Line UMK A. V. Grachev. Physics (7-9)

Line UMK A. V. Grachev. Physics (10-11) (basic, advanced)

Brownian motion

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1. Particles

We know that all matter is made up of a huge number of very, very small particles that are in continuous and random motion. How did we know this? How were scientists able to learn about the existence of particles so small that no optical microscope can see? And even more so, how did they manage to find out that these particles are in continuous and random motion? Two phenomena helped scientists to understand this - Brownian motion and diffusion. We will discuss these phenomena in more detail.

2. Brownian motion

The English scientist Robert Brown was not a physicist or chemist. He was a botanist. And he did not expect at all that he would discover such an important phenomenon for physicists and chemists. And he could not even suspect that in his rather simple experiments he would observe the result of the chaotic movement of molecules. And it was exactly like that.

What were these experiments? They were almost the same as what students do in biology classes when they try to examine, for example, plant cells with a microscope. Robert Brown wanted to examine plant pollen under a microscope. Looking at grains of pollen in a drop of water, he noticed that the grains were not at rest, but were constantly twitching, as if they were alive. He probably thought so at first, but being a scientist, of course, he rejected this thought. He failed to understand why these pollen grains behave in such a strange way, but he described everything he saw, and this description fell into the hands of physicists, who immediately realized that they had visual evidence of the continuous and random movement of particles.

This movement, described by Brown, is explained as follows: pollen grains are large enough so that we can see them with an ordinary microscope, but we do not see water molecules, but, at the same time, pollen grains are small enough that due to impacts along them, water molecules surrounding them from all sides, they shifted first in one direction, then in the other. That is, this chaotic “dance” of pollen grains in a drop of water showed that water molecules continuously and randomly hit the pollen grains from different sides and displace them. Since then, the continuous and chaotic movement of small solid particles in a liquid or gas has been called brownian motion. The most important feature of this movement is that it is continuous, that is, it never stops.

3. Diffusion

Diffusion is another example of clear evidence of the continuous and random movement of molecules. And it lies in the fact that gaseous substances, liquids and even solids, although much more slowly, can self-mix with each other. For example, smells various substances spread in the air even in the absence of wind precisely due to this self-mixing. Or here's another example - if you throw a few crystals of potassium permanganate into a glass of water and wait about a day without stirring the water, then we will see that all the water in the glass will be colored evenly. This is due to the continuous movement of molecules that change places, and substances gradually mix on their own without external influence.

The book is addressed to high school students, students, teachers and teachers of physics, as well as to all those who want to understand what is happening in the world around us, and to cultivate a scientific view of all the diversity of natural phenomena. Each section of the book is, in fact, a set physical tasks, by solving which the reader will strengthen his understanding of physical laws and learn to apply them in cases of practical interest.

4. Properties of Brownian motion and diffusion

When physicists began to look more closely at the phenomenon described by Robert Brown, they noticed that, like diffusion, this process can be accelerated by increasing the temperature. That is, in hot water and coloring with potassium permanganate will occur faster, and the movement of small solid particles, for example, graphite chips or the same pollen grains, occurs with greater intensity. This confirmed the fact that the speed of the chaotic movement of molecules directly depends on temperature. Without going into details, we list the factors on which both the intensity of Brownian motion and the rate of diffusion can depend:

1) on temperature;

2) on the kind of substance in which these processes occur;

3) from the state of aggregation.

That is, at equal to the temperature diffusion of gaseous substances proceeds much faster than liquids, not to mention the diffusion of solids, which occurs so slowly that its result, and even then very insignificant, can be noticed or at very high temperatures, or for a very long time - years or even decades.

5. Practical application

Diffusion, even without practical application, is of great importance not only for humans, but also for all life on Earth: it is thanks to diffusion that oxygen enters our blood through the lungs, it is through diffusion that plants extract water from the soil, absorb carbon dioxide from the atmosphere and release it in it. oxygen, and fish breathe oxygen in the water, which from the atmosphere through diffusion enters the water.

The phenomenon of diffusion is also used in many areas of technology, and it is diffusion in solids. For example, there is such a process - diffusion welding. In this process, the parts are very strongly pressed against each other, heated up to 800 ° C and, through diffusion, they are connected to each other. It is thanks to diffusion that the earth's atmosphere, consisting of a large number of different gases, is not divided into separate layers in composition, but is approximately homogeneous everywhere - and if it were otherwise, we would hardly be able to breathe.

There are a huge number of examples of the impact of diffusion on our lives and on all of nature, which any of you can find if you want. But little can be said about the application of Brownian motion, except that the theory itself, which describes this motion, can be applied to other seemingly completely unrelated phenomena, phenomena. For example, this theory is used to describe random processes, using a large amount of data and statistics - such as price changes. Brownian motion theory is used to create realistic computer graphics. Interestingly, a person lost in the forest moves in much the same way as Brownian particles - wanders from side to side, repeatedly crossing its trajectory.

1) When telling the class about Brownian motion and diffusion, it is necessary to emphasize that these phenomena do not prove the existence of molecules, but prove the fact of their motion and that it is disorderly - chaotic.

2) Be sure to pay special attention to the fact that this is a continuous movement dependent on temperature, that is, a thermal movement that can never stop.

3) Demonstrate diffusion using water and potassium permanganate by instructing the most inquisitive children to conduct a similar experiment at home and taking photographs of water with potassium permanganate every hour or two during the day (on the weekend, the children will do this with pleasure, and they will send you a photo). It is better if in such an experiment there are two containers with water - cold and hot, so that you can clearly demonstrate the dependence of the diffusion rate on temperature.

4) Try to measure the diffusion rate in the classroom using, for example, a deodorant - at one end of the classroom we spray a small amount of aerosol, and 3-5 meters from this place, the student with a stopwatch fixes the time after which he will smell. It is both fun and interesting, and will be remembered by children for a long time!

5) Discuss with the children the concept of chaos and the fact that even in chaotic processes, scientists find some patterns.

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

Brownian motion is the most obvious experimental confirmation of the ideas of the molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation interval is large enough so that the forces acting on the particle from the molecules of the medium change their direction many times, then the average square of the projection of its displacement on some axis (in the absence of other external forces) is proportional to time.
When deriving Einstein's law, it is assumed that particle displacements in any direction are equally probable and that the inertia of a Brownian particle can be neglected compared to the influence of friction forces (this is acceptable for sufficiently long times). The formula for the coefficient D is based on the application of the Stokes law for the hydrodynamic resistance to the motion of a sphere of radius a in a viscous fluid. The relationships for and D were experimentally confirmed by the measurements of J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant k and the Avogadro constant NA are experimentally determined. In addition to the translational Brownian motion, there is also a rotational Brownian motion - random rotation of a Brownian particle under the influence of impacts of the molecules of the medium. For rotational Brownian motion, the rms angular displacement of a particle is proportional to the observation time. These relationships were also confirmed by Perrin's experiments, although this effect is much more difficult to observe than translational Brownian motion.

The essence of the phenomenon

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

Brownian motion theory

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

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

Brownian motion as non-Markovian
random process

The theory of Brownian motion, well developed over the last century, is approximate. And although in most cases of practical importance the existing theory gives satisfactory results, in some cases it may require clarification. Thus, the experimental work carried out at the beginning of the 21st century in Polytechnic University Lausanne, the University of Texas and the European Molecular Biology Laboratory in Heidelberg (under the direction of S. Janey) showed the difference in the behavior of a Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable with increasing particle sizes. The studies also touched upon the analysis of the movement of the surrounding particles of the medium and showed a significant mutual influence of the movement of the Brownian particle and the movement of the particles of the medium caused by it on each other, that is, the presence of a "memory" in the Brownian particle, or, in other words, its dependence statistical characteristics in the future from the entire prehistory of its behavior in the past. This fact was not taken into account in the Einstein-Smoluchowski theory.
The process of Brownian motion of a particle in a viscous medium, generally speaking, belongs to the class of non-Markov processes, and for its more accurate description it is necessary to use integral stochastic equations.

When observing a suspension of flower pollen in water under a microscope, Brown observed a chaotic movement of particles that arises "not from the movement of a liquid and not from its evaporation." Visible only under a microscope, suspended particles of 1 μm or less in size performed disordered independent movements, describing complex zigzag trajectories. Brownian motion does not weaken with time and does not depend on chemical properties environment, its intensity increases with increasing temperature of the medium and with a decrease in its viscosity and particle size. Even a qualitative explanation of the causes of Brownian motion was possible only 50 years later, when the cause of Brownian motion began to be associated with the impact of liquid molecules on the surface of a particle suspended in it.

The first quantitative theory of Brownian motion was given by A. Einstein and M. Smoluchowski in 1905-06. based on molecular kinetic theory. It was shown that random walks of Brownian particles are associated with their participation in thermal motion along with the molecules of the medium in which they are suspended. Particles have on average the same kinetic energy, but due to the greater mass they have a lower speed. The theory of Brownian motion explains the random motion of a particle by the action of random forces from molecules and friction forces. According to this theory, the molecules of a liquid or gas are in constant thermal motion, and the impulses of different molecules are not the same in magnitude and direction. If the surface of a particle placed in such a medium is small, as is the case for a Brownian particle, then the impacts experienced by the particle from the surrounding molecules will not be exactly compensated. Therefore, as a result of the “bombardment” by molecules, a Brownian particle begins to move randomly, changing the magnitude and direction of its speed approximately 10 14 times per second. It followed from this theory that, by measuring the displacement of a particle for certain time and knowing its radius and the viscosity of the liquid, you can calculate the Avogadro number.

When observing Brownian motion, the position of a particle is fixed at regular intervals. The shorter the time intervals, the more broken the particle's trajectory will look.

The patterns of Brownian motion serve as a clear confirmation of the fundamental provisions of the molecular kinetic theory. It was finally established that the thermal form of the motion of matter is due to the chaotic motion of atoms or molecules that make up macroscopic bodies.

The theory of Brownian motion played an important role in substantiating statistical mechanics; it is the basis for the kinetic theory of coagulation of aqueous solutions. In addition, it also has practical significance in metrology, since Brownian motion is considered as the main factor limiting accuracy. measuring instruments. For example, the limit of accuracy of readings of a mirror galvanometer is determined by the trembling of the mirror, like a Brownian particle bombarded by air molecules. The laws of Brownian motion determine the random movement of electrons, causing noise in electrical circuits. Dielectric losses in dielectrics are explained by random movements of the dipole molecules that make up the dielectric. Random movements of ions in electrolyte solutions increase their electrical resistance.

Brownian motion

In the second half of the 20th century, a serious discussion about the nature of atoms flared up in scientific circles. On one side were irrefutable authorities such as Ernst Mach (cm. shock waves), who argued that atoms are simply mathematical functions, which successfully describe the observed physical phenomena and having no real physical basis. On the other hand, scientists of the new wave - in particular, Ludwig Boltzmann ( cm. Boltzmann constant) - insisted that atoms are physical realities. And neither of the two sides was aware that already decades before the start of their dispute, experimental results had been obtained that once and for all decided the question in favor of the existence of atoms as a physical reality - however, they were obtained in the discipline of natural science adjacent to physics by the botanist Robert Brown.

Back in the summer of 1827, Brown, while studying the behavior of pollen under a microscope (he studied an aqueous suspension of plant pollen Clarkia pulchella), suddenly discovered that individual spores make absolutely chaotic impulsive movements. He determined for certain that these movements were in no way connected with the eddies and currents of water, or with its evaporation, after which, having described the nature of the movement of particles, he honestly signed his own impotence to explain the origin of this chaotic movement. However, being a meticulous experimenter, Brown found that such a chaotic movement is characteristic of any microscopic particles, be it plant pollen, mineral suspensions, or any crushed substance in general.

It was only in 1905 that none other than Albert Einstein realized for the first time that this mysterious, at first glance, phenomenon serves as the best experimental confirmation of the correctness of the atomic theory of the structure of matter. He explained it something like this: a spore suspended in water is subjected to constant “bombardment” by randomly moving water molecules. On average, molecules act on it from all sides with equal intensity and at regular intervals. However, no matter how small the dispute, due to purely random deviations, it first receives an impulse from the side of the molecule that hit it from one side, then from the side of the molecule that hit it from the other, etc. As a result of averaging such collisions, it turns out that that at some point the particle "twitches" in one direction, then, if on the other side it was "pushed" by more molecules, in the other, etc. Using the laws mathematical statistics and molecular kinetic theory of gases, Einstein derived an equation describing the dependence of the mean square displacement of a Brownian particle on macroscopic parameters. ( Interesting fact: in one of the volumes of the German journal "Annals of Physics" ( Annalen der Physik) in 1905, three articles by Einstein were published: an article with a theoretical explanation of Brownian motion, an article on the foundations of the special theory of relativity, and, finally, an article describing the theory of the photoelectric effect. It was for the latter that Albert Einstein was awarded Nobel Prize in physics in 1921.)

In 1908, the French physicist Jean-Baptiste Perrin (Jean-Baptiste Perrin, 1870-1942) conducted a brilliant series of experiments that confirmed the correctness of Einstein's explanation of the phenomenon of Brownian motion. It became finally clear that the observed "chaotic" motion of Brownian particles is a consequence of intermolecular collisions. Since "useful mathematical conventions" (according to Mach) cannot lead to observable and perfectly real movements physical particles, it became finally clear that the debate about the reality of atoms is over: they exist in nature. As a “bonus game”, Perrin got the formula derived by Einstein, which allowed the Frenchman to analyze and estimate the average number of atoms and / or molecules colliding with a particle suspended in a liquid over a given period of time and, using this indicator, calculate the molar numbers of various liquids. This idea was based on the fact that every this moment time, the acceleration of a suspended particle depends on the number of collisions with the molecules of the medium ( cm. Newton's laws of mechanics), and hence on the number of molecules per unit volume of liquid. And this is nothing but Avogadro's number (cm. Avogadro's law) is one of the fundamental constants that determine the structure of our world.

Brownian motion is the chaotic and random movement of small particles, usually molecules in different liquids or gases. The reason for the emergence of Brownian motion is the collision of some (smaller particles) with other particles (already larger ones). What is the history of the discovery of Brownian motion, its significance in physics, and in particular in atomic and molecular theory? What examples of Brownian motion are there in real life? Read about all this further in our article.

Discovery of Brownian motion

The discoverer of the Brownian movement was the English botanist Robert Brown (1773-1858), in fact it was in his honor that it was called "Brownian". In 1827, Robert Brown was engaged in active research on the pollen of various plants. He was especially interested in what part pollen takes in the reproduction of plants. And just like that, observing the movement of pollen in vegetable juice, the scientist noticed that small particles now and then make random tortuous movements.

Brown's observation was confirmed by other scientists. In particular, it was noticed that the particles tend to accelerate with an increase in temperature, as well as with a decrease in the size of the particles themselves. And with an increase in the viscosity of the medium in which they were located, their movement, on the contrary, slowed down.

Robert Brown, discoverer of Brownian motion.

At first, Robert Brown thought that he was observing the movement, even the “dance” of some living microorganisms, because the pollen itself is, in fact, the male sex cells of plants. But particles of dead plants had a similar movement, and even plants dried a hundred years ago in herbariums. The scientist was even more surprised when he began to study inanimate matter: small particles of coal, soot, and even dust particles of London air. Then glass, various and varied minerals fell under the researcher's microscope. And everywhere these "active molecules" were seen, being in constant and chaotic movement.

This is interesting: you yourself can observe Brownian motion with your own eyes, for this you will need a not powerful microscope (after all, during the life of Robert Brown there were no powerful modern microscopes yet). When viewed through this microscope, for example, smoke in a blackened box and illuminated by a side beam of light, one can see small pieces of soot and ash that will constantly bounce back and forth. This is Brownian motion.

Brownian motion and atomic-molecular theory

The movement discovered by Brown soon became very famous in scientific circles. The discoverer himself showed it with pleasure to many of his colleagues. However, for many years, neither Robert Brown himself nor his colleagues could explain the causes of the Brownian movement, why it occurs at all. Moreover, the Brownian movement was completely erratic and defied any logic.

His explanation was given only at the end of the 19th century, and it was not immediately accepted by the scientific community. In 1863, the German mathematician Ludwig Christian Wiener suggested that Brownian motion is due to oscillatory movements some invisible atoms. In fact, this was the first explanation of this strange phenomenon, connected with the properties of atoms and molecules, the first attempt to penetrate the mystery of the structure of matter with the help of Brownian motion. In particular, Wiener tried to measure the dependence of the speed of particles on their size.

Subsequently, Wiener's ideas were developed by other scientists, among them was the famous Scottish physicist and chemist William Ramsay. It was he who managed to prove that the cause of the Brownian motion of small particles is the impact on them of even smaller particles, which are no longer visible in an ordinary microscope, just as the waves shaking a distant boat are not visible from the shore, although the movement of the boat itself can be seen quite clearly.

Thus Brownian motion became one of the constituent parts atomic-molecular theory and at the same time an important proof of the fact that all matter consists of the smallest particles: atoms and molecules. It's hard to believe, but even at the beginning of the 20th century, some scientists denied the atomic-molecular theory, and did not believe in the existence of molecules and atoms. Scientific works Ramsay associated with Brownian motion dealt a crushing blow to the opponents of atomism, and made all scientists finally convinced that you see for yourself, atoms and molecules exist, and their action can be seen with your own eyes.

Brownian motion theory

Despite the outward disorder of the chaotic motion of particles, they nevertheless tried to describe their random movements by mathematical formulas. Thus was born the theory of Brownian motion.

By the way, one of those who developed this theory was the Polish physicist and mathematician Marian Smoluchowski, who at that time worked at Lviv University and lived in hometown the author of this article, in the beautiful Ukrainian city of Lviv.

Lviv University, now the University. I. Frank.

In parallel with Smoluchovsky, the theory of Brownian motion was studied by one of the luminaries of world science - the famous Albert Einstein, who at that time was still a young and well-known worker in the Patent Office of the Swiss city of Bern.

As a result, both scientists created their own theory, which can also be called the Smoluchowski-Einstein theory. In particular, a mathematical formula was formed, according to which the average value of the squared displacement of a Brownian particle ( s 2) over time t is directly proportional to the temperature T and inversely proportional to the fluid viscosity n, particle size r and constant .

N A: s 2 = 2RTt/6ph rN A - this is what this formula looks like.

R in the formula 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 was done by the French physicist Jean Baptiste Perrin.

To observe Brownian particles, Perrin used the latest at that time ultramicroscope, through which the smallest particles of matter were already visible. In his experiments, the scientist, armed with a stopwatch, noted the positions of certain Brownian particles at regular intervals (for example, after 30 seconds). Then, connecting the positions of the particles with straight lines, various intricate trajectories of their movement were obtained. All this was sketched on a special graphed sheet.

This is what the drawings looked like.

Compiling Einstein's theoretical formula with his observations, Perrin was able to obtain the most accurate value of the Avogadro number for that time: 6.8 . 10 23

With his experiments, he confirmed the theoretical conclusions of Einstein and Smoluchowski.

Brownian motion and diffusion

The movement of particles during Brownian motion is outwardly very similar to the motion of particles during - the mutual penetration of molecules of different substances under the influence of temperature. Then what is the difference between Brownian motion and diffusion? In fact, both diffusion and Brownian motion occur due to the random thermal motion of molecules, and as a result are described by similar mathematical rules.

The difference between them is that during diffusion, a molecule always moves in a straight line until it collides with another molecule, after which it changes the trajectory of its movement. A Brownian particle does not "free-flight", but experiences very small and frequent "jitters", as it were, as a result of which it randomly moves here and there. Figuratively speaking, a Brownian particle is like an empty can of beer lying around in a square where a large crowd of people has gathered. People scurry back and forth, touch the jar with their feet and it flies randomly in different directions like a Brownian particle. And the movement of the people themselves in the crowd is already more characteristic of the movement of particles during diffusion.

If you look at the micro level, then the reason for the movement of a Brownian particle is its collision with smaller particles, while during diffusion, particles collide with other similar particles.

Both diffusion and Brownian motion occur under the influence of temperature. As the temperature decreases, both the speed of particles during Brownian motion and the speed of particle motion during diffusion slow down.

Examples of Brownian motion in real life

The theory of Brownian motion, these random walks, has a practical implementation in our real life. For example, why does a person who gets lost in the forest periodically return to the same place? Because it does not move in circles, but approximately in the same way as a Brownian particle usually moves. Therefore, he crosses his own path himself many times.

Therefore, having no clear guidelines and directions of movement, a lost person is likened to a Brownian particle that performs chaotic movements. But in order to get out of the forest, you need to have clear guidelines, develop a system, instead of performing various senseless actions. In a word, you should not behave in life like a Brownian particle, rushing from side to side, but you should know your direction, goal and vocation, have dreams, courage and perseverance to achieve them. This is how we smoothly moved from physics to philosophy. This concludes this article.

Brownian motion, video

And finally, an educational video on the topic of our article.

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