Results of the stern experiment. Speeds of gas molecules
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 by speed 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 onto the coordinate axes obey certain laws.
During collisions, the speeds 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 significantly greater than the average energy value at a given temperature. The speed of such a molecule will be high, but it will still have a finite value, since the maximum possible speed is the speed of light - 3·10 8 m/s. Consequently, 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 are very small ones.
As theory and experiments show, the distribution of molecules by speed is not random, but quite definite. Let us determine how many molecules, or what part of the molecules have velocities that lie in a certain interval near a given speed.
Let a given mass of gas contain N molecules, while dN molecules have velocities ranging from υ before υ +dυ. Obviously this is the number of molecules dN proportional to the total number of molecules N and the value of the specified speed interval dυ
Where a- proportionality coefficient.
It is also obvious that dN depends on the speed υ , since in intervals of the same size, but at different absolute values of 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) must 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 a unit velocity interval near the velocity υ . More precisely, the distribution function has the meaning of the probability of any gas molecule having a speed contained in unit interval near speed υ . That's why they call her probability density.
Integrating (2) over all speed values from 0 to we obtain
(5)
From (5) it follows that
(6)
Equation (6) is called normalization condition functions. It determines the probability that a molecule has one of the speed values from 0 to . The speed of the molecule has some meaning: this event is reliable and its probability is equal to one.
Function f(υ ) was found by Maxwell in 1859. She was named Maxwell distribution:
(7)
Where A– coefficient that does not depend on speed, m– molecular mass, T– 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)
When increasing υ the factor in (8) changes faster than it grows υ 2. Therefore, the distribution function (8) starts at the origin, reaches a maximum at a certain speed 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, you can graphically find the relative number of molecules whose velocities lie in a given velocity range from υ before dυ(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 range from υ before dυ
(9)
From (8) it is also clear 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 its own devices, then after a certain period of time it will return to equilibrium. This period of time is called relaxation time. It is different for different systems. If the gas is in an equilibrium state, then the distribution of molecules by speed does not change over time. The speeds of individual molecules are constantly changing, but the number of molecules dN, whose speeds lie in the range from υ before dυ remains constant all the time.
The Maxwellian velocity distribution of molecules is always established when the system reaches a state of equilibrium. The movement of gas molecules is chaotic. The exact definition of the randomness of thermal motion is as follows: the movement of molecules is completely chaotic if the velocities of the molecules are distributed according to Maxwell. It follows that the temperature is determined by the average kinetic energy namely chaotic movements. No matter how high the speed of a strong wind is, it will not make it “hot”. Even the strongest wind can be both cold and warm, because the temperature of the gas is determined not by the directional speed of the wind, but by the speed of the chaotic movement of molecules.
From the graph of the distribution function (Fig. 1) it is clear 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).
Let us 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 we find the speed that corresponds to the maximum of the distribution function from the condition:
From the last equation:
(11)
Where R– universal gas constant, μ - molar mass.
Taking into account (11) from (8) we 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 its reduced form. Let's introduce the relative speed:
Where υ – given speed, υ n- the most probable speed. Taking this into account, equation (9) takes the form:
(13)
(13) is a universal equation. In this form, the distribution function does not depend on the type of gas or temperature.
Curve f(υ ) is asymmetrical. From the graph (Fig. 1) it is clear that most of the molecules have speeds greater than υ n. The asymmetry of the curve means that the arithmetic mean speed of the molecules is not equal 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)
The average square of the speed of molecules is obtained 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 obtain:
From the last expression we find the root mean square speed:
(16)
Comparing (11), (15) and (16), we can conclude that and equally depend on temperature and differ only in numerical values: (Fig. 2).
Fig.2. Maxwell distribution over absolute velocity values
The Maxwell distribution is valid for gases in a state of equilibrium; the number of molecules under consideration must be sufficiently large. For a small number of molecules, significant deviations from the Maxwell distribution (fluctuations) can be observed.
The first experimental determination of molecular velocities was carried out by Stern in 1920. Stern's device consisted of two cylinders of different radii mounted on the same axis. The air from the cylinders was pumped out to a deep vacuum. A platinum thread coated with a thin layer of silver was stretched along the axis. When an electric current was passed through the filament, it heated to a high temperature (~1200 o C), which led to the evaporation of silver atoms.
A narrow longitudinal slit was made in the wall of the inner cylinder, through which moving silver atoms passed. Deposited on the inner surface of the outer cylinder, they formed a clearly visible thin strip directly opposite the slot.
The cylinders began to rotate at a constant angular velocity ω. Now the atoms that passed through the slit no longer settled directly opposite the slit, but were displaced by a certain distance, since during their flight the outer cylinder had time to rotate through a certain angle. When the cylinders rotated at a constant speed, the position of the strip formed by 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 provided approximate information about the nature of the velocity distribution of molecules.
The Maxwell distribution has been more accurately 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 diagram of this experiment is shown in Fig. 3.
Fig.3. Diagram of Lammert's experiment
1 - rapidly rotating disks, 2 - narrow slits, 3 - oven, 4 - collimator, 5 - trajectory of molecules, 6 - detector
Two disks 1, mounted on a common axis, had radial slots 2, shifted relative to each other at an angle φ . Opposite the slots there was furnace 3, in which fusible metal was heated to a high temperature. Heated metal atoms, in this case mercury, flew out of the furnace and, using collimator 4, were directed in the required direction. The presence of two slits in the collimator ensured the movement of particles between the disks along a straight path 5. Next, the atoms that passed through the slits in the disks were recorded using a detector 6. The entire described installation was placed in deep vacuum.
When the disks rotated with a constant angular velocity ω, only atoms that had a certain speed passed through their slits freely υ . For atoms passing through both slits the equality must be satisfied:
where Δ t 1 - time of flight of molecules between disks, Δ t 2 - time to turn the disks at an angle φ . Then:
By changing the angular speed of rotation of the disks, it was possible to isolate molecules with a certain speed from the beam υ , and from the intensity recorded by the detector, judge their relative content in the beam.
In this way, it was possible to experimentally verify Maxwell's law of molecular velocity distribution.
In the second half of the nineteenth century, the study of Brownian (chaotic) motion of molecules aroused keen interest among many theoretical physicists of that time. The substance developed by the Scottish scientist James, although it was generally accepted in European scientific circles, existed only in a hypothetical form. There was no practical confirmation of it then. The movement of molecules remained inaccessible to direct observation, and measuring their speed seemed simply an insoluble 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 a practical justification and proof of the validity of one of the most progressive theories of that time - molecular kinetics.
By the beginning of the twentieth century, world science had reached a sufficient level of development for the emergence of real possibilities for experimental verification of Maxwell's theory. The German physicist Otto Stern in 1920, using the molecular beam method, 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 assessment of atoms, which followed from the hypothetical assumptions made by Maxwell. True, Stern’s experience could only provide very approximate information about the very nature of the speed 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 rotation speed of the unit and the angle between the holes, Lammert was able to isolate individual molecules from the beam that have different speed characteristics. But it was Stern’s experience that laid the foundation for experimental research in the field of molecular kinetic theory.
In 1920, the first experimental installation necessary for conducting experiments of this kind was created. It consisted of a pair of cylinders designed personally by Stern. A thin platinum rod coated with silver was placed inside the device, which evaporated when the axis was heated with electricity. Under vacuum conditions that were created inside the installation, a narrow beam of silver atoms passed through a longitudinal slit 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 the atoms reached the surface, it managed to rotate 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, he, together with Walter Gerlach, 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 power at its core. Under the influence of the magnetic field generated by this powerful component, they were deflected according to the orientation of their own magnetic spin.
Year. The experiment was one of the first practical proofs of the validity of the molecular kinetic theory of the structure of matter. It directly measured the speed of thermal motion of molecules and confirmed the presence of a distribution of gas molecules by speed.
To conduct the experiment, Stern prepared a device consisting of two cylinders of different radii, the axis of which coincided and a platinum wire coated with a layer of silver was placed on it. A sufficiently low pressure was maintained in the space inside the cylinders through continuous pumping of air. When an electric current was passed through the wire, the melting point of silver was reached, due to which the silver began to evaporate and the silver atoms flew to the inner surface of the small cylinder uniformly and rectilinearly at a speed v, determined by the heating temperature of the platinum wire, i.e., the melting point of silver. A narrow slit was made in the inner cylinder, through which atoms could fly further without hindrance. The walls of the cylinders were specially cooled, which contributed to the settling of atoms falling on them. In this state, a fairly clear narrow strip of silver plaque formed on the inner surface of the large cylinder, located directly opposite the slit of the small cylinder. Then the entire system began to rotate with a certain sufficiently high angular velocity ω . In this case, the plaque band shifted in the direction opposite to the direction of rotation and lost its clarity. By measuring the displacement s the darkest part of the strip from its position when the system was at rest, Stern determined the flight time, after which he found the speed of movement of the molecules:
,
Where s- stripe offset, l- the distance between the cylinders, and u- the speed of movement of the points of the outer cylinder.
The speed of movement of silver atoms found in this way coincided with the speed calculated according to the laws of molecular kinetic theory, and the fact that the resulting strip was blurred testified to the fact that the speeds of the atoms are different and distributed according to a certain law - Maxwell’s distribution law: atoms, those moving faster shifted relative to the strip obtained at rest by shorter distances than those moving more slowly.
Write a review of the article "Stern Experience"
Literature
- Brief dictionary of physical terms / Comp. A. I. Bolsun, rector. M. A. Elyashevich. - Mn. : Higher School, 1979. - P. 388. - 416 p. - 30,000 copies.
Links
- Landsberg. Elementary physics textbook. Volume 1. Mechanics. Heat. Molecular physics. - 12th ed. - M.: FIZMATLIT, 2001. - ISBN 5-9221-0135-8.
- Internet school Prosveshchenie.ru.(Russian) (inaccessible link - story) . Retrieved April 5, 2008.
- Stern experience- article from the Great Soviet Encyclopedia.
Excerpt characterizing the Stern Experiment
So he lay now on his bed, leaning his heavy, large, disfigured head on his plump arm, and thought, with one eye open, peering into the darkness.Since Bennigsen, who corresponded with the sovereign and had the most power in the headquarters, avoided him, Kutuzov was calmer in the sense that he and his troops would not be forced to again participate in useless offensive actions. The lesson of the Tarutino battle and its eve, painfully memorable for Kutuzov, should also have had an effect, he thought.
“They must understand that we can only lose by acting offensively. Patience and time, these are my heroes!” – thought Kutuzov. He knew not to pick an apple while it was green. It will fall on its own when it is ripe, but if you pick it green, you will spoil the apple and the tree, and you will set your teeth on edge. He, as an experienced hunter, knew that the animal was wounded, wounded as only the entire Russian force could wound, but whether it was fatal or not was a question that had not yet been clarified. Now, according to the dispatches of Lauriston and Berthelemy and according to the reports of the partisans, Kutuzov almost knew that he was mortally wounded. But more evidence was needed, we had to wait.
“They want to run and see how they killed him. Wait and see. All maneuvers, all attacks! - he thought. - For what? Everyone will excel. There's definitely something fun about fighting. They are like children from whom you can’t get any sense, as was the case, because everyone wants to prove how they can fight. That's not the point now.
And what skillful maneuvers all these offer me! It seems to them that when they invented two or three accidents (he remembered the general plan from St. Petersburg), they invented them all. And they all have no number!”
The unresolved question of whether the wound inflicted in Borodino was fatal or not fatal had been hanging over Kutuzov’s head for a whole month. On the one hand, the French occupied Moscow. On the other hand, undoubtedly with his whole being Kutuzov felt that that terrible blow, in which he, together with all the Russian people, strained all his strength, should have been fatal. But in any case, proof was needed, and he had been waiting for it for a month, and the more time passed, the more impatient he became. Lying on his bed on his sleepless nights, he did the very thing that these young generals did, the very thing for which he reproached them. He came up with all possible contingencies in which this certain, already accomplished death of Napoleon would be expressed. He came up with these contingencies in the same way as young people, but with the only difference that he did not base anything on these assumptions and that he saw not two or three, but thousands. The further he thought, the more of them appeared. He came up with all kinds of movements of the Napoleonic army, all or parts of it - towards St. Petersburg, against it, bypassing it, he came up with (which he was most afraid of) and the chance that Napoleon would fight against him with his own weapons, that he would remain in Moscow , waiting for him. Kutuzov even dreamed up the movement of Napoleon’s army back to Medyn and Yukhnov, but one thing he could not foresee was what happened, that crazy, convulsive rushing of Napoleon’s army during the first eleven days of his speech from Moscow - the throwing that made it possible something that Kutuzov still did not dare to think about even then: the complete extermination of the French. Dorokhov's reports about Broussier's division, news from the partisans about the disasters of Napoleon's army, rumors about preparations for departure from Moscow - everything confirmed the assumption that the French army was defeated and was about to flee; but these were only assumptions that seemed important to young people, but not to Kutuzov. With his sixty years of experience, he knew what weight should be attributed to rumors, he knew how capable people who want something are of grouping all the news so that they seem to confirm what they want, and he knew how in this case they willingly miss everything that contradicts. And the more Kutuzov wanted this, the less he allowed himself to believe it. This question occupied all his mental strength. Everything else was for him just the usual fulfillment of life. Such habitual fulfillment and subordination of life were his conversations with staff, letters to m me Stael, which he wrote from Tarutin, reading novels, distributing awards, correspondence with St. Petersburg, etc. n. But the death of the French, foreseen by him alone, was his spiritual, only desire.
From formulas
we obtain a formula for calculating the root mean square speed of movement of molecules of a monatomic gas:
where R is the universal gas constant.
Thus depends on the temperature and nature of the gas. So, at 0°C for hydrogen it is equal to 1800 m/s. for nitrogen - 500 m/s.
O. Stern was the first to experimentally determine the speed of molecules. In the chamber from which air has been evacuated, there are two coaxial cylinders 1 and 2 (Fig. 1), which can rotate around an axis at a constant angular velocity.
A silver-plated platinum wire is stretched along the axis, through which an electric current is passed. It heats up and the silver evaporates. Silver atoms enter cylinder 1 through slot 4 in the wall of cylinder 2 and settle on its inner surface, leaving a trace in the form of a narrow strip parallel to the slot. If the cylinders are stationary, then the strip is located opposite the slot (point B in Fig. 2, a) and has the same thickness.
When the cylinder rotates uniformly with angular velocity, the strip moves in the direction opposite to the rotation by a distance s relative to point B (Fig. 2, b). Point B of cylinder 1 has shifted by this distance in time t, which is necessary for the silver atoms to travel a distance equal to R - r, where R and r are the radii of cylinders 1 and 2.
where is the linear speed of points on the surface of cylinder 1. Hence
Speed of silver atoms
Knowing R, r, and having measured s experimentally, using this formula one can calculate the average speed of movement of silver atoms. In the Stern experiment. This value coincides with the theoretical value of the root mean square speed of molecules. This serves as experimental proof of the validity of formula (1), and consequently, formula (3).
In Stern's experiment, it was discovered that the width of the strip on the surface of a rotating cylinder is much larger than the geometric image of the slit and its thickness is not the same in different places (Fig. 3, a). This can only be explained by the fact that the silver atoms move at different speeds. Atoms flying at a certain speed reach point B'. Atoms flying faster end up at a point lying in Figure 2 above point B’, and atoms flying slower end up below point B’. Thus, each point in the image corresponds to a certain speed, which can be easily determined from experience. This explains the fact that the thickness of the layer of silver atoms deposited on the surface of the cylinder is not the same everywhere. The greatest thickness is in the middle part of the layer, and at the edges the thickness decreases.
Studying the cross-sectional shape of a strip of deposited silver using a microscope showed that it has a shape approximately corresponding to that shown in Figure 3, b. Based on the thickness of the deposited layer, one can judge the velocity distribution of silver atoms.
Let us divide the entire range of experimentally measured velocities of silver atoms into small ones. Let be one of the speeds of this interval. Using the density of the layer, we calculate the number of atoms having a speed in the range from , and plot the function
where N is the total number of silver atoms deposited on the surface of the cylinder. We obtain the curve shown in Figure 4. It is called the velocity distribution function of molecules.
The area of the shaded area is
those. equal to the relative number of atoms having a speed within
We see that the numbers of particles with speeds from different intervals are sharply different. There is some speed, around the value of which are the speeds at which the largest number of molecules move. It is called the most probable speed, and it corresponds to the maximum in Figure 4. This curve corresponds well to the curve obtained by J. Maxwell, who, using the statistical method, theoretically proved that in gases that are in a state of thermodynamic equilibrium, a certain value is established that does not change with time, the distribution of molecules by speed, which obeys a well-defined statistical law, graphically depicted by the curve. The most probable speed, as Maxwell showed, depends on the temperature of the gas and the mass of its molecules according to the formula
1 - platinum wire with a layer of silver applied to it; 2 - slit forming a beam of silver atoms; 3 - plate on which silver atoms are deposited; P and P1 are the positions of the deposited silver strips when the device is stationary and when the device is rotating.
To conduct the experiment, Stern prepared a device consisting of two cylinders of different radii, the axis of which coincided and a platinum wire coated with a layer of silver was placed on it. A sufficiently low pressure was maintained in the space inside the cylinders through continuous pumping of air. When an electric current was passed through the wire, the melting point of silver was reached, due to which the silver began to evaporate and the silver atoms flew to the inner surface of the small cylinder uniformly and rectilinearly at a speed v (\displaystyle v), determined by the heating temperature of the platinum wire, that is, the melting point of silver. A narrow slit was made in the inner cylinder, through which atoms could fly further without hindrance. The walls of the cylinders were specially cooled, which contributed to the settling of atoms falling on them. In this state, a fairly clear narrow strip of silver plaque formed on the inner surface of the large cylinder, located directly opposite the slit of the small cylinder. Then the entire system began to rotate with a certain sufficiently high angular velocity ω (\displaystyle \omega ). In this case, the plaque band shifted in the direction opposite to the direction of rotation and lost its clarity. By measuring the displacement s (\displaystyle s) the darkest part of the strip from its position when the system was at rest, Stern determined the flight time, after which he found the speed of movement of the molecules:
t = s u = l v ⇒ v = u l s = ω R b i g (R b i g − R s m a l l) s (\displaystyle t=(\frac (s)(u))=(\frac (l)(v))\Rightarrow v =(\frac (ul)(s))=(\frac (\omega R_(big)(R_(big)-R_(small)))(s))),Where s (\displaystyle s)- stripe offset, l (\displaystyle l)- the distance between the cylinders, and u (\displaystyle u)- the speed of movement of the points of the outer cylinder.
The speed of movement of silver atoms found in this way (584 m/s) coincided with the speed calculated according to the laws of molecular kinetic theory, and the fact that the resulting strip was blurred testified to the fact that the velocities of the atoms are different and distributed according to some law - Maxwell's distribution law: atoms that moved faster were displaced relative to the strip obtained at rest by smaller distances than those that moved more slowly. At the same time, the experience provided only approximate information about the nature of the Maxwell distribution; more accurate experimental confirmation dates back to 1930 (
Articles on the topic
