One of the most convincing evidence of the reality of the movement of molecules is the phenomenon of the so-called Brownian motion, discovered in 1827 by the English botanist Brown while studying tiny spores suspended in water. He discovered, when examined under a microscope with high magnification, that these spores were in continuous disorderly movement, as if performing a wild fantastic dance.
Further experiments showed that these movements are not associated with the biological origin of the particles or with any movement of the liquid. Similar movements are performed by any small particles suspended in a liquid or gas. This kind of random movement occurs, for example, in smoke particles in still air. This random movement of particles suspended in a liquid or gas is called Brownian motion.
Special studies have shown that the nature of Brownian motion depends on the properties of the liquid or gas in which the particles are suspended, but does not depend on the properties of the substance of the particles themselves. The speed of movement of Brownian particles increases with increasing temperature and with decreasing particle size.
All these patterns are easy to explain if we accept that the movements of suspended particles arise as a result of the impacts they experience from the moving molecules of the liquid or gas in which they are located.
Of course, each Brownian particle is subject to such impacts from all sides. Given the complete disorder of molecular movements, one would seem to expect that the number of impacts striking a particle from any direction should be exactly equal to the number of impacts from the opposite direction,
so all these shocks must completely compensate each other and the particles must remain motionless.
This is exactly what happens if the particles are not too small. But when we are dealing with microscopic particles cm), the situation is different. Indeed, from the fact that molecular movements are chaotic, it only follows that on average the number of impacts in different directions is the same. But in a statistical system such as a liquid or gas, deviations from the average values are inevitable. Such deviations from the average values of certain quantities, which occur in a small volume or over short periods of time, are called fluctuations. If there is a body of normal size in a liquid or gas, then the number of shocks that it experiences from the molecules is so great that it is impossible to notice either individual shocks or a random predominance of shocks in one direction over shocks in other directions. For small particles, the total number of shocks they experience is relatively small, so that the predominance of the number of impacts in one direction or another becomes noticeable, and it is thanks to such fluctuations in the number of impacts that those characteristic, as if convulsive movements of suspended particles arise, which are called Brownian motion .
It is clear that the movements of Brownian particles are not molecular movements: we see not the result of the impact of one molecule, but the result of the predominance of the number of impacts in one direction over the number of impacts in the opposite direction. Brownian motion only very clearly reveals the very existence of random molecular movements.
Thus, Brownian motion is explained by the fact that due to the random difference in the number of impacts of molecules on a particle from different directions, a certain resultant force of a certain direction arises. Since fluctuations are usually short-term, after a short period of time the direction of the resultant will change, and with it the direction of movement of the particle will change. Hence the observed chaotic nature of Brownian motions, reflecting the chaotic nature of molecular motion.
We will now supplement the above qualitative explanation of Brownian motion with a quantitative consideration of this phenomenon. Its quantitative theory was first given by Einstein and, independently, by Smoluchowski (1905). We will present here a simpler derivation than those of these authors of the basic relationship of this theory.
Due to incomplete compensation of the impacts of molecules, a Brownian particle is acted upon, as we have seen, by a certain resulting force under the influence of which the particle moves. In addition to this force, the particle is acted upon by a frictional force caused by the viscosity of the medium and directed against the force
For simplicity, we assume that the particle has the shape of a sphere of radius a. Then the friction force can be expressed by the Stokes formula:
where is the coefficient internal friction liquid (or gas), the speed of particle movement. The equation of particle motion (Newton’s second law) therefore has the form:
Here is the mass of the particle, its radius vector relative to an arbitrary coordinate system, the speed of the particle and the resultant of the forces caused by the impacts of molecules.
Let's consider the projection of the radius vector onto one of the coordinate axes, for example, onto the axis For this component, equation (7.1) will be rewritten as:
where is the component of the resulting force along the axis
Our task is to find the displacement x of a Brownian particle that it receives under the influence of molecular impacts. Each particle constantly undergoes collisions with molecules, after which it changes the direction of its movement. Different particles receive displacements that differ in both magnitude and direction. The probable value of the sum of the displacements of all particles is zero, since the displacements can have both a positive and a negative sign with equal probability. The average value of the particle displacement projection x will therefore be equal to zero. However, the average value of the squared displacement will not be equal to zero, i.e. the value x since it does not change its sign when the sign of x changes. Let us therefore transform equation (7.2) so that it includes the quantity. To do this, we multiply both sides of this equation by
We use obvious identities:
Substituting these expressions into (7.3), we obtain:
This equality is valid for any particle and therefore it is also valid for the average values of the quantities included in it,
if averaging is carried out over a sufficiently large number of particles. Therefore you can write:
where is the average value of the square of the particle's displacement, the average value of the square of its velocity. As for the average value of the quantity included in the equality, it is equal to zero, since for a large number of particles both positive and negative values are equally often taken. Equation (7.2) therefore takes the form:
The value in this equation represents the average value of the square of the velocity projections on the axis. Since the movements of particles are completely chaotic, the average values of the squares of the velocity projections along all three coordinate axes must be equal to each other, i.e.
It is also obvious that the sum of these quantities must be equal to the average value of the square of the particle velocity
Hence,
Thus, the expression of interest to us, included in (7.4), is equal to:
The quantity is the average kinetic energy of a Brownian particle. Colliding with molecules of a liquid or gas, Brownian particles exchange energy with them and are in thermal equilibrium with the medium in which they move. Therefore, the average kinetic energy of the translational motion of a Brownian particle must be equal to the average kinetic energy of molecules
liquid (or gas), which, as we know, is equal to
and therefore
The fact that the average kinetic energy of a Brownian particle is equal (as for a gas molecule!) is of fundamental importance. Indeed, the basic equation (3.1) we derived earlier is valid for any particles that do not interact with each other and perform chaotic movements. Whether these are molecules invisible to the eye or much larger Brownian particles containing billions of molecules is indifferent. From a molecular kinetic point of view, a Brownian particle can be treated as a giant molecule. Therefore, the expression for the average kinetic energy of such a particle must be the same as for the molecule. The velocities of Brownian particles, of course, are incomparably lower, corresponding to their greater mass.
Let us now return to equation (7.4) and, taking into account (7.5), rewrite it
This equation is easy to integrate. Having denoted we get:
and after separating the variables, our equation becomes:
Integrating the left side of this equation from 0 to and the right side from to we get:
The value, as can be easily seen, is negligibly small under normal experimental conditions. Indeed, the dimensions of Brownian particles do not exceed cm, the viscosity of the liquid is usually close to the viscosity of water, i.e., approximately equal (in the system of units, the density of the substance of the particles is of the order of unity. Bearing in mind that the mass of the particle is equal to , we find that the exponent at is , that the value can be neglected. Consequently, if the period of time between successive observations of a Brownian particle exceeds which, of course, always occurs, then
For finite time intervals and corresponding displacements, equation (7.6) can be rewritten as:
The average value of the squared displacement of a Brownian particle over a period of time along the X axis, or any other axis, is proportional to this period of time.
Formula (7.7) allows you to calculate the average value of the square of displacements, and the average is taken over all particles participating in the phenomenon. But this formula is also valid for the average value of the square of many successive movements of a single particle over equal periods of time. From an experimental point of view, it is more convenient to observe precisely the movements of a single particle. Such observations were made by Perrin in 1909.
Perrin observed the movement of particles through a microscope, the eyepiece of which was equipped with a grid of mutually perpendicular lines that served as a coordinate system. Using a grid, Perrin marked on it the sequential positions of one of his favorite particles at certain time intervals (for example, 30 s). Having then connected the points marking the positions of the particle on the grid, he obtained a picture similar to that shown in Fig. 7. This figure shows both the displacements of the particle and their projections onto the axis
It should be borne in mind that the motion of a particle is much more complex than can be judged from Fig. 7, since positions are marked here at not too short intervals of time (about 30 s). If we reduce these gaps, it turns out that each straight line segment in the figure will unfold into the same complex zigzag trajectory as the entire figure. 7.
Since the constant can be determined from the situation equation.
Perrin's experiments had great importance for the final substantiation of the molecular kinetic theory.
Brownian motion
From Brownian motion (encyclopedia Elements)
In the second half of the twentieth century, a serious debate 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 that successfully describe observable physical phenomena and have no real physical basis. On the other hand, scientists of the new wave - in particular, Ludwig Boltzmann ( cm. Boltzmann's constant)—insisted that atoms were physical realities. And neither of the two sides realized that already decades before the start of their dispute, experimental results had been obtained that once and for all resolved the issue 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 flower pollen under a microscope (he studied the aqueous suspension of plant pollen Clarkia pulchella), suddenly discovered that individual spores make absolutely chaotic impulse movements. He determined for certain that these movements were in no way connected with the turbulence and currents of water, or with its evaporation, after which, having described the nature of the movement of particles, he honestly admitted his own powerlessness to explain the origin of this chaotic movement. However, being a meticulous experimenter, Brown established that such chaotic movement is characteristic of any microscopic particles - be it plant pollen, suspended minerals, or any crushed substance in general.
It was only in 1905 that none other than Albert Einstein first realized that this seemingly mysterious phenomenon served 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 chaotically moving water molecules. On average, molecules act on it from all sides with equal intensity and at equal intervals of time. However, no matter how small the spore is, due to purely random deviations, first it receives an impulse from the molecule that hit it on one side, then from the side of the molecule that hit it on the other, etc. As a result of averaging such collisions, it turns out that that at some moment the particle “twitches” in one direction, then, if on the other side it is “pushed” by more molecules, in the other, etc. Using the laws of mathematical statistics and the molecular kinetic theory of gases, Einstein derived the equation, describing the dependence of the root-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 (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 completely real movements of physical particles, it became finally clear that the debate about the reality of atoms is over: they exist in nature. As a “prize game,” Perrin received a 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 in 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 therefore on the number of molecules per unit volume of liquid. And this is nothing more than Avogadro's number (cm. Avogadro's Law) is one of the fundamental constants that determine the structure of our world.
From Brownian motion In any environment there are constant microscopic pressure fluctuations. They, acting on particles placed in the environment, lead to their random movements. This chaotic movement of tiny particles in a liquid or gas is called Brownian motion, and the particle itself is called Brownian.Brownian motion Brownian motion
(Brownian motion), the random movement of tiny particles suspended in a liquid or gas under the influence of molecular impacts environment; discovered by R. Brown.
BROWNIAN MOTIONBROWNIAN MOTION (Brownian motion), random movement of tiny particles suspended in a liquid or gas, occurring under the influence of impacts from environmental molecules; discovered by R. Brown (cm. BROWN Robert (botanist) in 1827
When observing a suspension of flower pollen in water under a microscope, Brown observed a chaotic movement of particles arising “not from the movement of the liquid or from its evaporation.” Suspended particles 1 µm in size or less, visible only under a microscope, performed disordered independent movements, describing complex zigzag trajectories. Brownian motion does not weaken with time and does not depend on the chemical properties of the medium; its intensity increases with increasing temperature of the medium and with a decrease in its viscosity and particle size. Even a qualitative explanation of the causes of Brownian motion was possible only 50 years later, when the cause of Brownian motion began to be associated with impacts of liquid molecules on the surface of a particle suspended in it.
The first quantitative theory of Brownian motion was given by A. Einstein (cm. EINSTEIN Albert) and M. Smoluchowski (cm. SMOLUCHOWSKI Marian) in 1905-06 based on molecular kinetic theory. It was shown that random walks of Brownian particles are associated with their participation in thermal motion along with the molecules of the medium in which they are suspended. Particles have on average the same kinetic energy, but due to their larger mass they have a lower speed. The theory of Brownian motion explains the random movements of a particle by the action of random forces from molecules and friction forces. According to this theory, the molecules of a liquid or gas are in constant thermal motion, and the impulses of different molecules are not the same in magnitude and direction. If the surface of a particle placed in such a medium is small, as is the case for a Brownian particle, then the impacts experienced by the particle from the molecules surrounding it will not be exactly compensated. Therefore, as a result of “bombardment” by molecules, the Brownian particle comes into random motion, changing the magnitude and direction of its speed approximately 10 14 times per second. From this theory it followed that by measuring the displacement of a particle over a certain time and knowing its radius and the viscosity of the liquid, one can calculate Avogadro’s number (cm. AVOGADRO CONSTANT).
The conclusions of the theory of Brownian motion were confirmed by measurements by J. Perrin (cm. PERRIN Jean Baptiste) and T. Svedberg (cm. Svedberg Theodor) in 1906. Based on these relations, the Boltzmann constant was experimentally determined (cm. BOLZMANN CONSTANT) and Avogadro's constant.
When observing Brownian motion, the position of the particle is recorded at regular intervals. The shorter the time intervals, the more broken the trajectory of the particle will look.
The laws of Brownian motion serve as a clear confirmation of the fundamental principles of molecular kinetic theory. It was finally established that the thermal form of motion of matter is due to the chaotic movement of atoms or molecules that make up macroscopic bodies.
The theory of Brownian motion played important role in the substantiation of statistical mechanics, the kinetic theory of coagulation is based on it aqueous solutions. In addition, she also has practical significance in metrology, since Brownian motion is considered as the main factor limiting the accuracy of measuring instruments. For example, the limit of accuracy of the readings of a mirror galvanometer is determined by the vibration of the mirror, like a Brownian particle bombarded by air molecules. The laws of Brownian motion determine the random movement of electrons, 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.
encyclopedic Dictionary. 2009 .
See what “Brownian motion” is in other dictionaries:
- (Brownian motion), the random movement of small particles suspended in a liquid or gas, occurring under the influence of impacts from environmental molecules. Explored in 1827 by England. scientist R. Brown (Brown; R. Brown), whom he observed through a microscope... ... Physical encyclopedia
BROWNIAN MOTION- (Brown), the movement of tiny particles suspended in a liquid, occurring under the influence of collisions between these particles and the molecules of the liquid. It was first noticed under an English microscope. botanist Brown in 1827. If in sight... ... Great Medical Encyclopedia
- (Brownian motion) random movement of tiny particles suspended in a liquid or gas under the influence of impacts from environmental molecules; discovered by R. Brown... Big Encyclopedic Dictionary
BROWNIAN MOTION, disordered, zigzag movement of particles suspended in a flow (liquid or gas). It is caused by the uneven bombardment of larger particles from different sides by smaller molecules of a moving flow. This… … Scientific and technical encyclopedic dictionary
Brownian motion- – oscillatory, rotational or forward movement particles of the dispersed phase under the influence of thermal movement of molecules of the dispersion medium. general chemistry: textbook / A. V. Zholnin ... Chemical terms
BROWNIAN MOTION- random movement of tiny particles suspended in a liquid or gas, under the influence of impacts from environmental molecules in thermal motion; plays an important role in some physical chem. processes, limits accuracy... ... Big Polytechnic Encyclopedia
Brownian motion- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN Brownian motion ... Technical Translator's Guide
This article or section needs revision. Please improve the article in accordance with the rules for writing articles... Wikipedia
Continuous chaotic movement of microscopic particles suspended in a gas or liquid, caused by the thermal movement of environmental molecules. This phenomenon was first described in 1827 by the Scottish botanist R. Brown, who studied under... ... Collier's Encyclopedia
More correct is Brownian motion, the random movement of small (several micrometers or less in size) particles suspended in a liquid or gas, occurring under the influence of shocks from the molecules of the environment. Discovered by R. Brown in 1827.… … Great Soviet Encyclopedia
Books
- Brownian motion of a vibrator, Yu.A. Krutkov, Reproduced in the original author's spelling of the 1935 edition (publishing house Izvestia of the USSR Academy of Sciences). IN… Category: Mathematics Publisher:
Brownian motion- random movement of microscopic visible particles of a solid substance suspended in a liquid or gas, caused by the thermal movement of the particles of the liquid or gas. Brownian motion never stops. Brownian motion is related to thermal motion, but these concepts should not be confused. Brownian motion is a consequence and evidence of the existence of thermal motion.
Brownian motion is the most clear experimental confirmation of the concepts of molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation period is large enough for the forces acting on the particle from the molecules of the medium to change their direction many times, then the average square of the projection of its displacement on any axis (in the absence of other external forces) is proportional to time.
When deriving Einstein's law, it is assumed that particle displacements in any direction are equally probable and that the inertia of a Brownian particle can be neglected compared to the influence of friction forces (this is acceptable for sufficiently long times). Formula for coefficient D is based on the application of Stokes' law for hydrodynamic resistance to the motion of a sphere of radius A in a viscous fluid. The relationships for A and D were experimentally confirmed by measurements by J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant was experimentally determined k and Avogadro's constant N A. In addition to translational Brownian motion, there is also rotational Brownian motion - the random rotation of a Brownian particle under the influence of impacts of molecules of the medium. For rotational Brownian motion, the root mean square angular displacement of the particle is proportional to the observation time. These relationships were also confirmed by Perrin's experiments, although this effect is much more difficult to observe than translational Brownian motion.
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Brownian motion occurs due to the fact that all liquids and gases consist of atoms or molecules - tiny particles that are in constant chaotic thermal motion, and therefore continuously push the Brownian particle from different directions. It was found that large particles with sizes greater than 5 µm practically do not participate in Brownian motion (they are stationary or sediment), smaller particles (less than 3 µm) move forward along very complex trajectories or rotate. When a large body is immersed in a medium, the shocks occurring in huge quantities are averaged and form a constant pressure. If a large body is surrounded by a medium on all sides, then the pressure is practically balanced, only the lifting force of Archimedes remains - such a body smoothly floats up or sinks. If the body is small, like a Brownian particle, then pressure fluctuations become noticeable, which create a noticeable randomly varying force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in the medium.
Opening
Brownian motion theory
Construction of the classical theory
D = R T 6 N A π a ξ , (\displaystyle D=(\frac (RT)(6N_(A)\pi a\xi )),)
Where D (\displaystyle D)- diffusion coefficient, R (\displaystyle R)- universal gas constant, T (\displaystyle T)- absolute temperature, N A (\displaystyle N_(A))- Avogadro's constant, a (\displaystyle a)- particle radius, ξ (\displaystyle \xi )- dynamic viscosity.
Experimental confirmation
Einstein's formula was confirmed by the experiments of Jean Perrin and his students in 1908-1909. As Brownian particles, they used grains of resin from the mastic tree and gum, the thick milky sap of trees of the genus Garcinia. The validity of the formula was established for various particle sizes - from 0.212 microns to 5.5 microns, for various solutions (sugar solution, glycerin) in which the particles moved.
Brownian motion as a non-Markov random process
The theory of Brownian motion, well developed over the last century, is an approximate one. And although in most practically important cases the existing theory gives satisfactory results, in some cases it may require clarification. Thus, experimental work carried out at the beginning of the 21st century in Polytechnic University Lausanne, the University of Texas and the European Molecular Biology Laboratory in Heidelberg (under the leadership of S. Jeney) 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 surrounding particles of the medium and showed a significant mutual influence of the movement of the Brownian particle and the movement of the particles of the medium caused by it on each other, that is, the presence of “memory” of the Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory her past behavior. This fact was not taken into account in the Einstein-Smoluchowski theory.
The process of Brownian motion of a particle in a viscous medium, generally speaking, belongs to the class of non-Markov processes, and for a more accurate description it is necessary to use integral stochastic equations.
Thermal movement
Any substance consists of tiny particles - molecules. Molecule- is the smallest particle of a given substance that retains all of it Chemical properties. Molecules are located discretely in space, i.e. at certain distances from each other, and are in a state of continuous disorderly (chaotic) movement .
Since bodies consist of a large number of molecules and the movement of molecules is random, it is impossible to say exactly how many impacts one or another molecule will experience from others. Therefore, they say that the position of the molecule and its speed at each moment of time are random. However, this does not mean that the movement of molecules does not obey certain laws. In particular, although the speeds of molecules at some point in time are different, most of them have speed values close to some specific value. Usually, when speaking about the speed of movement of molecules, they mean average speed (v$cp).
It is impossible to single out any specific direction in which all molecules move. The movement of molecules never stops. We can say that it is continuous. Such continuous chaotic movement of atoms and molecules is called -. This name is determined by the fact that the speed of movement of molecules depends on body temperature. The higher the average speed of movement of the molecules of a body, the higher its temperature. Conversely, the higher the body temperature, the greater the average speed of molecular movement.
The movement of liquid molecules was discovered by observing Brownian motion - the movement of very small particles of solid matter suspended in it. Each particle continuously makes abrupt movements in arbitrary directions, describing trajectories in the form of a broken line. This behavior of particles can be explained by considering that they experience impacts from liquid molecules simultaneously from different sides. The difference in the number of these impacts from opposite directions leads to the movement of the particle, since its mass is commensurate with the masses of the molecules themselves. The movement of such particles was first discovered in 1827 by the English botanist Brown, observing pollen particles in water under a microscope, which is why it was called - Brownian motion.
