In the world around us, various kinds of physical phenomena occur that are directly related to change in body temperature. Since childhood, we have known that cold water, when heated, first becomes barely warm and only after a certain time becomes hot.
With such words as “cold”, “hot”, “warm”, we define different degrees of “heating” of bodies, or, in the language of physics, different temperatures of bodies. The temperature of warm water is slightly higher than the temperature of cool water. If you compare the temperature of summer and winter air, the difference in temperature is obvious.
Body temperature is measured using a thermometer and expressed in degrees Celsius (°C).
As is known, diffusion occurs faster at higher temperatures. It follows from this that the speed of movement of molecules and temperature are deeply interrelated. If you increase the temperature, the speed of movement of molecules will increase, if you decrease it, it will decrease.
Thus, we conclude: body temperature directly depends on the speed of movement of molecules.
Hot water consists of exactly the same molecules as cold water. The difference between them is only in the speed of movement of the molecules.
Phenomena that relate to heating or cooling of bodies and temperature changes are called thermal. These include heating or cooling air, melting metal, and melting snow.
Molecules, or atoms, which are the basis of all bodies, are in endless chaotic motion. The number of such molecules and atoms in the bodies around us is enormous. A volume equal to 1 cm³ of water contains approximately 3.34 · 10²² molecules. Any molecule has a very complex trajectory of movement. For example, gas particles moving at high speeds in different directions can collide with each other and with the walls of the container. Thus, they change their speed and continue moving again.
Figure 1 shows the random movement of paint particles dissolved in water.
Thus, we draw another conclusion: The chaotic movement of particles that make up bodies is called thermal motion.
Chaoticity is the most important feature of thermal motion. One of the most important proofs of molecular motion is diffusion and Brownian motion.(Brownian motion is the movement of tiny solid particles in a liquid under the influence of molecular impacts. As observation shows, Brownian motion cannot stop).
In liquids, molecules can vibrate, rotate, and move relative to other molecules. If we take solids, then their molecules and atoms vibrate around certain average positions.
Absolutely all molecules of the body participate in the thermal movement of molecules and atoms, which is why with a change in thermal movement, the state of the body itself and its various properties also change. Thus, if you increase the temperature of ice, it begins to melt, taking on a completely different form - ice becomes liquid. If, on the contrary, you lower the temperature of, for example, mercury, then it will change its properties and turn from a liquid into a solid.
T 
The body temperature directly depends on the average kinetic energy of the molecules. We draw an obvious conclusion: the higher the temperature of a body, the greater the average kinetic energy of its molecules. And, conversely, as the body temperature decreases, the average kinetic energy of its molecules decreases.
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"THERMAL MOTION OF MOLECULES"
KSTU. Caf. Physicists. Gaisin N.K., Kazantsev S.A., Minkin V.S., Samigullin F.M.
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The nature of the thermal motion of molecules in different states. Average energies of molecules in different phases. Distribution of molecules by speed.
Diffusion. Diffusion coefficient.
Simulation of molecular motion using a computer.
Exercise. Observation and analysis: 1-trajectories of molecular motion in threestates of aggregation, 2 graphs of molecular velocity distribution, 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. Distribution of molecules by speed.
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 the various states. This is due to the fact that intermolecular interaction forces tend to bring molecules closer together, and thermal chaotic movement prevents this, and such two trends in different states of aggregation make significantly different contributions to the nature of the movement of molecules. To quantitatively analyze the influence of various contributions, the value of the total average energy of the molecule and the contribution of the kinetic and potential components to this energy are usually considered.
In gases, the average distance between molecules is larger than their sizes, 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 during expansion the molecular system can do work on external systems. As a result, the molecules are distributed evenly in space, spend more time at large distances (Fig. 1a) and move uniformly and rectilinearly without interaction. The interaction of molecules is short-term and occurs only when they collide, which leads to a significant change in the trajectory of movement.
In solids, the average distance between molecules is comparable to their sizes, therefore the attractive forces are very strong and even the relatively high intensity of movement does not allow the molecules to move apart over long distances. In this case, the negative potential energy of interaction is much greater than the kinetic energy, so 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 perform oscillatory movements around certain average positions, called crystal lattice nodes (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 large distances from each other after some time. In liquids, the negative potential energy of interaction is comparable in magnitude to kinetic energy, so the total energy of the molecule is close to zero, which allows the liquid to easily deform and occupy the available volume without separation under the influence 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 average positions, which also move chaotically 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 connecting the thermodynamic parameters of the substance; 2- a certain radial function characterizing the distribution of molecules in space; 3-Maxwell function characterizing the velocity distribution of molecules ( Fig.2).
With each act of interaction of molecules with each other, their speeds change and as a result, after some time, an equilibrium state is established in which the number of molecules dN having speed in a certain range of values dV remains constant and is determined by the Maxwell function F(V) according to the relations
dN= N F(V) V, F(V)=4V 2 (m/2kT) 3/2 exp(-mV 2 /2kT).
The form of this function is shown in Fig. 2, it significantly depends on temperature T and is characterized by the presence of a maximum, which indicates the presence of the most probable speed V ver. As can be seen from the graphs (Fig. 2), there are molecules in the substance with any speeds, but the number of molecules with speeds in the dV range around the most likely one will be the greatest. The Maxwellian velocity distribution of molecules is characteristic of 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 movement of molecules in a substance, diffusion occurs. Diffusion is the phenomenon of 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 movement of molecules is especially intense and possible over long distances.
Phenomenologically, diffusion is described by Fick's law, which establishes a connection 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 component i transferred per unit time through a unit cross-sectional area perpendicular to the direction of flow of the substance, n i is the numerical density of component i, D i – diffusion coefficient, V 0 – hydrodynamic speed of matter. The diffusion coefficient in the SI system has the dimension m 2 With -1 . The minus sign in Fick's formula indicates that the diffusion flow is directed opposite to the direction of increase in the concentration of the substance. The Fick equation describes only a 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 average free path and 
i is the arithmetic average speed of translational motion of gas molecules of type i, d i is the effective diameter, m i is the mass of molecules, n i is the numerical density, p is 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 movement as in gases. According to the well-known theory of Frenkel, molecules in liquids, as in solids, undergo random vibrations around equilibrium positions. These positions can be thought of as potential wells created by surrounding molecules. In crystals, molecules cannot leave their equilibrium positions, and therefore we can assume that there are practically no translational movements of molecules in them. In liquids such positions are not constant. From time to time, molecules change their equilibrium positions, remaining in a dense environment of other molecules.
The diffusion of molecules in one-component liquids, caused by their thermal motion in the absence of concentration gradients, is usually called self-diffusion of molecules. In order for molecules, having overcome the interaction with surrounding molecules, to make a transition to a new position, energy is needed. The minimum energy that is required for a molecule to leave a temporary potential well is called activation energy. A molecule that has received such energy is called activated. Molecules that vibrate randomly are activated by collisions with surrounding molecules. The activation energy in liquids is much lower than in crystals. Therefore, 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
- coefficient weakly dependent on temperature, E– activation energy.
To obtain the formula for the diffusion coefficient for a liquid, consider the diffusion flow through a certain surface of area s. During thermal motion, molecules pass through this surface in both forward and reverse directions. Therefore, the specific diffusion flux can be expressed as 
, where the signs correspond to the forward and reverse directions of the axis X. Let's find the quantities J+ and J-- . Obviously, only those molecules that are located from it at a distance no further than the average length of the molecular jump can pass through a selected surface in one jump without deflection δ
. Let's construct a cylinder on both sides of the surface with a base area s. Through the surface s Only those molecules that are contained in the volume of the cylinder will pass through δ
s. However, not all molecules will pass, but only those whose jumps are directed along the axis X. If we assume that molecules move with equal probability along the x, y and z, then only 1/6 of the total number of molecules in the cylinder will pass through the cross section in a given direction. Then the number of molecules passing through the surface s in the forward direction N + in one jump will be expressed as 
, Where
n 1 – number of molecules per unit volume at a distance δ
to the left of the surface s. Similar reasoning about the passage of molecules through a surface s in the opposite direction will result in the expression 
, Where
n 2 – 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 apart at an average distance
δ
can be written as n 1 -n 2 =nx. Then we get 
. Comparing this formula with Fick's law for the case when V 0 =0, we find
,
where 
, Where 
- a coefficient that weakly depends on temperature; this formula for liquids and dense gases gives the 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, equally probable movements (walks). Over some fairly large period of time, the molecules can describe a long trajectory, but they will shift from their original position to an insignificant distance. Let us consider a collection of molecules in the form of randomly moving particles, select a certain molecule from this collection and assume that at the initial moment of time it is at the origin of the coordinate system. Then at regular intervals Δt we will mark the radius vectors of its location r(t i )
. The vector of movement of the molecule between ( i-1m i–th moments of time will be expressed in the form 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 a point with a radius vector r(t To )
, which is expressed as the vector sum of the displacements 
r(t To )
=
r i. The squared displacement of the particle during this time will be 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 molecules of the population under consideration; 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 common in the double sum, so its statistical average is equal to zero. Then the mean square of particle displacement will be 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 per time t k equal 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 proven 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 with concentration gradients, the fluxes of individual components can be determined if the diffusion coefficients of all components in the mixture are known. They are found experimentally using radioactive labeling methods or the nuclear magnetic resonance method, in which the mean square displacement of “labeled” molecules can be determined.
@ 3_Modeling the movement of molecules using a computer.
Modern computer technology has enormous 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 simulating molecular motion. This method is widely used in gases, liquids, crystals and polymers. It comes down 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 in arbitrary conditions, similar to real ones. In this respect, it can be likened to a real experiment, which is why such a simulation is sometimes called a numerical experiment. The significance of these “experiments” is that they make it possible to monitor changes in several macroscopic parameters over time that characterize a system of particles, and by averaging them over time or over the number of particles, 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 individual particle.
The modeling algorithm consists of several stages. First, 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) CT, 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.
Next, knowing the interaction potential of particles (for example, the Lennard-Jones potential) and, accordingly, the force of intermolecular interaction, the resulting instantaneous forces acting on each particle from all other particles are calculated, and the instantaneous accelerations of particles caused by , the action of these forces. Knowing the accelerations, as well as the initial coordinates and velocities, the velocities and coordinates of particles are calculated at the end of a given short period of time t(usually 10 -14 s). With an average particle speed of about 10 3 m/s, the displacement of particles in such a short period of time is on the order of 10 -11 m, which is significantly less than their size.
Consecutive repetition of such calculations with storage of instantaneous forces, velocities and coordinates of particles allows one to know the coordinates and velocities of the entire system of particles over a sufficiently large period of time. The limited volume is taken into account by special boundary conditions. Either it is believed 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 believed that a given cell is surrounded on all sides by the same cells and, if a particle leaves a given cell, then at the same time a particle identical to it enters from the opposite cell. Thus, the number of particles and their total energy in the cell volume do not change. Due to the mathematically random nature of the initial distribution of particles in velocities and coordinates, some time is required (relaxation time –10 -12 - 10 -11 s), during which an equilibrium state of particles is established in the system in velocities (Maxwellian velocity distribution) and in coordinates (distribution according to the radial distribution function).
The values of macroscopic parameters characterizing the system are calculated by averaging them along the trajectory or over the velocities of the particles. For example, the pressure on the walls of a vessel can be obtained by averaging changes in the momenta of particles colliding with the boundaries of the cell. By averaging the number of particles in spherical layers located at various distances r from a selected molecule, the radial distribution function can be determined. From the mean squares of particle displacements over a given time, the self-diffusion coefficients of molecules can be calculated. Other required characteristics are determined in a similar way.
Naturally, processes occurring in a system of particles in a short time are calculated by a computer over a considerable period of time. The computer time spent on calculations can amount to tens or even hundreds of hours. This depends on the number of particles selected 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, increasing the time of observation of the process of their movements to 10 -9 s; the accuracy of calculating the characteristics of the systems under study allows not only to clarify theoretical positions, but also to use them in practice.
@ 4_Exercise. Observation and analysis: 1-trajectories of movement of molecules in three states of aggregation, 2-graphs of the distribution of molecules by speed, 3-radial distribution functions, 4-self-diffusion coefficients.
In this exercise, a computer program simulates using molecular dynamics the motion of argon atoms (with the Lennard-Jones interaction potential) in three states of aggregation: dense gas, liquid, solid. To complete this exercise, you must enter the MD-L4.EXE program, sequentially view and execute the proposed menu items.
The program menu contains four items:
1 OPERATING INSTRUCTIONS
2 SELECTION OF PARAMETERS OF SIMULATED STATES,
3 SIMULATION OF PARTICLE DYNAMICS,
4 END OF WORK.
In point 1-<<ИНСТРУКЦИЯ ДЛЯ РAБОТЫ>> tells about the program and the methodology for working with the program. It is necessary to note and remember: 1) This program provides work in two modes to perform two types of work necessary when modeling molecular motion in different phases; 2) The simulation results are displayed on two screens, switching between which is done by simultaneously pressing keys Alt+1 And Alt+2 , stopping the program and exiting to the menu is done by simultaneously pressing the keys Ctrl And S; 3) To execute a program correctly, you must follow its messages and execute them correctly.
In step 2 the program operates in 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 states of aggregation: reduced pressure P*=Pd 3 /e and reduced total energy of one particle U*=u/e. 3here: n-number density, u-internal energy of one particle, k-Boltzmann constant, P-pressure, T-temperature, d-effective diameter of the particle, e-depth of the potential well. To calculate, it is necessary to consider the phase diagram in coordinates n*, T* (n*=nd 3 - reduced numerical density, T*=kT/e - reduced temperature) and enter n*, T*. On this phase diagram you need to find the regions: dense gas, liquid, solid states and enter n*, T* for three points in each of these regions. 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, selected by you and calculated by the program, into Table N1; for these points you will simulate the movement of argon atoms.
In menu item 3 the program operates in mode<<МОДЕЛИРОВAНИЕ ДИНAМИКИ ЧAСТИЦ>>, it allows you to consider the picture of the movement of molecules in different states of aggregation and calculate a number of thermodynamic parameters by averaging. After selecting (using an additional menu) the type of modeled state of aggregation (dense gas, liquid, solid), the program will offer you the parameters of this state included in the program; if you have selected other parameters, they can be changed at this stage according to Table N1 ( for this purpose upon 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 about the system and fluctuation graphs for: 1-temperature, 2-potential energy of the particle, 3-kinetic energy, 4-total energy of the particle. In addition, instantaneous additional numerical information is displayed in the creeping 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, t i (c)-computer time for calculating 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 particle velocity distribution against the background of the Maxwell distribution (Vver - the most probable speed, given and average temperatures); 2-graph of the radial distribution function, 3-graph of the dependence of the mean square of particle displacement on time and the value of the self-diffusion coefficient.
After starting the program, you need to observe changes in characteristics and wait until the potential and kinetic fluctuations become sufficiently small (5-10%). This state can be considered equilibrium; it is achieved automatically by the program by carrying out dynamics for 2.10 -12 s, after which the radial distribution function and the velocity distribution function will correspond to equilibrium. 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. Carry out similar calculations for three temperatures in each state of aggregation; for the last temperature, sketch the velocity distribution function and the radial distribution function.
After finishing work through step 4-<<КОНЕЦ РAБОТЫ>> you need to return to working with the manual.
Prepare Table N1, Table N2 in your notebook.
Table N1. Parameters of three simulated phase states of argon.
The nature of thermal motion in crystals. Crystal structure is an equilibrium state of a system of atoms that corresponds to a minimum of potential energy. At rest, the sum of the forces acting on each atom of the crystal from other atoms is zero. Atoms in crystals vibrate around fixed equilibrium positions. The nature of these thermal fluctuations is very complex. The particle interacts with neighboring particles, that is, vibrations are transmitted from atom to atom and propagate in the crystal in the form of a wave.
Due to the fact that each atom is strongly connected with its neighbors, it cannot move on its own, alone - it forces its neighbors to move in time with itself. As a result, microscopic movement in a crystal must be imagined not as the movement of individual atoms, but as certain collective, synchronous vibrations of a large number of atoms. Such vibrations are called phonons. It is phonons that, as physicists say, are the true degrees of freedom in a crystalline solid. In terms of phonons, one can describe sound waves, the heat capacity of a crystal, the superconductivity of some materials, and, finally, a wide variety of microscopic phenomena in a crystal.
Incoherent, i.e. uncorrelated, independent phonons are always present in a crystal. They have very different wavelengths, propagate in very different directions, overlap each other - and as a result only lead to small, chaotic vibrations of individual atoms. However, if we now create a large number of coherent phonons (that is, phonons of the same type - with the same wavelength, moving in the same direction in the same phase), we will get a monochromatic deformation wave propagating throughout the crystal. Each vibration corresponds to one phonon state with momentum and energy, k is the wave vector
So, the vibrations of the atoms of the crystal are replaced by the propagation of a system of sound waves in the substance, the quanta of which are phonons. The phonon spin is zero (in units). A phonon is a boson and is described by Bose-Einstein statistics. Phonons and their interaction with electrons play a fundamental role in modern ideas about the physics of superconductors, thermal conductivity processes, and scattering processes in solids. The model of a metal crystal can be represented as a set of harmonically interacting oscillators, and the greatest contribution to their average energy is made by low-frequency oscillations corresponding to elastic waves, the quanta of which are phonons. sound waves Spinbosons Bose-Einstein statistics interaction with electrons of thermal conductivity superconductors
According to quantum mechanics, lattice vibrations can be associated with quasiparticles - phonons. The minimum portion of energy that a crystal lattice can absorb or emit during thermal vibrations corresponds in this figure to a transition from one energy level to another. It is equal to h ν and is the phonon energy. Thus, an analogy can be drawn between light and thermal vibrations of a crystal lattice - elastic waves are considered as the propagation of certain quasi-elastic particles - phonons.
A phonon, unlike ordinary particles, can only exist in a certain medium that is in a state of thermal excitation. It is impossible to imagine a phonon that would propagate in a vacuum, since it describes the quantum nature of thermal vibrations of the lattice and is forever locked in the crystal. The corpuscular aspect of small vibrations of atoms in a crystal lattice leads to the concept of a phonon, and the propagation of elastic thermal waves in a crystal can be considered as the transfer of phonons.
The theory of thermal waves in a crystal was developed by Debye. The quantum nature of thermal waves, i.e. their discreteness manifests itself at a temperature called the characteristic Debye temperature, where is the maximum frequency of thermal vibrations of particles, k is Boltzmann’s constant. The quantity is called the Debye energy. For most solids, the Debye temperature is 100 K. Therefore, almost all solids under ordinary conditions do not exhibit quantum features. The Debye temperature is one of the most important characteristics of a crystal.
In solid state physics, the concept of a phonon gas, i.e., a large number of independent quasiparticles located in the volume of a solid body, is widely used. When thermal energy is absorbed by a solid body, the intensity of atomic vibrations increases. The internal energy of a solid consists of the energy of the ground state of the lattice and the energy of phonons. According to Debye's theory, the excited state of the lattice can be represented as an ideal gas of phonons moving freely in the volume of the crystal. In a certain temperature range, a phonon gas is similar to an ideal gas.
Heat capacity of the crystal. Classical theory. The heat capacity of a solid body with a volume V means the value U - internal energy, which is the sum of the vibrational motion of particles located at the nodes of the crystal lattice, and the potential energy of their interaction.
According to classical statistical mechanics, the average energy of a harmonic oscillator is equal, and it accounts for the kinetic energy and the same amount for the potential energy. A mole of a substance in a crystal lattice contains N A free particles, has 3N A degrees of freedom and has energy
Then, in a crystal, the heat capacity at constant volume differs little from the heat capacity at constant pressure, so we can simply put it and talk about the heat capacity of a solid. This statement is called the law of Dulong and Petit. The law is true in a certain temperature range and is not valid at low temperatures.
Heat capacity of the crystal. Quantum theory. Einstein's model. Einstein identified a crystal lattice of N atoms with a system of 3N independent harmonic oscillators. Assuming that the distribution of oscillators over states with different energies obeys Boltzmann’s law, we can find the average oscillator energy. Heat capacity of the crystal. Quantum theory. Debye model. At low temperatures, Einstein's model only qualitatively predicts the change in heat capacity. The discrepancy between the experimental data and Einstein's theory was eliminated by Debye. He took into account that a solid body has a whole spectrum of frequencies. Einstein's idea that all oscillators have the same oscillation frequency is overly simplistic.
A new phenomenon in condensed matter, the “jumping” of phonons from one solid body to another through a void, is described. Due to it, a sound wave can overcome thin vacuum gaps, and heat can be transferred through a vacuum billions of times more efficiently than with ordinary thermal radiation.
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One of the most important parameters characterizing a molecule is the minimum potential interaction energy. The attractive forces acting between molecules tend to condense the substance, i.e., bring its molecules closer to a distance r 0, when their potential interaction energy is minimal and equal to, but this approach is prevented by the chaotic thermal movement of the molecules. The intensity of this movement is determined by the average kinetic energy of the molecule, which is of the order kT, Where k– Boltzmann constant. The aggregative states of a substance significantly depend on the ratio of quantities and kT.
Let us assume that the temperature of the system of molecules under consideration is so high that
kT>> In this case, intense chaotic thermal motion prevents the forces of attraction from connecting molecules into aggregates of several particles that come close to a distance r 0: during collisions, the large kinetic energy of molecules will easily break these aggregates into component molecules and, thus, the probability of the formation of stable aggregates will be arbitrarily small. Under these circumstances, the molecules in question will obviously be in a gaseous state.
If the temperature of the particle system is very low, i.e. kT << молекулам, действующими силами притяжения, тепловое движение не может помешать приблизиться друг к другу на расстояние близкое к r 0 in a specific order. In this case, the system of particles will be in a solid state, and the small kinetic energy of thermal motion will force the molecules to perform random small vibrations around certain equilibrium positions (crystal lattice nodes).
Finally, at the temperature of the particle system, determined from the approximate equality kT≈ kinetic energy of thermal motion of molecules, the value of which is approximately equal to the potential energy of attraction, will not be able to move the molecule to a distance significantly exceeding r 0 . Under these conditions, the substance will be in a liquid aggregate state.
Thus, a substance, depending on its temperature and the size inherent in a given type of its constituent molecules, will be in a gaseous, solid or liquid state.
Under normal conditions, the distance between molecules in a gas is tens of times (see example 1.1) greater than their sizes; Most of the time they move in a straight line without interaction, and only a much smaller part of the time, when they are at close distances from other molecules, interact with them, changing the direction of their movement. Thus, in the gaseous state, the movement of a molecule looks as shown schematically in Fig. 7, A.
In the solid state, each molecule (atom) of a substance is in an equilibrium position (crystal lattice node), near which it makes small vibrations, and the direction (for example, ah" in Fig. 7, b) and the amplitude of these oscillations randomly changes (for example, in the direction bb") after a time significantly greater than the period of these oscillations; The vibration frequencies of molecules are generally not the same. The vibrations of an individual molecule of a solid body are shown in general terms in Fig. 7, b.
The molecules of a solid are packed so tightly that the distance between them is approximately equal to their diameter, i.e. distance r 0 in Fig. 3. It is known that the density of the liquid state is approximately 10% less than the density of the solid state, all other things being equal. Therefore, the distance between the molecules of the liquid state is slightly greater r 0 . Considering that in the liquid state molecules also have greater kinetic energy of thermal motion, it should be expected that, unlike the solid state, they can, performing oscillatory motion, quite easily change their location, moving a distance not significantly exceeding the diameter of the molecule. The trajectory of a liquid molecule looks approximately as shown schematically in Fig. 7, V. Thus, the motion of a molecule in a liquid combines translational motion, as occurs in a gas, with vibrational motion, as occurs in a solid.
[Physics test 24] Forces of intermolecular interaction. Aggregate state of matter. The nature of the thermal motion of molecules in solid, liquid, gaseous bodies and its change with increasing temperature. Thermal expansion of bodies. Linear expansion of solids when heated. Volumetric thermal expansion of solids and liquids. Transitions between states of aggregation. Heat of phase transition. Phase equilibrium. Heat balance equation.
Forces of intermolecular interaction.
Intermolecular interaction is electrical in nature. Between themthere are forces of attraction and repulsion, which quickly decrease with increasingdistances between molecules.Repulsive forces actonly at very short distances.Practical behavior of matter andits physical statedetermined by what isdominant: forces of attractionor chaotic thermal motion.Forces dominate in solidsinteractions, so theyretains its shape.
Aggregate state of matter.
- the ability (solid) or inability (liquid, gas, plasma) to maintain volume and shape,
- the presence or absence of long-range (solid) and short-range order (liquid), and other properties.
Thermal motion in solids is mainly vibrational. At high
temperatures, intense thermal movement prevents molecules from approaching each other - gaseous
state, the movement of molecules is translational and rotational. . In gases less than 1% volume
accounts for the volume of the molecules themselves. At intermediate temperatures
molecules will continuously move in space, exchanging places, however
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 in nature (at the moment when they
jump to a new equilibrium position).
Thermal expansion of bodies.
The thermal movement of molecules explains the phenomenon of thermal expansion of bodies. At
heating, the amplitude of the vibrational motion of molecules increases, which leads to
increase in body size.
Linear expansion of solids when heated.
The linear expansion of a solid body is described by the formula: L=L0(1+at), where a is the coefficient of linear expansion ~10^-5 K^-1.
Volumetric thermal expansion of solids and liquids.
The volumetric expansion of bodies is described by a similar formula: V = V0(1+Bt), B is the coefficient of volumetric expansion, and B = 3a.
Transitions between states of aggregation.
A substance can be in solid, liquid, or gaseous states. These
states are called aggregate states of matter. The substance can pass from
one state to another. A characteristic feature of the transformation of matter is
the possibility of the existence of stable inhomogeneous systems when a substance can
is in several states of aggregation at once. When describing such systems
use a broader concept of phase of matter. For example, carbon in solid
state of aggregation can be in two different phases - diamond and graphite. Phase
called the totality of all parts of the system, which in the absence of external
impact 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 another, then they speak of phase equilibrium.
Heat of phase transition.
Heat of phase transition- the amount of heat that must be imparted to a substance (or removed from it) during an equilibrium isobaric-isothermal transition of a substance from one phase to another (first-order phase transition - boiling, melting, crystallization, polymorphic transformation, etc.).
For phase transitions of the second kind, the heat of phase transformation is zero.
An equilibrium phase transition at a given pressure occurs at a constant temperature—the phase transition temperature. The heat of phase transition is equal to the product of the phase transition temperature and the difference in entropies in the two phases between which the transition occurs.
Phase equilibrium.