The speed of movement of gas molecules, Stern's experiment. Stern's experiment - experimental confirmation of the theory
In the second half of the nineteenth century, the study of the Brownian (chaotic) motion of molecules aroused keen interest among many theoretical physicists of that time. Although the substance developed by the Scottish scientist James was generally recognized in European scientific circles, it existed only in a hypothetical form. There was no practical confirmation of it then. The movement of molecules remained inaccessible to direct observation, and the measurement of their speed seemed simply an unsolvable scientific problem.
That is why experiments capable of proving in practice the very fact of the molecular structure of matter and determining the speed of movement of its invisible particles were initially perceived as fundamental. The decisive importance of such experiments for physical science was obvious, since it made it possible to obtain practical substantiation and proof of the validity of one of the most progressive theories of that time - molecular kinetic theory.
By the beginning of the twentieth century, world science had reached a sufficient level of development for the emergence of real opportunities for experimental verification of Maxwell's theory. The German physicist Otto Stern in 1920, using the method of molecular beams, which was invented by the Frenchman Louis Dunoyer in 1911, was able to measure the speed of movement of gas molecules of silver. Stern's experiment irrefutably proved the validity of the law. The results of this experiment confirmed the correctness of the estimate of atoms, which followed from the hypothetical assumptions made by Maxwell. True, Stern's experiment was able to give only very approximate information about the very nature of the velocity gradation. Science had to wait another nine years for more detailed information.
Lammert was able to verify the distribution law with greater accuracy in 1929, who somewhat improved Stern's experiment by passing a molecular beam through a pair of rotating disks that had radial holes and were shifted relative to each other by a certain angle. By changing the speed of rotation of the aggregate and the angle between the holes, Lammert was able to isolate individual molecules from the beam, which have different speed indicators. But it was Stern's experiment that laid the foundation for experimental research in the field of molecular-kinetic theory.
In 1920, the first experimental setup was created, which is necessary for conducting experiments of this kind. It consisted of a pair of cylinders designed personally by Stern. A thin platinum rod with a silver coating was placed inside the device, which evaporated when the axis was heated by electricity. Under vacuum conditions that were created inside the facility, a narrow beam of silver atoms passed through a longitudinal slot cut on the surface of the cylinders and settled on a special external screen. Of course, the unit was in motion, and during the time that the atoms reached the surface, it managed to turn through a certain angle. In this way, Stern determined the speed of their movement.
But this is not the only scientific achievement of Otto Stern. A year later, together with Walter Gerlach, he conducted an experiment that confirmed the presence of spin in atoms and proved the fact of their spatial quantization. The Stern-Gerlach experiment required the creation of a special experimental setup with a powerful one at its core. Under the influence of the magnetic field generated by this powerful component, they deviated according to the orientation of their own magnetic spin.
Municipal educational institution gymnasium No. 1
Central District of Volgograd
Physics lesson on the topic
The movement of molecules. Experimental determination of the velocities of the molecules
Grade 10
Prepared by: physics teacher of the highest category
Petrukhin
Marina Anatolievna.
UMK: N. S. Purysheva,
N. E. Vazheevskaya,
D. A. Isaev
"Physics - 10", a workbook for this textbook and a multimedia supplement to the textbook.
Volgograd, 2015
Related lesson
The movement of molecules.
Experimental determination of the velocities of the molecules
Grade 10
annotation.
Understanding the most important issues of modern physics is impossible without some, at least the most elementary ideas about statistical regularities. Considering a gas as a system consisting of a huge number of particles allows us to give an intelligible idea of the probability, the statistical nature of the regularities of such systems, of statistical distributions indicating the probability with which the particles of the system have one or another value of the parameters that determine their state, and on based on this to state the main provisions of the classical theory of gases. One of the lessons that allow us to form this idea is the presented lesson on the teaching materials of the Drofa publishing house: a physics textbook by N. S. Purysheva, N. E. Vazheevskaya, D. A. Isaev, a workbook for this textbook and a multimedia supplement to textbook.
Explanatory note.
This lesson can be carried out in the process of studying the topic “Fundamentals of the MKT structure of matter” in grade 10.
The new lesson material allows students to deepen their knowledge of the basics of the kinetic theory of gases and use it in solving problems to determine the velocities of the molecules of various gases.
Each stage of the lesson is accompanied by a thematic slide of the multimedia application and a video clip.
The purpose of the lesson:
Activity: the formation of new ways of activity in students (the ability to ask and answer effective questions; discussion of problem situations; the ability to evaluate their activities and their knowledge).
Lesson objectives:
Tutorial: the formation of the ability to analyze, compare, transfer knowledge to new situations, plan one's activities when constructing an answer, completing tasks and searching activities through physical concepts (most probable speed, average speed, root mean square speed), activate the mental activity of students.
Nurturing: fostering discipline in the performance of group tasks, creating conditions for positive motivation in the study of physics, using a variety of methods of activity, reporting interesting information; to cultivate a sense of respect for the interlocutor, an individual culture of communication.
Developing: to develop the ability to build independent statements in oral speech based on the learned educational material, the development of logical thinking, the development of the ability to use a unified mathematical approach for a quantitative description of physical phenomena based on molecular representations when solving problems.
Lesson type: lesson learning new material.
Teaching methods: heuristic, explanatory - illustrative, problematic, demonstrations and practical tasks, solving the problem of physical content.
Expected results:
be able to draw conclusions based on experiment;
develop rules of discussion and follow them;
understand the meaning of the issues discussed and show interest in this topic.
Preparatory stage: knowledge of the basic equations, dependencies on this topic (each student has a theoretical block on the topic in the form of a lecture - abstract)
Equipment: a device for demonstrating Stern's experience;
a computer and a projector to demonstrate the presentation and video clip "Stern's Experience".
Stages of the lesson.
Organizational stage (greeting, checking readiness for the lesson, emotional mood), (1 minute)
The stage of goal setting, lesson objectives and problems about the method of measuring the speed of molecules, (4 minutes)
The stage of studying new educational material, showing presentation slides with students' comments, which allows you to create a visual impression of the topic, activate visual memory (check the level of assimilation of the system of concepts on this topic), (20 minutes)
The stage of consolidating acquired knowledge in solving problems (application of knowledge in practice, their secondary comprehension), (8 minutes)
The stage of summarizing and summing up the results of the lesson (to analyze the success of mastering knowledge and methods of activity), (4 minutes)
Information about homework (aimed at further development of knowledge), (1 minute)
Reflection, (2 minutes)
Lesson script.
Activities of a physics teacher
Student activities
organizational stage.
Hello guys! I am glad to welcome you to the lesson, where we will continue to open the pages in the knowledge of the classical theory of gases. Interesting discoveries lie ahead. Greet each other.
Then let's get started...
Goal setting and motivation.
In the last lesson, we got acquainted with the basic provisions of the molecular-kinetic theory of an ideal gas. Participating in continuous chaotic motion, the molecules constantly collide with each other, while the number of colliding particles their speed are different at each point in time.
What do you think is the topic of the lesson “expecting” us today?
Yes, indeed, the goal that we set for ourselves today: we will get acquainted with one of the methods for determining the speed of molecules - the method of molecular beams, proposed by the German physicist Otto Stern in 1920.
We opened notebooks, wrote down the date and the topic of today's lesson: The movement of molecules. Experimental determination of the velocities of molecular motion.
Recall what is the speed of thermal motion of molecules?
Let us calculate the speed of Ag molecules during evaporation from the surface, T = 1500K.
Let me remind you that the speed of sound is 330m/s, and the speed of silver molecules is 588m/s, compare.
Let us calculate the speed of hydrogen molecules H 2 at a temperature close to absolute zero T=28K.
For example: the speed of a passenger plane is 900m/s, the speed of the Moon around the Earth is 1000m/s.
And now put yourself in the place of scientists of the 19th century, when these data were obtained, doubts arose about the correctness of the kinetic theory itself. After all, it is known that smells spread rather slowly: it takes time of the order of tens of seconds for the smell of perfume spilled in one corner of the room to spread to another corner.
Therefore, the question arises: what is the actual speed of the molecules?
When the smell of perfume spreads, does something interfere with the perfume molecules?
How does this affect the speed of the directed movement of molecules?
Let us calculate the speed of hydrogen molecules H 2 at a temperature close to room temperature T=293K.
Then what is the speed? What?
But how to measure it, to determine its value in practice? Let's solve the following problem:
Let there be 1 molecule. It is necessary to determine the free path velocity of the molecules. How do molecules move between collisions?
Let the molecule travel 1 meter, we will find the time at a hydrogen speed of 1911 m / s, it turned out to be 0.00052 s.
As you can see, the time is very short.
There is a problem again!
The stage of studying new educational material.
It is impossible to solve this problem in school conditions; Otto Stern (1888-1970) did it for us in 1920, replacing the translational motion with a rotational one.
Let's watch a short video and then discuss some issues.
What was the installation used by O. Stern?
How was the experience carried out?
Velocity values were obtained close to the velocity calculated by the formula:
, 
,
where 
is the linear speed of points on the surface of cylinder B.
, then
, which is in agreement with molecular-kinetic theory. The speed of the molecules coincides with the calculated one obtained on the basis of the MKT, this was one of the confirmations of its validity.
From the experience of O. Stern, it was found that at a temperature of 120 0 C, the velocities of most silver atoms lie in the range from 500 m/s to 625 m/s. When the conditions of the experiment change, for example, the temperature of the substance from which the wire is made, other values of the velocities are obtained, but the nature of the distribution of atoms in the deposited layer does not change.
Why is the strip of silver displaced and blurred along the edges in Stern's experiment, moreover, is it non-uniform in thickness?
What conclusion can be drawn about the distribution of atoms and molecules by velocities?
Consider table No. 12 of the textbook on page 98 for nitrogen molecules. What can be seen from the table?
The English physicist D.K. Maxwell also considered it improbable that all molecules move at the same speed. In his opinion, at any given temperature, most molecules have velocities that lie within fairly narrow limits, but some molecules can move at a greater or lesser speed. Moreover, the scientist believed that in every volume of gas at a given temperature there are molecules that have both very small and very high velocities. Colliding with each other, some molecules increase the speed, while others decrease. But if the gas is in a stationary state, then the number of molecules with a particular velocity remains constant. Based on this idea, D. Maxwell investigated the question of the velocity distribution of molecules in a gas in a stationary state.
He established this dependence long before the experiments of O. Stern. The results of D.K. Maxwell's work were universally recognized, but they were not confirmed experimentally. This was done by O. Stern.
Think? What is the merit of O. Stern?
Consider Fig. 64 on page 99 of the textbook and examine the nature of the velocity distribution of molecules.
The form of the distribution function of molecules according to the speed of movement, which D. Maxwell determined theoretically, qualitatively coincided with the profile of the deposit of silver atoms on a brass plate in the experiment of O. Stern.
The study of the profile of a strip of silver allowed the scientist to conclude the existence most likely average speed particle motion (i.e., the speed at which the largest number of molecules move).
Where does the maximum of the distribution curve shift with increasing temperature?
In addition to the most probable and average speeds, the movement of molecules is characterized by the average square of the speed:
, and the square root of this value is the mean square speed.
Let's look again at how cognition took place when studying the question of the velocities of molecular motion?
The stage of consolidating the acquired knowledge in solving problems.
Let's make mathematical calculations and check the theory in a specific situation.
Task #1
What speed did a silver vapor molecule have if its angular displacement in Stern's experiment was 5.4º at a device rotation frequency of 150 sˉ¹? The distance between the inner and outer cylinders is 2 cm.
The stage of generalization and summing up the lesson
Today at the lesson we got acquainted with one of the methods for determining the speed of molecules - the method of molecular beams, proposed by the German physicist Otto Stern.
What is the significance of O. Stern's experience in the development of ideas about the structure of matter?
Information about homework.
Reflection.
In the course of our lesson, you showed yourself to be observant theorists, able not only to notice everything new and interesting around you, but also to independently conduct scientific research.
Our lesson has come to an end.
Let's answer the question: "What did you like about the lesson?" and “What do you remember about the lesson?”
And in conclusion, I want to quote the words of Virey:
“All discoveries in the sciences and in philosophy often result from generalizations or from applications of a fact to other similar facts”
Thanks guys for the collaboration. I was glad to meet you. See you!
Lesson topic: Determining the speed of molecules.
(students write the number and topic of the lesson in their notebooks)
(answers from several students)
, on the other side
, knowing that 
, hence
, or 
, where
is the universal gas constant, 
8,31 
Speed of silver molecules supersonic.
590m/s, the same!!! Can not be!
What speed to find and measure?
Air molecules interfere.
She is decreasing.
We got a great speed, and nothing prevents the molecules from moving?
The speed of the free path of molecules.
Evenly.
How to measure it?
(watch video)
The installation consisted of: a platinum thread coated with a thin layer of silver, which was located along the axis inside a cylinder with a radius 
and outer cylinder 
. Air has been evacuated from the cylinder by a pump.
When an electric current was passed through the wire, it was heated to a temperature higher than the melting point of silver 961.9 0 C. The walls of the outer cylinder were cooled so that the silver molecules would better settle on the way of the screen. The installation was brought into rotation with an angular velocity of 2500–2700 rpm.
During the rotation of the device, the silver strip took on a different appearance, because if all the atoms emitted from the thread had the same speed, then the image of the slit on the screen would not change in shape and size, but would only slightly shift to the side. The blurring of the silver strip indicates that the atoms flying out of the red-hot filament move at different speeds. Atoms moving fast move less than atoms moving slower.
The velocity distribution of atoms and molecules is a certain pattern that characterizes their movement.
The table shows that the largest number of nitrogen molecules have velocities from 300m/s to 500m/s.
91% of the molecules have velocities included in the range from 100m/s to 700m/s.
9% of the molecules have velocities less than 100m/s and greater than 700m/s.
O. Stern, using the method of molecular beams invented by the French physicist Louis Dunoyer (1911), measured the speed of gas molecules and experimentally confirmed the velocity distribution of gas molecules obtained by D. K. Maxwell. The results of Stern's experiment confirmed the correctness of the estimate of the average velocity of atoms, which follows from the Maxwell distribution.
From the graph, it was possible to determine the shift for the middle of the slit image and, accordingly, calculate average speed
movement of atoms.
At T 2 T 1, the maximum of the distribution curve shifts to the region of high velocities.
N. S. Purysheva, N. E. Vazheevskaya, D. A. Isaev, textbook "Physics - 10", workbook for this textbook.
Physics: 3800 tasks for schoolchildren and university applicants. - M .: Bustard, 2000.
Rymkevich A.P. Collection of problems in physics. 10-11 cells. – M.: Bustard, 2010.
L. A. Kirik “Independent and control work in physics”. Grade 10. M.: Ileksa, Kharkov: Gymnasium, 1999.
Encyclopedia for children. Technics. Moscow: Avanta+, 1999.
Encyclopedia for children. Physics. Ch. I. M.: Avanta +, 1999.
Encyclopedia for children. Physics. Ch. P. M.: Avanta+, 1999.
Physical experiment at school. / Comp. G. P. Mansvetova, V. F. Gudkov. - M.: Enlightenment, 1981.
Glazunov A. T. Technique in the course of physics in high school. M.: Education, 1977.
Initially, it was hypothesized that the molecules move at different speeds.
These velocities are related to temperature and there is a certain law of distribution of molecules in terms of velocities, which followed from observations, in particular, of Brownian motion.
The experiment is one of the fundamental physical experiments. At present, the atomic and molecular theory has been confirmed by numerous experiments and is generally recognized.
Reflection of educational actions.
Today I found out...
It was interesting…
It was difficult…
I realized that... I learned...
I was surprised...
Used Books:
Electronic Applications:
L. Ya. Borevsky "Course of physics of the XXI century", basic + for schoolchildren and applicants. MediaHouse. 2004
Interactive physics course for grades 7-11. LLC "Physikon", 2004. Russian version "Live Physics", Institute of New Technologies
Physics, X-XI classes. Multimedia course-M.: Russobit Publishing LLC.-2004 (http://www. russobit-m. ru/)
open physics. At 2 o'clock (CD) / Ed. CM. Goat. - M .: OOO "Fizikon". - 2002 (http://www.physicon.ru/.)
The study of diffusion and Brownian motion allows one to gain some idea of the speed of the chaotic motion of gas molecules. One of the simplest and most illustrative experiments for its determination is the experiment of O. Stern, performed by him in 1920. The essence of this experiment is as follows.
On a horizontal table, which can rotate around the O axis (Fig. 3.2), cylindrical surfaces A and B are fixed perpendicular to the table. Surface B is solid, and surface A has a narrow slot parallel to the O axis. A platinum silver-plated wire is located vertically along the O axis, which is connected to the electrical circuit. When a current is passed through the wire, it heats up and the silver evaporates from its surface. Silver molecules fly in all directions and mainly settle on the inner side of the cylindrical surface A. Only a narrow beam of silver molecules flies through the slit in this
surface and settles in the area M on the surface B. The width of the plaque in M is determined by the width of the gap in the surface A. To prevent silver molecules from scattering upon collisions with air molecules, the entire installation is covered with a cap, from which air is pumped out. The narrower the gap in the surface A, the narrower the plaque in the area M and the more accurately the velocity of the molecules can be determined.
The very definition of speed is based on the following idea. If the entire installation is brought into rotation around the O axis with a constant angular velocity, then during the time during which the molecule will fly from the slot to the surface B, the latter will have time to turn and the plaque will shift from the M region to the K region. Consequently, the flight time of the molecule along the radius and the time displacement of point M of surface B by the same distance. Since the molecule flies uniformly, then
where is the desired speed, is the radius of the cylindrical surface A. Since the linear speed of the points of the surface B is south, the time can be expressed by another formula:
In this way,
Since they remain constant during the experiment and are determined in advance, by measuring it is possible to find the speed of the molecule. In Stern's experiment, it turned out to be close to 500 m/s.
Since the plaque in region K is blurred, it can be concluded that the silver molecules fly towards the surface B at different speeds. The average values of the velocities of molecules can be mathematically expressed by the formula
As an example, we note that at 0 °C, the average velocity of hydrogen molecules is 1840 m/s, and that of nitrogen is 493 m/s. The change in the thickness of the plaque in the region K gives an idea of the distribution of molecules according to their speeds of motion. It turns out that a small number of molecules have speeds that are several times higher than the average speed.
(Think about where in Fig. 3.2 the molecules left a trace, the speed of which is greater than the average speed, and how the position of the plaque will change if the current in the wire O is increased.)
The assumption that the molecules of a body can have any speed was first theoretically proved in 1856 by an English physicist J. Maxwell. He believed that the speed of molecules at a given moment of time is random, and therefore their distribution over speeds is statistical in nature ( Maxwell distribution).
The nature of the distribution of molecules by velocities established by him is graphically represented by the curve shown in fig. 1.17. The presence of a maximum (bump) in it indicates that the velocities of most molecules fall within a certain interval. It is asymmetric, since there are fewer molecules with high speeds than with small ones.
Fast molecules determine the course of many physical processes under ordinary conditions. For example, thanks to them, the evaporation of liquids occurs, because at room temperature most molecules do not have enough energy to break bonds with other molecules (it is much higher (3 / 2) . kT), and for molecules with high speeds it is sufficient.
 |
| Rice. 1.18. O. Stern's experience |
The distribution of molecules according to Maxwell's velocities for a long time remained experimentally unconfirmed, and only in 1920 the German scientist O. stern was able to experimentally measure speed of thermal motion of mo-lecules.
On a horizontal table that could rotate around a vertical axis (Fig. 1.18), there were two coaxial cylinders A and B. from which air was pumped out to a pressure of the order of 10 -8 Pa. Along the axis of the cylinders there was a platinum wire C coated with a thin layer of silver. When an electric current passed through the wire, it heated up, and silver intensively evaporated from its surface, which mainly settled on the inner surface of cylinder A. Some of the silver molecules passed through a narrow slot in cylinder A outward, falling on the surface. cylinder B. If the cylinders did not rotate, the silver molecules, moving in a straight line, settled opposite the slot in the circumference of point D. When the system was set in motion with an angular velocity of about 2500-2700 rpm, the image of the slot shifted to point E, and its edges were “blurred”, forming a hillock with gentle slopes.
In science Stern's experience finally confirmed the validity of the molecular-kinetic theory.
Bearing in mind that the displacement l =v. t = ω R A t, and the flight time of the molecules t = (R B -R A) /v, we get:
l =ω(R B -R A)R A /v.
As can be seen from the formula, the displacement of a molecule from point D depends on the speed of its movement. Velocity Calculations of Silver Molecules from Data Stern experience at a coil temperature of about 1200 °C, they gave values ranging from 560 to 640 m/s, which was in good agreement with the theoretically determined average molecular velocity of 584 m/s.
The average speed of thermal motion of gas molecules can be found using the equation p=nm0v̅ 2 x:
E = (3 / 2). kT = m 0 v̅ 2 / 2.
Hence, the average square of the speed of the translational motion of the molecule is equal to:
v̅ 2 = 3kT /m 0 , or v =√(v̅ 2) =√(3 kT /m0). material from the site
The square root of the mean square of the speed of a molecule is called mean square speed.
Given that k \u003d R / N A and m 0 \u003d M / N A, from the formula v =√(3 kT /m0) we get:
v =√ (3RT/M).
Using this formula, you can calculate the root-mean-square velocity of molecules for any gas. For example, at 20°C ( T= 293K) for oxygen it is 478 m/s, for air - 502 m/s, for hydrogen - 1911 m/s. Even at such significant speeds (approximately equal to the speed of propagation of sound in a given gas), the movement of gas molecules is not so rapid, since numerous collisions occur between them. Therefore, the trajectory of the motion of a molecule resembles the trajectory of the motion of a Brownian particle.
The root-mean-square velocity of a molecule does not differ significantly from the average velocity of its thermal motion - it is approximately 1.2 times greater.
On this page, material on the topics:
Abstract about the experience of stern
Molecule speed lesson
Vimiryuvannya swidkosti ruhu molecules doslid stern summary of the lesson
The essence of the stern experience
Stern physicist's experience
Questions about this item:
Lecture 5
As a result of numerous collisions of gas molecules with each other (~10 9 collisions per 1 second) and with the walls of the vessel, a certain statistical distribution of molecules in terms of velocities is established. In this case, all directions of the molecular velocity vectors turn out to be equally probable, and the velocity modules and their projections on the coordinate axes obey certain regularities.
During collisions, the velocities of molecules change randomly. It may turn out that one of the molecules in a series of collisions will receive energy from other molecules and its energy will be much greater than the average value of the energy at a given temperature. The speed of such a molecule will be large, but, nevertheless, it will have a finite value, since the maximum possible speed is the speed of light - 3·10 8 m/s. Therefore, the speed of a molecule can generally have values from 0 to some υ max. It can be argued that very high speeds compared to average values are rare, as well as very small ones.
As theory and experiments show, the distribution of molecules in terms of velocities is not random, but quite definite. Let us determine how many molecules, or what part of the molecules has velocities lying in a certain interval near the given speed.
Let a given mass of gas contain N molecules, while dN molecules have velocities ranging from υ before υ +dv. Obviously, this is the number of molecules dN proportional to the total number of molecules N and the value of the specified speed interval dv
where a- coefficient of proportionality.
It is also obvious that dN also depends on the speed υ , since in the same intervals, but at different absolute values of the speed, the number of molecules will be different (example: compare the number of people living at the age of 20-21 years and 99-100 years). This means that the coefficient a in formula (1) should be a function of speed.
Taking this into account, we rewrite (1) in the form
(2)
From (2) we get
(3)
Function f(υ ) is called the distribution function. Its physical meaning follows from formula (3)
if (4)
Hence, f(υ ) is equal to the relative fraction of molecules whose velocities are contained in the unit interval of velocities near the velocity υ . More precisely, the distribution function has the meaning of the probability for any gas molecule to have a velocity contained in unit interval near speed υ . Therefore it is called probability density.
Integrating (2) over all velocities from 0 to we obtain
(5)
From (5) it follows that
(6)
Equation (6) is called normalization condition functions. It determines the probability that the molecule has one of the speed values from 0 to . The speed of the molecule has some meaning: this event is certain and its probability is equal to one.
Function f(υ ) was found by Maxwell in 1859. She was named Maxwell distribution:
(7)
where A is a coefficient that does not depend on speed, m is the mass of the molecule, T is the gas temperature. Using the normalization condition (6), we can determine the coefficient A:
Taking this integral, we get A:
Taking into account the coefficient A the Maxwell distribution function has the form:
(8)
With increasing υ the factor in (8) changes faster than it grows υ 2. Therefore, the distribution function (8) begins at the origin of coordinates, reaches a maximum at a certain velocity value, then decreases, asymptotically approaching zero (Fig. 1).
Fig.1. Maxwellian distribution of molecules
by speed. T 2 > T 1
Using the Maxwell distribution curve, one can graphically find the relative number of molecules whose velocities lie in a given interval of velocities from υ before dv(Fig. 1, area of the shaded strip).
Obviously, the entire area under the curve gives the total number of molecules N. From equation (2), taking into account (8), we find the number of molecules whose velocities lie in the interval from υ before dv
(9)
From (8) it is also seen that the specific form of the distribution function depends on the type of gas (the mass of the molecule m) and temperature and does not depend on the pressure and volume of the gas.
If an isolated system is taken out of equilibrium and left to itself, then after a certain period of time it will return to a state of equilibrium. This period of time is called relaxation time. It is different for different systems. If the gas is in equilibrium, then the velocity distribution of molecules does not change with time. The speeds of individual molecules are constantly changing, but the number of molecules dN, whose velocities lie in the interval from υ before dv remains constant all the time.
The Maxwellian velocity distribution of molecules is always established when the system comes to equilibrium. The movement of gas molecules is chaotic. The exact definition of the randomness of thermal motions is as follows: the movement of molecules is completely random if the velocities of the molecules are distributed according to Maxwell. It follows that the temperature is determined by the average kinetic energy chaotic movements. No matter how great the speed of a strong wind, it will not make it "hot". The wind, even the strongest, can be both cold and warm, because the gas temperature is determined not by the directed wind speed, but by the speed of the chaotic movement of molecules.
From the graph of the distribution function (Fig. 1) it can be seen that the number of molecules whose velocities lie in the same intervals d υ , but near different speeds υ , more if the speed υ approaches the speed that corresponds to the maximum of the function f(υ ). This speed υ n is called the most probable (most probable).
We differentiate (8) and equate the derivative to zero:
Because 
,
then the last equality is satisfied when:
(10)
Equation (10) is satisfied when:
AND 
The first two roots correspond to the minimum values of the function. Then the speed that corresponds to the maximum of the distribution function can be found from the condition:
From the last equation:
(11)
where R is the universal gas constant, μ - molar mass.
Taking into account (11), from (8) one can obtain the maximum value of the distribution function
(12)
From (11) and (12) it follows that with increasing T or when decreasing m curve maximum f(υ ) shifts to the right and becomes smaller, but the area under the curve remains constant (Fig. 1).
To solve many problems, it is convenient to use the Maxwell distribution in the reduced form. Let's introduce the relative speed:
where υ - this speed υ n- the most incredible speed. With this in mind, equation (9) takes the form:
(13)
(13) is a universal equation. In this form, the distribution function does not depend on either the type of gas or the temperature.
Curve f(υ ) is asymmetric. From the graph (Fig. 1) it can be seen that most of the molecules have velocities greater than υ n. The asymmetry of the curve means that the arithmetic mean velocity of the molecules is not equal to υ n. The arithmetic average speed is equal to the sum of the speeds of all molecules, divided by their number:
Let us take into account that according to (2)
(14)
Substituting into (14) the value f(υ ) from (8) we obtain the arithmetic average speed:
(15)
We obtain the average square of the speed of molecules by calculating the ratio of the sum of the squares of the speeds of all molecules to their number:
After substitution f(υ ) from (8) we get:
From the last expression we find the mean square speed:
(16)
Comparing (11), (15) and (16), we can conclude that, and are equally dependent on temperature and differ only in numerical values: (Fig. 2).
Fig.2. Maxwell distribution by absolute values of velocities
The Maxwell distribution is valid for gases in equilibrium, the considered number of molecules must be large enough. For a small number of molecules, significant deviations from the Maxwell distribution (fluctuations) can be observed.
The first experimental determination of the velocities of molecules was carried out by stern in 1920. Stern's device consisted of two cylinders of different radii, fixed on the same axis. The air from the cylinders was evacuated to a deep vacuum. A platinum thread covered with a thin layer of silver was stretched along the axis. When an electric current was passed through the filament, it was heated to a high temperature (~1200 o C), which led to the evaporation of silver atoms.
A narrow longitudinal slot was made in the wall of the inner cylinder, through which moving silver atoms passed. Settling on the inner surface of the outer cylinder, they formed a well-observed thin strip directly opposite the slit.
The cylinders began to rotate with a constant angular velocity ω. Now the atoms that passed through the slit no longer settled directly opposite the slit, but were displaced over a certain distance, since during their flight the outer cylinder had time to turn through a certain angle. When the cylinders rotated at a constant speed, the position of the strip formed by the atoms on the outer cylinder shifted by a certain distance l.
Particles settle at point 1 when the installation is stationary; when the installation rotates, particles settle at point 2.
The obtained speed values confirmed Maxwell's theory. However, this method gave approximate information about the nature of the distribution of molecules over velocities.
More accurately, the Maxwell distribution was verified by experiments Lammert, Easterman, Eldridge and Costa. These experiments quite accurately confirmed Maxwell's theory.
Direct measurements of the velocity of mercury atoms in a beam were made in 1929 Lammert. A simplified scheme of this experiment is shown in Fig. 3.
Fig.3. Scheme of Lammert's experiment
1 - rapidly rotating disks, 2 - narrow slits, 3 - oven, 4 - collimator, 5 - molecular trajectory, 6 - detector
Two disks 1, mounted on a common axis, had radial slots 2, shifted relative to each other by an angle φ . Opposite the slots was furnace 3, in which low-melting metal was heated to a high temperature. Heated metal atoms, in this case mercury, flew out of the furnace and were directed in the required direction with the help of collimator 4. The presence of two slits in the collimator ensured the movement of particles between the disks along a rectilinear trajectory 5. Further, the atoms that passed through the slits in the disks were recorded using detector 6. The entire described setup was placed in a deep vacuum.
When the disks rotated with a constant angular velocity ω, only atoms that had a certain speed passed through their slots without hindrance υ . For atoms passing through both slits, the equality must hold:
where ∆ t 1 - time of flight of molecules between disks, Δ t 2 - the time of rotation of the disks at an angle φ . Then:
By changing the angular velocity of rotation of the disks, it was possible to separate molecules from the beam with a certain speed υ , and according to the intensity recorded by the detector, to judge their relative content in the beam.
In this way, it was possible to experimentally verify the Maxwellian law of the distribution of molecules with respect to velocities.
