As a rule, we cognize the world through the so-called macroscopic bodies (Greek "macro" - large). These are all the bodies that surround us: houses, cars, water in a glass, water in the ocean, etc. We were interested in what happens to these bodies and around them. Now we will also be interested in what happens inside the bodies. The section of physics called MKT will help us answer this question.
MKT - molecular-kinetic theory. It explains the physical phenomena and properties of bodies in terms of their internal microscopic structure. This theory is based on three statements:
All bodies are made up of small particles, between which there are gaps.
The particles of bodies are constantly and randomly moving.
Particles of bodies interact with each other: they attract and repel.
These statements are called the basic provisions of the ILC. All of them are confirmed by numerous experiments.
In the macroscopic approach, we are interested in the bodies themselves: their dimensions, volume, mass, energy, and so on. Take a look at the picture on the left. For example, macroscopically studying water splashes, we will measure their size, volume, mass.
In the microscopic approach, we are also interested in dimensions, volume, mass and energy. However, not the bodies themselves, but the particles of which they are composed: molecules, ions and atoms. This is exactly what the top picture symbolizes. But one should not think that molecules, ions and atoms can be seen through a magnifying glass. This drawing is just an artistic hyperbole. These particles can be seen only with the help of special, so-called electron microscopes.
The MKT has not always been a scientific theory. Originating before our era, the molecular (or, as it was called before, atomic) theory remained only a convenient hypothesis for more than two thousand years! And only in the XX century it turns into a full-fledged physical theory. Here is how the famous physicist E. Rutherford says about this:
“Not a single physicist or chemist can close his eyes to the enormous role that the atomic hypothesis currently plays in science. ... By the end of the 19th century, its ideas permeated a very large area of physics and chemistry. The idea of atoms became more and more concrete. ... The simplicity and usefulness of atomic views in explaining the most diverse phenomena of physics and chemistry naturally raised the authority of this theory in the eyes of scientists. There was a tendency to consider the atomic hypothesis no longer as a useful working hypothesis, for which it is very difficult to find direct and convincing evidence, but as one of the firmly established facts of nature.
But there was also no shortage of scientists and philosophers who pointed out the groundlessness of this theory, on which, however, so much was built. We can agree with the usefulness of the idea of molecules for explaining these experiments, but what confidence do we have that atoms really exist, and are not just a fiction, the fruit of our imagination? It must be said, however, that this lack of direct evidence has by no means shaken the faith of the vast majority of men of science in the granular structure of matter.
The denial of the atomic theory has never contributed and will never contribute to the discovery of new facts. The great advantage of the atomic theory is that it gives us, so to speak, a tangible concrete idea of matter, which not only serves us to explain many phenomena, but also renders us enormous services as a working hypothesis.
There are two methods for studying the properties of a substance: molecular-kinetic and thermodynamic.
The molecular-kinetic theory interprets the properties of bodies that are directly observed in experiment (pressure, temperature, etc.) as the total result of the action of molecules. At the same time, she uses statistical method, being interested not in the movement of individual molecules, but only in the average values that characterize the movement of a huge collection of particles. Hence its other name - statistical physics.
Thermodynamics studies the macroscopic properties of bodies without being interested in their microscopic picture. Thermodynamics is based on several fundamental laws (called the principles of thermodynamics), established on the basis of a generalization of a large set of experimental facts. Thermodynamics and molecular-kinetic theory complement each other, forming essentially a single whole.
DEFINITION
Atom - the smallest particle of a given chemical element. All atoms that exist in nature are represented in Mendeleev's periodic system of elements.
Atoms are combined into a molecule by chemical bonds based on electrical interaction. The number of atoms in a molecule can be different. A molecule can consist of one, two, three, or even several hundred atoms.
DEFINITION
Molecule- the smallest particle of a given substance that has its chemical properties.
Molecular Kinetic Theory- the doctrine of the structure and properties of matter based on the concept of the existence of atoms and molecules.
The founder of the molecular kinetic theory is M.V. Lomonosov (1711-1765), who formulated its main provisions and applied them to explain various thermal phenomena.
Basic Provisions of Molecular Kinetic Theory
The main provisions of the ICT:
- all bodies in nature consist of the smallest particles (atoms and molecules);
- particles are in continuous chaotic motion, which is called thermal;
- particles interact with each other: forces of attraction and repulsion act between the particles, which depend on the distance between the particles.
The molecular kinetic theory is confirmed by many phenomena.
The mixing of various liquids, the dissolution of solids in liquids, is explained by the mixing of molecules of various kinds. In this case, the volume of the mixture may differ from the total volume of its constituent components. what does it say about different sizes molecular compounds.
DEFINITION
Diffusion- the phenomenon of penetration of two or more adjoining substances into each other.
Diffusion proceeds most intensively in gases. The spread of odors is due to diffusion. Diffusion indicates that the molecules are in constant chaotic motion. Also, the phenomenon of diffusion indicates that there are gaps between the molecules, i.e. matter is discrete.
DEFINITION
Brownian motion- thermal motion of the smallest microscopic particles suspended in a liquid or gas.
This phenomenon was first observed by the English botanist R. Brown in 1827. While observing flower pollen suspended in water through a microscope, he saw that each pollen particle makes rapid random movements, moving over a certain distance. As a result of individual movements, each pollen particle moved along a zigzag trajectory (Fig. 1a).
Fig.1. Brownian motion: a) trajectories of motion of individual particles suspended in a liquid; b) transfer of momentum by liquid molecules to a suspended particle.
Further research brownian motion in various liquids and with various solid particles showed that this movement becomes the more intense, the smaller the particle size and the higher the temperature of the experiment. This movement never stops and does not depend on any external causes.
R. Brown could not explain the observed phenomenon. The theory of Brownian motion was built by A. Einstein in 1905 and received experimental confirmation in the experiments of the French physicist J. Perrin (1900-1911).
Liquid molecules that are in constant chaotic motion, when colliding with a suspended particle, transfer some impulse to it (Fig. 1, b). In the case of a large particle, the number of molecules incident on it from all sides is large, their impacts are compensated at each moment of time, and the particle remains practically motionless. If the particle size is very small, then the impacts of the molecules are not compensated - on the one hand, a larger number of molecules can hit it than on the other, as a result of which the particle will begin to move. It is precisely such a movement under the influence of random impacts of molecules that Brownian particles perform. Although Brownian particles are billions of times larger than the mass of individual molecules and move at very low speeds (compared to the speeds of molecules), their movement can still be observed under a microscope.
Examples of problem solving
EXAMPLE 1
EXAMPLE 2
MOLECULAR-KINETIC THEORY
a branch of molecular physics that considers many properties of substances based on the ideas of the rapid chaotic movement of a huge number of atoms and molecules that make up these substances. The molecular kinetic theory focuses not on the differences between individual types of atoms and molecules, but on the common features that exist in their behavior. Even the ancient Greek philosophers, who were the first to express atomistic ideas, believed that atoms are in continuous motion. D. Bernoulli tried to give a quantitative analysis of this movement in 1738. A fundamental contribution to the development of molecular kinetic theory was made in the period from 1850 to 1900 by R. Clausius in Germany, L. Boltzmann in Austria and J. Maxwell in England. These same physicists laid the foundations of statistical mechanics, a more abstract discipline that studies the same subject as molecular kinetic theory, but without building detailed, and therefore less general models. The deepening of the statistical approach at the beginning of the 20th century. associated mainly with the name of the American physicist J. Gibbs, who is considered one of the founders of statistical mechanics. Revolutionary ideas were also introduced into this science by M. Planck and A. Einstein. In the mid-1920s, classical mechanics finally gave way to new, quantum mechanics. It gave impetus to the development of statistical mechanics, which continues to this day.
MOLECULAR-KINETIC THEORY OF HEAT
It is known that heated bodies, cooling down, give part of their heat to colder bodies. Until the 19th century it was believed that heat is a kind of liquid (caloric) flowing from one body to another. One of the main achievements of 19th century physics It became that heat began to be considered simply as one of the forms of energy, namely, the kinetic energy of atoms and molecules. This idea applies to all substances - solid, liquid and gaseous. Particles of a heated body move faster than a cold one. For example, the sun's rays, by heating our skin, cause its molecules to oscillate faster, and we feel these vibrations as heat. In the cold wind, air molecules, colliding with molecules on the surface of our body, take energy from them, and we feel cold. In all cases when heat is transferred from one body to another, the movement of particles in the first of them slows down, in the second it accelerates, and the energy of the particles of the second body increases exactly as much as the energy of the particles of the first decreases. Many thermal phenomena familiar to us can be directly explained using molecular kinetic theory. Since heat is generated by the random movement of molecules, it is possible to increase the body temperature (increase the heat reserve in it) not due to heat supply, but, for example, using friction: the molecules of rubbing surfaces, colliding with each other, begin to move more intensively, and the temperature of the surfaces rises . For the same reason, a piece of iron heats up when it is struck with a hammer. Another thermal phenomenon is an increase in the pressure of gases when heated. With an increase in temperature, the speed of movement of molecules increases, they more often and more strongly hit the walls of the vessel in which the gas is located, which manifests itself in an increase in pressure. The gradual evaporation of liquids is explained by the fact that their molecules one after another pass into the air, while the fastest of them disappear first, and those that remain have, on average, less energy. That is why when liquids evaporate from a wet surface, it cools. The mathematical apparatus built on the molecular kinetic theory makes it possible to analyze these and many other effects based on the equations of motion of molecules and general provisions probability theory. Let's assume that we have raised a rubber ball to a certain height and then released it from our hands. The ball will hit the floor and then bounce several times, each time to a lower height than before, since on impact some of its kinetic energy is converted into heat. Such an impact is called partially elastic. A piece of lead does not bounce off the floor at all - at the first blow, all its kinetic energy is converted into heat, and the temperature of the piece of lead and the floor rises slightly. Such an impact is called absolutely inelastic. An impact in which all the kinetic energy of the body is conserved without being converted into heat is called perfectly elastic. In gases, when atoms and molecules collide with each other, only their velocities are exchanged (we do not consider here the case when, as a result of collisions, gas particles interact - enter into chemical reactions); the total kinetic energy of the entire set of atoms and molecules cannot be converted into heat, since it already is. The continuous movement of atoms and molecules of matter is called thermal motion. In liquids and solids, the picture is more complex: in addition to kinetic energy, it is necessary to take into account the potential energy of particle interaction.
Thermal motion in air. If the air is cooled to a very low temperature, it will turn into a liquid, while the volume of liquid formed will be very small. For example, when liquefying 1200 cm3 of atmospheric air, 2 cm3 of liquid air are obtained. The main assumption of the atomic theory is that the sizes of atoms and molecules almost do not change when the aggregate state of matter changes. Consequently, in atmospheric air, the molecules must be at distances from each other that are much greater than in a liquid. Indeed, out of 1200 cm3 of atmospheric air, more than 1198 cm3 is occupied by empty space. Air molecules move randomly in this space at very high speeds, constantly colliding with each other like billiard balls.
The pressure of a gas or steam. Let us consider a rectangular vessel, the unit volume of which contains n gas molecules of mass m each. We will be interested only in those molecules that hit one of the walls of the vessel. Let us choose the x axis so that it is perpendicular to this wall and consider a molecule whose velocity component v along the chosen axis is equal to vx. When a molecule hits the vessel wall, its momentum in the direction of the x axis will change by -2mvx. In accordance with Newton's third law, the momentum transferred to the wall will be the same. It can be shown that if all molecules move at the same speed, then (1/2) nvx molecules collide with a unit wall area of 1 c. To see this, let us consider a gas boundary layer near one of the walls filled with molecules with the same values of v and vx (Fig. 1). Let us assume that the thickness of this layer is so small that most of the molecules pass through it without collision. Molecule A will reach the wall at time t = l /vx; by this time, exactly half of the molecules from the boundary layer will hit the wall (the other half is moving away from the wall). Their number is determined by the gas density and the volume of the boundary layer with area A and thickness l: N = (1/2) nAl. Then the number of molecules that hit a single area in 1 s will be N/At = (1/2) nvx, and the total momentum transferred to this area in 1 s will be (1/2) nvx ×2mvx = nmvx2. In fact, the vx component is not the same for different molecules, so the value of vx2 should be replaced by its average value
and">
. If the molecules move randomly, then the average of all vx is equal to the average of vy and vz, so that
and
where is the average value of v2 for all molecules. The impacts of molecules against the wall follow one after another so quickly that the sequence of transmitted impulses is perceived as a constant pressure P. The value of P can be found if we remember that pressure is a force acting per unit area, and force, in turn, is a speed momentum changes. Therefore, P is equal to the rate of change of momentum per unit area, i.e.
We will obtain the same relation if, instead of the random movement of molecules in all directions, we consider the movement of one sixth of their number perpendicular to each of the six faces of a rectangular vessel, assuming that each molecule has kinetic energy
Boyle's Law - Mariotte. In formula (1), n denotes not the total number of molecules, but the number of molecules per unit volume. If the same number of molecules are placed in half the volume (without changing the temperature), then the value of n doubles, and the pressure doubles, if v2 does not depend on density. In other words, at a constant temperature, the pressure of a gas is inversely proportional to its volume. The English physicist R. Boyle and the French physicist E. Mariotte experimentally established that when low pressures this statement is true for any gas. Thus, the Boyle-Mariotte law can be explained by making the reasonable assumption that, at low pressures, the velocity of molecules does not depend on n.
Dalton's Law. If there is a mixture of gases in the vessel, i.e. there are several different varieties molecules, then the momentum imparted to the wall by molecules of each kind does not depend on whether molecules of other kinds are present. Thus, according to molecular kinetic theory, the pressure of a mixture of two or more ideal gases is equal to the sum of the pressures that each of the gases would create if it occupied the entire volume. This is the law of Dalton, which is subject to gas mixtures at low pressures.
Molecular speeds. Formula (1) makes it possible to estimate the average velocity of gas molecules. Thus, the atmospheric pressure at sea level is approximately 106 dynes/cm2 (0.1 MPa), and the mass of 1 cm3 of air is 0.0013 g. Substituting these values into formula (1), we obtain a very large value for the velocity of molecules:
At high altitudes, where the atmosphere is very rarefied, air molecules can travel great distances per second without colliding with each other. At the surface of the Earth, a different picture is observed: for 1 s, each molecule collides with other molecules on average approx. 800 million times. It describes a highly broken trajectory, and in the absence of air currents, after one second, with a high probability, it will be only 1-2 cm away from where it was at the beginning of this second.
Avogadro's law. As we have said, air at room temperature has a density of approximately 0.0013 g/cm3 and creates a pressure of 106 dynes/cm2. Hydrogen gas, which has a density of only 0.00008 g/cm3 at room temperature, also produces a pressure of 106 dynes/cm2. According to formula (1), the gas pressure is proportional to the number of molecules per unit volume and their average kinetic energy. In 1811, the Italian physicist A. Avogadro put forward a hypothesis according to which equal volumes of different gases at the same temperature and pressure contain the same number molecules. If this hypothesis is correct, then from relation (1) we obtain that for different gases under the above conditions, the value (1/2) mv2 is the same, i.e. the average kinetic energy of the molecules is the same. This conclusion is in good agreement with the molecular kinetic theory
(see also HEAT).
The mass of 1 cm3 of hydrogen is small not because there are fewer molecules in a given volume, but because the mass of each hydrogen molecule is several times less than the mass of a molecule of nitrogen or oxygen - the gases that make up the air. It has been established that the number of molecules of any gas in 1 cm3 at 0 ° C and normal atmospheric pressure is 2.687 * 10 19.
Average free path. An important quantity in the molecular kinetic theory of gases is the average distance traveled by a molecule between two collisions. This value is called the mean free path and is denoted by L. It can be calculated as follows. Imagine that the molecules are spheres of radius r; then their centers during the collision will be at a distance of 2r from each other. In its motion, the molecule "hit" all the molecules within the cross section area p (2r)2 and, moving over a distance L, it "hit" all the molecules in the volume 4pr2L, so that the average number of molecules it would collide with would be 4pr2Ln . To find L, you need to take this number equal to 1, whence
From this relation, one can directly find the radius of the molecule if the value L is known (it can be found from measurements of gas viscosity; see below). The value of r turns out to be of the order of 10-8 cm, which is consistent with the results of other measurements, and L for typical gases under normal conditions is from 100 to 200 molecular diameters. The table shows the L values for atmospheric air at different heights above sea level.
SPEED DISTRIBUTION OF MOLECULES
In the middle of the 19th century there was not only the development of molecular-kinetic theory, but also the formation of thermodynamics. Some concepts of thermodynamics turned out to be useful for the molecular-kinetic theory as well - these are, first of all, absolute temperature and entropy.
Thermal balance. In thermodynamics, the properties of substances are considered mainly on the basis of the idea that any system tends to a state with the highest entropy and, having reached such a state, cannot spontaneously leave it. Such a representation is consistent with the molecular-kinetic description of the behavior of a gas. The set of gas molecules has a certain total energy, which can be distributed among individual molecules in a huge number of ways. Whatever the initial distribution of energy, if the gas is left to itself, then the energy will quickly be redistributed and the gas will come to a state of thermal equilibrium, i.e. to the state with the highest entropy. Let us try to formulate this statement more strictly. Let N (E) dE be the number of gas molecules with kinetic energy in the range from E to E + dE. Regardless of the initial distribution of energy, the gas, left to itself, will come to a state of thermal equilibrium with a characteristic function N (E) corresponding to the steady temperature. Instead of energies, one can consider the velocities of molecules. Let f (v) dv denote the number of molecules with velocities ranging from v to v + dv. In a gas, there will always be a certain number of molecules with velocities in the range from v to v + dv. Already a moment later, none of these molecules will have a speed lying in the specified interval, since they will all undergo one or more collisions. But on the other hand, other molecules with velocities previously significantly different from v will, as a result of collisions, acquire velocities ranging from v to v + dv. If the gas is in a stationary state, then the number of molecules that acquire the speed v, after a sufficiently long period of time, will be equal to the number of molecules whose speed will cease to be equal to v. Only in this case the function n (v) can remain constant. This number, of course, depends on the velocity distribution of gas molecules. The form of this distribution in a gas at rest was established by Maxwell: if there are N molecules in total, then the number of molecules with velocities in the interval from v to v + dv is equal to
where the parameter b depends on the temperature (see below).
gas laws. The above estimates for the average speed of air molecules at sea level corresponded to ordinary temperature. According to the molecular kinetic theory, the kinetic energy of all gas molecules is the heat that it possesses. At a higher temperature, the molecules move faster and the gas contains more heat. As follows from formula (1), if the volume of a gas is constant, then its pressure increases with increasing temperature. This is how all gases behave (Charles law). If the gas is heated at constant pressure, it will expand. It has been established that at low pressure for any gas of volume V containing N molecules, the product of pressure and volume is proportional to the absolute temperature:
where T is the absolute temperature, k is a constant. It follows from Avogadro's law that the value of k is the same for all gases. It is called the Boltzmann constant and is equal to 1.38 * 10 -14 erg / K. Comparing expressions (1) and (3), it is easy to see that the total energy of the translational motion of N molecules, equal to (1/2) Nmv2, is proportional to the absolute temperature and is equal to
On the other hand, integrating expression (2), we obtain that the total energy of the translational motion of N molecules is 3Nm /4b 2. Hence
By substituting expression (5) into formula (2), one can find the distribution of molecules over velocities at any temperature T. The molecules of many common gases, such as nitrogen and oxygen (the main components of atmospheric air), consist of two atoms, and their molecule resembles a dumbbell in shape . Each such molecule not only moves forward with great speed, but also rotates very quickly. In addition to translational energy, N molecules have rotational energy NkT, so the total energy of N molecules is (5/2) NkT.
Experimental verification of the Maxwell distribution. In 1929 it became possible to directly find the velocity distribution of gas molecules. If a small hole is made in the wall of a vessel containing a gas or vapor at a certain temperature, or a narrow slit is cut, then the molecules will fly out through them, each at its own speed. If the hole leads into another vessel, from which the air is pumped out, then most molecules will have time to fly a distance of several centimeters before the first collision. In the setup shown schematically in Fig. 2, there is a vessel V containing a gas or vapor, the molecules of which escape through the slot S1; S2 and S3 - slots in the transverse plates; W1 and W2 are two disks mounted on a common shaft R. Several radial slots are cut into each disk. The slit S3 is located in such a way that, if there were no disks, the molecules flying out of the slit S1 and passing through the slit S2 would fly through the slit S3 and hit the detector D. If one of the slits of the disk W1 is opposite the slit S2, then the molecules through the slots S1 and S2, they will also pass through the slot of the disk W1, but they will be delayed by the disk W2, mounted on the shaft R so that its slots do not coincide with the slots of the disk W1. If the disks are stationary or rotate slowly, then the molecules from the vessel V do not enter the detector D. If the disks rotate rapidly at a constant speed, then some of the molecules pass through both disks. It is not difficult to understand which molecules will be able to overcome both obstacles - those that will overcome the distance from W1 to W2 in the time required to shift the disk slot W2 to the desired angle. For example, if all the slots of the disk W2 are rotated by 2° relative to the slots of the disk W1, then the molecules that fly from W1 to W2 during the rotation of the disk W2 by 2° will enter the detector. By changing the frequency of rotation of the shaft with disks, it is possible to measure the velocities of molecules emitted from the vessel V and plot their distribution. The distribution thus obtained agrees well with Maxwell's.
Brownian motion. In the 19th century the method of measuring molecular velocities described above was not yet known, but one phenomenon made it possible to observe the incessant thermal motion of molecules in a liquid. The Scottish botanist R. Brown (in the previous transcription - Brown) in 1827, observing under a microscope the particles of pollen suspended in water, found that they do not stand still, but move all the time, as if something is pushing them into one , then to the other side. Later it was suggested that the chaotic motion of particles is caused by the continuous thermal motion of the molecules of the liquid, and precise studies of the motion, called Brownian, confirmed the correctness of this hypothesis.
(see BROWNIAN MOVEMENT).
Heat capacity of gas or steam. The amount of heat required to raise the temperature of a certain amount of a substance by 1 degree is called its heat capacity. It follows from formula (4) that if the gas temperature is increased at a constant volume from T to T + 1, then the energy of translational motion will increase by (3/2) Nk. All the thermal energy of a monatomic gas is the energy of translational motion. Therefore, the heat capacity of such a gas at constant volume is Cv = (3/2) Nk, and the heat capacity per molecule is (3/2) k. The heat capacity of N diatomic molecules, which also have the energy of rotational motion kT, is equal to Cv = (5/2) Nk, and one molecule accounts for (5/2) k. In both cases, the heat capacity does not depend on temperature, and the heat energy is given by
Saturated steam pressure. If you pour a little water into a large closed vessel that contains air but no water vapor, then some of it will immediately evaporate and vapor particles will begin to spread throughout the vessel. If the volume of the vessel is very large compared to the volume of water, then evaporation will continue until all the water turns into steam. If enough water is poured in, then not all of it will evaporate; the evaporation rate will gradually decrease and eventually the process will stop - the volume of the vessel will be saturated with water vapor. From the standpoint of molecular-kinetic theory, this is explained as follows. From time to time, one or another water molecule located in a liquid medium near the surface receives enough energy from neighboring molecules to escape into the vapor-air medium. Here it collides with other similar molecules and with air molecules, describing a very intricate zigzag trajectory. In its movement, it also hits the walls of the vessel and the surface of the water; however, it can bounce off the water or be absorbed by it. While water is evaporating, the number of vapor molecules captured by it from the vapor-air medium remains less than the number of molecules leaving the water. But there comes a moment when these quantities are equalized - an equilibrium is established, and the vapor pressure reaches saturation. In this state, the number of molecules per unit volume of vapor above the liquid remains constant (of course, if the temperature is constant). The same picture is observed for solids, but for most bodies the vapor pressure becomes noticeable only at high temperatures.
VIBRATIONS OF ATOMS IN SOLID BODIES AND LIQUIDS
Looking under a microscope at a well-preserved ancient Greek or Roman gemma, one can see that its details remain as clear as they seem to have been when the gemma had just passed from the hands of the craftsman who made it. It is clear that over a huge period of time, only very few atoms were able to "escape" from the surface of the stone from which the gem is made - otherwise, its details would lose clarity over time. Most atoms solid body can perform only oscillatory motions relative to some fixed position, and with an increase in temperature, the average frequency of these oscillations and their amplitude only increase. When a substance begins to melt, the behavior of its molecules becomes similar to the behavior of liquid molecules. If in a solid body each particle oscillates in a small volume occupying a fixed position in space, then in a liquid this volume itself moves slowly and randomly, and the oscillating particle moves with it.
THERMAL CONDUCTIVITY OF GAS
In any unevenly heated body, heat is transferred from its warmer parts to colder parts. This phenomenon is called thermal conductivity. Using molecular kinetic theory, one can find the rate at which a gas conducts heat. Let us consider a gas enclosed in a rectangular vessel, the upper surface of which has a higher temperature than the lower one. The temperature of the gas in the vessel gradually decreases when moving from the upper layers to the lower ones - there is a temperature gradient in the gas. Let us consider a thin horizontal layer of gas AB, which has a temperature T (Fig. 3), and an adjacent layer CD with a slightly higher temperature, T ў. Let the distance between AB and CD be equal to the mean free path L. According to formula (4), the average energy of a molecule in the AB layer is proportional to the temperature T, and in the CD layer it is proportional to the temperature T. Let us consider a molecule from the AB layer that collides with another molecule in point A, after which it moves without collisions to point C. With a high probability it will fall into the layer CD with an energy corresponding to the layer AB. Conversely, a molecule from the CD layer moving without collisions from point D to point B layer AB with a higher energy, corresponding to the layer CD from which it flew out. It is clear that during such collisions more energy is transferred from CD to AB than from AB to CD - there is a continuous flow of heat from the warmer to the colder layer. The same picture observed for all layers in the gas.
The rate of heat propagation can be calculated with fairly good accuracy, even if we neglect the fact that the mean free path of some molecules is greater, while others are less than the average. Consider the plane FG, parallel to the planes AB and CD and passing in the middle between them (Fig. 3), and select a unit area of this plane. If there are n molecules per unit volume moving at an average speed c, then in 1 s (1/2) nc molecules will cross FG from the bottom up and transfer the energy (1/2) ncE ; the same number of molecules will cross FG from top to bottom and transfer the energy (1/2) ncEў, where E and Еў are the average energies of molecules at temperatures T and T". Thus, if both flows of molecules moved perpendicular to the FG plane, then the difference of the transferred energies would be equal to (1/2) nc (E "- E). But the molecules intersect FG at all possible angles, and to take this into account, the indicated value should be multiplied by 2/3. Using relation (6), we obtain
Where Cv is the heat capacity of n molecules contained in a unit volume. When moving from CD to AB, which are at a distance L from each other, the temperature decreases by (T "- T) and if dT / dz is the temperature gradient in the direction perpendicular to the FG plane, then
Substituting the temperature difference, expressed through the gradient, into formula (7), we obtain that the total energy transferred through a unit area in 1 s is equal to
The value of K, described by the expression K = (1/3)CvcL,
is called the thermal conductivity of the gas.
GAS VISCOSITY
If you measure the speed of the river at different depths, you will find that near the bottom the water is almost motionless, and the closer to the surface, the faster it moves. Thus, in the river flow there is a velocity gradient similar to the temperature gradient discussed above; at the same time, due to the viscosity, each higher layer carries with it the neighboring one lying under it. This pattern is observed not only in liquids, but also in gases. Using the molecular-kinetic theory, we will try to determine the viscosity of the gas. Let us assume that the gas flows from left to right and that in the horizontal layer CD in Fig. 3, the flow velocity is greater than in layer AB located directly below CD. Let, as before, the distance between the planes be equal to the mean free path. Gas molecules move rapidly throughout the volume along chaotic trajectories, but this chaotic movement is superimposed by the directed movement of the gas. Let u be the gas flow velocity in the AB layer (in the direction from A to B), and u" be a slightly higher velocity in the CD layer (in the direction from C to D). In addition to the momentum due to random motion, the molecule in the AB layer has momentum mu, and in the CD layer - by the momentum mu. Molecules passing from AB to CD without collisions transfer to the CD layer the momentum mu corresponding to the AB layer, while particles falling from CD into AB mix with molecules from AB and bring with them momentum mu". Consequently, a momentum equal to
Since the rate of change of momentum is equal to force, we have obtained an expression for the force per unit area with which one layer acts on another: the slower layer slows down the faster one, and the latter, on the contrary, dragging the slower layer along with it, accelerates it. Similar forces act between adjacent layers in the entire volume of the flowing gas. If du/dz is the velocity gradient in the gas in the direction perpendicular to FG, then
The value nm in formula (8) is the mass of gas per unit volume; if we denote this quantity by r, then the force per unit area will be equal to
where the coefficient (1/3)rLc is the viscosity of the gas. Two conclusions follow from the last two sections of the article. The first is that the ratio of viscosity to thermal conductivity is r/Cv. The second follows from the expression for L given earlier and consists in the fact that the viscosity of a gas depends only on its temperature and does not depend on pressure and density. The correctness of both conclusions has been confirmed experimentally with high accuracy.
see also
HEAT ;
STATISTICAL MECHANICS;
THERMODYNAMICS.
LITERATURE
Hirschfeld J., Curtiss C., Byrd R. Molecular theory gases and liquids. M., 1961 Frenkel Ya.I. Kinetic theory of liquids. L., 1975 Kikoin A.K., Kikoin I.K. Molecular physics. M., 1976
Collier Encyclopedia. - Open society. 2000 .
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- (abbreviated as MKT) the theory of the 19th century, which considered the structure of matter, mainly gases, from the point of view of three main approximately correct provisions: all bodies are composed of particles: atoms, molecules and ions; particles are in continuous ... ... Wikipedia
- (abbreviated as MKT) a theory that considers the structure of matter from the point of view of three main approximately correct provisions: all bodies consist of particles whose size can be neglected: atoms, molecules and ions; particles are in continuous ... ... Wikipedia
The main provisions of the ICT:
1. All substances consist of the smallest particles: molecules, atoms or ions.
2. These particles are in continuous chaotic motion, the speed of which determines the temperature of the substance.
3. Between the particles there are forces of attraction and repulsion, the nature of which depends on the distance between them.
An ideal gas is a gas whose interaction between molecules is negligible.
The main differences between an ideal gas and a real one are: particles of an ideal gas are very small balls, practically material points; there are no forces of intermolecular interaction between the particles; particle collisions are absolutely elastic. A real gas is a gas that is not described by the Clapeyron-Mendeleev equation of state for an ideal gas. Dependences between its parameters show that the molecules in a real gas interact with each other and occupy a certain volume. The state of a real gas is often described in practice by the generalized Mendeleev-Clapeyron equation.
2 Status parameters and functions. The equation of state for an ideal gas.
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The pressure is due to the interaction of the molecules of the working fluid with the surface and is numerically equal to the force acting on the unit surface area of the body along the normal to the latter.
Temperature is a physical quantity that characterizes the degree of heating of a body. From the point of view of molecular kinetic concepts, temperature is a measure of the intensity of the thermal motion of molecules.
Specific volume v is the volume per unit mass of a substance. If a homogeneous body of mass M occupies a volume v, then by definition v= V/M. In the SI system, the unit of specific volume is 1 m3/kg. There is an obvious relationship between the specific volume of a substance and its density:
If all thermodynamic parameters are constant in time and are the same at all points of the system, then such a state of the system is called equilibrium.
For an equilibrium thermodynamic system, there is a functional relationship between the state parameters, which is called the equation of state
Clapeyron - Mendeleev equation
3 Mixtures of gases. Apparent molecular weight. Gas constant of a mixture of gases.
A mixture of gases is a mechanical combination of gases that do not interact with each other. chemical reaction gases. The main law that determines the behavior of a gas mixture is Dalton's law: the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of all its constituent components: Partial pressure pi is the pressure that a gas would have if it alone occupied the entire volume of the mixture at the same temperature . The gas constant of a mixture is defined as: - the apparent (average) molecular weight of the mixture. With a volumetric composition, with a mass composition:.-universal gas constant.
4 The first law of thermodynamics.
The first law of thermodynamics is the law of conservation of energy, written using thermodynamic concepts (analytical formulation: a perpetual motion machine of the 1st kind is impossible):
Energy. Under the internal energy in thermodynamics understand the kinetic energy of the movement of molecules, the potential energy of their interaction and zero (energy of movement of particles inside the molecule at T=0K). The kinetic energy of molecules is a function of temperature, the value of potential energy depends on the average distance between molecules and, consequently, on the volume V occupied by the gas, that is, it is a function of V. Therefore, the internal energy U is a function of the state of the body.
Heat. The energy transferred from one body to another due to the difference in temperature is called heat. Heat can be transferred either through direct contact between bodies (thermal conduction, convection), or at a distance (by radiation), and in all cases this process is possible only if there is a temperature difference between the bodies.
Job. The energy transferred from one body to another when the volume of these bodies changes or moves in space is called work. With a finite change in volume, the work against the forces of external pressure, called the work of expansion, is equal to The work of changing the volume is equivalent to the area under the process curve in the diagram p, v.
Internal energy is a property of the system itself, it characterizes the state of the system. Heat and work are the energy characteristics of the processes of mechanical and thermal interactions of a system with environment. They characterize the amounts of energy that are transferred to the system or given away by it through its boundaries in a certain process.
The atoms or molecules that make up a gas move freely at a considerable distance from each other and interact only when they collide with each other (hereinafter, in order not to repeat myself, I will only mention "molecules", meaning by this "molecules or atoms"). Therefore, the molecule moves in a straight line only in the intervals between collisions, changing the direction of motion after each such interaction with another molecule. The average length of a rectilinear segment of the motion of a gas molecule is called average free path. The higher the density of the gas (and hence the smaller the average distance between molecules), the shorter the average free path between collisions.
In the second half of the 19th century, such a seemingly simple picture of the atomic-molecular structure of gases, through the efforts of a number of theoretical physicists, developed into a powerful and fairly universal theory. The new theory was based on the idea of the relationship between measurable macroscopic indicators of the state of the gas (temperature, pressure and volume) with microscopic characteristics - the number, mass and speed of movement of molecules. Since molecules are constantly in motion and, as a result, have kinetic energy, this theory is called molecular kinetic theory gases.
Take, for example, pressure. At any moment in time, the molecules hit the walls of the vessel and, with each impact, transmit to them a certain impulse of force, which is extremely small in itself, but the total effect of millions of molecules produces a significant force effect on the walls, which we perceive as pressure. For example, when you inflate a car tire, you move atmospheric air molecules into the closed volume of the tire in addition to the number of molecules already inside it; as a result, the concentration of molecules inside the tire is higher than outside, they hit the walls more often, the pressure inside the tire is higher than atmospheric pressure, and the tire becomes inflated and elastic.
The meaning of the theory is that, from the average free path of molecules, we can calculate the frequency of their collisions with the walls of the vessel. That is, having information about the speed of movement of molecules, it is possible to calculate the characteristics of the gas that can be directly measured. In other words, the molecular-kinetic theory gives us a direct connection between the world of molecules and atoms and the tangible macrocosm.
The same applies to the understanding of temperature within the framework of this theory. The higher the temperature, the higher the average velocity of the gas molecules. This relationship is described by the following equation:
1/2mv 2 = kT
where m is the mass of one gas molecule, v - average speed of thermal motion of molecules, T - gas temperature (in Kelvin), and k is the Boltzmann constant. The basic equation of molecular kinetic theory defines a direct relationship between the molecular characteristics of a gas (left) and measurable macroscopic characteristics (right). The temperature of the gas is directly proportional to the square of the average velocity of the molecules.
The molecular kinetic theory also gives a fairly definite answer to the question of the deviations of the velocities of individual molecules from the mean value. Each collision between gas molecules leads to a redistribution of energy between them: too fast molecules slow down, too slow ones accelerate, which leads to averaging. At any moment, countless millions of such collisions are taking place in the gas. Nevertheless, it turned out that at a given temperature of a gas in a stable state, the average number of molecules with a certain speed v or energy E, does not change. This happens because, from a statistical point of view, the probability that a molecule with energy E changes its energy and goes into a similar energy state, is equal to the probability that another molecule, on the contrary, goes into a state with energy E. Thus, although each individual molecule has an energy E only sporadically, the average number of molecules with energy E remains unchanged. (We see a similar situation in human society. No one stays seventeen for more than one year—thank God!—yet, on average, the percentage of seventeens in a stable human community stays pretty much the same.)
This idea of the average distribution of molecules over velocities and its rigorous formulation belongs to James Clark Maxwell - the same outstanding theorist also owns a rigorous description of electromagnetic fields ( cm. Maxwell's equations). It was he who deduced the distribution of molecules in terms of velocities at a given temperature (see figure). Most of the molecules are in the energy state corresponding to the peak Maxwell distributions and average velocity, however, in fact, the velocities of the molecules vary within fairly large limits.
