MKT is easy!

"Nothing exists but atoms and empty space..." - Democritus
"Any body can divide indefinitely" - Aristotle

The main provisions of the molecular kinetic theory (MKT)

Purpose of the ICB- this is an explanation of the structure and properties of various macroscopic bodies and thermal phenomena occurring in them, by the movement and interaction of the particles that make up the bodies.
macroscopic bodies- These are large bodies, consisting of a huge number of molecules.
thermal phenomena- phenomena associated with heating and cooling bodies.

Main statements of the ILC

1. A substance consists of particles (molecules and atoms).
2. There are gaps between the particles.
3. Particles move randomly and continuously.
4. Particles interact with each other (attract and repel).

MKT confirmation:

1. experimental
- mechanical crushing of the substance; dissolution of a substance in water; compression and expansion of gases; evaporation; body deformation; diffusion; Brigman's experiment: oil is poured into a vessel, a piston presses on the oil from above, at a pressure of 10,000 atm, the oil begins to seep through the walls of a steel vessel;

Diffusion; Brownian motion of particles in a liquid under the impact of molecules;

Poor compressibility of solid and liquid bodies; significant efforts to break solids; coalescence of liquid droplets;

2. straight
- photography, particle size determination.

Brownian motion

Brownian motion is the thermal motion of suspended particles in a liquid (or gas).

Brownian motion has become evidence of the continuous and chaotic (thermal) motion of the molecules of matter.
- discovered by the English botanist R. Brown in 1827
- A theoretical explanation based on the MKT was given by A. Einstein in 1905.
- experimentally confirmed by the French physicist J. Perrin.

Mass and size of molecules

Particle sizes

The diameter of any atom is about cm.

Number of molecules in a substance

where V is the volume of the substance, Vo is the volume of one molecule

Mass of one molecule

where m is the mass of the substance,
N is the number of molecules in the substance

Mass unit in SI: [m]= 1 kg

In atomic physics, mass is usually measured in atomic mass units (a.m.u.).
Conventionally, it is considered to be 1 a.m.u. :

Relative molecular weight of a substance

For the convenience of calculations, a quantity is introduced - the relative molecular weight of the substance.
The mass of a molecule of any substance can be compared with 1/12 of the mass of a carbon molecule.

where the numerator is the mass of the molecule and the denominator is 1/12 of the mass of the carbon atom

This quantity is dimensionless, i.e. has no units

Relative atomic mass of a chemical element

where the numerator is the mass of the atom and the denominator is 1/12 of the mass of the carbon atom

The quantity is dimensionless, i.e. has no units

The relative atomic mass of each chemical element is given in the periodic table.

Another way to determine the relative molecular weight of a substance

The relative molecular mass of a substance is equal to the sum of the relative atomic masses of the chemical elements that make up the molecule of the substance.
We take the relative atomic mass of any chemical element from the periodic table!)

Amount of substance

The amount of substance (ν) determines the relative number of molecules in the body.

where N is the number of molecules in the body and Na is Avogadro's constant

Unit of measurement of the amount of a substance in the SI system: [ν] = 1 mol

1 mol- this is the amount of a substance that contains as many molecules (or atoms) as there are atoms in carbon with a mass of 0.012 kg.

Remember!
1 mole of any substance contains the same number of atoms or molecules!

But!
The same amount of a substance for different substances has a different mass!

Avogadro constant

The number of atoms in 1 mole of any substance is called Avogadro's number or Avogadro's constant:

Molar mass

Molar mass (M) is the mass of a substance taken in one mole, or otherwise, it is the mass of one mole of a substance.

Molecule mass
- Avogadro's constant

Molar mass unit: [M]=1 kg/mol.

Formulas for solving problems

These formulas are obtained by substituting the above formulas.

The mass of any amount of matter

Definition 1

Molecular Kinetic Theory- this is the doctrine of the structure and properties of matter, based on the idea of ​​the existence of atoms and molecules, as the smallest particles of chemical substances.

The main provisions of the molecular-kinetic theory of the molecule:

1. All substances can be in liquid, solid and gaseous state. They are formed from particles that are made up of atoms. Elementary molecules can have a complex structure, that is, they can contain several atoms. Molecules and atoms are electrically neutral particles that, under certain conditions, acquire an additional electric charge and turn into positive or negative ions.
2. Atoms and molecules move continuously.
3. Particles with an electrical nature of force interact with each other.

The main provisions of the MKT and their examples have been listed above. Between particles there is a small gravitational influence.

Figure 3. one . one . The trajectory of a Brownian particle.

Definition 2

The Brownian motion of molecules and atoms confirms the existence of the main provisions of the molecular kinetic theory and substantiates it experimentally. This thermal movement of particles occurs with molecules suspended in a liquid or gas.

Experimental substantiation of the main provisions of the molecular kinetic theory

In 1827, R. Brown discovered this movement, which was due to random impacts and movements of molecules. Since the process was chaotic, the blows could not balance each other. Hence the conclusion that the speed of a Brownian particle cannot be constant, it is constantly changing, and the direction movement is depicted as a zigzag, shown in Figure 3. one . one .

A. Einstein spoke about Brownian motion in 1905. His theory was confirmed in the experiments of J. Perrin in 1908 - 1911.

Definition 3

Consequence from Einstein's theory: offset square< r 2 >of the Brownian particle relative to the initial position, averaged over many Brownian particles, is proportional to the observation time t .

Expression< r 2 >= D t explains the diffusion law. According to the theory, we have that D increases monotonically with increasing temperature. Random motion is visible in the presence of diffusion.

Definition 4

Diffusion- this is the definition of the phenomenon of penetration of two or more contiguous substances into each other.

This process occurs rapidly in an inhomogeneous gas. Thanks to diffusion examples with different densities, a homogeneous mixture can be obtained. When oxygen O 2 and hydrogen H 2 are in the same vessel with a partition, when it is removed, the gases begin to mix, forming a dangerous mixture. The process is possible when hydrogen is at the top and oxygen is at the bottom.

Interpenetration processes also occur in liquids, but much more slowly. If we dissolve a solid, sugar, in water, we get a homogeneous solution, which is a clear example of diffusion processes in liquids. Under real conditions, mixing in liquids and gases is masked by rapid mixing processes, for example, when convection currents occur.

Diffusion of solids is distinguished by its slow speed. If the interaction surface of metals is cleaned, then it can be seen that over a long period of time, atoms of another metal will appear in each of them.

Definition 5

Diffusion and Brownian motion are considered related phenomena.

With the interpenetration of particles of both substances, the movement is random, that is, there is a chaotic thermal movement of molecules.

The forces acting between two molecules depend on the distance between them. Molecules have both positive and negative charges. At large distances, forces of intermolecular attraction predominate, at small distances, repulsive forces prevail.

Picture 3 . 1 . 2 shows the dependence of the resulting force F and potential energy E p of the interaction between molecules on the distance between their centers. At a distance r = r 0, the interaction force vanishes. This distance is conditionally taken as the diameter of the molecule. At r = r 0 the potential energy of interaction is minimal.

Definition 6

To move two molecules apart with distance r 0 , E 0 should be reported, called binding energy or potential well depth.

Figure 3. one . 2.The power of interaction F and potential energy of interaction E p two molecules. F > 0- repulsive force F< 0 - force of gravity.

Since molecules are small in size, simple monatomic ones can be no more than 10 - 10 m. Complex ones can reach sizes hundreds of times larger.

Definition 7

The random random movement of molecules is called thermal movement.

As the temperature increases, the kinetic energy of thermal motion increases. At low temperatures, the average kinetic energy, in most cases, is less than the potential well depth E 0 . This case shows that the molecules flow into a liquid or solid with an average distance between them r 0 . If the temperature rises, then the average kinetic energy of the molecule exceeds E 0, then they fly apart and form a gaseous substance.

In solids, molecules move randomly around fixed centers, that is, equilibrium positions. In space, it can be distributed in an irregular manner (in amorphous bodies) or with the formation of ordered bulk structures (crystalline bodies).

Aggregate states of substances

The freedom of thermal motion of molecules is seen in liquids, since they do not have binding to centers, which allows movement throughout the volume. This explains its fluidity.

Definition 8

If the molecules are close, they can form ordered structures with several molecules. This phenomenon has been named close order. distant order characteristic of crystalline bodies.

The distance in gases between molecules is much larger, so the acting forces are small, and their movements go along a straight line, waiting for the next collision. The value of 10 - 8 m is the average distance between air molecules under normal conditions. Since the interaction of forces is weak, the gases expand and can fill any volume of the vessel. When their interaction tends to zero, then one speaks of the representation of an ideal gas.

Kinetic model of an ideal gas

In microns, the amount of matter is considered proportional to the number of particles.

Definition 9

mole- this is the amount of a substance containing as many particles (molecules) as there are atoms in 0, 012 to g of carbon C 12. A carbon molecule is made up of one atom. It follows that 1 mole of a substance has the same number of molecules. This number is called permanent Avogadro N A: N A \u003d 6, 02 ċ 1023 mol - 1.

Formula for determining the amount of a substance ν is written as the ratio N of the number of particles to the Avogadro constant N A: ν = N N A .

Definition 10

The mass of one mole of a substance call the molar mass M. It is fixed in the form of the formula M \u003d N A ċ m 0.

The expression of the molar mass is made in kilograms per mole (k g / mol b).

Definition 11

If the substance has one atom in its composition, then it is appropriate to speak of the atomic mass of the particle. The unit of an atom is 1 12 masses of the carbon isotope C 12, called atomic mass unit and written as ( a. eat.): 1 a. e. m. \u003d 1, 66 ċ 10 - 27 to g.

This value coincides with the mass of the proton and neutron.

Definition 12

The ratio of the mass of an atom or molecule of a given substance to 1 12 of the mass of a carbon atom is called relative mass.

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Molecular Kinetic Theory(abbreviated MKT) - a theory that arose in the 19th century and considers the structure of matter, mainly gases, from the point of view of three main approximately correct provisions:

All bodies are made up of particles. atoms, molecules and ions;

particles are in continuous chaotic movement (thermal);

particles interact with each other absolutely elastic collisions.

The MKT has become one of the most successful physical theories and has been confirmed by a number of experimental facts. The main evidence of the provisions of the ICT were:

Diffusion

Brownian motion

Change aggregate states substances

Based on the MCT, a number of branches of modern physics have been developed, in particular, physical kinetics and statistical mechanics. In these branches of physics, not only molecular (atomic or ionic) systems are studied, which are not only in "thermal" motion, and interact not only through absolutely elastic collisions. The term molecular-kinetic theory is practically not used in modern theoretical physics, although it is found in textbooks for general physics courses.

Ideal gas - mathematical model gas, which assumes that: 1) potential energy interactions molecules can be neglected compared to kinetic energy; 2) the total volume of gas molecules is negligible. Between molecules there are no forces of attraction or repulsion, collisions of particles between themselves and with the walls of the vessel absolutely elastic, and the interaction time between molecules is negligible compared to the average time between collisions. In the extended model of an ideal gas, the particles of which it is composed also have a shape in the form of elastic spheres or ellipsoids, which allows taking into account the energy of not only translational, but also rotational-oscillatory motion, as well as not only central, but also non-central collisions of particles, etc.

There are classical ideal gas (its properties are derived from the laws of classical mechanics and are described Boltzmann statistics) and quantum ideal gas (properties are determined by the laws of quantum mechanics, described by statisticians Fermi - Dirac or Bose - Einstein)

Classical ideal gas

The volume of an ideal gas depends linearly on temperature at constant pressure

The properties of an ideal gas based on molecular kinetic concepts are determined based on the physical model of an ideal gas, in which the following assumptions are made:

In this case, the gas particles move independently of each other, the gas pressure on the wall is equal to the total momentum transferred during the collision of particles with the wall per unit time, internal energy- the sum of energies of gas particles.

According to the equivalent formulation, an ideal gas is one that simultaneously obeys Boyle's Law - Mariotte and Gay Lussac , that is:

where is pressure and is absolute temperature. The properties of an ideal gas are described the Mendeleev-Clapeyron equation

,

where - , - weight, - molar mass.

where - particle concentration, - Boltzmann's constant.

For any ideal gas, Mayer's ratio:

where - universal gas constant, - molar heat capacity at constant pressure, - molar heat capacity at constant volume.

Statistical calculation of the distribution of velocities of molecules was performed by Maxwell.

Consider the result obtained by Maxwell in the form of a graph.

Gas molecules constantly collide as they move. The speed of each molecule changes upon collision. It can rise and fall. However, the RMS speed remains unchanged. This is explained by the fact that in a gas at a certain temperature, a certain stationary velocity distribution of molecules does not change with time, which obeys a certain statistical law. The speed of an individual molecule can change over time, but the proportion of molecules with speeds in a certain range of speeds remains unchanged.

It is impossible to raise the question: how many molecules have a certain speed. The fact is that, although the number of molecules is very large in any even small volume, but the number of speed values ​​is arbitrarily large (as numbers in a sequential series), and it may happen that not a single molecule has a given speed.

Rice. 3.3

The problem of the distribution of molecules over velocities should be formulated as follows. Let in unit volume n molecules. What proportion of molecules has velocities from v 1 to v 1 + ∆ v? This is a statistical task.

Based on Stern's experience, it can be expected that the largest number of molecules will have some average speed, and the proportion of fast and slow molecules is not very large. The necessary measurements showed that the fraction of molecules , referred to the velocity interval Δ v, i.e. , has the form shown in Fig. 3.3. Maxwell in 1859 theoretically determined this function on the basis of probability theory. Since then, it has been called the velocity distribution function of molecules or Maxwell's law.

Let us derive the velocity distribution function of ideal gas molecules

- speed interval near the speed .

is the number of molecules whose velocities lie in the interval
.

is the number of molecules in the considered volume.

- angle of molecules whose velocities belong to the interval
.

is the fraction of molecules in a unit velocity interval near the velocity .

- Maxwell's formula.

Using Maxwell's statistical methods, we obtain the following formula:

.

is the mass of one molecule,
is the Boltzmann constant.

The most probable speed is determined from the condition
.

Solving we get
;
.

Denote b/w
.

Then
.

Let us calculate the fraction of molecules in a given range of velocities near a given speed in a given direction.

.

.

is the proportion of molecules that have velocities in the interval
,
,
.

Developing Maxwell's ideas, Boltzmann calculated the velocity distribution of molecules in a force field. In contrast to the Maxwell distribution, the Boltzmann distribution uses the sum of kinetic and potential energies instead of the kinetic energy of molecules.

In the Maxwell distribution:
.

In the Boltzmann distribution:
.

In a gravitational field

.

The formula for the concentration of ideal gas molecules is:

and respectively.

is the Boltzmann distribution.

is the concentration of molecules at the Earth's surface.

- concentration of molecules at height .

Heat capacity.

The heat capacity of a body is a physical quantity equal to the ratio

,
.

Heat capacity of one mole - molar heat capacity

.

Because
- process function
, then
.

Considering

;

;



.

- Mayer's formula.

That. the problem of calculating the heat capacity is reduced to finding .

.

For one mole:

, hence
.

Diatomic gas (O 2, N 2, Cl 2, CO, etc.).

(hard dumbbell model).

Total number of degrees of freedom:

.

Then
, then

;
.

This means that the heat capacity must be constant. However, experience shows that the heat capacity depends on temperature.

When the temperature is lowered, first the vibrational degrees of freedom are "frozen" and then the rotational degrees of freedom.

According to the laws of quantum mechanics, the energy of a harmonic oscillator with a classical frequency can only take on a discrete set of values

Polyatomic gases (H 2 O, CH 4, C 4 H 10 O, etc.).

;
;
;

Let's compare theoretical data with experimental ones.

It's clear that 2 atomic gases equals , but changes at low temperatures contrary to the heat capacity theory.

Such a course of the curve from testifies to the "freezing" of the degrees of freedom. On the contrary, at high temperatures, additional degrees of freedom are connected  these data cast doubt on the uniform distribution theorem. Modern physics makes it possible to explain the dependence from using quantum concepts.

Quantum statistics has eliminated the difficulties in explaining the dependence of the heat capacity of gases (in particular, diatomic gases) on temperature. According to the provisions of quantum mechanics, the energy of the rotational motion of molecules and the energy of vibrations of atoms can only take on discrete values. If the energy of thermal motion is much less than the difference between the energies of neighboring energy levels (), then during the collision of molecules, rotational and vibrational degrees of freedom are practically not excited. Therefore, at low temperatures, the behavior of a diatomic gas is similar to that of a monatomic gas. Since the difference between neighboring rotational energy levels is much smaller than between neighboring vibrational levels ( ), then with increasing temperature, rotational degrees of freedom are first excited. As a result, the heat capacity increases. With a further increase in temperature, vibrational degrees of freedom are also excited, and a further increase in heat capacity occurs. A. Einstein, approximately believed that the vibrations of the atoms of the crystal lattice are independent. Using the model of a crystal as a set of harmonic oscillators independently oscillating with the same frequency, he created a qualitative quantum theory of the heat capacity of a crystal lattice. This theory was subsequently developed by Debye, who took into account that the vibrations of atoms in a crystal lattice are not independent. Having considered the continuous frequency spectrum of oscillators, Debye showed that the main contribution to the average energy of a quantum oscillator is made by oscillations at low frequencies corresponding to elastic waves. Thermal excitation of a solid can be described as elastic waves propagating in a crystal. According to the corpuscular-wave dualism of the properties of matter, elastic waves in a crystal are compared with quasiparticles-phonons that have energy. A phonon is an energy quantum of an elastic wave, which is an elementary excitation that behaves like a microparticle. Just as the quantization of electromagnetic radiation led to the idea of ​​photons, so the quantization of elastic waves (as a result of thermal vibrations of the molecules of solids) led to the idea of ​​phonons. The energy of the crystal lattice is the sum of the energy of the phonon gas. Quasiparticles (in particular, phonons) are very different from ordinary microparticles (electrons, protons, neutrons, etc.), since they are associated with the collective motion of many particles of the system.

Phonons cannot arise in a vacuum, they exist only in a crystal.

The momentum of a phonon has a peculiar property: when phonons collide in a crystal, their momentum can be transferred to the crystal lattice in discrete portions - the momentum is not conserved in this case. Therefore, in the case of phonons, one speaks of a quasi-momentum.

Phonons have zero spin and are bosons, and therefore the phonon gas obeys Bose–Einstein statistics.

Phonons can be emitted and absorbed, but their number is not kept constant.

The application of Bose–Einstein statistics to a phonon gas (a gas of independent Bose particles) led Debye to the following quantitative conclusion. At high temperatures, which are much higher than the characteristic Debye temperature (classical region), the heat capacity of solids is described by the Dulong and Petit law, according to which the molar heat capacity of chemically simple bodies in the crystalline state is the same and does not depend on temperature. At low temperatures, when (quantum region), the heat capacity is proportional to the third power of the thermodynamic temperature: The characteristic Debye temperature is: , where is the limiting frequency of elastic vibrations of the crystal lattice.

The central concept of this topic is the concept of the molecule; the complexity of its assimilation by schoolchildren is due to the fact that the molecule is an object that is not directly observable. Therefore, the teacher must convince tenth-graders of the reality of the microcosm, of the possibility of its knowledge. In this regard, much attention is paid to the consideration of experiments that prove the existence and motion of molecules and allow one to calculate their main characteristics (the classical experiments of Perrin, Rayleigh, and Stern). In addition, it is advisable to familiarize students with the calculation methods for determining the characteristics of molecules. When considering evidence for the existence and movement of molecules, students are told about Brown's observations of the random movement of small suspended particles, which did not stop during the entire time of observation. At that time, a correct explanation of the cause of this movement was not given, and only after almost 80 years A. Einstein and M. Smoluchovsky built, and J. Perrin experimentally confirmed the theory of Brownian movement. From the consideration of Brown's experiments, it is necessary to draw the following conclusions: a) the motion of Brownian particles is caused by impacts of the molecules of the substance in which these particles are suspended; b) Brownian motion is continuous and random, it depends on the properties of the substance in which the particles are suspended; c) the movement of Brownian particles makes it possible to judge the movement of the molecules of the medium in which these particles are located; d) Brownian motion proves the existence of molecules, their motion and the continuous and chaotic nature of this motion. Confirmation of this nature of the movement of molecules was obtained in the experiment of the French physicist Dunoyer (1911), who showed that gas molecules move in different directions and in the absence of collisions their movement is rectilinear. At present, no one doubts the fact of the existence of molecules. Advances in technology have made it possible to directly observe large molecules. It is advisable to accompany the story about Brownian motion with a demonstration of a model of Brownian motion in vertical projection using a projection lamp or a codoscope, as well as showing the film fragment "Brownian motion" from the film "Molecules and Molecular Motion". In addition, it is useful to observe Brownian motion in liquids using a microscope. The drug is made from a mixture of equal parts of two solutions: a 1% sulfuric acid solution and a 2% aqueous solution of hyposulfite. As a result of the reaction, sulfur particles are formed, which are suspended in solution. Two drops of this mixture are placed on a glass slide and the behavior of the sulfur particles is observed. The preparation can be made from a highly diluted solution of milk in water or from a solution of watercolor paint in water. When discussing the issue of the size of molecules, the essence of R. Rayleigh's experiment is considered, which is as follows: a drop of olive oil is placed on the surface of water poured into a large vessel. The drop spreads over the surface of the water and forms a round film. Rayleigh suggested that when the drop stops spreading, its thickness becomes equal to the diameter of one molecule. Experiments show that the molecules of various substances have different sizes, but to estimate the size of the molecules they take a value equal to 10 -10 m. A similar experiment can be done in the class. To demonstrate the calculation method for determining the size of molecules, an example is given of calculating the diameters of molecules of various substances from their densities and the Avogadro constant. It is difficult for schoolchildren to imagine the small sizes of molecules; therefore, it is useful to give a number of examples of a comparative nature. For example, if all dimensions are increased so many times that the molecule is visible (i.e., up to 0.1 mm), then a grain of sand would turn into a hundred-meter rock, an ant would increase to the size of an ocean ship, a person would have a height of 1700 km. The number of molecules in the amount of substance 1 mol can be determined from the results of the experiment with a monomolecular layer. Knowing the diameter of the molecule, you can find its volume and the volume of the amount of substance 1 mol, which is equal to where p is the density of the liquid. From here, the Avogadro constant is determined. The calculation method consists in determining the number of molecules in the amount of 1 mol of a substance from the known values ​​of the molar mass and the mass of one molecule of the substance. The value of the Avogadro constant, according to modern data, is 6.022169 * 10 23 mol -1. Students can be introduced to the calculation method for determining the Avogadro constant by suggesting that it be calculated from the values ​​of the molar masses of various substances. Schoolchildren should be introduced to the Loschmidt number, which shows how many molecules are contained in a unit volume of gas under normal conditions (it is equal to 2.68799 * 10 -25 m -3). Tenth graders can independently determine the Loschmidt number for several gases and show that it is the same in all cases. By giving examples, you can give the guys an idea of ​​how large the number of molecules in a unit volume is. If a rubber balloon were to be pierced so thin that 1,000,000 molecules would escape through it every second, then approximately 30 billion molecules would be needed. years for all the molecules to come out. One of the methods for determining the mass of molecules is based on the experiment of Perrin, who proceeded from the fact that drops of resin in water behave in the same way as molecules in the atmosphere. Perrin counted the number of droplets in different layers of the emulsion, highlighting layers with a thickness of 0.0001 cm using a microscope. The height at which there are two times fewer such droplets than at the bottom was equal to h = 3 * 10 -5 m. The mass of one drop of resin turned out to be equal to M \u003d 8.5 * 10 -18 kg. If our atmosphere consisted only of oxygen molecules, then at an altitude of H = 5 km, the oxygen density would be half that at the Earth's surface. The proportion m/M=h/H is recorded, from which the mass of an oxygen molecule m=5.1*10 -26 kg is found. Students are offered to independently calculate the mass of a hydrogen molecule, the density of which is half that of the Earth's surface, at a height of H = 80 km. At present, the values ​​of the masses of molecules have been refined. For example, oxygen is set to 5.31*10 -26 kg, and hydrogen is set to 0.33*10 -26 kg. When discussing the issue of the speeds of movement of molecules, students are introduced to the classical experiment of Stern. When explaining the experiment, it is advisable to create its model using the "Rotating disk with accessories" device. Several matches are fixed on the edge of the disk in a vertical position, in the center of the disk - a tube with a groove. When the disk is stationary, the ball lowered into the tube, rolling down the chute, knocks down one of the matches. Then the disk is brought into rotation at a certain speed, fixed by the tachometer. The newly launched ball will deviate from the original direction of movement (relative to the disk) and knock down a match located at some distance from the first one. Knowing this distance, the radius of the disk and the speed of the ball on the rim of the disk, it is possible to determine the speed of the ball along the radius. After that, it is advisable to consider the essence of Stern's experiment and the design of its installation, using the film fragment "Stern's Experiment" as an illustration. When discussing the results of Stern's experiment, attention is drawn to the fact that there is a certain distribution of molecules over velocities, as evidenced by the presence of a strip of deposited atoms of a certain width, and the thickness of this strip is different. In addition, it is important to note that molecules moving at high speed settle closer to the place opposite the gap. The greatest number of molecules has the most probable speed. It is necessary to inform students that, theoretically, the law of the distribution of molecules according to velocities was discovered by J. K. Maxwell. The velocity distribution of molecules can be modeled on the Galton board. The question of the interaction of molecules was already studied by schoolchildren in the 7th grade; in the 10th grade, knowledge on this issue is deepened and expanded. It is necessary to emphasize the following points: a) intermolecular interaction has an electromagnetic nature; b) intermolecular interaction is characterized by forces of attraction and repulsion; c) the forces of intermolecular interaction act at distances not greater than 2-3 molecular diameters, and at this distance only the attractive force is noticeable, the repulsive forces are practically equal to zero; d) as the distance between the molecules decreases, the interaction forces increase, and the repulsive force grows faster (in proportion to r -9) than the attractive force (in proportion to r -7 ). Therefore, when the distance between the molecules decreases, the attractive force first prevails, then at a certain distance r o the attractive force is equal to the repulsive force, and with further approach, the repulsive force prevails. It is expedient to illustrate all of the above with a graph of dependence on distance, first of the attractive force, the repulsive force, and then the resultant force. It is useful to construct a graph of the potential energy of interaction, which can later be used when considering the aggregate states of matter. Tenth-graders' attention is drawn to the fact that the state of stable equilibrium of interacting particles corresponds to the equality of the resultant forces of interaction to zero and the smallest value of their mutual potential energy. In a solid body, the interaction energy of particles (binding energy) is much greater than the kinetic energy of their thermal motion, so the motion of solid body particles is vibrations relative to the nodes of the crystal lattice. If the kinetic energy of the thermal motion of molecules is much greater than the potential energy of their interaction, then the motion of the molecules is completely random and the substance exists in a gaseous state. If the kinetic energy thermal particle motion is comparable to the potential energy of their interaction, then the substance is in a liquid state.

1.1. Thermodynamic parameters. @

A mentally selected macroscopic system, considered by the methods of thermodynamics, is called a thermodynamic system. All bodies not included in the system under study are called the external environment. The state of the system is set by thermodynamic parameters (or, in other words, state parameters) - a set of physical quantities that characterize the properties of the system. Usually, pressure p, temperature T and specific volume v are chosen as the main parameters. There are two types of thermodynamic parameters: extensive and intensive. Extensive parameters are proportional to the amount of matter in the system, while intensive parameters do not depend on the amount of matter and the mass of the system. Intensive parameters are pressure, temperature, specific volume, etc., and extensive parameters are volume, energy, entropy.

The volume is proportional to the amount of substance in the system. In calculations, it is more convenient to operate with the specific volume v - this is a value equal to the ratio of the volume to the mass of the system, that is, the volume per unit mass v = V / m = 1/ρ, where ρ is the density of the substance.

Pressure is a physical quantity where dF n is the projection of the force on the normal to the surface with an area of ​​dS.

Temperature is a physical quantity that characterizes the energy of a macroscopic system in a state of thermodynamic equilibrium. The temperature of the system is a measure of the intensity of thermal motion and the interaction of the particles that form the system. This is the molecular-kinetic meaning of temperature. Currently, there are two temperature scales - thermodynamic (graded in Kelvin (K)) and International practical (graded in degrees Celsius (˚С)). 1˚С = 1K. The relationship between the thermodynamic temperature T and the temperature according to the International Practical Scale is: T = t + 273.15˚С.

Any change in the state of a thermodynamic system, characterized by a change in its parameters, is called a thermodynamic process. A thermodynamic process is called equilibrium if the system passes through a series of infinitely close equilibrium states. An equilibrium state is a state in which the system eventually comes under constant external conditions and then remains in this state for an arbitrarily long time. The real process of changing the state of the system will be the closer to the equilibrium, the slower it takes place.

1. 2. The equation of state of an ideal gas. @

The physical model of an ideal gas is widely used in molecular kinetic theory. This is a substance in a gaseous state for which the following conditions are met:

1. The intrinsic volume of gas molecules is negligible compared to the volume of the vessel.

2. There are no interactions between gas molecules, except for random collisions.

3. Collisions of gas molecules between themselves and with the walls of the vessel are absolutely elastic.

The ideal gas model can be used in the study of real gases, because they under conditions close to normal (pressure p 0 = 1.013∙10 5 Pa, temperature T 0 = 273.15 K) behave similarly to an ideal gas. For example, air at T=230K and p=p0/50 is similar to the ideal gas model in all three criteria.

The behavior of ideal gases is described by a number of laws.

Avogadro's Law: Moles of any gas at the same temperature and pressure occupy the same volume. Under normal conditions, this volume is equal to V M =22.4∙10 -3 m 3 /mol. One mole of various substances contains the same number of molecules, called the Avogadro number N A = 6.022∙10 23 mol -1.

Boyle's law - Mariotte: for a given mass of gas at a constant temperature, the product of gas pressure and its volume is a constant value pV = const at T = const and m = const.

Charles' law: the pressure of a given mass of gas at constant volume changes linearly with temperature p=p 0 (1+αt) at V = const and m = const.

Gay-Lussac's law: the volume of a given mass of gas at constant pressure changes linearly with temperature V = V 0 (1 + αt) at p = const and m = const. In these equations, t is the temperature on the Celsius scale, p 0 and V 0 are pressure and volume at 0 ° C, the coefficient α \u003d 1 / 273.15 K -1.

The French physicist and engineer B. Clapeyron and the Russian scientist D. I. Mendeleev, combining Avogadro's law and the laws of ideal gases of Boyle - Mariotte, Charles and Gay - Lussac, derived the equation of state of an ideal gas - an equation that links together all three thermodynamic parameters of the system: for one mole of gas pV M = RT and for an arbitrary mass of gas

It can be obtained if we take into account that k \u003d R / N A \u003d 1.38 ∙ 10 -23 J / K is the Boltzmann constant, and n \u003d N A / V M is the concentration of gas molecules.

To calculate the pressure in a mixture of different gases, Dalton's law is used: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases included in it: p \u003d p 1 + p 2 + ... + p n. Partial pressure is the pressure that a gas in a gas mixture would produce if it alone occupied a volume equal to the volume of the mixture at the same temperature. To calculate the partial pressure of an ideal gas, the Mendeleev–Clapeyron equation is used.

1. 3. The basic equation of the molecular-kinetic theory of ideal gases and its consequences. @

Consider a monatomic ideal gas occupying a certain volume V (Fig. 1.1.) Let the number of collisions between molecules be negligible compared to the number of collisions with the vessel walls. Let us single out some elementary area ΔS on the wall of the vessel and calculate the pressure exerted on this area. With each collision, a molecule with a mass m 0 moving perpendicular to the site with a speed υ transfers to it a momentum, which is the difference in the momenta of the molecule before and after the collision:

m 0 υ -(-m 0 υ) = 2m 0 υ.

During the time Δt, the area ΔS will reach only those molecules that are enclosed in the volume of the cylinder with the base ΔS and length υΔt. This number of molecules will be nυΔSΔt, where n is the concentration of molecules. However, it must be taken into account that the molecules actually move towards the site at different angles and have different velocities, and the velocity of the molecules changes with each collision. To simplify the calculations, the chaotic motion of molecules is replaced by motion along three mutually perpendicular coordinate axes, so that at any time 1/3 of the molecules move along each of them, with half - 1/6 - moving in one direction, half in the opposite direction. Then the number of impacts of molecules moving in a given direction on the platform ΔS will be nυΔSΔt /6. When colliding with the platform, these molecules will transfer momentum to it.

In this case, when the force acting per unit area is constant, for the gas pressure on the vessel wall we can write p = F/ΔS = ΔP/ΔSΔt = nm 0 υ 2 /3. Molecules in the vessel move with a variety of speeds υ 1, υ 2…. υ n, their total number is N. Therefore, it is necessary to consider the root-mean-square velocity, which characterizes the entire set of molecules:

The above equation is the basic equation of the molecular kinetic theory of ideal gases. Since m 0 ‹υ kv › 2 /2 is the average energy of the translational motion of the molecule ‹ ε post ›, the equation can be rewritten as:
where E is the total kinetic energy of the translational motion of all gas molecules. Thus, the pressure is equal to two-thirds of the energy of the translational motion of the molecules contained in a unit volume of gas.
Let us also find the kinetic energy of the translational motion of one molecule ‹ ε post ›, taking into account

k \u003d R / N A we get:

Hence it follows that the average kinetic energy of the chaotic translational motion of ideal gas molecules is proportional to its absolute temperature and depends only on it, i.e. temperature is a quantitative measure of the energy of the thermal motion of molecules. At the same temperature, the average kinetic energies of the molecules of any gas are the same. At T=0K ‹ε post › = 0 and the translational motion of gas molecules stops, however, analysis of various processes shows that T = 0K is an unattainable temperature.

4. Taking into account that ‹ε post › = 3kT/2, р = 2n‹ ε post ›/3, we get from here: р = nkT.

We have obtained the already familiar version of the Mendeleev-Clapeyron equation, derived in this case from the concepts of molecular-kinetic theory by the statistical method. The last equation means that at the same temperature and pressure, all gases contain the same number of molecules per unit volume.

1. 4. Barometric formula. @

When deriving the basic equation of the molecular kinetic theory, it was assumed that if external forces do not act on the gas molecules, then the molecules are uniformly distributed over the volume. However, the molecules of any gas are in the potential gravitational field of the Earth. Gravity, on the one hand, and thermal motion of molecules, on the other hand, lead to a certain stationary state of the gas, in which the concentration of gas molecules and its pressure decrease with height. Let us derive the law of change in gas pressure with height, assuming that the gravitational field is uniform, the temperature is constant, and the mass of all molecules is the same. If the atmospheric pressure at height h is equal to p, then at height h + dh it is equal to p + dp (Fig. 1.2). When dh > 0, dр< 0, т.к. давление с высотой убывает. Разность давлений р и (р + dр) равна гидростатическому давлению столба газа авсd, заключенного в объеме цилиндра высотой dh и площадью с основанием равным единице. Это запишется в следующем виде: p- (p+dp) = gρdh, - dp = gρdh или dp = ‑gρdh, где ρ – плотность газа на высоте h. Воспользуемся уравнением состояния идеального газа рV = mRT/M и выразим плотность ρ=m/V=pM/RT. Подставим это выражение в формулу для dр:

dp = - pMgdh/RT or dp/p = - Mgdh/RT

Integration of this equation gives the following result: Here C is a constant and in this case it is convenient to denote the constant of integration as lnC. Potentiating the resulting expression, we find that

This expression is called the barometric formula. It allows you to find atmospheric pressure as a function of altitude, or altitude if the pressure is known.

Figure 1.3 shows the dependence of pressure on altitude. An instrument for determining altitude above sea level is called an altimeter or altimeter. It is a barometer calibrated in terms of altitude.

1. 5. Boltzmann's law on the distribution of particles in an external potential field. @

here n is the concentration of molecules at a height h, n 0 is the same at the Earth's surface. Since M \u003d m 0 N A, where m 0 is the mass of one molecule, and R \u003d k N A, then we get P \u003d m 0 gh - this is the potential energy of one molecule in the gravitational field. Since kT~‹ε post ›, then the concentration of molecules at a certain height depends on the ratio P and ‹ε post ›

The resulting expression is called the Boltzmann distribution for the external potential field. It follows from it that at a constant temperature the density of a gas (with which the concentration is related) is greater where the potential energy of its molecules is less.

1. 6. Maxwell's distribution of ideal gas molecules over velocities. @

When deriving the basic equation of the molecular kinetic theory, it was noted that molecules have different velocities. As a result of multiple collisions, the velocity of each molecule changes with time in absolute value and in direction. Due to the randomness of the thermal motion of molecules, all directions are equally probable, and the mean square velocity remains constant. We can write down

The constancy of ‹υ kv › is explained by the fact that a stationary velocity distribution of molecules that does not change with time is established in the gas, which obeys a certain statistical law. This law was theoretically derived by D.K. Maxwell. He calculated the function f(u), called the velocity distribution function of molecules. If we divide the range of all possible velocities of molecules into small intervals equal to du, then for each interval of speed there will be a certain number of molecules dN(u) that have a speed enclosed in this interval (Fig.1.4.).

The function f(v) determines the relative number of molecules whose velocities lie in the range from u to u+ du. This number is dN(u)/N= f(u)du. Applying the methods of probability theory, Maxwell found the form for the function f(u)

This expression is the law on the distribution of molecules of an ideal gas in terms of velocities. The specific form of the function depends on the type of gas, the mass of its molecules and temperature (Fig. 1.5). The function f(u)=0 at u=0 and reaches a maximum at some value of u in, and then asymptotically tends to zero. The curve is asymmetric about the maximum. The relative number of molecules dN(u)/N whose velocities lie in the interval du and equal to f(u)du is found as the area of ​​the shaded strip with base dv and height f(u) shown in Fig. 1.4. The entire area bounded by the f (u) curve and the abscissa axis is equal to one, because if you sum up all the fractions of molecules with all possible speeds, you get one. As shown in Fig. 1.5, with increasing temperature, the distribution curve shifts to the right, i.e. the number of fast molecules increases, but the area under the curve remains constant, because N = const.

The speed u at which the function f(u) reaches its maximum is called the most probable speed. From the condition that the first derivative of the function f(v) ′ = 0 is equal to zero, it follows that

An experiment conducted by the German physicist O. Stern experimentally confirmed the validity of the Maxwell distribution (Figure 1.5.). The Stern device consists of two coaxial cylinders. A platinum wire coated with a layer of silver passes along the axis of the inner cylinder with a slot. If current is passed through the wire, it heats up and the silver evaporates. Silver atoms, flying out through the slot, fall on the inner surface of the second cylinder. If the device rotates, then the silver atoms will not settle against the gap, but will be displaced from the point O for a certain distance. The study of the amount of sediment makes it possible to estimate the distribution of molecules by velocities. It turned out that the distribution corresponds to the Maxwellian one.

DEFINITION

The equation underlying the molecular kinetic theory connects macroscopic quantities describing (for example, pressure) with the parameters of its molecules (and their velocities). This equation looks like:

Here, is the mass of a gas molecule, is the concentration of such particles per unit volume, and is the averaged square of the molecular velocity.

The basic equation of the MKT clearly explains how an ideal gas creates on the vessel walls surrounding it. Molecules all the time hit the wall, acting on it with a certain force F. Here it should be remembered: when a molecule hits an object, a force -F acts on it, as a result of which the molecule “bounces” from the wall. In this case, we consider the collisions of molecules with the wall to be absolutely elastic: the mechanical energy of the molecules and the wall is completely conserved without passing into . This means that only the molecules change during collisions, and the heating of the molecules and the wall does not occur.

Knowing that the collision with the wall was elastic, we can predict how the velocity of the molecule will change after the collision. The velocity modulus will remain the same as before the collision, and the direction of motion will change to the opposite with respect to the Ox axis (we assume that Ox is the axis that is perpendicular to the wall).

There are a lot of gas molecules, they move randomly and often hit the wall. Having found the geometric sum of forces with which each molecule acts on the wall, we find out the gas pressure force. To average the velocities of molecules, it is necessary to use statistical methods. That is why the basic MKT equation uses the averaged square of the molecular velocity , and not the square of the averaged velocity : the averaged velocity of randomly moving molecules is equal to zero, and in this case we would not get any pressure.

Now the physical meaning of the equation is clear: the more molecules are contained in the volume, the heavier they are and the faster they move, the more pressure they create on the walls of the vessel.

Basic MKT equation for the ideal gas model

It should be noted that the basic MKT equation was derived for the ideal gas model with the appropriate assumptions:

1. Collisions of molecules with surrounding objects are absolutely elastic. For real gases, this is not entirely true; some of the molecules still pass into the internal energy of the molecules and the wall.
2. The forces of interaction between molecules can be neglected. If the real gas is at high pressure and relatively low temperature, these forces become very significant.
3. We consider molecules to be material points, neglecting their size. However, the dimensions of the molecules of real gases affect the distance between the molecules themselves and the wall.
4. And, finally, the main equation of the MKT considers a homogeneous gas - and in reality we often deal with mixtures of gases. Such as, .

However, for rarefied gases, this equation gives very accurate results. In addition, many real gases at room temperature and at pressures close to atmospheric are very similar in properties to an ideal gas.

As is known from the laws, the kinetic energy of any body or particle. Replacing the product of the mass of each of the particles and the square of their speed in the equation we wrote down, we can represent it as:

Also, the kinetic energy of gas molecules is expressed by the formula , which is often used in problems. Here k is Boltzmann's constant, establishing the relationship between temperature and energy. k=1.38 10 -23 J/K.

The basic equation of the MKT underlies thermodynamics. It is also used in practice in astronautics, cryogenics and neutron physics.

Examples of problem solving

EXAMPLE 1

Exercise	Determine the speed of movement of air particles under normal conditions.
Solution	We use the basic MKT equation, considering air as a homogeneous gas. Since air is actually a mixture of gases, the solution to the problem will not be absolutely accurate. Gas pressure: We can notice that the product is a gas, since n is the concentration of air molecules (the reciprocal of volume), and m is the mass of the molecule. Then the previous equation becomes: Under normal conditions, the pressure is 10 5 Pa, the air density is 1.29 kg / m 3 - these data can be taken from the reference literature. From the previous expression we obtain air molecules:
Answer	m/s

EXAMPLE 2

Exercise	Determine the concentration of homogeneous gas molecules at a temperature of 300 K and 1 MPa. Consider the gas to be ideal.
Solution	Let's start the solution of the problem with the basic equation of the MKT: , as well as any material particles: . Then our calculation formula will take a slightly different form: