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 no longer used in modern theoretical physics, although it is found in textbooks on the course of general physics.
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 makes it possible to take 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 when the particles collide 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 , i.e:
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, the number of speed values is arbitrarily large (like numbers in a sequential series), and it may happen that not a single molecule has a given speed.
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Rice. 3.3Based 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 rotational degrees of freedom are first excited with increasing temperature. 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 elastic wave energy quantum, 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. 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 were increased so many times that the molecule was 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 method for determining the mass of molecules is based on the experience 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 speed 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. All of the above is expediently illustrated by 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 motion of particles is comparable to the potential energy of their interaction, then the substance is in a liquid state.
Matter is made up of particles.
Molecule is the smallest particle of a substance that has its basic chemical properties.
A molecule is made up of atoms. Atom- the smallest particle of a substance that does not divide in chemical reactions.
Many molecules are made up of two or more atoms held together by chemical bonds. Some molecules are made up of hundreds of thousands of atoms.
The second position of the molecular kinetic theory
Molecules are in continuous chaotic motion. This movement does not depend on external influences. The movement occurs in an unpredictable direction due to the collision of molecules. The proof is Brownian motion particles (discovered by R. Brown in 1827). Particles are placed in a liquid or gas and their unpredictable movement is observed due to collisions with the molecules of the substance.
Brownian motion
The proof of chaotic motion is diffusion- the penetration of molecules of one substance into the gaps between the molecules of another substance. For example, we feel the smell of an air freshener not only in the place where it was sprayed, but it gradually mixes with air molecules throughout the room.
Aggregate state of matter
AT gases the average distance between molecules is hundreds of times greater than their size. Molecules generally move progressively and uniformly. After collisions, they begin to rotate.
AT liquids the distance between molecules is much smaller. Molecules perform vibrational and translational motion. Molecules at short intervals jump into new equilibrium positions (we observe the fluidity of a liquid).
AT solid Molecules in bodies oscillate and very rarely move (only with increasing temperature).
The third position of the molecular kinetic theory
There are interaction forces between molecules that are electromagnetic in nature. These forces make it possible to explain the emergence of elastic forces. When a substance is compressed, the molecules approach each other, a repulsive force arises between them, when external forces move the molecules away from each other (stretch the substance), an attractive force arises between them.
Matter density
This is a scalar value, which is determined by the formula
Density of substances - known tabular values
Chemical characteristics of the substance
Avogadro constant N A- number of atoms contained in 12 g of carbon isotope
§ 2. Molecular physics. Thermodynamics
Main provisions of molecular kinetic theory(MKT) are as follows.1. Substances are made up of atoms and molecules.
2. Atoms and molecules are in continuous chaotic motion.
3. Atoms and molecules interact with each other with forces of attraction and repulsion
The nature of the movement and interaction of molecules can be different, in this regard, it is customary to distinguish 3 states of aggregation of matter: solid, liquid and gaseous. The interaction between molecules is strongest in solids. In them, the molecules are located in the so-called nodes of the crystal lattice, i.e. in positions where the forces of attraction and repulsion between molecules are equal. The motion of molecules in solids is reduced to oscillatory motion around these equilibrium positions. In liquids, the situation differs in that, having fluctuated around some equilibrium positions, the molecules often change them. In gases, the molecules are far from each other, so the interaction forces between them are very small and the molecules move forward, occasionally colliding with each other and with the walls of the vessel in which they are located.
Relative molecular weight M r call the ratio of the mass m o of a molecule to 1/12 of the mass of a carbon atom moc:
The amount of a substance in molecular physics is usually measured in moles.
Molem ν called the amount of a substance that contains the same number of atoms or molecules (structural units) as they are contained in 12 g of carbon. This number of atoms in 12 g of carbon is called Avogadro's number:
Molar mass M = M r 10 −3 kg/mol is the mass of one mole of a substance. The number of moles in a substance can be calculated using the formula
The basic equation of the molecular kinetic theory of an ideal gas is:
where m0 is the mass of the molecule; n- concentration of molecules; Ṽ
is the root mean square velocity of the molecules.
2.1. Gas laws
The equation of state of an ideal gas is the Mendeleev-Clapeyron equation:Isothermal process(Boyle-Mariotte law):
For a given mass of gas at a constant temperature, the product of pressure and its volume is a constant value:
In coordinates p − V isotherm is a hyperbola, and in coordinates V − T and p − T- straight (see fig. 4) 
Isochoric process(Charles law):
For a given mass of gas with a constant volume, the ratio of pressure to temperature in degrees Kelvin is a constant value (see Fig. 5). 
isobaric process(Gay-Lussac's law):
For a given mass of gas at constant pressure, the ratio of gas volume to temperature in degrees Kelvin is a constant value (see Fig. 6). 
Dalton's law:
If a vessel contains a mixture of several gases, then the pressure of the mixture is equal to the sum of the partial pressures, i.e. the pressures that each gas would create in the absence of the others.
2.2. Elements of thermodynamics
Internal energy of the body is equal to the sum of the kinetic energies of the random movement of all molecules relative to the center of mass of the body and the potential energies of the interaction of all molecules with each other.Internal energy of an ideal gas is the sum of the kinetic energies of the random movement of its molecules; Since the molecules of an ideal gas do not interact with each other, their potential energy vanishes.
For an ideal monatomic gas, the internal energy
The amount of heat Q called a quantitative measure of the change in internal energy during heat transfer without doing work.
Specific heat is the amount of heat that 1 kg of a substance receives or gives off when its temperature changes by 1 K
Work in thermodynamics:
work during isobaric expansion of a gas is equal to the product of the gas pressure and the change in its volume:
The law of conservation of energy in thermal processes (the first law of thermodynamics):
the change in the internal energy of the system during its transition from one state to another is equal to the sum of the work of external forces and the amount of heat transferred to the system:
Applying the first law of thermodynamics to isoprocesses:
a) isothermal process T = const ⇒ ∆T = 0.
In this case, the change in the internal energy of an ideal gas
Hence: Q=A.
All the heat transferred to the gas is spent on doing work against external forces;
b) isochoric process V = const ⇒ ∆V = 0.
In this case, the work of the gas
Hence, ∆U = Q.
All the heat transferred to the gas is spent on increasing its internal energy;
in) isobaric process p = const ⇒ ∆p = 0.
In this case:
adiabatic A process that occurs without heat exchange with the environment is called:
In this case A = −∆U, i.e. the change in the internal energy of the gas occurs due to the work of the gas on external bodies.
As the gas expands, it does positive work. The work A performed by external bodies on the gas differs from the work of the gas only in sign:
The amount of heat required to heat up a body in a solid or liquid state within one state of aggregation, calculated by the formula
where c is the specific heat of the body, m is the mass of the body, t 1 is the initial temperature, t 2 is the final temperature.
The amount of heat required to melt the body at the melting point, calculated by the formula
where λ is the specific heat of fusion, m is the mass of the body.
The amount of heat required for evaporation, is calculated by the formula
where r is the specific heat of vaporization, m is the mass of the body.
In order to convert part of this energy into mechanical energy, heat engines are most often used. Heat engine efficiency The ratio of the work A done by the engine to the amount of heat received from the heater is called:
The French engineer S. Carnot came up with an ideal heat engine with an ideal gas as a working fluid. The efficiency of such a machine 
Air, which is a mixture of gases, contains water vapor along with other gases. Their content is usually characterized by the term "humidity". Distinguish between absolute and relative humidity.
absolute humidity called the density of water vapor in the air ρ ([ρ] = g/m 3). You can characterize absolute humidity by the partial pressure of water vapor - p([p] = mm Hg; Pa).
Relative humidity (ϕ)- the ratio of the density of water vapor present in the air to the density of the water vapor that would have to be contained in the air at that temperature in order for the vapor to be saturated. You can measure relative humidity as the ratio of the partial pressure of water vapor (p) to that partial pressure (p 0) that saturated steam has at this temperature: 
Molecular kinetic theory describes the behavior and properties of a special ideal object called ideal gas. This physical model is based on the molecular structure of matter. The creation of molecular theory is associated with the works of R. Clausius, J. Maxwell, D. Joule and L. Boltzmann.
Ideal gas. Molecular-kinetic theory of ideal gas is built on the following assumptions:
atoms and molecules can be considered as material points in continuous motion;
the intrinsic volume of gas molecules is negligible compared to the volume of the vessel;
all atoms and molecules are distinguishable, that is, it is possible in principle to follow the movement of each particle;
before the collision of gas molecules between them, there are no interaction forces, and the collisions of molecules between themselves and with the walls of the vessel are assumed to be absolutely elastic;
the motion of each atom or molecule of a gas is described by the laws of classical mechanics.
The laws obtained for an ideal gas can be used in the study of real gases. For this, experimental models of an ideal gas are created, in which the properties of a real gas are close to those of an ideal gas (for example, at low pressures and high temperatures).
Ideal gas laws
Boyle-Mariotte law:
for a given mass of gas at a constant temperature, the product of the gas pressure and its volume is a constant value: pV = const , (1.1)
at T = const , m = const .
Curve showing the relationship between quantities R and V, characterizes the properties of a substance at a constant temperature, and is called isotherm this is a hyperbola (Fig. 1.1.), and the process proceeding at a constant temperature is called isothermal.
Gay-Lussac's laws:
The volume of a given mass of gas at constant pressure varies linearly with temperature
V = V 0 (1 + t ) at P = const , m = const . (1.2)
p = p 0 (1 + t ) at V = const , m = const . (1.3)
In equations (1.2) and (1.3), temperature is expressed on the Celsius scale, pressure and volume - at
0 С, while
.
A process that takes place at constant pressure is called isobaric, it can be represented as a linear function (Fig. 1.2.).
A process that takes place at constant volume is called isochoric(Fig. 1.3.).
It follows from equations (1.2) and (1.3) that isobars and isochores intersect the temperature axis at the point t =1/ \u003d - 273.15 С . If we move the origin to this point, then we move on to the Kelvin scale.
Introducing into formulas (1.2) and (1.3) thermodynamic temperature, the laws of Gay-Lussac can be given a more convenient form:
V = V 0 (1+t) = = V 0 = =V 0 T;
p = p 0 (1+t) = p 0 = p 0 T;
at
p=const, m=const
;
(1.4)
at V = const, m = const
,
(1.5)
where indices 1 and 2 refer to arbitrary states lying on the same isobar or isochore .
Avogadro's law:
moles of any gases at the same temperatures and pressures occupy the same volumes.
Under normal conditions, this volume is equal to V,0 \u003d 22.4110 -3 m 3 / mol . By definition, one mole of different substances contains the same number of molecules, equal to constant Avogadro:N A = 6,02210 23 mol -1 .
Dalton's law:
the pressure of a mixture of different ideal gases is equal to the sum of the partial pressures R 1 , R 2 , R 3 … R n, gases included in it:
p = p 1 + p 2 + R 3 + …+ p n .
Partial pressure – This 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.
Ideal gas equation of state
(Clapeyron-Mendeleev equation)
There is a definite relationship between temperature, volume and pressure. This relationship can be represented by a functional dependency:
f(p, V, T)= 0.
In turn, each of the variables ( p, v, t) is a function of two other variables. The type of functional dependence for each phase state of a substance (solid, liquid, gaseous) is found experimentally. This is a very laborious process and the equation of state has been established only for gases that are in a rarefied state, and in an approximate form for some compressed gases. For substances that are not in a gaseous state, this problem has not yet been solved.
The French physicist B. Clapeyron brought ideal gas equation of state, by combining the laws of Boyle-Mariotte, Gay-Lussac, Charles:
. (1.6)
Expression (1.6) is the Clapeyron equation, where AT is the gas constant. It is different for different gases.
DI. Mendeleev combined Clapeyron's equation with Avogadro's law, referring equation (1.6) to one mole and using the molar volume V . According to Avogadro's law, for the same R and T moles of all gases occupy the same molar volume V .
.
Therefore, the constant AT will be the same for all ideal gases. This constant is usually denoted R and equal to R=
8,31
.
Clapeyron-Mendeleev equation has the following form:
p V . = R T.
From equation (1.7) for one mole of gas, one can go to to the Clapeyron-Mendeleev equation for an arbitrary mass of gas:
, (1.7)
where
–
molar mass
(mass of one mole of substance, kg/mol); m
mass of gas;
- amount of matter .
More often, another form of the ideal gas equation of state is used, introducing Boltzmann's constant: 
.
Then equation (1.7) looks like this:
,
(1.8)
where
–
concentration of molecules (number of molecules per unit volume). It follows from this expression that the pressure of an ideal gas is directly proportional to the concentration of its molecules or the density of the gas. At the same temperatures and pressures, all gases contain the same number of molecules per unit volume. The number of molecules contained in 1 m 3 under normal conditions is called
Loschmidt number:
N L = 2.68 10 25 m -3.
Basic equation of molecular kinetic
theory of ideal gases
The most important task The kinetic theory of gases is the theoretical calculation of the pressure of an ideal gas based on molecular kinetic concepts. The basic equation of the molecular kinetic theory of ideal gases is derived using statistical methods.
It is assumed that the gas molecules move randomly, the number of mutual collisions between the gas molecules is negligible compared to the number of impacts on the walls of the vessel, and these collisions are absolutely elastic. On the wall of the vessel, some elementary area S and calculate the pressure that the gas molecules will exert on this area.
It is necessary to take into account the fact that the molecules can actually move towards the site at different angles and can have different velocities, which, moreover, can change with each collision. In theoretical calculations, the chaotic motion of molecules is idealized, they are replaced by motion along three mutually perpendicular directions.
If we consider a vessel in the form of a cube, in which N gas molecules in six directions, it is easy to see that at any moment 1/3 of the number of all molecules moves along each of them, and half of them (that is, 1/6 of the number of all molecules) moves in one direction, and the second half (also 1/6) - in the opposite direction. With each collision, an individual molecule moving perpendicular to the site, reflecting, transfers momentum to it, while its momentum (momentum) changes by the amount
R 1 =m 0 v – (– m 0 v) = 2 m 0 v.
The number of impacts of molecules moving in a given direction on the site will be equal to: N = 1/6 n Svt. When colliding with the platform, these molecules will transfer momentum to it.
P= N P 1 =2 m 0 vnSvt= m 0 v 2 nSt,
where n is the concentration of molecules. Then the pressure that the gas exerts on the wall of the vessel will be equal to:
p = 
=
nm 0
v 2
.
(1.9)
However, gas molecules move at different speeds: v 1 , v 2 , …,v n, so the velocities must be averaged. The sum of the squares of the velocities of the gas molecules, divided by their number, determines the root mean square velocity:
.
Equation (1.9) will take the form:
(1.10)
expression (1.10) is called the basic equation of molecular kinetic theory ideal gases.
Given that 
, we get:
p V = N 
=E,
(1.11)
where E is the total kinetic energy of the translational motion of all gas molecules. Therefore, the gas pressure is directly proportional to the kinetic energy of the translational motion of the gas molecules.
For one mole of gas m =, and the Clapeyron-Mendeleev equation has the following form:
p V . = R T,
and since it follows from (1.11) that p V . = v sq. 2 , we get:
R.T.= v sq. 2 .
Hence, the root-mean-square velocity of gas molecules is equal to
v
sq.
=
=
=
,
where k = R/N A = 1.3810 -23 J/K – Boltzmann's constant. From here you can find the mean square velocity of oxygen molecules at room temperature - 480 m/s, hydrogen - 1900 m/s.
Molecular-kinetic meaning of temperature
Temperature is a quantitative measure of how hot a body is. To clarify the physical meaning of the absolute thermodynamic temperature T Let's compare the basic equation of the molecular-kinetic theory of gases (1.14) with the Clapeyron-Mendeleev equation p V = R.T.
Equating the right parts of these equations, we find the average value of the kinetic energy 0 of one molecule ( = N/N A , k=R/N A):
.
The most important conclusion of the molecular kinetic theory follows from this equation: the average kinetic energy of the translational motion of one molecule of an ideal gas depends only on temperature, while it is directly proportional to thermodynamic temperature. Thus, the thermodynamic temperature scale acquires a direct physical meaning: at T= 0 the kinetic energy of ideal gas molecules is zero. Therefore, based on this theory, the translational motion of the gas molecules will stop and its pressure will become equal to zero.
Theory of equilibrium properties of an ideal gas
Number of degrees of freedom of molecules. The molecular-kinetic theory of ideal gases leads to a very important consequence: gas molecules move randomly, and the average kinetic energy of the translational motion of the molecule is determined solely by temperature.
The kinetic energy of molecular motion is not exhausted by the kinetic forward motion energy: it also consists of kinetic energies rotation and fluctuations molecules. In order to calculate the energy going into all types of molecular motion, it is necessary to define number of degrees of freedom.
Under number of degrees of freedom (i) of the body is implied the number of independent coordinates that must be entered to determine the position of the body in space.
H 
For example, a material point has three degrees of freedom, since its position in space is determined by three coordinates: x, y and z. Therefore, a monatomic molecule has three degrees of freedom of translational motion.
D 
a buchatomic molecule has 5 degrees of freedom (Fig. 1.4): 3 degrees of freedom of translational motion and 2 degrees of freedom of rotational motion.
Molecules of three or more atoms have 6 degrees of freedom: 3 degrees of freedom of translational motion and 3 degrees of freedom of rotational motion (Fig. 1.5).
Each gas molecule has a certain number of degrees of freedom, three of which correspond to its translational motion.
Regulation on the equal distribution of energy
by degrees of freedom
The basic premise of the molecular-kinetic theory of gases is the assumption of complete randomness in the motion of molecules. This applies to both oscillatory and rotational movements, and not just translational. It is assumed that all directions of motion of molecules in a gas are equally probable. Therefore, we can assume that for each degree of freedom of a molecule, on average, there is the same amount of energy - this is the position on the equipartition of energy over degrees of freedom. The energy per one degree of freedom of a molecule is:
. (1.12)
If the molecule has i degrees of freedom, then for each degree of freedom there is on average:
.
(1.13)
Internal energy of an ideal gas
If we attribute the total supply of internal energy of the gas to one mole, then we obtain its value by multiplying by the Avogadro number:
.
(1.14)
It follows that the internal energy of one mole of an ideal gas depends only on the temperature and the number of degrees of freedom of the gas molecules.
Maxwell and Boltzmann distributions
Distribution of molecules of an ideal gas according to the velocities and energies of thermal motion (Maxwell distribution). At a constant gas temperature, all directions of molecular motion are assumed to be equally probable. In this case, the root-mean-square velocity of each molecule remains constant and is equal to
.
This is explained by the fact that in an ideal gas, which is in a state of equilibrium, a certain stationary velocity distribution of molecules that does not change with time is established. this distribution is subject to a certain statistical law, which was theoretically derived by J. Maxwell. Maxwell's law is described by the function
,
that is the function f(v) determines the relative number of molecules 
, whose velocities lie in the interval from v
before v+dv. Applying the methods of probability theory, Maxwell found the law of distribution of molecules of an ideal gas in terms of velocities:
. (1.15)
The distribution function is shown graphically in fig. 1.6. The area bounded by the distribution curve and the x-axis is equal to one. This means that the function f(v) satisfies the normalization condition:
.
With 
velocity at which the distribution function of ideal gas molecules in terms of velocities f(v) is maximum, is called most likely
speed
v B .
Values v = 0 and v = correspond to the minima of expression (1.15). The most probable speed can be found by differentiating expression (1.23) and equating it to zero:
=
=
1,41
With an increase in temperature, the maximum of the function will shift to the right (Fig. 1.6), that is, with an increase in temperature, the most probable speed also increases, however, the area bounded by the curve remains unchanged. It should be noted that in gases and at low temperatures there is always a small number of molecules that move at high speeds. The presence of such "hot" molecules is of great importance in the course of many processes.
Arithmetic average speed molecules is determined by the formula
.
Root mean square speed
=
1,73
.
The ratio of these velocities does not depend on temperature or on the type of gas.
Distribution function of molecules by thermal motion energies. This function can be obtained by substituting the value of kinetic energy instead of velocity into the distribution equation of molecules (1.15):
.
Having integrated the expression over the energy values from 
before 
, we get average kinetic energy ideal gas molecules:
.
barometric formula. Boltzmann distribution. When deriving the basic equation of the molecular kinetic theory of gases and the Maxwell distribution of molecules by velocities, it was assumed that the molecules of an ideal gas are not affected by external forces, therefore the molecules are uniformly distributed throughout the volume. However, the molecules of any gas are in the Earth's gravitational field. When deriving the law of dependence of pressure on height, it is assumed that the gravitational field is uniform, the temperature is constant and the mass of all molecules is the same:
. (1.16)
Expression (1.16) is called barometric formula. It allows you to find the atmospheric pressure depending on the altitude, or by measuring the pressure, you can find the altitude. As h 1 is the height above sea level, where the pressure is considered normal, then the expression can be modified:
.
The barometric formula can be converted using the expression p = nkT:
,
G 
de n
–
concentration of molecules at altitude h,
m 0
gh=P–
potential energy of a molecule in a gravitational field. At constant temperature, the density of the gas is greater where the potential energy of the molecule is lower. Graphically, the law of decrease in the number of particles per unit volume with height looks as shown in Fig. 1.7.
For an arbitrary external potential field, we write the following general expression
,
