in 1827 while conducting research on plant pollen, is the random movement of microscopic particles (Brownian particles) solid(dust particles, plant pollen particles, etc.) caused by the thermal movement of liquid (or gas) particles. Brownian motion is a consequence and evidence of the existence of thermal motion.

Once he looked under a microscope at elongated cytoplasmic grains isolated from pollen cells of a North American plant, 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.”

Plant cell nuclei

Brown was the first to identify the nucleus in a plant cell and published this information in 1831. He gave it the name "Nucleus", or "Areola". The first term became generally accepted and has been preserved to the present day, but the second did not receive widespread use and was forgotten. It is very important that Brown insisted on the constant presence of a nucleus in all living cells.

Brown's discovery.

In 1827 Brown conducted research on plant pollen. He was particularly interested in how pollen participates in the process of fertilization. Once, under a microscope, he examined elongated cytoplasmic grains suspended in water from pollen cells of the North American plant Clarkia pulchella. 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 wondered if these were the “elementary molecules of living beings” that the famous French naturalist Georges Buffon (1707–1788), author of the 36-volume Natural History, spoke about. 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."

Brownian motion and atomic-molecular theory.

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, in the Proceedings of the National Academy of Sciences of the United States, a 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 - 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, pushed (or pulled) in different directions by many ants, should move in approximately the same way. 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 On 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...

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 the atom, nor the 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. Thus, 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.

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 (s2) during time t is directly proportional to the temperature T and inversely proportional to the viscosity of the liquid h, the particle size r and Avogadro's constant

NA: s2 = 2RTt/6phrNA,

Where R is the 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 which are shown in the figure (they are taken from Perrin's book Atomy, 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.
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 fairly accurate value for Avogadro’s number for that time: 6.8.1023. Perrin also used a microscope to study the vertical distribution of Brownian particles (see 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 could be followed, 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 is clear why the ball turned out: the MnO4– ions formed during the dissolution of the crystal go into solution (the gel is mainly water) and, as a result of diffusion, move evenly in all directions, while gravity has practically 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 a molecule is u, then, moving in a straight line, it will cover a distance L = ut in time t, 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 such figures it is clear that, due to chaotic motion, 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 the molecule flies from one collision to another, the average 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 in 10,000 seconds (less than three hours) distance L = 5000 km, and will shift from the original position is only s = 0.7 m (70 cm), which is why substances move so slowly due to diffusion, even in gases.

Having walked for the first time to the nearest lantern (to the right or to the left), the sailor will find himself at a distance s1 = ± 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: s12 = l2. After some time, the sailor, having already completed N “wanderings”, will be at a distance

SN = from start. And having passed again (in one direction) to the nearest lamp, at a distance sN+1 = sN ± l, or, using the square of the displacement, s2N+1 = s2N ±2sN l + l2. If the sailor repeats this movement many times (from N to N + 1), then as a result of averaging (he takes the Nth step to the right or to the left with equal probability), the term ±2sNl will be reduced, so that (angle brackets indicate the averaged value).

Since s12 = l2, then

S22 = s12 + l2 = 2l2, s32 = s22 + l2 = 3ll2, etc., i.e. s2N = Nl2 or sN =l. The total distance traveled L can be written both as the product of the sailor’s speed and the travel time (L = ut), and as the product of the number of wanderings and the distance between the lanterns (L = Nl), therefore, ut = Nl, whence N = ut/l and finally sN = . Thus, we obtain the dependence of the displacement of the sailor (as well as the molecule or Brownian particle) on time. For example, if there are 10 m between the lamps and a sailor walks at a speed of 1 m/s, then in an hour he common path will be L = 3600 m = 3.6 km, while the displacement from the zero point in the same time will be equal to only s = 190 m. In three hours it will cover L = 10.8 km, and will shift by s = 330 m and etc.

The product ul 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.

Robert Brown was born on December 21, 1773 in the family of a Protestant minister. He studied at Marischal College at the University of Aberdeen, then at the University of Edinburgh, where he studied medicine and botany. In 1795 he joined the Northern Regiment of the Scottish Militia as an assistant surgeon, with whom he was in Ireland. Here Brown collected local plants and met the English naturalist Joseph Banks (1743-1820), on whose recommendation he was appointed botanist on an expedition sent in 1801 on the ship Investigator to explore the coast of Australia. In 1805 Brown returned to England, bringing with him about 4,000 species of Australian plants, many birds and minerals for the Banks collection.

In 1810-1820. Brown was in charge of the Linnean Library and the extensive collections of Banks, who was then president of the Royal Society of London. In 1820 he became librarian and curator of the botanical department of the British Museum, where his collections were transferred after Banks' death. From 1849 to 1853, Robert Brown was president of the Linnean Society in London.

The scientist’s morphological and embryological studies had great importance to build a natural plant system. Brown discovered the embryo sac in the ovule (1825), showed that the ovules in conifers and cycads are not enclosed in the ovary, which established the main difference between angiosperms and gymnosperms; He discovered archegonia in the ovules of coniferous plants. Brown was the first to correctly describe the nucleus in plant cells (1831).

In 1827, the scientist 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 elongated cytoplasmic grains isolated from pollen cells of the North American plant Clarkia pulchella, 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. This discovery was later named after him (