=

Where \u003d 0.001 kg / mol - the molar mass of hydrogen. That's why

=

2.4.2. Determine the average kinetic energy of the translational motion of one air molecule under normal conditions. The concentration of molecules under normal conditions n 0 \u003d 2.7 * 10 25 m -3

Analysis and decision. From the basic equation of the molecular - kinetic theory of gases

J

2.4.3. Find the average kinetic energy rotational motion of one oxygen molecule at a temperature T = 350K, as well as the kinetic energy of the rotational motion of all molecules contained in m = 4g of oxygen.

Analysis and decision.

It is known that each degree of freedom of a gas molecule has the same average energy, expressed by the formula

=

where k is the Boltzmann constant, T is the absolute temperature of the gas.

Since two degrees of freedom are attributed to the rotational motion of a diatomic molecule (the oxygen molecule is diatomic), the average energy of the rotational motion of an oxygen molecule is expressed by the formula

=

Considering that k = 1.38 * 10 -23 J / K and T = 350K, we get

\u003d 1.38 * 10 -23 * 350 J \u003d 4.83 * 10 -21 J.

The kinetic energy of the rotational motion of all gas molecules is determined by the equality

w = N(1)

The number of all gas molecules can be calculated by the formula

N = N A  (2)

where N A is the Avogadro number,  is the number of kilomoles of gas.

Considering that the number of kilomoles

where m is the mass of gas, is the mass of one kilomole of gas, then formula (2) takes the form N = N A

Substituting this expression for N into formula (1) we obtain

w = N A (3)

Let us express the quantities included in this formula in SI units and substitute into formula (3):

2.4.4. Calculate the specific heat capacities at a constant volume C V and at a constant pressure of neon and hydrogen, taking these gases as ideal.

Analysis and decision.

The specific heat capacities of ideal gases are expressed by the formulas:

With V = (1)

C p =
(2)

where i is the number of degrees of freedom of a gas molecule, - molar mass.

For neon (monatomic gas) i = 3 and \u003d 20 * 10 -3 kg / mol.

Calculating according to formulas (1) and (2), we obtain: С V =
J/kg*k

C p =
J/kg*k

For hydrogen (diatomic gas) i = 3 and \u003d 2 * 10 -3 kg / mol. Calculating by the same formulas, we get:

With V =
J/kg*k

C p =
J/kg*k

2.4.5. Find the root-mean-square velocity, the average kinetic energy of translational motion and the average total kinetic energy of helium and nitrogen molecules at a temperature of t = 27 0 C. Determine the total energy of all molecules of 100 g of each of the gases.

Analysis and decision.

The average kinetic energy of the translational motion of one molecule of any gas is uniquely determined by its thermodynamic temperature:

= (1)

where k \u003d 1.38 * 10 -23 J / K - Boltzmann's constant.

However, the mean square velocity of gas molecules depends on the mass of its molecules:

(2)

where m 0 is the mass of one molecule.

The average total energy of a molecule depends not only on temperature, but also on the structure of molecules, i.e., on the number i of degrees of freedom: = ikT/2

The total kinetic energy of all molecules, which is equal to its internal energy for an ideal gas, can be found as the product to the number of all molecules:

Obviously, N = N А m/ (5)

where m is the mass of the entire gas, the ratio m/ determines the number of moles, and N A is Avogadro's constant. Expression (4), taking into account the Clapeyron-Mendeleev equation, will allow us to calculate the total energy of all gas molecules.

According to equality (1)< W о п >\u003d 6.2 * 10 -21 J, and the average energy of the translational motion of one molecule and helium and nitrogen are the same.

The root mean square speed is found by the formula

, where R \u003d 8.31 J / k mol

For helium V kv \u003d 13.7 * 10 2 m / s

For nitrogen V kv \u003d 5.17 * 10 2 m / s

Helium is a monatomic gas, therefore, i = 3, then< W о п >\u003d W o \u003d 6.2 * 10 -21 J.

Nitrogen is a diatomic gas, hence i = 5 and< W о п >\u003d 5/2 kT \u003d 10.4 * 10 -21 J.

The total energy of all molecules after substituting expressions (3) and (5) into (4) has the form

W = kT
=

For helium W = 93.5 kJ, for nitrogen W = 22.3 kJ.

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1st level of difficulty.

Lesson type: combined.

Total lesson time: 1 hour 10 minutes.

(t = 2–3 min.)

(Slide 1)

EC 0. Goal setting:

Didactic goal of the module:

(Slide 2)

UE 1. Updating knowledge

Private didactic goal:

Exercise 1.

For students D - type: Fill in the table, indicating the designation (symbol) of the physical quantity and its unit of measurement.

Result evaluation: 1 point

For students And - type: Think over the logical connections between formulas (branches).

Make your own “physical tree”.

Result score: 1 point.

Task 2.

(Slide 3)

Generalized algorithm for solving a typical problem:

For students I - type:

Task number 1.

1. Determine the number of atoms in 1 m 3 of copper. The density of copper is 9000 kg/m 3 .
2. Use a generalized algorithm for solving problems of this type; apply it to the solution of this problem by describing the step-by-step actions that you performed.

Result score: 1 point.

Students D - type:

Task number 1.

1. The mass of a silver strip obtained during the rotation of the cylinder during a physical experiment is 0.2 g. Find the number of silver atoms contained in it.
2. Describe the step-by-step actions you performed to solve the problem. Compare the steps you have outlined with the actions of a generalized algorithm for solving problems of this type.

Result score: 1 point.

UE 2. The physical model of gas is an ideal gas.

(Slide 4)

Private didactic goal:

1. Formulate the concept of “ideal gas”.
2. Formation of scientific outlook.

Part 1. When studying phenomena in nature and technical practice, it is impossible to take into account all the factors influencing the course of a particular phenomenon. However, from experience it is always possible to establish the most important of them. Then all other factors that do not have a decisive influence can be neglected. On this basis, it is created idealized (simplified)) representation of such a phenomenon. The model created on this basis helps to study the actual processes and predict their course in various cases. Consider one of these idealized concepts.

(Slide 5)

F. O.- Name the properties of gases.
– Explain these properties on the basis of MKT.
How is pressure defined? Units in SI?

The physical properties of a gas are determined by the chaotic motion of its molecules, and the interaction of molecules does not have a significant effect on its properties, and the interaction has the nature of a collision, and the attraction of molecules can be neglected. Most of the time, gas molecules move as free particles.

(Slide 6)

This allows us to introduce the concept of an ideal gas, in which:

1. attraction forces are completely absent;
2. the interaction between molecules is not taken into account at all;
3. molecules are considered free.

Exercise 1.

Cards with a task for each student I, D - type .

Type I students:

1. After carefully studying §63 p. 153, find the definition of an ideal gas in the text. Learn it. (1 point)
2. Try to answer the question: “Why is the kinetic energy of a rarefied gas much greater than the potential energy of interaction?” (1 point)

D-type students:

1. Find in the text § 63 p.15 the definition of an ideal gas. Learn it. (1 point)
2. Write the wording in your notebook. (1 point)
3. Using the periodic table, name the gases that best fit the concept of an “ideal gas”. (1 point)

UE3. Gas pressure in the MKT.

Private didactic goal:

1. Prove that despite the change in pressure, p 0 ≈ const.

1. What do gas molecules exert on the walls of the vessel during their movement?
2. When will the gas pressure be greater?
3. What is the impact force of one molecule? Can a manometer record the impact force of a single molecule? Why?
4. Make a conclusion why the average value of pressure p 0 remains a certain value.

Gas molecules, hitting the wall of the vessel, exert pressure on it. The magnitude of this pressure is the greater, the greater the average kinetic energy of the translational motion of gas molecules and their number per unit volume.

Exercise 1.

Cards with a task for each student I, D - type .

Students I, D - type:

Conclude: Why does the average value of gas pressure p 0 in a closed vessel remain practically unchanged?

Result score: 1 point.

Teacher explanations (IT, IE, ID, DT, DE, DD):

The occurrence of gas pressure can be explained using a simple mechanical model.

(Slide 8)

EC 4. Average values ​​of the modulus of velocities of individual molecules.

(Slide 9)

Private didactic goal:

Introduce the concept of “average value of speed”, “average value of the square of speed”.

Exercise 1.

Cards with a task for each student I, D - type.

Students I - type:

Read §64 pp. 154–156 carefully.

1. Find answers to the following questions in the text:
   
   
   
2. Write down the answers in your notebook.

D-type students:

Study § 64 pp. 154–156. (1 point)

1. Answer the questions:
   1.1. What does the average speed of all particles depend on?
   1.2. What is the mean square of speed?
   1.3. Velocity Projection Mean Square Formula.
2. Write down the answers in your notebook.

Generalization of the teacher (IT, IE, ID, DT, DE, DD):

(Slide 10, 11)

The velocities of molecules vary randomly, but the mean square of the speed is a well-defined value. In the same way, the growth of students in the class is not the same, but its average value is a certain value.

Task 2.

Cards with a task for each student I, D - type.

Students I - type:

D-type students:

UE5. Output control.

Private didactic goal: Check the assimilation of educational elements; evaluate your knowledge.

Cards with a task for each student I, D - type .

Exercise 1.

Students I, D - type

Discuss which of the following properties of real gases are not taken into account, and which are taken into account in the ideal gas model.

1. In a rarefied gas, the volume that would be occupied by gas molecules with their dense “packing” (intrinsic volume) is negligible compared to the entire volume occupied by the gas. Therefore, the intrinsic volume of molecules in the model of an ideal gas..
2. In a vessel containing a large number of molecules, the motion of the molecules can be considered completely chaotic. This fact in the ideal gas model….
3. The molecules of an ideal gas are, on average, at such distances from each other at which the cohesive forces between the molecules are very small. These forces are in a mole of an ideal gas….
4. Collisions of molecules with each other can be considered absolutely elastic. These are the properties in the ideal gas model….
5. The motion of gas molecules obeys the laws of Newtonian mechanics. This fact in the ideal gas model ….
   A) not taken into account
   B) taken into account (taken into account)

Task 2.

– Explanations are given for each of the expressions for the velocities of molecules (1–3) (A–B). Find them.

A) According to the rule of addition of vectors and the Pythagorean theorem, the square of the speed υ any molecule can be written as follows: υ 2 = υ x 2 + υ y 2

B) the directions Ox, Oy and Oz are equal due to the random movement of molecules.

C) with a large number (N) of randomly moving particles, the moduli of the velocities of individual molecules are different.

Evaluation of the result: check yourself on the code and evaluate. For each correct answer - 1 point.

UE6. Summarizing.

Private didactic goal: Fill in the checklist; evaluate your knowledge.

Control sheet (IT, IE, ID, DT, DE, DD):

Fill out the control sheet. Calculate points for completing assignments. Give yourself a final score:

16–18 points - “5”;
13–15 points - “4”;
9–12 points – “test”;
less than 9 points - "failure".

Give the checklist to the teacher.

Differentiated homework:

"Record": Find in the table “Periodic system of elements D.I. Mendeleev” chemical elements that are closest in their properties to an ideal gas. Explain your choice.

“Failure”: § 63–64.

(Slide 12).

Internet resources:

Lesson outline.

M5. Ideal gas in MKT. The average value of the square of the speed of molecules.

1st level of difficulty.

Lesson type: combined.

Total lesson time: 1 hour 10 minutes.

Stage 1. Organizational moment (number, topic, organizational issues).

(t = 2-3 min.)

(slide 1)

UE 0 . Goal setting:

Didactic goal of the module:

(slide 2)

Acquaintance with the theory of sufficiently rarefied gases.

Proof that the average speed of molecules depends on

movement of all particles.

Stage 2 . Repetition. (t = 10-15 min.)

UE 1 . Knowledge update

Private didactic goal:

Updating of basic knowledge on the topics of module M1-M4.

Finding out the degree of assimilation of educational material by students in order to further eliminate gaps.

Exercise 1.

D-type students : Fill in the table, indicating the designation (symbol) of the physical quantity and its unit of measurement.

Physical quantity

Designation

Unit

(SI)

Molar mass

Amount of substance

Avogadro constant

Matter density

Mass of matter

Number of molecules (atoms)

Relative molecular weight

Result score: 1 point.

I-type learners : Think over logical connections between formulas (branches).

Make your own “physical tree”.

Result score: 1 point.

Task 2.

(slide 3)

Generalized algorithm for solving a typical problem:

m = m 0 N

Make a numerical calculation.

Cards with a task for each student I, D-type.

I-type students :

Task number 1.

1. Determine the number of atoms in 1 m 3 of copper. The density of copper is 9000 kg/m 3 .

2. Use a generalized algorithm for solving problems of this type; apply it to the solution of this problem by describing the step-by-step actions that you performed.

Result score: 1 point.

D-type students :

Task number 1.

The mass of a silver strip obtained during the rotation of the cylinder during a physical experiment is 0.2 g. Find the number of silver atoms contained in it.

Describe the step-by-step actions you performed to solve the problem. Compare the steps you have outlined with the actions of a generalized algorithm for solving problems of this type.

Result score: 1 point.

Stage 3. Basic. Presentation of educational material.(t = 30-35 min.)

UE 2. Physical model of gas - ideal gas

(slide 4)

Private didactic goal:

Define the term "ideal gas".

Formation of scientific outlook.

Teacher's explanation(IT, IE, ID, DT, DE, DD)

Part 1.

When studying phenomena in nature and technical practice, it is impossible to take into account all the factors influencing the course of a particular phenomenon. However, from experience it is always possible to establish the most important of them. Then all other factors that do not have a decisive influence can be neglected. On this basis, it is created idealized

(simplified) representation of such a phenomenon. The model created on this basis helps to study the actual processes and predict their course in various cases. Consider one of these idealized concepts.

(slide 5):

F. O.- Name the properties of gases.

Explain these properties on the basis of MKT.

How is pressure defined? Units in SI?

The physical properties of a gas are determined by the chaotic motion of its molecules, and the interaction of molecules does not have a significant effect on its properties, and the interaction has the nature of a collision, and the attraction of molecules can be neglected. Most of the time, gas molecules move as free particles.

(slide 6):

This allows us to introduce the concept of an ideal gas, in which:

attraction forces are completely absent;

the interaction between molecules is not taken into account at all;

molecules are considered free.

Exercise 1.

Cards with a task for each student I, D-type .

Type I students:

After carefully studying §63 p. 153, find the definition of an ideal gas in the text. Learn it. (1 point)

(1 point)

Try to answer the question: “Why is the kinetic energy of a rarefied gas much greater than the potential energy of interaction?” (1 point).

D-type students :

Find in the text § 63 p.15 the definition of an ideal gas. Learn it.

(1 point).

Write the wording in your notebook.

(1 point).

Using the periodic table, name the gases that best fit the concept of "ideal gas". (1 point).

UE3 . Gas pressure in the MKT.

Private didactic goal:

Prove that despite the change in pressure, p 0 ≈ const.

Teacher's explanation(IT, IE, ID, DT, DE, DD):

F.O.:

What do gas molecules exert on the walls of the vessel during their movement?

When will the gas pressure be greater?

What is the impact force of one molecule? Can a manometer record the impact force of a single molecule? Why?

Make a conclusion why the average value of pressure p 0 remains a certain value.

(slide 7)

Gas molecules, hitting the wall of the vessel, exert pressure on it. The magnitude of this pressure is the greater, the greater the average kinetic energy of the translational motion of gas molecules and their number per unit volume.

P

p0

0 t

Exercise 1.

Cards with a task for each student I, D-type .

Students I, D-type :

Conclude:Why is the average gas pressure p 0 remains practically unchanged in a closed vessel?

Result score: 1 point.

Teacher's explanation(IT, IE, ID, DT, DE, DD):

The occurrence of gas pressure can be explained using a simple mechanical model.

(slide 8)

Part 2.

EC 4 . Average values ​​of the modulus of velocities of individual molecules.

(slide 9)

Private didactic goal:

Introduce the concept of "average value of speed", "average value of the square of speed".

Exercise 1.

I-type students :

Read carefully § 64 pp. 154-156.

Find answers to the following questions in the text:

Write down the answers in your notebook.

(1 point)

D-type students :

Study § 64 pp. 154-156. (1 point).

Answer the questions:

1.1. What does the average speed of all particles depend on?

1.2. What is the mean square of speed?

1.3. Velocity Projection Mean Square Formula.

Write down the answers in your notebook.

(1 point).

Teacher generalization(IT, IE, ID, DT, DE, DD):

(slide 10, slide 11)

The velocities of molecules vary randomly, but the mean square of the speed is a well-defined value. In the same way, the growth of students in the class is not the same, but its average value is a certain value.

υ 2 = + +

= 1

Task 2.

Cards with a task for each student I, D - type.

I-type students :

After carefully examining the table, try to understand the essence of the distribution of silver atoms in terms of speeds.

f (υ) =

(1 point)

Try to mentally change the speed interval (reduce it). Explain what will happen to the schedule? How to change the broken line bounding the chart rectangles from above? (2 points)

Speed ​​range, m/s

Fraction of atoms, %

Speed ​​range, m/s

Fraction of atoms, %

0-100

1,4

600-700

9,2

100-200

8,1

700-800

4,8

200-300

16,7

800-900

2,0

300-400

21,5

900-1000

0,6

400-500

20,3

over 1000

0,3

500-600

15,1

D-type students :

Study the table of the distribution of silver atoms by speed

Plot the speed distribution of silver atoms

f (υ) =

(1 point)

Decrease the speed range. Explain what will happen to the schedule? How to change the broken line bounding the chart rectangles from above?

(2 points)

Task No. 2. When conducting the Stern experiment, the silver strip turns out to be somewhat blurry, since at a given temperature the velocities of the atoms are not the same. According to the determination of the thickness of the silver layer in different places of the strip, it is possible to calculate the fraction of atoms with velocities lying in one or another range of velocities out of their total number. As a result of the measurements, the following table was obtained:

Speed ​​range, m/s

Fraction of atoms, %

Speed ​​range, m/s

Fraction of atoms, %

0-100

1,4

600-700

9,2

100-200

8,1

700-800

4,8

200-300

16,7

800-900

2,0

300-400

21,5

900-1000

0,6

400-500

20,3

over 1000

0,3

500-600

15,1

4 - stage. Control of knowledge and skills of students.(t = 8-10min.)

UE5. Output control.

Private didactic goal: Check the assimilation of educational elements; evaluate your knowledge.

Cards with a task for each student I, D - type .

Exercise 1.

Students I, D-type

Discuss which of the following properties of real gases are not taken into account, and which are taken into account in the ideal gas model.

In a rarefied gas, the volume that would be occupied by gas molecules in their dense "packing" (intrinsic volume) is negligible compared to the entire volume occupied by the gas. Therefore, the intrinsic volume of molecules in the model of an ideal gas..

In a vessel containing a large number of molecules, the motion of the molecules can be considered completely chaotic. This fact in the ideal gas model….

The molecules of an ideal gas are, on average, at such distances from each other at which the cohesive forces between the molecules are very small. These forces are in a mole of an ideal gas….

Collisions of molecules with each other can be considered absolutely elastic. These are the properties in the ideal gas model….

The motion of gas molecules obeys the laws of Newtonian mechanics. This fact in the ideal gas model ….

A) not taken into account

B) taken into account (taken into account)

Task 2.

Explanations are given for each of the expressions for the velocities of molecules (1-3) (A-B). Find them.

A) according to the rule of addition of vectors and the Pythagorean theorem, the square of the speed υ any molecule can be written as follows: υ 2 = υ x 2 + υ y 2

B) the directions Ox, Oy and Oz are equal due to the random movement of molecules.

C) with a large number (N) of randomly moving particles, the moduli of the velocities of individual molecules are different.

Evaluation of the result: check yourself on the code and evaluate. For each correct answer - 1 point.

Stage 5 Summarizing.(t=5 min.)

UE6. Summarizing.

Private didactic goal: Fill out the control sheet; evaluate your knowledge.

Control sheet (IT, IE, ID, DT, DE, DD):

Fill out the control sheet. Calculate points for completing assignments. Give yourself a final score:

16-18 points - "5";

13-15 points - "4";

9-12 points - "test";

less than 9 points - "failure".

Give the checklist to the teacher.

Learning element

Tasks (question)

Total points

1

2

UE1

UE2

UE3

UE4

UE5

Total

18

Grade

….

Differentiated homework:

"Record": Find V table "Periodic system of elements D.I. Mendeleev" chemical elements that are closest in their properties to an ideal gas. Explain your choice.

"Mistake":§63-64.

(slide 12).

• Before proceeding to the calculation of gas pressure using the molecular kinetic theory, let us consider in more detail the simple laws related to the average values ​​of the rates of thermal motion of molecules.

Averages

Let us assume that the gas molecules move randomly. The speed of any molecule can be either very large or very small. The direction of movement of molecules changes randomly when they collide with each other. This was discussed in Chapter 2. The observation of Brownian motion serves as evidence for the participation of molecules in chaotic motion.

However, although the movement of individual molecules is chaotic, the behavior of all molecules as a whole reveals simple patterns. First, if any direction is arbitrarily singled out in a gas, then the average number of molecules moving in that direction must be equal to the average number of molecules moving in the opposite direction. After all, chaos in the movement of molecules means that none of the directions of movement is predominant. All of them are equal.

In the same way, the average number of people walking down the street of the city in one direction and the other, on average over a sufficiently large period of time (or for a sufficiently large group of people) is the same. Of course, if you exclude special cases like a street procession.

Secondly, simple laws are valid for the arithmetic mean velocity of molecules. Let there be N molecules. The projections of the velocities of these molecules on the X axis can take on various values: v 1x , v 2x , v 3x , ..., v Nx , and each projection can be either positive or negative. The arithmetic mean of the projection of the velocity x on a given direction X is equal to the sum of the projections of the velocities of all molecules divided by their number:

Because of the chaos in the motion of molecules, positive velocity projections occur just as often as negative ones. Therefore, the average value of the velocity projection on a given direction X is zero: x = 0. If this were not the case, then the gas would move as a whole.

The average value of the velocity projection modulus | x | is a well-defined value other than zero. Let's explain this with an example. The growth of students in one class is not the same, but the average value of growth is a certain value. To find it, you need to add up the height of all students and divide this sum by their number (Fig. 4.2).

Rice. 4.2

Average squared speed

We will be interested in the mean square of the velocity projection. It is found in the same way as the square of the speed modulus (see expression (4.1.2)):

Molecular velocities take on a continuous series of values. It is practically impossible to determine the exact values ​​of the velocities and calculate the average value (statistical average) using formula (4.3.2). Let's define it a little differently, more realistically. Denote by n 1 the number of molecules in a volume of 1 cm 3 with velocity projections close to v 1x ; through n 2 - the number of molecules in the same volume, but with velocities close to v kx , etc. (1) The number of molecules with velocities close to the maximum v kx , denoted by n k great). In this case, the condition n 1 + n 2 + ... + n i + ... + n k = n must be satisfied, where n is the concentration of molecules. Then for the average value of the square of the velocity projection, instead of formula (4.3.2), we can write the following equivalent formula:

Since the X direction is no different from the Y and Z directions (again, due to the chaos in the movement of molecules), the equalities are valid

« Physics - Grade 10 "

Remember what a physical model is.
Is it possible to determine the speed of one molecule?

Ideal gas.

In a gas at ordinary pressures, the distance between the molecules is many times greater than their size. In this case, the interaction forces of molecules are negligible and the kinetic energy of molecules is much greater than the potential energy of interaction. Gas molecules can be thought of as material points or very small solid balls. Instead of real gas, between whose molecules interaction forces act, we will consider it model - ideal gas.

Ideal gas- This is a theoretical model of a gas that does not take into account the size of molecules (they are considered material points) and their interaction with each other (except in cases of direct collision).

Naturally, when molecules of an ideal gas collide, a repulsive force acts on them. Since, according to the model, we can consider gas molecules as material points, we neglect the sizes of molecules, assuming that the volume they occupy is much less than the volume of the vessel.

In a physical model, only those properties of a real system are taken into account, the consideration of which is absolutely necessary to explain the studied patterns of behavior of this system.

No model can convey all the properties of the system. Now we have to solve the problem: to calculate, using the molecular-kinetic theory, the pressure of an ideal gas on the walls of a vessel. For this problem, the ideal gas model turns out to be quite satisfactory. It leads to results that are confirmed by experience.

Gas pressure in molecular-kinetic theory.

Let the gas be in a closed vessel. The pressure gauge shows the gas pressure p 0 . How does this pressure arise?

Each gas molecule, hitting the wall, acts on it with a certain force for a short period of time. As a result of random impacts against the wall, the pressure changes rapidly with time, approximately as shown in Figure 9.1. However, the effects caused by the impacts of individual molecules are so weak that they are not recorded by the manometer. The pressure gauge records the time-average force acting on each unit area of ​​the surface of its sensitive element - the membrane. Despite small changes in pressure, the average value of pressure p 0 practically turns out to be a quite definite value, since there are a lot of impacts on the wall, and the masses of molecules are very small.

The average pressure has a certain value in both gas and liquid. But there are always slight random deviations from this average. The smaller the surface area of ​​the body, the more noticeable the relative changes in the pressure force acting on this area. So, for example, if a section of the surface of a body has a size of the order of several diameters of a molecule, then the pressure force acting on it changes abruptly from zero to a certain value when the molecule hits this section.

The average value of the square of the speed of molecules.

To calculate the average pressure, you need to know the value of the average velocity of molecules (more precisely, the average value of the square of the velocity). This is not an easy question. You are used to the fact that each particle has speed. The average speed of the molecules depends on what are the speeds of movement of all molecules.

How does the definition of the average velocity of a body in mechanics differ from the determination of the average velocity of gas molecules?

From the very beginning, one must give up trying to follow the movement of all the molecules that make up the gas. There are too many of them, and they move very difficult. We don't need to know how each molecule moves. We must find out what result the movement of all gas molecules leads to.

The nature of the motion of the entire set of gas molecules is known from experience. Molecules participate in random (thermal) motion. This means that the speed of any molecule can be either very large or very small. The direction of movement of molecules constantly changes when they collide with each other.

The speeds of individual molecules can be anything, but the average value of the modulus of these speeds is quite definite.

In the future, we will need the average value of not the speed itself, but the square of the speed - root mean square speed. The average kinetic energy of molecules depends on this value. And the average kinetic energy of molecules, as we will soon see, is of great importance in the entire molecular-kinetic theory. Let us designate the moduli of velocities of individual gas molecules as υ 1 , υ 2 , υ 3 , ... , υ N . The average value of the square of the speed is determined by the following formula:

where N is the number of molecules in the gas.

But the square of the modulus of any vector is equal to the sum of the squares of its projections on the coordinate axes OX, OY, OZ.

From the course of mechanics it is known that when moving on a plane υ 2 = υ 2 x + υ 2 y. In the case when the body is moving in space, the square of the speed is equal to:

υ 2 \u003d υ 2 x + υ 2 y + υ 2 z. (9.2)

The average values ​​of υ 2 x , υ 2 y and υ 2 z can be determined using formulas similar to formula (9.1). Between the average value and the average values ​​of the squares of the projections, there is the same relationship as the ratio (9.2):

Indeed, equality (9.2) is valid for each molecule. Adding these equalities for individual molecules and dividing both sides of the resulting equation by the number of molecules N, we arrive at formula (9.3).

>Attention! Since the directions of the three axes OX, OY and OZ are equal due to the random movement of molecules, the average values ​​of the squares of the velocity projections are equal to each other:

Taking into account relation (9.4), we substitute into formula (9.3) instead of and . Then for the average square of the velocity projection onto the OX axis we obtain

i.e. the mean square of the velocity projection is equal to the mean square of the velocity itself. The multiplier appears due to the three-dimensionality of space and, accordingly, the existence of three projections for any vector.

The speeds of molecules vary randomly, but the mean square of the speed is a well-defined value.