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

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

Brownian motion

We have got new format! You can now listen to the article.

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

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

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

where D- coefficient diffusion of a Brownian particle. For spherical particle radius a: (T- abs. temp-ra, - dynamic. medium viscosity). When deriving Einstein's law, it is assumed that particle displacements in any direction are equally probable and that the inertia of a Brownian particle can be neglected compared to the effect of friction forces (this is acceptable for sufficiently large ones). F-la for the coefficient. D based on the application stokes law for hydrodynamic resistance to the movement of a sphere with a radius a in a viscous liquid. Relations for and D were experimentally confirmed by the measurements of J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant is experimentally determined k and Avogadro constant N A.

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

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

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

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

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

Brownian motion

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 that successfully describe the observed physical phenomena and have 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, it would go to 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 rms 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 the 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 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 “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 at each given moment of 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- chaotic movement of microscopic particles of solid matter visible suspended in a liquid or gas, caused by the thermal movement of particles of a liquid or gas. Brownian motion never stops. Brownian motion is related to thermal motion, but these concepts should not be confused. Brownian motion is a consequence and evidence of the existence of thermal motion.

Brownian motion is the most obvious experimental confirmation of the ideas of the molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation interval is large enough so that the forces acting on the particle from the molecules of the medium change their direction many times, then the average square of the projection of its displacement on any axis (in the absence of other external forces) is proportional to time.

When deriving Einstein's law, it is assumed that particle displacements in any direction are equally probable and that the inertia of a Brownian particle can be neglected compared to the effect 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 ratios for A and D were experimentally confirmed by the measurements of J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant is experimentally determined k and the Avogadro constant N A. 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.

Encyclopedic YouTube

• 1 / 5

  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, then the shocks that occur in large numbers are averaged and form a constant pressure. If a large body is surrounded by a medium from all sides, then the pressure is practically balanced, only the lifting force Archimedes remains - such a body smoothly floats up or sinks. If the body is small, like a Brownian particle, then pressure fluctuations become noticeable, which create a noticeable randomly changing force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in a medium.

  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)) is the Avogadro 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 the resin of the mastic tree and gummigut, the thick milky juice 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-Markovian random process

  The theory of Brownian motion, well developed over the last century, is approximate. And although in most cases of practical importance the existing theory gives satisfactory results, in some cases it may require clarification. Thus, experimental work carried out at the beginning of the 21st century at the Polytechnic University of Lausanne, the University of Texas and the European Molecular Biology Laboratory in Heidelberg (under the direction of S. Dzheney) showed the difference in the behavior of a Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable when increase in particle size. The studies also touched upon the analysis of the movement of the surrounding particles of the medium and showed a significant mutual influence of the movement of the Brownian particle and the movement of the particles of the medium caused by it on each other, that is, the presence of a "memory" in the Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory her behavior in the past. This fact was not taken into account in the Einstein-Smoluchowski theory.

  The process of Brownian motion of a particle in a viscous medium, generally speaking, belongs to the class of non-Markovian processes, and for its more accurate description it is necessary to use integral stochastic equations.

  What is Brownian motion

  This movement is characterized by the following features:

  • continues indefinitely without any visible change,
  • the intensity of motion of Brownian particles depends on their size, but does not depend on their nature,
  • intensity increases with increasing temperature,
  • the intensity increases with decreasing viscosity of the liquid or gas.

  Brownian motion is not molecular motion, but serves as direct evidence for the existence of molecules and the chaotic nature of their thermal motion.

  The essence of Brownian motion

  The essence of this movement is as follows. A particle together with liquid or gas molecules form one statistical system. In accordance with the theorem on the uniform distribution of energy over degrees of freedom, each degree of freedom accounts for 1/2kT of energy. The energy 2/3kT per three translational degrees of freedom of a particle leads to the motion of its center of mass, which is observed under a microscope in the form of particle trembling. If a Brownian particle is sufficiently rigid, then another 3/2kT of energy is accounted for by its rotational degrees of freedom. Therefore, with its trembling, it also experiences constant changes in orientation in space.

  It is possible to explain Brownian motion in the following way: the cause of Brownian motion is fluctuations of pressure, which is exerted on the surface of a small particle by the molecules of the medium. Force and pressure change in modulus and direction, as a result of which the particle is in random motion.

  The motion of a Brownian particle is a random process. The probability (dw) that a Brownian particle, which was in a homogeneous isotropic medium at the initial time (t=0) at the origin, will shift along an arbitrarily directed (at t$>$0) axis Ox so that its coordinate will lie in the interval from x to x+dx is equal to:

  where $\triangle x$ is a small change in the particle's coordinate due to fluctuation.

  Consider the position of a Brownian particle at some fixed time intervals. We place the origin of coordinates at the point where the particle was at t=0. Let $\overrightarrow(q_i)$ denote the vector that characterizes the movement of the particle between (i-1) and i observations. After n observations, the particle will move from the zero position to the point with the radius vector $\overrightarrow(r_n)$. Wherein:

  \[\overrightarrow(r_n)=\sum\limits^n_(i=1)(\overrightarrow(q_i))\left(2\right).\]

  The movement of the particle occurs along a complex broken line all the time of observation.

  Let's find the average square of the removal of the particle from the beginning after n steps in a large series of experiments:

  \[\left\langle r^2_n\right\rangle =\left\langle \sum\limits^n_(i,j=1)(q_iq_j)\right\rangle =\sum\limits^n_(i=1) (\left\langle (q_i)^2\right\rangle )+\sum\limits^n_(i\ne j)(\left\langle q_iq_j\right\rangle )\left(3\right)\]

  where $\left\langle q^2_i\right\rangle $ is the mean square of the particle displacement at the i-th step in a series of experiments (it is the same for all steps and is equal to some positive value a2), $\left\langle q_iq_j\ right\rangle $- is the average value of the dot product when i-th step on displacement at the j-th step in various experiments. These quantities are independent of each other, both positive and negative values ​​of the scalar product are equally common. Therefore, we assume that $\left\langle q_iq_j\right\rangle $=0 for $\ i\ne j$. Then we have from (3):

  \[\left\langle r^2_n\right\rangle =a^2n=\frac(a^2)(\triangle t)t=\alpha t=\left\langle r^2\right\rangle \left( 4\right),\]

  where $\triangle t$ is the time interval between observations; t=$\triangle tn$ - the time during which the mean square of particle removal became equal to $\left\langle r^2\right\rangle .$ We get that the particle is moving away from the origin. It is essential that the average square of the removal grows in proportion to the first power of time. $\alpha \ $- can be found experimentally, or theoretically, as will be shown in example 1.

  The Brownian particle moves not only forward, but also rotating. The average value of the rotation angle $\triangle \varphi $ of a Brownian particle over time t is:

  \[(\triangle \varphi )^2=2D_(vr)t(5),\]

  where $D_(vr)$ is the rotational diffusion coefficient. For a spherical Brownian particle of radius - a $D_(vr)\ $ is equal to:

  where $\eta $ is the viscosity coefficient of the medium.

  Brownian motion limits accuracy measuring instruments. The limit of accuracy of a mirror galvanometer is determined by the trembling of the mirror, like a Brownian particle that is hit by air molecules. The random movement of electrons causes noise in electrical networks.

  Example 1

  Task: In order to mathematically fully characterize Brownian motion, you need to find $\alpha $ in the formula $\left\langle r^2_n\right\rangle =\alpha t$. Consider the viscosity coefficient of the liquid known and equal to b, the temperature of the liquid T.

  Let us write down the equation of motion of a Brownian particle in projection onto the Ox axis:

  where m is the mass of the particle, $F_x$ is the random force acting on the particle, $b\dot(x)$ is the term of the equation characterizing the friction force acting on the particle in the fluid.

  Equations for quantities related to other coordinate axes have a similar form.

  We multiply both sides of equation (1.1) by x, and transform the terms $\ddot(x)x\ and\ \dot(x)x$:

  \[\ddot(x)x=\ddot(\left(\frac(x^2)(2)\right))-(\dot(x))^2,\dot(x)x=(\frac (x^2)(2)\)(1.2)\]

  Then equation (1.1) is reduced to the form:

  \[\frac(m)(2)(\ddot(x^2))-m(\dot(x))^2=-\frac(b)(2)\left(\dot(x^2) \right)+F_xx\ (1.3)\]

  We average both sides of this equation over an ensemble of Brownian particles, taking into account that the average of the time derivative is equal to the derivative of the average value, since this is averaging over an ensemble of particles, and, therefore, we rearrange it by the operation of differentiation with respect to time. As a result of averaging (1.3), we obtain:

  \[\frac(m)(2)\left(\left\langle \ddot(x^2)\right\rangle \right)-\left\langle m(\dot(x))^2\right\rangle =-\frac(b)(2)\left(\dot(\left\langle x^2\right\rangle )\right)+\left\langle F_xx\right\rangle \ \left(1.4\right). \]

  Since the deviations of a Brownian particle in any direction are equally probable, then:

  \[\left\langle x^2\right\rangle =\left\langle y^2\right\rangle =\left\langle z^2\right\rangle =\frac(\left\langle r^2\right \rangle )(3)\left(1.5\right)\]

  Using $\left\langle r^2_n\right\rangle =a^2n=\frac(a^2)(\triangle t)t=\alpha t=\left\langle r^2\right\rangle $, we get $\left\langle x^2\right\rangle =\frac(\alpha t)(3)$, hence: $\dot(\left\langle x^2\right\rangle )=\frac(\alpha ) (3)$, $\left\langle \ddot(x^2)\right\rangle =0$

  Due to the random nature of the force $F_x$ and the particle coordinate x and their independence from each other, the equality $\left\langle F_xx\right\rangle =0$ must hold, then (1.5) reduces to the equality:

  \[\left\langle m(\dot(\left(x\right)))^2\right\rangle =\frac(\alpha b)(6)\left(1.6\right).\]

  According to the theorem on the uniform distribution of energy over degrees of freedom:

  \[\left\langle m(\dot(\left(x\right)))^2\right\rangle =kT\left(1.7\right).\] \[\frac(\alpha b)(6) =kT\to \alpha =\frac(6kT)(b).\]

  Thus, we obtain a formula for solving the problem of Brownian motion:

  \[\left\langle r^2\right\rangle =\frac(6kT)(b)t\]

  Answer: The formula $\left\langle r^2\right\rangle =\frac(6kT)(b)t$ solves the problem of the Brownian motion of suspended particles.

  Example 2

  Task: Gummigut particles of spherical shape with radius r participate in Brownian motion in gas. Density of gummigut $\rho $. Find the root-mean-square velocity of gum particles at temperature T.

  The root-mean-square velocity of molecules is:

  \[\left\langle v^2\right\rangle =\sqrt(\frac(3kT)(m_0))\left(2.1\right)\]

  A Brownian particle is in equilibrium with the matter in which it is located, and we can calculate its root-mean-square velocity using the formula for the velocity of gas molecules, which, in turn, move the Brownian particle. First, let's find the mass of the particle:

  \[\left\langle v^2\right\rangle =\sqrt(\frac(9kT)(4\pi R^3\rho ))\]

  Answer: The speed of a particle of gum suspended in a gas can be found as $\left\langle v^2\right\rangle =\sqrt(\frac(9kT)(4\pi R^3\rho ))$.