The nature of thermal motion in crystals. Phonons. Aggregate states of matter. The nature of thermal motion in these states. Features of thermal motion in various aggregate states of matter In gases, the thermal motion of molecules has the character

"HEAT MOTION OF MOLECULES"

KSTU. Dept. Physics. Gaisin N.K., Kazantsev S.A., Minkin V.S., Samigullin F.M.

To move through the text you can use:

1- keypress PgDn, PgUp,,  to move through pages and lines;

2- pressing the left mouse button on the selectedtext to go to the required section;

3- click the left mouse button on the selected icon@ to go to heading

The nature of the thermal motion of molecules in different states. Average energies of molecules in different phases. Velocity distribution of molecules.

Diffusion. diffusion coefficient.

Simulation of the movement of molecules using a computer.

The exercise. Observation and analysis: 1-trajectories of motion of molecules in threestates of aggregation, 2-graphs of the distribution of molecules by velocities, 3-radial distribution functions, 4-diffusion coefficients.

@ 1. The nature of the thermal motion of molecules in different states. Average energies of molecules in different phases. Velocity distribution of molecules.

As you know, molecules and atoms in matter are constantly in motion, which has a random, chaotic character. Nevertheless, in each state of aggregation there are characteristic features of this movement, which largely determine the properties of various states. This is due to the fact that the intermolecular forces of interaction tend to bring the molecules closer together, while thermal chaotic motion prevents this, and such two tendencies in different states of aggregation give significantly different contributions to the nature of the motion of molecules. For a quantitative analysis of the influence of various contributions, one usually considers the value of the total average energy of the molecule and the contribution to this energy of the kinetic and potential components.

In gases, the average distance between molecules is greater than their size, the attractive forces are small, and the intensity of movement is significant, which does not allow the molecules to unite for a long time, and in the absence of a vessel, the molecules tend to fill all the available space. In gases, the potential energy of interaction is negative, the kinetic energy is large, so the total energy of the molecule is positive, and when expanding, the molecular system can do work on external systems. As a result, the molecules are distributed uniformly in space, spend more time at large distances (Fig. 1a) and move uniformly and rectilinearly without interaction. The interaction of molecules is of a short-term nature and occurs only when they collide, which leads to a significant change in the trajectory of motion.

In solids, the average distance between molecules is comparable to their size; therefore, the forces of attraction are very large, and even a relatively high intensity of motion does not allow the molecules to disperse over long distances. In this case, the negative potential energy of interaction is much greater than the kinetic energy; therefore, the total energy of the molecule is also negative, and significant work must be done to destroy the solid. Molecules in a solid are located at strictly defined distances from each other and oscillate around certain average positions, called the nodes of the crystal lattice (Fig. 1c).

In liquids, the distance between molecules is comparable to their size, the attractive forces are large, but the intensity of thermal motion is also large, which allows the molecules to move away from each other over long distances after some time. In liquids, the negative potential energy of interaction is comparable in magnitude to the kinetic energy, so the total energy of the molecule is close to zero, which allows the liquid to be easily deformed and occupy the available volume without separation under the action of even weak external forces. Molecules in a liquid are on average at certain distances close to each other and perform vibration-like movements around their average positions, which also move randomly in space (Fig. 1b).

Rice. 1. The nature of the movement of molecules in gases (a), liquids (b) and solids (c)

As a result of the interaction between molecules, the molecular system after some time, called the relaxation time, comes to an equilibrium state, characterized by: 1- a certain equation of state that relates the thermodynamic parameters of the substance; 2- a certain radial function characterizing the distribution of molecules in space; 3- Maxwell function characterizing the distribution of molecules by velocities ( Fig.2).

With each act of interaction of molecules with each other, their velocities change and, as a result, after some time, an equilibrium state is established in which the number of molecules dN, having a speed in a certain range of values dV remains constant and is determined by the Maxwell function F(V) according to the ratios

dN= N F(V) V, F(V)=4V 2 (m/2kT) 3/2 exp(-mV 2 /2kT).

The form of this function is shown in Fig. 2, it significantly depends on the temperature T and is characterized by the presence of a maximum, which indicates the presence of the most probable velocity V ver. As can be seen from the graphs (Fig. 2), there are molecules in the substance with any velocities, but the number of molecules with velocities in the dV range near the most probable will be the largest. The Maxwellian velocity distribution of molecules is typical for all states of aggregation, but the relaxation time to such a distribution is different for them, this is due to the difference in the interaction time of molecules in different phases.

Rice. 2. Maxwellian velocity distribution of molecules.

@ 2_Diffusion. diffusion coefficient.

Due to the thermal motion of molecules in a substance, diffusion occurs. Diffusion is the transfer of a substance from one part of the volume it occupies to another. This phenomenon is most pronounced in gases and liquids, in which the thermal motion of molecules is especially intense and possible over long distances.

Phenomenologically, diffusion is described by Fick's law, which establishes a relationship between the specific flux J i component i and the concentration gradient of this component of the substance n i

Specific diffusion flux J i is the number of molecules of the i component transferred per unit of time through the unit area of the cross section perpendicular to the direction of the flow of the substance, n i is the number density of the i component, D i is the diffusion coefficient, V 0 is the hydrodynamic velocity of matter. The diffusion coefficient in the SI system has the dimension m 2 With -1 . The minus sign in the Fick formula indicates that the diffusion flux is directed opposite to the direction of growth of the substance concentration. The Fick equation describes only the stationary diffusion process, in which the concentration, its gradient and diffusion flux do not depend on time.

The mechanism of diffusion in gases is discussed in detail in the section on molecular physics. The molecular-kinetic theory of gases leads to the well-known expression for the diffusion coefficient

where i is the mean free path and i - arithmetic mean translational velocity of gas molecules of type i, d i - effective diameter, m i - mass of molecules, n i - number density, p - pressure. This formula is observed in a fairly wide range of pressures and temperatures for non-dense gases and gives a value of the order of 10 -5 m 2 /s.

However, the diffusion of molecules in liquids differs significantly from diffusion in gases, this is due to the difference in the nature of the movement of molecules in these phases. The density of a substance in the liquid state is thousands of times greater than its density in the gaseous state. Therefore, in liquids, each molecule sits in a dense environment of neighboring molecules and does not have the freedom of translational movements as in gases. According to Frenkel's well-known theory, molecules in liquids, as well as in solids, perform random oscillations around equilibrium positions. These positions can be considered as potential wells created by the surrounding molecules. In crystals, molecules cannot leave their equilibrium positions, and therefore it can be considered that there are practically no translational movements of molecules in them. In liquids, these positions are not permanent. From time to time, molecules change their equilibrium positions, remaining in a dense environment of other molecules.

Diffusion of molecules in one-component liquids, due to their thermal motion in the absence of concentration gradients, is usually called self-diffusion of molecules. In order for the molecules, having overcome the interaction with the surrounding molecules, to make a transition to a new position, energy is needed. The minimum energy required for a molecule to leave the temporary potential well is called the activation energy. A molecule that has received such energy is called activated. Molecules that perform random vibrations are activated as a result of collisions with surrounding molecules. The activation energy in liquids is much less than in crystals. Therefore, the transitions of molecules in liquids from one place to another are much more frequent than in crystals. The number of activated molecules is determined by the Boltzmann distribution, and the frequency of transitions (jumps㿹 of molecules to new positions , is determined by the approximate formula
, where 0 is a coefficient weakly dependent on temperature, E is the activation energy.

To obtain the formula for the diffusion coefficient for a liquid, consider the diffusion flux through a certain surface with an area s. During thermal motion, molecules pass through this surface in both forward and backward directions. Therefore, the specific diffusion flux can be expressed as
, where the signs correspond to the forward and reverse directions of the axis X. Find the quantities J+ and J-- . Obviously, only those molecules that are at a distance from it at a distance not further than the average length of the jump of molecules can pass through the selected surface in one jump without deflection. δ . Construct on both sides of the surface a cylinder with a base area s. Through the surface s only those molecules that are enclosed in the volume of the cylinder will pass δ s. However, not all molecules will pass, but only those of them whose jumps are directed along the axis X. If we assume that the molecules move with equal probability along the axes x, y and z, then only 1/6 of the total number of molecules in the cylinder will pass through the cross section in this direction. Then the number of molecules passing in one jump through the surface s in the forward direction N + will be expressed as
, where n 1 - the number of molecules per unit volume at a distance δ to the left of the surface s. A similar argument about the passage of molecules through a surface s in the opposite direction will lead to the expression
, where n 2 - the number of molecules per unit volume at a distance δ to the right of the surface s. Then the diffusion fluxes can be found as and . The total flow will be expressed as

, where n 1 -n 2 is the difference in the concentrations of molecules in layers spaced from each other at an average distance δ can be written as n 1 -n 2 =nx. Then we get
. Comparing this formula with Fick's law for the case when V 0 =0, we find

where
, where
- coefficient weakly dependent on temperature, this formula for liquids and dense gases gives a value for D about 10 -9 m 2 / s.

The phenomenon of self-diffusion of molecules can also be analyzed by considering the thermal translational motion of molecules as a series of random, equiprobable displacements (wanderings). For some sufficiently long period of time, the molecules can describe a long trajectory, but they will shift from their original position by an insignificant distance. Consider a set of molecules in the form of randomly moving particles, select a certain molecule from this set and assume that it is at the origin of the coordinate system at the initial moment of time. Further at regular intervals Δt we will mark the radius vectors of its location r(t i ) . Molecule displacement vector between ( i-1m i–m moments of time will be expressed as Δ r i = r(t i )- r(t i -1 ). By the time t To = k Δt the molecule will be displaced from the initial point of observation to the point with the radius vector r(t To ) , which is expressed as the vector sum of the offsets  r(t To ) =  r i. The square of the displacement of the particle during this time is expressed as

r (t To ) = (Δ r i) 2 =
(Δ r i Δ r j) + Δ r i 2 .

Let us average the resulting expression over all the molecules of the considered set, then, due to the independence of the displacements of molecules at different time intervals, both positive and negative values of the scalar product are equally often found in the double sum, therefore its statistical average is zero. Then the average square of particle displacement is written as< r 2 (t k)> = <Δ r i 2>. in liquid<Δ r i 2 > should be considered equal to the mean square of the molecular jump δ 2 , and the number of jumps in time t k equal to  t k . Then< r 2 (t k)>= t k δ 2 . Comparing this expression with the formula for D, we obtain the well-known Einstein relation, from which the molecular-kinetic meaning of the diffusion coefficient becomes clear D

< r 2 (t)> = 6Dt.

It can be proved that the diffusion coefficients in the Einstein and Fick formulas are identical. For a single-component system, this coefficient is called the self-diffusion coefficient; in the case of diffusion in multicomponent mixtures, if they have concentration gradients, the fluxes of individual components can be determined if the diffusion coefficients of all components in the mixture are known. Experimentally, they are found by methods of radioactive labels or by the method of nuclear magnetic resonance, in which it is possible to determine the mean square of the displacement of "labeled" molecules.

@ 3_Modeling the movement of molecules using a computer.

Modern computer technology has a huge memory and high speed. Such qualities make them an indispensable tool for modeling a number of physical processes. In molecular physics, the method of molecular dynamics is widely developed - a method for modeling molecular motion. This method is widely used in gases, liquids, crystals and polymers. It is reduced to the numerical solution of the equations of the dynamics of particle motion in a limited volume of space, taking into account the interactions between them, and can simulate the behavior of molecules under arbitrary conditions, similar to real ones. In this respect, it can be likened to a real experiment, so such simulation is sometimes called a numerical experiment. The significance of these “experiments” lies in the fact that they make it possible to follow the change in time of several macroscopic parameters characterizing a system of particles, and averaging them over time or over the number of particles, to obtain the thermodynamic parameters of simulated real systems. In addition, they provide the ability to visualize molecular motion, allowing you to follow the trajectory of any single particle.

The simulation algorithm consists of several stages. Initially, a certain number of particles (within 10 2 -10 3) are randomly distributed in a certain limited volume (in a cell), randomly setting the initial velocities and coordinates of each particle. The initial velocities of the particles are set so that the average kinetic energy of the translational motion of the particles is equal to (3/2) kT, i.e. corresponded to the temperature of the experiment, and the initial coordinates are set in accordance with the average intermolecular distance of the simulated system.

Further, knowing the particle interaction potential (for example, the Lennard-Jones potential) and, accordingly, the intermolecular interaction force, the resulting instantaneous forces acting on each particle from all other particles are calculated, and the instantaneous particle accelerations caused by by the action of these forces. Knowing the accelerations, as well as the initial coordinates and velocities, the velocities and coordinates of the particles are calculated at the end of a given short time interval  t(usually 10 -14 s). With an average particle velocity of about 10 3 m/s, the displacement of particles over such a short period of time is about 10 -11 m, which is much less than their size.

Successive repetition of such calculations with memorization of the instantaneous forces, velocities and coordinates of the particles makes it possible to know the coordinates and velocities of the entire system of particles over a sufficiently long time interval. The limited volume is taken into account by special boundary conditions. Either it is assumed that at the boundary of a given volume, the particle experiences an absolutely elastic collision with the wall and returns to the volume again, or it is assumed that the given cell is surrounded on all sides by the same cells, and if the particle leaves the given cell, then the particle identical to it enters at the same time. from the opposite cell. Thus, the number of particles and their total energy in the volume of the cell do not change. Due to the mathematically random nature of the initial distribution of particles over velocities and coordinates, it takes some time (relaxation time -10 -12 - 10 -11 s), during which the equilibrium state of particles is established in the system in terms of velocities (Maxwellian distribution of velocities) and in coordinates (distribution according to the radial distribution function).

The values of macroscopic parameters characterizing the system are calculated by averaging them over the trajectory or over the particle velocities. For example, the pressure on the walls of a vessel can be obtained by averaging the changes in the momenta of particles colliding with the boundaries of the cell. By averaging the number of particles in spherical layers located at different distances r from the chosen molecule, one can determine the radial distribution function. From the mean squares of particle displacements over a given time, it is possible to calculate the coefficients of self-diffusion of molecules. Other desired characteristics are determined in a similar way.

Naturally, the processes occurring in a system of particles in a short time are calculated by a computer in a considerable time. Computer time spent on calculations can be tens or even hundreds of hours. This depends on the number of particles chosen in the cell and on the speed of the computer. Modern computers make it possible to simulate the dynamics of up to 10 4 particles, bringing the time of monitoring the process of their movements to 10 -9 s, the accuracy of calculating the characteristics of the systems under study allows not only to refine the theoretical provisions, but also to use them in practice.

@ 4_Exercise. Observation and analysis: 1-trajectories of the movement of molecules in three states of aggregation, 2-graphs of the distribution of molecules by velocities, 3-radial distribution functions, 4-self-diffusion coefficients.

In this exercise, the computer program simulates the movement of argon atoms (with the Lennard-Jones interaction potential) in three aggregate states using the molecular dynamics method: dense gas, liquid, solid. To perform this exercise, you must enter the MD-L4.EXE program, sequentially view and perform the proposed menu items.

The program menu contains four items:

1 INSTRUCTIONS FOR WORK,

2 SELECTION OF PARAMETERS OF SIMULATED STATES,

3 MODELING THE DYNAMICS OF PARTICLES,

4 END OF WORK.

In paragraph 1-<<ИНСТРУКЦИЯ ДЛЯ РAБОТЫ>> tells about the program and about the method of working with the program. It is necessary to note and remember: 1) This program provides for work in two modes to perform two types of work required when simulating molecular motion in different phases; 2) Simulation results are displayed on two screens, switching between which is done by pressing the keys simultaneously alt+1 and alt+2 , stop the program and exit to the menu by simultaneously pressing the keys ctrl and S; 3) For the correct execution of the program, it is necessary to follow its messages and execute them correctly.

In point 2, the program works in the mode<<ВЫБОР ПAРAМЕТРОВ МОДЕЛИРУЕМЫХ СОСТОЯНИЙ>> , which allows us to consider the phase diagram for a system of particles with the Lennard-Jones interaction potential and calculate the following parameters for various aggregate states: reduced pressure P*=Pd 3 /e and reduced total energy of one particle U*=u/e. 3des: n-number density, u-internal energy of one particle, k-Boltzmann constant, P-pressure, T-temperature, d-effective particle diameter, e-depth of the potential well. For the calculation, it is necessary to consider the phase diagram in the coordinates n*, T* (n*=nd 3 - reduced number density, T*=kT/e - reduced temperature) and enter n*, T*. On this phase diagram, you need to find areas: dense gas, liquid, solid states and enter n*, T* for three points in each of these areas. To analyze the influence of temperature, it is necessary to select points with different temperatures, but with the same densities (T* and n* can be taken from Table N1). Enter the thermodynamic parameters of these points for three states chosen by you and calculated by the program in Table N1, for these points you will simulate the motion of argon atoms.

In menu item 3, the program works in the mode<<МОДЕЛИРОВAНИЕ ДИНAМИКИ ЧAСТИЦ>> , it allows us to consider the picture of the motion of molecules in different states of aggregation and calculate a number of thermodynamic parameters by averaging. After selecting (using the additional menu) the type of aggregate state being simulated (dense gas, liquid, solid body), the program will offer you the parameters of this state embedded in the program, if you have chosen other parameters, then they can be changed at this stage according to Table N1 ( for this on request<<ВЫ БУДЕТЕ МЕНЯТЬ ПЛОТНОСТЬ И ТЕМПЕРAТУРУ? (Y/N)>> press Y, otherwise press N) . In this mode, information about the dynamics is displayed on two screens, to turn them on, you need to press alt and 1 or alt and 2 .

The first screen displays data on the system and graphs of fluctuations for: 1-temperature, 2-potential energy of the particle, 3-kinetic energy, 4-total energy of the particle. In addition, instant additional numerical information is displayed in the running line: Ni-current number of iteration steps, t(c)-physical time of dynamics simulation, EP+EK(J)-total energy of one particle, U*-reduced energy, T(K) - temperature, ti (c) - computer time for counting one step for one particle, P* - reduced pressure, Pv (Pa) - pressure (verial), P = nkT, dt (c) - time integration step.

The second screen displays particle trajectories and graphs of characteristics obtained by averaging the dynamic parameters of particle motion: 1-graphs of the particle velocity distribution against the background of the Maxwell distribution (Vver - the most probable velocity, given and average temperatures); 2-plot of the radial distribution function, 3-plot of the mean square particle displacement versus time and the value of the self-diffusion coefficient.

After starting the program, you need to observe the changes in the characteristics and wait for the moment when the fluctuations of the potential and kinetic become sufficiently small (5-10%). This state can be considered equilibrium, it is automatically achieved by the program by conducting the dynamics for 2.10 -12 s, after which the radial distribution function and the velocity distribution function will correspond to the equilibrium ones. After reaching the equilibrium state (after approximately 1.10 -11 s.), it is necessary to enter the required data from both screens into Table N2. Perform similar calculations for three temperatures in each state of aggregation; for the last temperature, draw the velocity distribution function and the radial distribution function.

After finishing work through point 4-<<КОНЕЦ РAБОТЫ>> it is necessary to return to work with the methodological manual.

Prepare in your notebook Table N1, Table N2.

Table N1. Parameters of three simulated phase states of argon.

"HEAT MOTION OF MOLECULES"

KSTU. Dept. Physics. Gaisin N.K., Kazantsev S.A., Minkin V.S., Samigullin F.M.

To move through the text you can use:

1- keypress PgDn, PgUp,,  to move through pages and lines;

2- pressing the left mouse button on the selectedtext to go to the required section;

3- click the left mouse button on the selected icon@ to go to heading

The nature of the thermal motion of molecules in different states. Average energies of molecules in different phases. Velocity distribution of molecules.

Diffusion. diffusion coefficient.

Simulation of the movement of molecules using a computer.

@ 1. The nature of the thermal motion of molecules in different states. Average energies of molecules in different phases. Velocity distribution of molecules.

Rice. 1. The nature of the movement of molecules in gases (a), liquids (b) and solids (c)

dN= N F(V) V, F(V)=4V 2 (m/2kT) 3/2 exp(-mV 2 /2kT).

Rice. 2. Maxwellian velocity distribution of molecules.

@ 2_Diffusion. diffusion coefficient.

The mechanism of diffusion in gases is discussed in detail in the section on molecular physics. The molecular-kinetic theory of gases leads to the well-known expression for the diffusion coefficient

where
, where
- coefficient weakly dependent on temperature, this formula for liquids and dense gases gives a value for D about 10 -9 m 2 / s.

r (t To ) = (Δ r i) 2 =
(Δ r i Δ r j) + Δ r i 2 .

< r 2 (t)> = 6Dt.

@ 3_Modeling the movement of molecules using a computer.

The program menu contains four items:

1 INSTRUCTIONS FOR WORK,

2 SELECTION OF PARAMETERS OF SIMULATED STATES,

3 MODELING THE DYNAMICS OF PARTICLES,

4 END OF WORK.

After finishing work through point 4-<<КОНЕЦ РAБОТЫ>> it is necessary to return to work with the methodological manual.

Prepare in your notebook Table N1, Table N2.

Table N1. Parameters of three simulated phase states of argon.

Thermal motion of molecules.
The most convincing fact is the Brownian motion of molecules. Brownian motion of molecules confirms the chaotic nature of thermal motion and the dependence of the intensity of this motion on temperature. For the first time, the random movement of small solid particles was observed by the English botanist R. Brown in 1827, examining solid particles suspended in water - spores of the club moss. Draw students' attention to the fact that the movement of disputes occurs along straight lines that make up a broken line. Since then, the motion of particles in a liquid or gas has been called Brownian. Carry out a standard demonstration experiment "Observation of Brownian motion" using a round box with two glasses.

By changing the temperature of a liquid or gas, for example, by increasing it, one can increase the intensity of Brownian motion. A Brownian particle moves under the influence of molecular impacts. The explanation for the Brownian motion of a particle is that the impacts of liquid or gas molecules on the particle do not cancel each other out. The quantitative theory of Brownian motion was developed by Albert Einstein in 1905. Einstein showed that the mean square of the displacement of a Brownian particle is proportional to the temperature of the medium, depends on the shape and size of the particle, and is directly proportional to the observation time. The French physicist J. Perrin conducted a series of experiments that quantitatively confirmed the theory of Brownian motion.

Calculation of the number of impacts on the vessel wall. Consider an ideal monatomic gas in equilibrium in a vessel of volume V. Let us single out molecules with a velocity from v to v + dv. Then the number of molecules moving in the direction of the angles  and  with these velocities will be equal to:

dN v,, = dN v d/4. (14.8)

Let us single out an elementary surface with area dP., which we will take as part of the vessel wall. For a unit of time, this area will be reached by molecules enclosed in an oblique cylinder with a base dП and a height v cos  (see Fig. 14.3). The number of intersections of the selected surface by the molecules chosen by us (the number of impacts on the wall) per unit time d v,, will be equal to the product of the concentration of molecules and the volume of this oblique cylinder:

d v,, = dП v cos  dN v,, /V, (14.9)
where V is the volume of the vessel containing the gas.

Integrating expression (14.9) over the angles within the solid angle 2, which corresponds to a change in the angles  and  in the range from 0 to /2 and from 0 to 2, respectively, we obtain a formula for calculating the total number of impacts of molecules with velocities from v to v + dv against the wall.

Integrating the expression over all velocities, we obtain that the number of impacts of molecules on a wall with area dP per unit time will be equal to:

. (14.11)

Taking into account the definition of the average speed, we obtain that the number of hits of molecules on the wall of a unit area per unit time will be equal to:

= N/V /4 = n /4.

The Boltzmann distribution, i.e. the distribution of particles in an external potential field, can be used to determine the constants used in molecular physics. One of the most important and famous experiments in this area is Perrin's work on Avogadro's number. Since gas molecules are not visible even through a microscope, much larger Brownian particles were used in the experiment. These particles were placed in a solution in which a buoyant force acted on them. In this case, the force of gravity acting on the Brownian particles decreased, and thus the distribution of particles along the height seemed to be stretched. This made it possible to observe this distribution under a microscope.

One of the difficulties was to obtain suspended particles of exactly the same size and shape. Perrin used particles of gum and mastic. Rubbing gummigut in water. Perrin received a bright yellow emulsion, in which, when observed under a microscope, many spherical granules could be distinguished. Instead of mechanical grinding, Perrin also treated gum or mastic with alcohol, which dissolved these substances. When such a solution was diluted with a large amount of water, an emulsion was obtained from the same spherical grains as during mechanical grinding of gum. To select grains of exactly the same size, Perrin subjected particles suspended in water to repeated centrifugation and in this way obtained a very homogeneous emulsion consisting of spherical particles with a radius of the order of a micrometer. Having processed 1 kg of gummigut, Perrin received a fraction containing several decigrams of grains of the desired size after a few months. With this fraction, the experiments described here were carried out.

When studying the emulsion, it was necessary to make measurements at negligible height differences - only a few hundredths of a millimeter. Therefore, the height distribution of particles was studied using a microscope. A very thin glass with a wide hole drilled in it was glued to a microscope slide (shown in the figure). In this way, a flat bath (Zeiss (1816-1886) cuvette) was obtained, the height of which was about 100 µm (0.1 mm). A drop of emulsion was placed in the center of the bath, which was immediately flattened with a cover slip. To avoid evaporation, the edges of the coverslip were covered with paraffin or varnish. Then the drug could be observed for several days or even weeks. The preparation was placed on the stage of the microscope carefully set in a horizontal position. The lens was of very high magnification with a shallow depth of focus, so that only particles inside a very thin horizontal layer with a thickness of the order of a micrometer could be seen at a time. The particles performed intense Brownian motion. By focusing the microscope on a certain horizontal emulsion layer, it was possible to count the number of particles in this layer. Then the microscope was focused on another layer, and again the number of visible Brownian particles was counted. In this way it was possible to determine the ratio of the concentrations of Brownian particles at different heights. The height difference was measured with a micrometer screw of the microscope.

Now let's move on to specific calculations. Since Brownian particles are in the field of gravity and Archimedes, the potential energy of such a particle

In this formula, p is the density of the gum, p is the density of the liquid, V is the volume of the gum particle. The reference point for the potential energy is chosen at the bottom of the cell, that is, at h = 0. We write the Boltzmann distribution for such a field in the form

n(h) = n0e kT = n0e kT . Recall that n is the number of particles per unit volume at height h, and n0 is the number of particles per unit volume at height h = 0.

The number of balls AN visible through the microscope at height h is equal to n(h)SAh, where S is the area of the visible part of the emulsion, and Ah is the depth of field of the microscope (in Perrin's experiment, this value was 1 μm). Then we write the ratio of the numbers of particles at two heights h1 and h2 as follows:

AN1 = ((p-p")Vg(h2 _ h1) - exp

Calculating the logarithm of both sides of the equation and performing simple calculations, we obtain the value of the Boltzmann constant, and then the Avogadro number:

k(p_p")Vg(h2 _ h1)

When working under various conditions and with various emulsions, Perrin obtained the values of the Avogadro constant in the range from 6.5 1023 to 7.2 1023 mol-1. This was one of the direct proofs of the molecular-kinetic theory, in the validity of which at that time not all scientists believed.

Average energy of molecules.

The atoms and molecules that make up various substances are in a state of continuous thermal motion.

The first feature of thermal motion is its randomness; no direction of molecular motion is distinguished from other directions. Let us explain this: if we follow the movement of one molecule, then over time, due to collisions with other molecules, the magnitude of the velocity and direction of movement of this molecule change completely randomly; further, if at some point in time we fix the speeds of movement of all molecules, then in direction these speeds turn out to be evenly scattered in space, and in magnitude they have a wide variety of values.

The second feature of thermal motion is the existence of energy exchange between molecules, as well as between different types of motion; the energy of the translational motion of molecules can be converted into the energy of their rotational or vibrational motion and vice versa.

The exchange of energy between molecules, as well as between different types of their thermal motion, occurs due to the interaction of molecules (collisions between them). At large distances, the interaction forces between molecules are very small and can be neglected; at small distances, these forces have a noticeable effect. In gases, molecules spend most of their time at comparatively large distances from one another; only for very short periods of time, being close enough to each other, they interact with each other, changing the speed of their movements and exchanging energies. Such short-term interactions of molecules are called collisions. There are two types of collisions between molecules:

1) collisions, or impacts, of the first kind, as a result of which only the velocities and kinetic energies of the colliding particles change; the composition or structure of the molecules themselves do not experience any change;

2) collisions, or impacts, of the second kind, as a result of which changes occur within the molecules, for example, their composition or the relative arrangement of atoms within these molecules changes. During these collisions, part of the kinetic energy of the molecules is spent on doing work against the forces acting inside the molecules. In some cases, on the contrary, a certain amount of energy can be released due to a decrease in the internal potential energy of molecules.

In what follows, we will have in mind only collisions of the first kind that occur between gas molecules. Energy exchange during thermal motions in solid and liquid bodies is a more complex process and is considered in special sections of physics. Collisions of the second kind are used to explain the electrical conductivity of gases and liquids, as well as the thermal radiation of bodies.

To describe each type of thermal motion of molecules (translational, rotational or vibrational), it is necessary to set a number of quantities. For example, for the translational motion of a molecule, it is necessary to know the magnitude and direction of its velocity. For this purpose, it is enough to indicate three quantities: the value of the speed and two angles and between the direction of the speed and the coordinate planes, or three projections of the speed on the coordinate axes: (Fig. 11.1, a). Note that these three quantities are independent: for given angles and can have any values, and vice versa, for a given, for example, angle, values and can be any. Similarly, setting a specific value does not impose any restrictions on the values in reverse. Thus, to describe the translational motion of a molecule in space, it is necessary to set three quantities independent of each other: and or The energy of the translational motion of a molecule will consist of three independent components:

To describe the rotational motion of a molecule around its axis, it is necessary to indicate the magnitude and direction of the angular velocity of rotation, i.e., again, three quantities independent of each other: and c or (Fig. II. 1, b). The energy of rotational motion of a molecule will also consist of three independent components:

where are the moments of inertia of the molecule about three mutually perpendicular coordinate axes. For a monatomic molecule, all these moments of inertia are very small, so the energy of its rotational motion is neglected. For a diatomic molecule (Fig. II.1, c), the energy of rotational motion about the axis passing through the centers of atoms is neglected, therefore, for example,

To describe the vibrational motion of atoms in a molecule, one must first divide this motion into simple vibrations that occur along certain directions. It is convenient to decompose a complex oscillation into simple rectilinear oscillations occurring in three mutually perpendicular directions. These oscillations are independent of each other, that is, the frequency and amplitude of oscillations in one of these directions can correspond to any frequency and amplitude of oscillations in other directions. If each of these rectilinear oscillations is harmonic, then it can be described using the formula

Thus, to describe an individual rectilinear vibration of atoms, it is necessary to set two quantities: the vibration frequency ω and the vibration amplitude. These two quantities are also independent of each other: at a given frequency, the vibration amplitude is not bound by any conditions, and vice versa. Consequently, to describe the complex vibrational motion of a molecule around a point (i.e., its equilibrium position), it is necessary to set six quantities independent of each other: three frequencies and vibration amplitudes in three mutually perpendicular directions.

Quantities independent of each other that determine the state of a given physical system are called the degrees of freedom of this system. When studying thermal motion in bodies (to calculate the energy of this motion), the number of degrees of freedom of each molecule of this body is determined. In this case, only those degrees of freedom between which an energy exchange occurs are counted. A monatomic gas molecule has three degrees of freedom of translational motion; a diatomic molecule has three degrees of freedom of translational and two degrees of freedom of rotational motion (the third degree of freedom, corresponding to rotation around an axis passing through the centers of atoms, is not taken into account). Molecules containing three

an atom and more, have three translational and three rotational degrees of freedom. If oscillatory motion also participates in the energy exchange, then two degrees of freedom are added for each independent rectilinear vibration.

Considering separately the translational, rotational and vibrational motions of molecules, one can find the average energy that falls on each degree of freedom of these types of motion. Let us first consider the translational motion of molecules: let us assume that a molecule has kinetic energy (the mass of a molecule). The sum is the energy of the translational motion of all molecules. Dividing by degrees of freedom, we obtain the average energy per one degree of freedom of the translational motion of molecules:

It is also possible to calculate the average energies per one degree of freedom of the rotational and oscillatory movements. If each molecule has translational degrees of freedom, rotational degrees of freedom, and vibrational degrees of freedom, then the total energy of thermal motion of all molecules will be equal to

Topic: Forces of intermolecular interaction. Aggregate

state of matter. The nature of the thermal motion of molecules in solid,

liquid and gaseous bodies and its change with increasing temperature.

Thermal expansion of tel. Phase transitions. Heat phase

transitions. Phase balance.

Intermolecular interaction is electrical in nature. Between them

forces of attraction and repulsion act, which quickly decrease with increasing

distances between molecules.

Repulsive forces act only at very small distances.

In practice, the behavior of a substance and its state of aggregation is determined by what is dominant: attractive forces or chaotic thermal motion.

Solids are dominated by interaction forces, so they retain their shape. The interaction forces depend on the shape and structure of the molecules, so there is no single law for their calculation.

However, if we imagine that the molecules have a spherical shape, the general nature of the dependence of the interaction forces on the distance between molecules –r is shown in Figure 1-a. Figure 1-b shows the dependence of the potential energy of the interaction of molecules on the distance between them. At a certain distance r0 (it is different for different substances) Fattract.= Fretract. The potential energy is minimal, at rr0 the repulsive forces predominate, and at rr0 it is vice versa.

Figure 1-c shows the transition of the kinetic energy of molecules into potential energy during their thermal motion (for example, vibrations). In all figures, the origin of coordinates is aligned with the center of one of the molecules. Approaching another molecule, its kinetic energy transforms into potential energy and reaches its maximum value at distances r=d. d is called the effective diameter of the molecules (the minimum distance that the centers of two molecules approach.

It is clear that the effective diameter depends, among other things, on temperature, since at a higher temperature the molecules can come closer together.

At low temperatures, when the kinetic energy of the molecules is small, they are attracted close and settled in a certain order - a solid state of aggregation.

Thermal motion in solids is mainly oscillatory. At high temperatures, intense thermal motion prevents the molecules from approaching each other - the gaseous state, the movement of molecules is translational and rotational .. In gases, less than 1% of the volume falls on the volume of the molecules themselves. At intermediate temperatures, the molecules will continuously move in space, exchanging places, but the distance between them is not much greater than d - liquid. The nature of the movement of molecules in a liquid is oscillatory and translational (at the moment when they jump to a new equilibrium position).

The thermal motion of molecules explains the phenomenon of thermal expansion of bodies. When heated, the amplitude of the vibrational motion of molecules increases, which leads to an increase in the size of the bodies.

The linear expansion of a rigid body is described by the formula:

l l 0 (1 t), where is the coefficient of linear expansion 10-5 K-1. The volumetric expansion of bodies is described by a similar formula: V V0 (1 t), is the coefficient of volumetric expansion, and =3.

The substance can be in solid, liquid, gaseous states. These states are called aggregate states of matter. Matter can change from one state to another. A characteristic feature of the transformation of a substance is the possibility of the existence of stable inhomogeneous systems, when a substance can be in several states of aggregation at once.

When describing such systems, a broader concept of the phase of matter is used. For example, carbon in a solid state of aggregation can be in two different phases - diamond and graphite. The phase is the totality of all parts of the system, which in the absence of external influence is physically homogeneous. If several phases of a substance at a given temperature and pressure exist in contact with each other, and at the same time the mass of one phase does not increase due to a decrease in the other, then one speaks of phase equilibrium.

The transition of a substance from one phase to another is called a phase transition. During a phase transition, an abrupt (occurring in a narrow temperature range) qualitative change in the properties of a substance occurs. These transitions are accompanied by an abrupt change in energy, density, and other parameters. There are phase transitions of the first and second kind. Phase transitions of the first kind include melting, solidification (crystallization), evaporation, condensation, and sublimation (evaporation from the surface of a solid body). Phase transitions of this kind are always associated with the release or absorption of heat, called the latent heat of the phase transition.

During phase transitions of the second kind, there is no abrupt change in energy and density. The heat of the phase transition is also equal to 0. Transformations during such transitions occur immediately in the entire volume as a result of a change in the crystal lattice at a certain temperature, which is called the Curie point.

Consider a transition of the first kind. When the body is heated, as noted, there is a thermal expansion of the body and, as a consequence, a decrease in the potential energy of particle interaction. A situation arises when, at a certain temperature, the relationship between potential and kinetic energies cannot ensure the equilibrium of the old phase state and the substance passes into a new phase.

Melting is the transition from a crystalline state to a liquid state. Q=m, specific heat of fusion, shows how much heat is needed to convert 1 kg of solid to liquid at the melting point, measured in J/kg. During crystallization, the released amount of heat is calculated using the same formula. Melting and crystallization occur at a specific temperature for a given substance, called the melting point.

Evaporation. Molecules in a liquid are bound by attractive forces, but some of the fastest molecules can leave the bulk of the liquid. In this case, the average kinetic energy of the remaining molecules decreases and the liquid cools. To maintain evaporation, it is necessary to supply heat: Q=rm, r is the specific heat of vaporization, which shows how much heat must be spent to transfer 1 kg of liquid to a gaseous state at a constant temperature.

Unit: J/kg. During condensation, heat is released.

The calorific value of the fuel is calculated by the formula: Q=qm.

Under conditions of mechanical and thermal equilibrium, the states of inhomogeneous systems are determined by setting pressure and temperature, since these parameters are the same for each part of the system. Experience shows that when two phases are in equilibrium, pressure and temperature are interconnected by a dependence that is a phase equilibrium curve.

The points lying on the curve describe an inhomogeneous system in which there are two phases. The points lying inside the regions describe homogeneous states of matter.

If the curves of all phase equilibria of one substance are built on a plane, then they will divide it into separate regions, and they themselves will converge at one point, which is called the triple point. This point describes the state of matter in which all three phases can coexist. In Figure 2, diagrams of the state of water are constructed.