Basic provisions of the ICT. Model ideal gas. Laws of Boyle-Mariotte, Gay-Lussac, Charles. Equation of Clapeyron - Mendeleev. Molecule and mole of a substance. Molecular and molar masses. Avogadro's number.

Basic equation of the MKT. Molecular-kinetic meaning of the concept of thermodynamic temperature.

Velocity distribution of ideal gas molecules (Maxwell distribution). Characteristic velocities of molecules. Distribution of ideal gas molecules in a potential force field (Boltzmann distribution). barometric formula.

The average number of collisions and the mean free path of molecules. Transfer phenomena: diffusion, internal friction, heat conduction.

Fundamentals of thermodynamics

Thermodynamic method of study common properties macroscopic systems. Internal energy as a thermodynamic function of the state of the system. The number of degrees of freedom of a molecule. The law of uniform distribution of energy over the degrees of freedom of molecules. First law of thermodynamics. The work of gas and the amount of heat. Specific and molar heat capacities. Mayer's equation.

Application of the first law of thermodynamics to isoprocesses. adiabatic process.

Thermal engines. Carnot cycle and its efficiency. The concept of entropy. The second law of thermodynamics.

Electrostatics

Electric charges and their properties. conservation law electric charge. Coulomb's law. electrostatic field. The intensity of the electrostatic field. The principle of superposition of electrostatic fields.

Tension vector flow. Gauss's theorem and its application to the calculation of electrostatic fields.

Potential and potential difference of the electrostatic field. equipotential surfaces. Relationship between tension and potential.

Dipole in an electrostatic field. Polarization of dielectrics. The dielectric constant of a substance. Electric field induction.

Conductors in an electrostatic field. Distribution of charges on the surface of conductors. Electric capacitance of a solitary conductor and a capacitor. Parallel and series connection of capacitors. Energy of a charged conductor and capacitor. Energy and energy density of the electrostatic field.

Constant electricity

Strength and current density. Third party forces. electromotive force and voltage. Ohm's law. conductor resistance. Series and parallel connection of conductors. Work and current power. Joule-Lenz law. Kirchhoff's rules for branched chains.

N O M E R A Z A D A H	7.11	7.12	7.13	7.14	7.15	7.16	7.17	7.18	7.19	7.20
7.1	7.2	7.3	7.4	7.5	7.6	7.7	7.8	7.9	7.10
6.11	6.12	6.13	6.14	6.15	6.16	6.17	6.18	6.19	6.20
6.1	6.2	6.3	6.4	6.5	6.6	6.7	6.8	6.9	6.10
5.1	5.2	5.3	5.4	5.5	5.6	5.7	5.8	5.9	5.10
4.1	4.2	4.3	4.4	4.5	4.6	4.7	4.8	4.9	4.10
3.21	3.22	3.23	3.24	3.25	3.26	3.27	3.28	3.29	3.30
3.11	3.12	3.13	3.14	3.15	3.16	3.17	3.18	3.19	3.20
3.1	3.2	3.3	3.4	3.5	3.6	3.7	3.8	3.9	3.10
2.31	2.32	2.33	2.34	2.35	2.36	2.37	2.38	2.39	2.40
2.21	2.22	2.23	2.24	2.25	2.26	2.27	2.28	2.29	2.30
2.11	2.12	2.13	2.14	2.15	2.16	2.17	2.18	2.19	2.20
2.1	2.2	2.3	2.4	2.5	2.6	2.7	2.8	2.9	2.10
1.11	1.12	1.13	1.14	1.15	1.16	1.17	1.18	1.19	1.20
1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	1.10
No. var

Elements of kinematics

Basic Formulas

Average and instant speed material point:

where is the movement of the point in time , is the radius vector that determines the position of the point.

For rectilinear uniform motion ():

where is the path traveled by the point in time .

Average and instantaneous acceleration of a material point:

Full acceleration in curvilinear motion:

where is the tangential component of acceleration directed tangentially to the trajectory; - normal component of acceleration directed to the center of curvature of the trajectory ( - radius of curvature of the trajectory at a given point).

The path and speed for the equally variable motion of a material point ():

where is the initial speed, "+" corresponds to uniformly accelerated motion, "-" - equally slow.

Angular speed:

Angular acceleration:

Angular velocity for uniform rotational motion of a rigid body:

where is the angle of rotation of the body, is the period of rotation; - frequency of rotation ( - the number of revolutions made by the body in time ).

Angle of rotation and angular velocity for uniform rotational motion of a rigid body ():

where is the initial angular velocity, "+" corresponds to uniformly accelerated rotation, "-" - uniformly retarded.

Relationship between linear and angular quantities:

where is the distance from the point to the instantaneous axis of rotation.

Examples of problem solving

Task 1. The dependence of the path traveled by the body on time is expressed by the equation ( = 2 m/s, = 3 m/s 2 , = 5 m/s 3). Write down expressions for speed and acceleration. Determine for the moment of time after the start of movement the distance traveled, speed and acceleration.

Given: ; ; ; ; .	Solution: To determine the dependence of the speed of the body on time, we determine the first derivative of the path with respect to time: , or after substitution . The distance traveled is defined as the difference.

Task 2. A body is thrown with speed at an angle to the horizontal. Taking the body as a material point, determine the normal and tangential acceleration of the body 1.2 s after the start of motion.

The projection remains constant in magnitude and direction during the movement of the point.

The projection onto the axis changes. At point C (Figure 1.1), the speed is directed horizontally, i.e. . This means that , where is the time during which the material point rises to the maximum height, or after substitution .

By the time 1.2 s the body will be on the descent. The total acceleration in the process of motion is directed vertically downwards and is equal to the free fall acceleration. The normal acceleration is equal to the projection of the free fall acceleration on the direction of the curvature radius, and the tangential acceleration is equal to the projection of the free fall acceleration on the direction of the motion velocity (see Fig.1.1).

From the triangles of speeds and accelerations we have:

where , ,

where is the speed at time

After substitution, we get:

Answer: , .

Task 3. The wheel of the car rotates uniformly. During 2 minutes it changed the rotation frequency from 240 to 60 min -1 . Determine: 1) angular acceleration wheels; 2) the number of complete revolutions made by the wheel during this time.

where are the angular velocities at the initial and final moments of time, respectively.

From equation (2) we get:

Angle of rotation . Therefore, expression (1) can be written as follows: .

From here: .

Answer: ; .

Task 4. The point moves along a circle with a radius in such a way that the time dependence of the radius rotation angle is given by the equation , where , . Determine by the end of the second second of rotation: a) angular velocity; b) linear speed; c) angular acceleration; d) normal acceleration; e) tangential acceleration.

Given: ; .	Solution: The dependence of the angular velocity on time is determined by taking the first derivative of the angle of rotation with respect to time, i.e. . For a moment in time , . Linear velocity of the point , or after substitution .
The dependence of the angular acceleration of a point on time is determined by the first derivative of the angular velocity with respect to time, i.e. . For a moment in time . Normal and tangential accelerations are determined by the formulas, respectively:
and . Answer: ; ; ; ; .

Control tasks

1.1. A body falls vertically from a height of 19.6 m with zero initial velocity. What path will the body cover: 1) for the first 0.1 from its movement, 2) for the last 0.1 from its movement? Think . Ignore air resistance.

1.2. A body falls vertically from a height of 19.6 m with zero initial velocity. How long will the body take to cover: 1) the first 1 m of its path, 2) the last 1 m of its path? Think . Ignore air resistance.

1.3. A body is thrown from a tower in a horizontal direction with an initial velocity of 10 m/s. Neglecting air resistance, determine for the moment of time = 2 s after the start of movement: 1) the speed of the body; 2) the radius of curvature of the trajectory. Think .

1.4. A stone is thrown horizontally with a speed of 5m/s. Determine the normal and tangential accelerations of the stone 1 s after the start of motion. Think . Ignore air resistance.

1.5. The material point begins to move along a circle with a radius = 2.5 cm with a constant tangential acceleration = 0.5 cm/s 2 . Determine: 1) the moment of time at which the acceleration vector forms an angle of 45 ° with the velocity vector; 2) the path traveled during this time by the moving point.

1.6. The dependence of the path traveled by the body on time is given by the equation , where =0.1m, =0.1m/s, =0.14m/s 2 , =0.01m/s 3 . 1) After how much time after the start of movement, the acceleration of the body will be equal to 1 m / s 2? 2) What is the average acceleration of the body for this period of time? after the start of the movement, the distance traveled, speed and acceleration. for this moment.

1.13. The disk rotates around a fixed axis so that the dependence of the angle of rotation of the disk radius on time is given by the equation ( = 0.1 rad / s 2). Determine the total acceleration of a point on the disk rim by the end of the second second after the start of movement, if at that moment the linear velocity of this point is 0.4 m/s.

1.14. A disk with a radius of 0.2 m rotates around a fixed axis so that the dependence of the angular velocity on time is given by the equation , where . Determine for points on the rim of the disk by the end of the first second after the start of movement, the total acceleration and the number of revolutions made by the disk in the first minute of movement.

1.15. A disk with a radius of 10 cm rotates so that the dependence of the angle of rotation of the disk radius on time is given by the equation ( = 2 rad, = 4 rad/s 3). Determine for points on the wheel rim: 1) normal acceleration at time 2 s; 2) tangential acceleration for the same moment; 3) the angle of rotation at which the full acceleration is 45° with the radius of the wheel.

1.16. The armature of the electric motor, having a rotation frequency of 50 s -1, after turning off the current, having made 628 revolutions, stopped. Determine the angular acceleration of the armature.

1.17. The wheel of the car rotates uniformly. During 2 minutes it changed the rotation frequency from 60 to 240 min -1 . Determine: 1) angular acceleration of the wheel; 2) the number of complete revolutions made by the wheel during this time.

1.18. The wheel, rotating uniformly accelerated, reached an angular velocity of 20 rad/s 10 revolutions after the start of rotation. Find the angular acceleration of the wheel.

1.19. The wheel after 1 min after the start of rotation acquires a speed corresponding to a frequency of 720 rpm. Find the angular acceleration of the wheel and the number of revolutions made by the wheel in that minute. The movement is considered to be uniformly accelerated.

1.20. The wheel, rotating equally slow, during braking, reduced the rotational speed in 1 min from 300 rpm to 180 rpm. Find the angular acceleration of the wheel and the number of revolutions made during this time.

The purpose of the lesson: Check students' knowledge and find out the degree of assimilation of the material of this topic.

During the classes

Organizing time.

Option -1 (1st level)

1. Calculate molecular weight oxygen - O₂. (Answer: 32 10 -3 kg/mol)

2. There is 80 g of oxygen, calculate the number of moles in it. (Answer: 2.5 moles)

3. Calculate the pressure of the gas on the walls of the cylinder, if it is known that it contains propane

(C3H4) with a volume of 3000 liters at a temperature of 300 K. The amount of substance of this gas is

140 mol. (Answer: 116kPa)

4. What is the reason for Brownian motion?

5. The figure shows the transition of an ideal gas from state 1 to state 2.

a) Give a name to the transition process. B) Show the graph of the process in PT and VT coordinates.

0 2 V

Option - 2 (1st level)

1. Calculate the molecular weight of water - H₂O. (Answer: 18 10-3kg/mol)

2. There are 200 g of water in a glass. Find the number of moles of water. (Answer: 11.1 moles)

3. The tank contains nitrogen weighing 4 kg at a temperature of 300 K and a pressure of 4 105 Pa.

Find the volume of nitrogen.

4. Why does gas occupy the entire volume provided to it?

5. The figure shows the transition of an ideal gas from state 1 to state 2.

a) Give a name to the transition process. B) Show the graph of the process in RT and VT coordinates Oh.

Option -1 (2nd level)

1. Determine the mass of 1022 nitrogen molecules.

Decision. m = m₀ N = M N/NA; m = 4.7 (kg)

2. Hydrogen temperature 25˚С. Calculate its density at normal atmospheric pressure.

Decision. ρ \u003d P M / RT \u003d 81 (g / cm³)

3. The flasks of electric lamps are filled with an inert gas at reduced pressure and temperature. Explain the reason.

4. In the RT coordinate system, a graph of the change in the state of an ideal gas is shown.

a) Give a name to each transition.

B) Draw the transitions in PV and VT coordinates.

5. Depending on the season, there is a difference in the mass of air that is inside the room. In summer, the air temperature is 40˚С, and in winter - 0˚С at normal atmospheric pressure. The molar mass of air is 29 10-3 kg/mol. Find the difference in the mass of air.

P V = m R T / M; m1 = P V M/R T1; m2 = P V M/R T2; ∆m = m₁ – m₂;

Δm = P V M/R (1/T1 – 1/T2); Δm = 8.2 (kg)

Option -2 (2nd level)

N = γ NA = m NA/M; N = 3.3 1012 (molecules)

2. Nitrogen is in a closed vessel with a capacity of 5 liters and has a mass of 5 g. It is heated from 20 ° C to 40 ° C. Calculate the nitrogen pressure before and after heating.

Decision. P1 V = mRT/M; P1 = mRT/VM; P1 = 8.7 (Pa)

P₁/P₂ = T₁/T₂; P₂ = P₁ T₂/T₁; P₂ = 9.3 104 (Pa)

3. Why are the chambers of automobile wheels pumped up to more pressure in winter than in summer?

4. In the RT coordinate system, a graph of the change in the state of an ideal gas is shown.

R 4 A) Give a name to each transition.

B) Draw transitions in coordinates

This manual includes tests for self-control, independent work, multi-level control work.
Suggested didactic materials compiled in full accordance with the structure and methodology of V. A. Kasyanov's textbooks “Physics. A basic level of. Grade 10" and "Physics. Deep level. Grade 10".

Task examples:

TS 1. Moving. Speed.
Uniform rectilinear motion
Option 1
1. Moving uniformly, a cyclist travels 40 m in 4 s. What distance will it cover when moving at the same speed in 20 s?
A. 30 m. B. 50 m. C. 200 m.
2. Figure 1 shows a graph of the movement of a motorcyclist. Determine from the graph the path traveled by the motorcyclist in the time interval from 2 to 4 s.
A. 6m. B. 2 m. C. 10 m.
3. Figure 2 shows the motion graphs of three bodies. Which of these graphs corresponds to movement with greater speed?
A. 1. B. 2. C. 3.
4. According to the motion graph shown in Figure 3, determine the speed of the body.
A. 1 m/s. B. 3 m/s. H. 9 m/s.
5. Two cars are moving along the road with constant speeds of 10 and 15 m/s. The initial distance between the cars is 1 km. Determine how long it will take for the second car to overtake the first.
A. 50 s. B. 80 s. V. 200 p.

Preface.
SELF-CHECK TESTS
TS-1. Move. Speed.
Uniform rectilinear motion.
TS-2. Rectilinear motion with constant acceleration
TS-3. Free fall. ballistic movement.
TS-4. Kinematics of periodic motion.
TS-5. Newton's laws.
TS-6. Forces in mechanics.
TS-7. Application of Newton's laws.
TS-8. Law of conservation of momentum.
TS-9. Force work. Power.
TS-10. Potential and kinetic energy.
TS-11. The law of conservation of mechanical energy.
TS-12. Movement of bodies in a gravitational field.
TS-13. Dynamics of free and forced oscillations.
TS-14. Relativistic mechanics.
TS-15. Molecular structure of matter.
TS-16. Temperature. Basic equation of molecular-kinetic theory.
TS-17. Clapeyron-Mendeleev equation. Isoprocesses.
TS-18. Internal energy. Gas work during isoprocesses. First law of thermodynamics.
TS-19. Thermal engines.
TS-20. Evaporation and condensation. Saturated steam. Air humidity. Boiling liquid.
TS-21. Surface tension. Wetting, capillarity.
TS-22. Crystallization and melting solids.
TS-23. Mechanical properties of solids.
TS-24. Mechanical and sound waves.
TS-25. The law of conservation of charge. Coulomb's law.
TS-26. The intensity of the electrostatic field.
TS-27. The work of the forces of the electrostatic field. The potential of the electrostatic field.
TS-28. Dielectrics and conductors in an electrostatic field.
TS-29. Capacitance of a solitary conductor and a capacitor. The energy of the electrostatic field.
INDEPENDENT WORKS
SR-1. Uniform rectilinear motion.
SR-2. Rectilinear motion with constant acceleration.
SR-3. Free fall. ballistic movement.
SR-4. Kinematics of periodic motion.
SR-5. Newton's laws.
SR-6. Forces in mechanics.
SR-7. Application of Newton's laws.
SR-8. Law of conservation of momentum.
SR-9. Force work. Power.
SR-9. Force work. Power.
SR-10. Potential and kinetic energy. Law of energy conservation.
SR-11. Absolutely inelastic and absolutely elastic collision.
SR-12. Movement of bodies in a gravitational field.
SR-13. Dynamics of free and forced oscillations.
SR-14. Relativistic mechanics.
SR-15. Molecular structure of matter.
SR-16. Temperature. Basic equation of molecular-kinetic theory.
SR-17. Clapeyron-Mendeleev equation. Isoprocesses.
SR-18. Internal energy. Gas work during isoprocesses.
SR-19. First law of thermodynamics.
SR-20. Thermal engines.
SR-21. Evaporation and condensation. Saturated steam. Air humidity.
SR-22. Surface tension. Wetting, capillarity.
SR-23. Crystallization and melting of solids. Mechanical properties of solids.
SR-24. Mechanical and sound waves.
SR-25. The law of conservation of charge. Coulomb's law.
SR-26. The intensity of the electrostatic field.
SR-27. The work of the forces of the electrostatic field. Potential.
SR-28. Dielectrics and conductors in an electrostatic field.
SR-29. Electrical capacity. Electrostatic field energy
TEST PAPERS
KR-1. Rectilinear movement.
KR-2. Free fall of bodies. ballistic movement.
KR-3. Kinematics of periodic motion.
KR-4. Newton's laws.
CR-5. Application of Newton's laws.
CR-6. Law of conservation of momentum.
CR-7. Law of energy conservation.
KR-8. Molecular kinetic theory ideal gas
CR-9. Thermodynamics.
KR-10. Aggregate states substances.
KR-11. Mechanical and sound waves.
KR-12. Forces of electromagnetic interaction of fixed charges.
KR-13. Energy of electromagnetic interaction of fixed charges.
ANSWERS
Tests for self-control.
Independent work.
Test papers.
Bibliography.

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• Physics, grade 10, basic level, textbook, Kasyanov V.A., 2014

Grade 10

Test No. 5

Option 1

25 m -3 .

3 -23

6 (m/s) 2 25 m -3 -26 kg?

25 m -3

3 -12 Pa?

Grade 10

Test No. 5

"Fundamentals of the Molecular Kinetic Theory of an Ideal Gas"

Option 2

5 m 3 18 molecules?

5 3 m/s.

21 J.

3 H 8

Grade 10

Test No. 5

"Fundamentals of the Molecular Kinetic Theory of an Ideal Gas"

Option 1

1. Determine the temperature of hydrogen and the mean square velocity of its molecules at a pressure of 100 kPa and a concentration of molecules of 10 25 m -3 .

2. A vessel shaped like a cube with a side of 1 m contains an ideal gas in the amount of 10-3 mol. Find the gas pressure if the mass of one molecule is 3 ∙ 10-23 r and the average speed of thermal motion of molecules is 500 m/s.

3. Under what pressure is the gas in the vessel, if the average square of the speed of its molecules is 10 6 (m/s) 2 , concentration of molecules 3 ∙ 10 25 m -3 , and the mass of each molecule is 5 ∙ 10-26 kg?

4. Concentration of gas molecules 4 ∙ 10 25 m -3 .Calculate the gas pressure at 290 K.

5. What is the number of molecules in a vessel with a volume of 5 m 3 at 300 K, if the gas pressure is 10-12 Pa?

Grade 10

Test No. 5

"Fundamentals of the Molecular Kinetic Theory of an Ideal Gas"

Option 2

1. What is the average speed of thermal motion of molecules if, at a pressure of 250 kPa, a gas weighing 8 kg occupies a volume of 15 m 3 ?

2. What pressure is produced by mercury vapor in a cylinder of a mercury lamp with a capacity of 3 10-5 m 3 at 300 K if it contains 10 18 molecules?

3. Determine the density of oxygen at a pressure of 1.3 ∙ 10 5 Pa, if the root-mean-square velocity of its molecules is 1.4 ∙ 10 3 m/s.

4. At what temperature is the average kinetic energy of gas molecules equal to 10.35 ∙ 10-21 J.

5. The 3000 l tank contains propane (C 3 H 8 ), the amount of substance of which is 140 mol, and the temperature is 300 K. What pressure does the gas exert on the walls of the vessel?

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: mechanical energy molecules and the wall is completely preserved 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 it's clear physical meaning equations: 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 quite significant.
3. Molecules count 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.
Decision	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.
Decision	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: