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1st level of difficulty.
Lesson type: combined.
Total lesson time: 1 hour 10 minutes.
Organizational moment (number, topic, organizational issues).(t = 2–3 min.)
(Slide 1)
EC 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 the motion of all particles.
UE 1. Updating knowledge
Private didactic goal:
- Updating of basic knowledge on the topics of module M1-M4.
- Finding out the degree of assimilation by students educational material to further fill the gaps.
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.
- 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.
3rd stage. Basic. Presentation of educational material.
(t = 30–35 min.)UE 2. The physical model of gas is an ideal gas.
(Slide 4)
Private didactic goal:
- Formulate the concept of “ideal gas”.
- Formation of scientific outlook.
(IT, IE, ID, DT, DE, DD)Teacher's explanation
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 real processes and to foresee their course in various occasions. 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)
- 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 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.
- 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.
Gas molecules, hitting the wall of the vessel, exert pressure on it. The value 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.
- Find answers to the following questions in the text:
- Write down the answers in your notebook.
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.
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:
| Task No. 2. When carrying out the Stern experiment, the silver strip turns out to be somewhat blurred, 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 fractions 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: |
4th stage. Control of knowledge and skills of students.
(t = 8–10 min.)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 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..
- 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.
B) at large numbers(N) randomly moving particles, the modules 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.
5th stage. Summarizing.
(t=5 min.)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.
| Learning element | Tasks (question) | Total points | |
| 1 | 2 | ||
| UE1 | 1 | 1 | 2 |
| UE2 | 3 | 3 | |
| UE3 | 1 | 1 | |
| UE4 | 1 | 3 | 4 |
| UE5 | 5 | 3 | 8 |
| Total | 18 | ||
| Grade | …. | ||
Differentiated homework:
"Record": Find in the table “ Periodic system 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:
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 
slightly different, more realistic. Denote by P 1
the number of molecules in a volume of 1 cm 3 with velocity projections close to v 1x ;
through P 2
- the number of molecules in the same volume, but with velocities close to v kx ,
etc.* The number of molecules with velocities close to the maximum v kx ,
denote by n k
(speed v k x
can be arbitrarily large). In this case, the following condition must be met: P 1
+ p 2
+
...
+ n i
+
... + n k = n, where P - 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:
* How these numbers can be determined will be discussed in §4.6.
Since the direction X no different from direction Y and Z (again due to the chaos in the movement of molecules), the equalities are true:
(4.3.4)
For each molecule, the square of the speed is:
The value of the mean square of the velocity, defined in the same way as the mean square of the velocity projection (see formulas (4.3.2) and (4.3.3)), is equal to the sum of the mean squares of its projections:
(4.3.5)
From expressions (4.3.4) and (4.3.5) it follows that
(4.3.6)
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, therefore, the existence of three projections for any vector.
The velocities of molecules vary randomly, but the average value of the projections of the velocity in any direction and the mean square of the velocity- well-defined values.
§ 4.4. Basic equation of molecular kinetic theory
Let us calculate the gas pressure using the molecular-kinetic theory. Based on the calculations performed, it will be possible to draw a very important conclusion about the relationship between the gas temperature and the average kinetic energy of molecules.
Let the gas be in a rectangular vessel with solid walls. The gas and the vessel have the same temperatures, that is, they are in a state of thermal equilibrium. We assume that collisions of molecules with walls are absolutely elastic. Under this condition, the kinetic energy of the molecules does not change as a result of the collision.
The requirement that collisions be perfectly elastic is not strictly necessary. It is not implemented exactly. Molecules can be reflected from the wall at different angles and with velocities not equal in absolute value to the velocities before the collision. But on the average, the kinetic energy of the molecules reflected by the wall will be equal to the kinetic energy of the falling molecules, if only thermal equilibrium exists. The calculation results do not depend on the detailed pattern of collisions of molecules with the wall. Therefore, it is quite acceptable to consider the collisions of molecules as similar to the collisions of elastic balls with an absolutely smooth solid wall.
Calculate the pressure of the gas on the wall of the vessel CD, having an area S and located perpendicular to the axis X (Fig. 4.3).
« 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 is a theoretical model of a gas that does not take into account the size of the molecules (they are considered material points) and their interaction with each other (with the exception of 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-averaged 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 area.
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?
At the outset, attempts to follow the movement of all the molecules that make up a gas must be abandoned. 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 very great importance throughout 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.
Determine chemical element. Carefully read the condition of the problem 1. Find it in the periodic table 2 .. Determine molar mass given substance (M) 3. Calculate the number of molecules N = (m/M) · N A 4.. Determine the mass of the molecule m = m 0 ·N 5. Evaluate the results. Determine the size of the molecules. 6. Algorithm for solving a typical problem
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14 Presentation for the lesson. Author: Podsosonnaya Oksana Viktorovna () Physics.10. physics teacher of the highest qualification category MKOU "Evening (shift) general education school 2 at a correctional colony" p. Chuguevka, Chuguevsky District, Primorsky Territory
Root mean square velocity of molecules - root-mean-square value of the velocity modules of all molecules of the considered amount of gas
Table of values of the mean square velocity of the molecules of some gases
In order to understand where we get this formula from, we derive the root mean square velocity of molecules. The derivation of the formula begins with the basic equation of molecular kinetic theory (MKT):
Where we have the amount of substance, for an easier proof, let's take 1 mole of substance for consideration, then we get:
If you look, then PV is two-thirds of the average kinetic energy of all molecules (and we have taken 1 mole of molecules):
Then, if we equate the right parts, we get that for 1 mole of gas, the average kinetic energy will be equal to:
But the average kinetic energy is also found as:
But now, if we equate the right parts and express the speed from them and take the square, Avogadro's number per molecule mass, we get the Molar mass, then we get the formula for the root-mean-square velocity of a gas molecule:
And if we paint the universal gas constant, as, and for one molar mass, then we will succeed?
In the formula we used:
Root mean square velocity of molecules
Boltzmann constant