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 these were 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 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 to explain it mysterious phenomenon, which was called "Brownian motion", neither Brown himself nor many other scientists could 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 waves rocking a distant boat are not visible from the shore, while the movements of the boat itself can be seen quite clearly. As they wrote in one of the articles of 1877, “... the law big 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. 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), laureate Nobel Prize, 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 such in a simple way the validity of the atomic-molecular doctrine is proved, therefore, a wide variety of alternative reasons have been put forward to explain the phenomenon. And this is quite in the spirit of science: until the cause of any 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, themselves experimental results 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 from the molecular-kinetic theory of the structure of matter, the existence of random movement 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 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 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 at another thing: so far in scientific journals(Nature, Science, Journal of Chemical Education) from time to time there are descriptions of new experiments on Brownian motion! 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 talk 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 blue vitriol, only in this case the ball will turn out not purple, but blue.
It is clear why the ball turned out: 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 practically does not affect 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 is 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. The 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 nitrogen or oxygen molecule 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 there are muddy ditches at both ends of the street where the lanterns end, 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 (sun, stars, highway noise or railway etc.) a person wanders in the forest, across the field in a snowstorm or in thick fog in circles, all the time 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 in complete silence (to exclude orientation by sounds) to go 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
What is Brownian motion?
Now you will get acquainted with the most obvious proof of the thermal motion of molecules (the second main position of the molecular kinetic theory). Be sure to try to look through a microscope and see how the so-called Brownian particles move.
Previously, you learned what diffusion, i.e., the mixing of gases, liquids and solids with their direct contact. This phenomenon can be explained by the random movement of molecules and the penetration of molecules of one substance into the space between the molecules of another substance. This can explain, for example, the fact that the volume of a mixture of water and alcohol is less than the volume of its components. But the most obvious evidence of the movement of molecules can be obtained by observing through a microscope the smallest particles of some kind suspended in water. solid. These particles move randomly, which is called Brownian.
This is the thermal movement of particles suspended in a liquid (or gas).
Observation of Brownian motion
The English botanist R. Brown (1773-1858) first observed this phenomenon in 1827, examining the moss spores suspended in water under a microscope. Later, he considered other small particles, including particles of stone from Egyptian pyramids. Now, to observe Brownian motion, particles of gummigut paint, which is insoluble in water, are used. These particles move randomly. The most striking and unusual thing for us is that this movement never stops. We are accustomed to the fact that any moving body sooner or later stops. Brown initially thought that the spores of the club moss showed signs of life.
thermal motion, and it cannot stop. As the temperature increases, its intensity increases. Figure 8.3 shows a diagram of the movement of Brownian particles. The positions of the particles marked with dots are determined at regular intervals of 30 s. These points are connected by straight lines. In reality, the particle trajectory is much more complicated.Brownian motion can also be observed in a gas. It is carried out by particles of dust or smoke suspended in the air.
The German physicist R. Pohl (1884-1976) colorfully describes the Brownian motion: “Few phenomena can captivate the observer as much as the Brownian motion. Here the observer is allowed to look behind the scenes of what happens in nature. Before him opens new world- non-stop hustle and bustle of a huge number of particles. The smallest particles fly quickly into the field of view of the microscope, almost instantly changing the direction of movement. Larger particles move more slowly, but they also constantly change direction. Large particles practically jostle in place. Their protrusions clearly show the rotation of particles around their axis, which constantly changes direction in space. Nowhere is there a trace of system or order. The dominance of blind chance - that's what a strong, overwhelming impression this picture makes on the observer.
At present, the concept Brownian motion used in a broader sense. For example, Brownian motion is the trembling of the arrows of sensitive measuring instruments, which occurs due to the thermal movement of the atoms of instrument parts and the environment.
Explanation of Brownian motion
Brownian motion can be explained only on the basis of molecular-kinetic theory. The reason for the Brownian motion of a particle is that the impacts of liquid molecules on the particle do not cancel each other out.. Figure 8.4 schematically shows the position of one Brownian particle and the molecules closest to it. When molecules move randomly, the impulses they transmit to a Brownian particle, for example, from the left and from the right, are not the same. Therefore, the resulting pressure force of liquid molecules on a Brownian particle is nonzero. This force causes a change in the motion of the particle.
The mean pressure has a certain value in both gas and liquid. But there are always slight random deviations from this average. The smaller the surface area of the body, the more noticeable the relative changes in the pressure force acting on given area. So, for example, if the area has a size of the order of several molecular diameters, then the pressure force acting on it changes abruptly from zero to a certain value when the molecule enters this area.
The molecular-kinetic theory of Brownian motion was created in 1905 by A. Einstein (1879-1955).
The construction of the theory of Brownian motion and its experimental confirmation by the French physicist J. Perrin finally completed the victory of the molecular-kinetic theory.
Perrin's experiments
The idea behind Perrin's experiments is as follows. It is known that the concentration of gas molecules in the atmosphere decreases with height. If there were no thermal motion, then all the molecules would fall to the Earth and the atmosphere would disappear. However, if there was no attraction to the Earth, then due to thermal motion, the molecules would leave the Earth, since the gas is capable of unlimited expansion. As a result of the action of these opposite factors, a certain distribution of molecules along the height is established, as mentioned above, i.e., the concentration of molecules decreases rather quickly with height. Moreover, the greater the mass of molecules, the faster their concentration decreases with height.
Brownian particles participate in thermal motion. Since their interaction is negligible, the aggregate of these particles in a gas or liquid can be considered as an ideal gas of very heavy molecules. Consequently, the concentration of Brownian particles in a gas or liquid in the Earth's gravitational field must decrease according to the same law as the concentration of gas molecules. This law is known.
Perrin, using a microscope of high magnification and a small depth of field (small depth of field), observed Brownian particles in very thin layers of liquid. Calculating the concentration of particles at different heights, he found that this concentration decreases with height according to the same law as the concentration of gas molecules. The difference is that due to the large mass of Brownian particles, the decrease occurs very quickly.
Moreover, counting Brownian particles at different heights allowed Perrin to determine Avogadro's constant in a completely new way. The value of this constant coincided with the known one.
All these facts testify to the correctness of the theory of Brownian motion and, accordingly, to the fact that Brownian particles participate in the thermal motion of molecules.
You have clearly seen the existence of thermal motion; We saw the chaotic movement going on. Molecules move even more randomly than Brownian particles.
The essence of the phenomenon
Now let's try to understand the essence of the phenomenon of Brownian motion. And it happens because all absolutely liquids and gases consist of atoms or molecules. But we also know that these smallest particles, being in continuous chaotic motion, constantly push the Brownian particle from different sides.
But here's what's interesting, scientists have proven that particles of larger sizes that exceed 5 microns remain motionless and almost do not participate in Brownian motion, which cannot be said about smaller particles. Particles with a size of less than 3 microns are able to move forward, making rotations or writing out complex trajectories.
When immersed in the environment of a large body, the tremors occurring in a huge number seem to reach an average level and maintain a constant pressure. In this case, the theory of Archimedes comes into play, since a large body surrounded by a medium on all sides balances the pressure and the remaining lifting force allows this body to float or sink.
But if the body has dimensions such as a Brownian particle, that is, completely imperceptible, then pressure deviations become noticeable, which contribute to the creation of a random force that leads to oscillations of these particles. It can be concluded that Brownian particles in the medium are in suspension, in contrast to large particles that sink or float.
Significance of Brownian motion
Let's try to figure out if Brownian motion in the natural environment has any meaning:
First, Brownian motion plays a significant role in plant nutrition from the soil;
Secondly, in human and animal organisms, the absorption of nutrients occurs through the walls of the digestive organs due to Brownian motion;
Thirdly, in the implementation of skin respiration;
And lastly, Brownian motion also matters in the distribution harmful substances in the air and in the water.
Homework
Read the questions carefully and give written answers to them:
1. Remember what is called diffusion?
2. What is the relationship between diffusion and thermal motion of molecules?
3. Define Brownian motion.
4. What do you think, is Brownian motion thermal, and justify your answer?
5. Will the nature of Brownian motion change when heated? If it changes, then how?
6. What instrument is used in the study of Brownian motion?
7. Does the pattern of Brownian motion change with increasing temperature, and how exactly?
8. Will there be any change in Brownian motion if the aqueous emulsion is replaced with glycerol?
G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics Grade 10
Brown's discovery.
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, under a microscope, he examined elongated cytoplasmic grains suspended in water isolated from the pollen cells of the North American plant Clarkia pulchella (pretty clarkia). 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 wondered 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, spoke about. 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. It is curious that in 1921 the Reports of the US National Academy of Sciences published a work 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 - 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 waves rocking a distant boat are not visible from the shore, while the movements of the boat itself can be seen quite clearly. 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 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 any 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 by the St. Petersburg professor of physics N.A. Gezehus, teacher of the famous academician A.F. Ioffe, was published in the Encyclopedic Dictionary of Brockhaus and Efron. 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 entitled 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 (s2) over time t is directly proportional to the temperature T and inversely proportional to the fluid viscosity h, the particle size r and the Avogadro constant
NA: s2 = 2RTt/6phrNA,
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. Connecting the points obtained 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.
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: 6.8.1023. Perrin also studied the vertical distribution of Brownian particles with a microscope (see AVOGADRO 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), 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 talk 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 MnO4– 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 is 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 a molecule is u, then, moving in a straight line, it will cover the 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 not fundamentally differ from the drawings obtained by Perrin. It can be seen from such figures that due to 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. The 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 nitrogen or oxygen molecule will fly in 10,000 seconds (less than three hours) a distance L = 5000 km, and will shift from the initial positions are 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 (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 there are muddy ditches at both ends of the street where the lanterns end, 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 s1 = ± l from the starting point. Since we are only interested in its removal from this point, but not the direction, we get rid of the signs by squaring this expression: s12 = l2. After some time, the sailor, having already made N “wanders”, will be at a distance
SN = from start. And having passed once again (to one of the sides) 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 left with equal probability), the term ±2sNl will decrease, 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 walks and the distance between the lamps (L = Nl), therefore, ut = Nl, whence N = ut/l and finally sN = . Thus, the dependence of the displacement of the sailor (and also of the molecule or Brownian particle) on time is obtained. For example, if there are 10 m between the lights and a sailor walks at a speed of 1 m/s, then in an hour his total path will be L = 3600 m = 3.6 km, while the displacement from the zero point in the same time will be only s = = 190 m. In three hours, it will travel L = 10.8 km, and move s = 330 m, and so on.
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 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, all the time 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 shift as a result of random walks, one must 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 in complete silence (to exclude orientation by sounds) to go 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...
The Scottish botanist Robert 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. 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, under a microscope, he examined elongated cytoplasmic grains suspended in water isolated from the pollen cells of the North American plant Clarkia pulchella (pretty clarkia). 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 sex cells of plants, but particles from dead plants, even from those dried a hundred years earlier in herbariums, also led. Then Brown wondered 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, spoke about. 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, “that 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.
For about 30 years, Brown's discovery did not attract the interest of physicists. The new phenomenon was not given of great importance, believing that it is due to the trembling of the drug, or analogous to the movement of dust particles, which is observed in the atmosphere when a ray of light falls on them, and which, as was known, is caused by the movement of air. But if the motions of Brownian particles were caused by some flows in the liquid, then such neighboring particles would move in concert, which contradicts the observational data.
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. 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. But Wiener's conclusions were complicated by the introduction of the concept of "atoms of the ether" in addition to the atoms of matter. In 1876, William Ramsay, and in 1877 the Belgian Jesuit priests Carbonel, Delso and Tirion, and finally, in 1888, Hui, clearly showed the thermal nature of Brownian motion [5].
“With a large area,” wrote Delso and Carbonel, “the impacts of molecules that cause pressure do not cause any shaking of the suspended body, because they together create uniform pressure on the body in all directions. But if the area is not sufficient to compensate for the unevenness, it is necessary to take into account the inequality of pressures and their continuous change from point to point. 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.
If this explanation is accepted, then the phenomenon of thermal motion of liquids, postulated by the kinetic theory, can be said to be proven ad oculos (visibly). Just as it is possible, without distinguishing the waves in the distance from the sea, this will explain the rocking of the boat on the horizon by waves, in the same way, without seeing the movement of molecules, one can judge it by the movement of particles suspended in the liquid.
This explanation of Brownian motion is not only important as a confirmation of the kinetic theory, it also has important theoretical implications. According to the law of conservation of energy, a change in the speed of a suspended particle must be accompanied by a change in temperature in the immediate vicinity of this particle: this temperature increases if the speed of the particle decreases, and decreases if the speed of the particle increases. Thus, the thermal equilibrium of a liquid is a statistical equilibrium.
An even more significant observation was made in 1888 by Huy: Brownian motion, strictly speaking, does not obey the second law of thermodynamics. Indeed, when a suspended particle spontaneously rises in a liquid, then part of the heat of its environment spontaneously turns into mechanical work, which is forbidden by the second law of thermodynamics. Observations, however, have shown that the particle rises less frequently, the heavier the particle. For particles of matter of ordinary sizes, this probability of such an uplift is practically zero.
Thus the second law of thermodynamics becomes a law of probability rather than a law of necessity. Previously, no experience has supported this statistical interpretation. It was enough to deny the existence of molecules, as was done, for example, by the school of energetics, which flourished under the leadership of Mach and Ostwald, for the second law of thermodynamics to become the law of necessity. But after the discovery of Brownian motion, a strict interpretation of the second law became already impossible: there was a real experience that showed that the second law of thermodynamics is constantly violated in nature, that a perpetual motion machine of the second kind is not only not excluded, but is constantly being realized right before our eyes.
Therefore, at the end of the last century, the study of Brownian motion acquired great theoretical significance and attracted the attention of many theoretical physicists, and in particular Einstein.
