Let there be n identical molecules in a state of random thermal motion at a certain temperature. After each act of collision between molecules, their speeds change randomly. As a result of an unimaginably large number of collisions, a stationary equilibrium state is established, when the number of molecules in a given velocity range remains constant.

As a result of each collision, the velocity projections of the molecules experience a random change by Δυ x , Δυ y , Δυ z , and the changes in each velocity projection are independent of each other. We will assume that force fields do not act on particles. Let us find under these conditions what is the number of particles d n out of the total n has a speed in the range from υ to υ+Δυ. At the same time, we cannot say anything definite about the exact value of the speed of one or another particle υ i, since it is impossible to follow the collisions and movements of each of the molecules either in experiment or in theory. Such detailed information unlikely to be of practical value.

The velocity distribution of ideal gas molecules was first obtained by the famous English scientist J. Maxwell in 1860 using the methods of probability theory.


The derivation of the formula for the velocity distribution function of molecules is in the textbook by Yu.I Tyurin et al. (Part 1) or I.V. Saveliev (vol. 1). We will use the results of this derivation.

Speed ​​is a vector quantity. For speed projections on the x-axis (x th component of the velocity) from (2.2.1) we have

Then

(2.3.1)

Where A 1 is a constant equal to

The graphic representation of the function is shown in Figure 2.2. It can be seen that the fraction of molecules with velocity is not equal to zero. At , (in this physical meaning constant A1).

The above expression and graph are valid for distribution of gas molecules over x-velocity components. It is obvious that also y- and z-velocity components can also be obtained:

Where, or

(2.3.2)

Formula (2.3.2) can be given a geometric interpretation: d nxyz is the number of molecules in a parallelepiped with sides dυ x, dυ y, dυ z, that is, in a volume d V\u003d dυ x dυ y dυ z (Fig. 2.3), located at a distance from the origin in the space of velocities.

This value (d nxyz) cannot depend on the direction of the velocity vector. Therefore, it is necessary to obtain the distribution function of molecules in terms of velocities, regardless of their direction, that is, in terms of the absolute value of the velocity.

If we collect together all the molecules per unit volume, whose velocities are in the range from υ to υ + dυ in all directions, and release them, then in one second they will find themselves in a spherical layer with a thickness of dυ and a radius of υ (Fig. 2.4). This spherical layer consists of those parallelepipeds, which were mentioned above.

The total number of molecules in the layer, as follows from (2.3.2)

Where is the fraction of all particles in a spherical layer of volume d V, whose velocities lie in the range from υ to υ+dυ.

For dv = 1 we get probability density , or velocity distribution function of molecules:

(2.3.4)

This function denotes the fraction of molecules of a unit volume of gas whose absolute velocities are contained in a unit velocity interval including the given velocity.

Denote: then from (2.3.4) we get:

(2.3.5)

The graph of this function is shown in Figure 2.5.

Conclusions:

Let us consider the limits of applicability of the classical description of particle velocity distribution. To do this, we use the Heisenberg uncertainty relation. According to this relation, the coordinates and momentum of a particle cannot simultaneously have a definite value. The classical description is possible if the following conditions are met:

Here, is Planck's constant, a fundamental constant that determines the scale of quantum (microscopic) processes.

Thus, if the particle is in the volume , then in this case it is possible to describe its motion on the basis of the laws of classical mechanics.

Most probable, rms and arithmetic mean velocities of gas molecules

Let us consider how the number of particles per unit interval of velocities changes with the absolute value of the velocity at a unit concentration of particles.

Plot of the Maxwell distribution function

,

Shown in Figure 2.6.

It can be seen from the graph that for “small” υ, i.e. at , we have ; then reaches a maximum A and then decreases exponentially.

The value of the speed, which accounts for the maximum dependence , called the most likely speed.

Let's find this speed from the condition of equality of the derivative .

Root mean square speed find using the ratio: Average arithmetic speed:
. .

Where is the number of molecules with speeds from υ to υ+dυ. If put here f(υ) and calculate, we get: In this form

Moreover

Maxwell's velocity distribution law and all the ensuing consequences are valid only for a gas in an equilibrium system. The law is statistical, and the better, the greater the number of molecules.

The molecules of any gas are in perpetual chaotic motion. The velocities of molecules can take on the most various meanings. Molecules collide, as a result of collisions, the velocities of the molecules change. In every this moment time, the speed of each individual molecule is random both in magnitude and direction.

But, if the gas is left to itself, then the various rates of thermal motion are distributed among the molecules of a given mass of gas at a given temperature according to a well-defined law, i.e. there is a distribution of molecules over velocities.

The law of distribution of molecules by speed was theoretically derived by Maxwell. Maxwell's law is expressed by the following formula:

where is the number of molecules whose velocities lie in the interval ; is the total number of molecules of a given mass of gas; is the base of the natural logarithm; – set value of speed from the interval ; is the most probable velocity of gas molecules at a given temperature.

Most likely speed is the speed close to which the largest number of molecules of a given mass of gas has. The value depends on the gas temperature.

Formula (10.6) gives the number of molecules whose velocities lie in a given velocity interval, regardless of the direction of the velocities.

If we raise a more particular question, namely, what is the number of molecules in a gas whose velocity components lie in the interval between and , and , and , then

or , (10.8)

where is the kinetic energy of a gas molecule; is the mass of the molecule; is the Boltzmann constant; - absolute temperature gas. Formulas (10.7) and (10.8) are also Maxwell distribution formulas. The velocity distribution curve of molecules corresponding to the distribution law (10.6) is shown in fig. 10.1. The abscissa shows the velocity values ​​that a single gas molecule can take.

The maximum of the curve corresponds to the most probable speed. The curve is asymmetric with respect to , because A gas contains a relatively small number of molecules with very high velocities.

Consider some interval , (Fig. 10.1). If it is small, then the area of ​​the shaded strip is close to the area of ​​the rectangle:

those. the area of ​​the shaded strip is the number of molecules whose velocities lie in the interval , . And the area under the entire curve is proportional to the total number of molecules of a given mass of gas.

Let us find at what value the curve will have a maximum. We find the maximum according to the usual rules of mathematics, equating to zero the first derivative with respect to:

Since , then .

Taking the derivative, we obtain that , i.e. the maximum of the curve corresponds to the most probable speed .

Maxwell theoretically found formulas by which the arithmetic mean speed can also be calculated. We list the speeds that can characterize the thermal motion of gas molecules.

1. Most likely speed. (10.9)

2. RMS speed:

3. Average arithmetic speed. (10.11)

All velocities are directly proportional and inversely proportional to , where is the mass of a mole of gas.

On fig. 10.1 graph I is built for temperature , and graph II - for temperature . It can be seen that with increasing temperature, the maximum of the curve shifts to the right, since as the temperature increases, the speed of the molecules increases. There are more fast molecules, the right branch of the curve rises, there are fewer slow molecules, the left branch goes steeper. And the whole curve goes down, because the area under the curve must remain the same, because the total number of gas molecules remained the same and, of course, could not change when the gas was heated.

Maxwell's law is a statistical law, i.e. a law that is valid for a very large number of molecules.

In addition, Maxwell's law does not take into account the external effect on the gas, i.e. there are no force fields acting on the gas.

10.4. Ideal gas in external field.
barometric formula. Boltzmann distribution

Consider a vertical column of air near the surface of the Earth (Fig. 10.2). If the height of the column is relatively small (does not exceed a few hundred meters), the density of the gas and the number of molecules per unit volume (concentration) will be approximately the same. However, if the height of the column is of the order of a kilometer or more, the uniformity of the distribution of molecules along the height is violated. gravity, which tends to concentrate molecules near the Earth's surface. As a result, air density and Atmosphere pressure will decrease with distance from the Earth's surface.

Let's define the law of change of pressure with height (find the barometric formula).

barometric formula shows how atmospheric pressure changes P from height h above the surface of the earth. Let the pressure near the surface of the Earth at a height . The pressure is known. It is required to find the change in pressure with height.

In the derivation, we assume that the temperature of the gas remains constant. Let's single out a cylindrical column of gas (air) with a cross section above the Earth's surface. Consider a layer of gas of infinitely small thickness , located at a height from the base of the column.

The difference of forces acting on the upper and lower base of the layer is equal to the weight of the gas contained in this layer, i.e.

The infinitely small mass of gas in the layer is calculated by the formula

where is the volume of the gas layer.

Then , where is the gas density; is the acceleration of gravity.

Pressure difference on both bases of the layer:

And still need to put a minus sign

because the minus sign has a physical meaning. It shows that gas pressure decreases with height. If you rise to a height, then the pressure of the gas will decrease by an amount.

We find the gas density from the Mendeleev-Clapeyron equation.

We substitute the expression in (10.12), we have

This differential equation with shared variables:

We integrate:

Get the barometric formula

On fig. 10.3 shows plots of pressure versus height for two temperatures T 1 and T 2 (T 2 > T one). With a change in gas temperature, the pressure P 0 at the Earth's surface remains unchanged, because it is equal to the weight of the one above earth's surface vertical column of gas of unit base area and unlimited in height. The weight of a gas does not depend on temperature.

From the barometric formula it is very easy to obtain the Boltzmann distribution for the case when the external force on the gas is the force of gravity.

The gas pressure at a height is directly proportional to the number of molecules per unit volume at this height, , is the concentration of molecules at a height , and , is the concentration of gas molecules at a height of .

Either or. (10.14)

Formula (10.14) is called the Boltzmann distribution for molecules in the gravity field.

On fig. 10.4 shows plots of concentrations of molecules with height for two temperatures T 1 and T 2 (T 2 >T 1) in the field of gravity. Molecule concentration n 0 at the Earth's surface decreases with increasing temperature ( n 0 (T 2) < n 0 (T 1)) due to the redistribution of molecules inside the gas column. Molecules with greater kinetic energy rise higher.

If , is the potential energy of a molecule at a height , then

Formula (10.15) is valid not only for the case when the molecules move in the gravity field. This formula expressing the Boltzmann distribution is valid for any force field with potential function :

The Perrin Experience (1870–1942).
Definition of Avogadro's number

The French physicist Perrin used the Boltzmann distribution to experimentally determine the Avogadro number.

The microscope was aimed at the top layer of the emulsion (Fig. 10.5), done through a microscope instant photo, counted the number of Brownian particles in the photograph. Next, the microscope tube was lowered by 0.01 mm, photographed again, and the number of Brownian particles in the photograph was counted. It turned out that there are more Brownian particles on the bottom of the vessel, less on the surface of the emulsion, and in general, the distribution of Brownian particles along the height corresponds to the Boltzmann distribution. Since gumballs are in a liquid (emulsion), their potential energy, taking into account the buoyancy force of Archimedes, can be written , where m 0 is the mass of the ball, m g is the mass of the volume of liquid displaced by the ball. Then the Boltzmann distribution can be written as .

If n 1 and n 2 – measured concentrations of particles at heights h 1 and h 2 , then ; , a .

Then we can define and .

the value

where and are the densities of the material of the balls and the emulsion.

Having determined experimentally Boltzmann's constant k Perrin obtained the value of Avogadro's number from the dependence. Exact value:

Topic 11
WORK, INTERNAL ENERGY AND HEAT.
THE FIRST ORIGIN OF THERMODYNAMICS

Thermodynamics is a science that studies the conditions of transformation various kinds energy into heat and vice versa, as well as the quantitative relationships observed in this case. Thermodynamics covers a wide range of phenomena observed in nature and technology. Special meaning it has for heat engineering, because provides the basis for the development of thermal and refrigeration machines. In thermodynamics, the word is often used body. In thermodynamics, air, water, mercury, any gas can be called a body, i.e. any substance that occupies a certain volume.

A thermodynamic system may include several bodies, but it may consist of one body, very often this body is an ideal gas.

A thermodynamic system is any set of considered bodies that can exchange energy with each other and with other bodies. For example, a thermodynamic system can be an ideal gas.

The state of a thermodynamic system is characterized by thermodynamic parameters. Thermodynamic parameters are quantities characterizing the state of the system. Thermodynamic parameters include such quantities as pressure, volume, temperature, substance density, etc. The parameters of the state of an ideal gas, for example, are the pressure P, volume V, temperature T. The equation that relates the parameters of the state of a thermodynamic system is called equation of state. For example, the Mendeleev-Clapeyron equation: .

The state of a thermodynamic system is called equilibrium, if all its parameters have a certain value and do not change with time under constant external conditions.

If the thermodynamic system is taken out of equilibrium and left to itself, then it returns to its original state. This process is called relaxation.

In thermodynamics, the regularities of various transitions of a system from one state to another are studied. Transition of a system from one state to another,which is accompanied by a change in at least one state parameter,called a process. The equation that determines the change in system parameters during the transition from one state to another is called the process equation.

Thermodynamics studies only the thermodynamically equilibrium states of bodies and slow processes, which are considered as equilibrium states that continuously follow each other. She learns general patterns transition of systems to the state of thermodynamic equilibrium.

Equilibrium processes– processes in which the rate of change of thermodynamic parameters is infinitely small, i.e. the change in thermodynamic parameters occurs over infinitely long times. This model, because All real processes are non-equilibrium.

An equilibrium process is a process that goes through a sequence of equilibrium states.

Non-equilibrium process– a process in which a change in thermodynamic parameters by a finite amount occurs in a finite time.

A non-equilibrium process cannot be graphically depicted.

In thermodynamics, a special method is used to study phenomena - thermodynamic method. Thermodynamics considers how a process proceeds.

Thermodynamics is based on two basic laws, which are a generalization of a vast amount of factual material. These laws gave rise to the whole science of thermodynamics and therefore were called the beginnings.

11.1. Internal energy of an ideal gas.
Number of degrees of freedom

Number of degrees of freedom called smallest number independent coordinates, which must be entered to determine the position of the body in space. is the number of degrees of freedom.

Consider monatomic gas. A molecule of such a gas can be considered a material point, the position of a material point (Fig. 11.1) in space is determined by three coordinates.

A molecule can move in three directions (Fig. 11.2).

Therefore, it has three translational degrees of freedom.

A molecule is a material point.

The energy of rotational motion, tk. the moment of inertia of a material point about the axis passing through the point is equal to zero

For a monatomic gas molecule, the number of degrees of freedom is .

Consider diatomic gas. In a diatomic molecule, each atom is taken as a material point and it is believed that the atoms are rigidly connected to each other, this is a dumbbell model of a diatomic molecule. Diatomic rigidly bound molecule(a set of two material points connected by a non-deformable bond), fig. 11.3.

The position of the center of mass of the molecule is given by three coordinates, (Fig. 11.4) these are three degrees of freedom, they determine forward movement molecules. But the molecule can also perform rotational movements around the axes and , these are two more degrees of freedom that determine rotation of the molecule. The rotation of the molecule around the axis is impossible, because material points cannot rotate around an axis passing through these points.

For a diatomic gas molecule, the number of degrees of freedom is .

Consider triatomic gas. The model of a molecule is three atoms (material points) rigidly connected to each other (Fig. 11.5).

A triatomic molecule is a rigidly bound molecule.

For a triatomic gas molecule, the number of degrees of freedom is .

For a polyatomic molecule, the number of degrees of freedom is .

For real molecules that do not have rigid bonds between atoms, it is also necessary to take into account the degrees of freedom of vibrational motion, then the number of degrees of freedom of a real molecule is

i= i act + i rotate + i fluctuations (11.1)

The law of uniform distribution of energy
by degrees of freedom (Boltzmann's law)

The law on the equipartition of energy over degrees of freedom states that if a system of particles is in a state of thermodynamic equilibrium, then the average kinetic energy of the chaotic movement of molecules per 1 degree of freedom translational and rotational movement is equal to

Therefore, a molecule having degrees of freedom has an energy

is the number of moles, where is the mass of the mole, and the internal energy of the gas is expressed by the formula

The internal energy of an ideal gas depends only on the temperature of the gas. The change in the internal energy of an ideal gas is determined by a change in temperature and does not depend on the process in which this change occurred.

Change in the internal energy of an ideal gas

where is the change in temperature.

The law of uniform distribution of energy applies to the oscillatory motion of atoms in a molecule. The vibrational degree of freedom accounts for not only kinetic energy, but also potential energy, and the average value of the kinetic energy per one degree of freedom is equal to the average value of the potential energy per one degree of freedom and is equal to

Therefore, if a molecule has the number of degrees of freedom
i= i act + i rotate + i vibrations, then the average total energy of the molecule: , and the internal energy of the mass gas:

11.2. Elemental work. Work of an ideal gas
with isoprocesses

If external forces do work on the system, then the work is negative.

Consider an ideal gas under a piston in a cylinder (Figure 11.6). The gas expands and the piston rises to an infinitesimal height. The force acting from the side of the gas on the piston is found by the formula

V P ( V) and straight lines passing through the ends of the segment parallel to the y-axis.

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, some statistical distribution molecules in speed. 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 higher 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

From (2) we get

Function f(υ ) is called the distribution function. Its physical meaning follows from formula (3)

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

From (5) it follows that

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:

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:

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

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. For various systems he is different. 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. Precise definition randomness of thermal motions is the following: 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 fast the speed strong wind, she won't make him "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:

then the last equality is satisfied when:

Equation (10) is satisfied when:

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:

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

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) 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)

Substituting into (14) the value f(υ ) from (8) we obtain the arithmetic average speed:

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:

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 passing through a thread electric current she warmed up 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 rotate at a constant speed, the position of the strip, formed by atoms on the outer cylinder, was displaced by some 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 to the necessary direction. 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.

When gas molecules collide, they change their velocities. The change in the speed of molecules occurs randomly. It is impossible to predict in advance what numerical speed a given molecule will have: this speed is random.

The distribution of molecules by velocity modules is described using the distribution function f(v):

where the ratio is equal to the fraction of molecules whose velocities lie in the range from v before v+dv. dv- interval width (Fig. 2).

Rice. 2. Velocity interval

Knowing the view f(v), you can find the number of molecules Δ N V of these molecules N, whose velocities fall within the interval of velocities from v before v + Δv. Attitude

gives the probability that the speed of a molecule will have a value within a given range of speeds dv.

Function f(v) must satisfy the normalization condition, that is, the condition must be satisfied:

The left side of expression (17.3) gives the probability that the molecule has a speed in the range from 0 to ∞. Since the speed of the molecule necessarily has some value, then the indicated probability is the probability of a certain event and, therefore, is equal to 1.

The distribution function was found theoretically by Maxwell. It looks like this:

where t 0 - the mass of the molecule.

Expression (16) is called the Maxwell distribution function.

From (16) it follows that the type of distribution of molecules in terms of velocities depends on the nature of the gas (the mass of the molecule) and temperature T. Pressure and volume do not affect the distribution of molecules over velocities.

Fig.3. Plot of the Maxwell distribution function

A schematic plot of the Maxwell distribution function is given in fig. 3. Let's analyze the chart.

1. At speeds tending to zero (v->0) and to infinity (v -> ∞) the distribution function also tends to zero. This means that very large and very small molecular velocities are unlikely.

2. Speed v B , corresponding to the maximum of the distribution function will be the most probable. This means that the main part of the molecules has velocities close to the probable one.

You can get a formula for calculating the most probable speed:

where k — Boltzmann's constant; t 0 - the mass of the molecule.

3. In accordance with the normalization condition (15), the area bounded by the curve f(v) and the abscissa axis is equal to one.

4. The distribution curve is asymmetric. This means that the proportion of molecules with velocities greater than the most probable more share molecules with velocities less than the most probable.

5. The shape of the curve depends on the temperature and nature of the gas. On fig. 4 shows the distribution function for the same gas at different temperatures. When heated, the maximum of the curve decreases and shifts to the right, since the proportion of "fast" molecules increases, and the proportion of "slow" molecules decreases. The area under both curves remains constant and equal to unity.


The law of distribution of molecules by velocities established by Maxwell and the consequences following from it are valid only for a gas in an equilibrium state. Maxwell's law is statistical, it can only be applied to a large number of particles

Rice. 4. Maxwell distributions at different temperatures

Using the Maxwell distribution function f(v), one can find a number of average values ​​characterizing the state of molecules.

Average arithmetic speed - the sum of the velocities of all molecules divided by the number of molecules:

Root mean square speed, which determines the average kinetic energy of molecules (see formula (10)), by definition is equal to

<v HF> = (19)

Calculation using the Maxwell distribution gives the following formulas:

<v HF> = (21)

Considering that the mass of one molecule is , where μ is the molar mass; N A - Avogadro's number, as well as the fact that kN A = R, then the expressions for the most probable, arithmetic mean, and mean square velocities can be rewritten as follows:

= ; (22)

<v HF> = . (24)

Comparing (22), (23) and (24), we can see that v B ,<v HF> equally depend on the gas temperature and molar mass, differing only in the factor. Their relationship looks like this:

<v B> : <v> : <v HF> = 1: 1,13: 1,22.

Maxwell distribution (distribution of gas moleculesby speed). In an equilibrium state, the gas parameters (pressure, volume and temperature) remain unchanged, but the microstates - the mutual arrangement of molecules, their velocities - are constantly changing. Because of huge amount molecules, it is practically impossible to determine the values ​​of their velocities at any moment, but it is possible, considering the speed of molecules as a continuous random variable, to indicate the distribution of molecules over velocities.

Let's isolate a single molecule. The randomness of the movement allows, for example, for the projection of speed x molecules take a normal distribution law. In this case, as shown by J.K. Maxwell, the probability density is written as follows:

where T 0 is the mass of the molecule, T is the thermodynamic temperature of the gas, k is the Boltzmann constant.

Similar expressions can be obtained for f( at ) and f( z ).

Based on formula (2.15), we can write down the probability that the molecule has a velocity projection lying in the interval from x before x + d X :

similar for other axes

Each of the conditions (2.29) and (2.30) reflects an independent event. Therefore, the probability that a molecule has a velocity whose projections simultaneously satisfy all conditions can be found by the probability multiplication theorem [see. (2.6)]:

Using (2.28), from (2.31) we obtain:

Note that from (2.32) one can obtain the Maxwellian probability distribution function of the absolute values ​​of the velocity (Maxwell velocity distribution):

and the probability that the velocity of the molecule has a value between before + d:

The graph of the function (2.33) is shown in Figure 2.5. Speed,corresponding to the maximum of the Maxwell curve is calledmost likely v. It can be determined using the maximum condition of the function:

The average speed of a molecule (mathematical expectation) can be found by the general rule [see. (2.20)]. Since the average value of the speed is determined, the integration limits are taken from 0 to  (mathematical details are omitted):

where M=t 0 N A is the molar mass of the gas, R = k N A is the universal gas constant, N A is Avogadro's number.

As the temperature increases, the maximum of the Maxwell curve shifts towards higher velocities and the distribution of molecules along is modified (Fig. 2.6; T 1 < Т 2 ). The Maxwell distribution allows you to calculate the number of molecules whose velocities lie in a certain interval . We get the corresponding formula.

Since the total number N molecules in a gas is usually large, then the probability d P can be expressed as the ratio of the number d N molecules whose velocities are contained in a certain interval d, to the total number N molecules:

From (2.34) and (2.37) it follows that

Formula (2.38) allows you to determine the number of molecules whose velocities lie in the range from i: to i> 2. To do this, we need to integrate (2.38):

or graphically calculate the area of ​​a curvilinear trapezoid ranging from 1 before 2 (Fig. 2.7).

If the speed interval d is sufficiently small, then the number of molecules whose velocities correspond to this interval can be calculated approximately using formula (2.38) or graphically as the area of ​​a rectangle with a base d.

To the question how many molecules have a speed equal to any particular value, a strange, at first glance, answer follows: if the speed is absolutely exactly given, then the speed interval is zero (d = 0) and from (2.38) we obtain zero, i.e., not a single molecule has a speed exactly equal to the predetermined one. This corresponds to one of the provisions of the theory of probability: for a continuous random variable, which is the speed, it is impossible to "guess" exactly its value, which has at least one molecule in the gas.

The velocity distribution of molecules has been confirmed by various experiments.

The Maxwell distribution can be considered as the distribution of molecules not only in terms of velocities, but also in terms of kinetic energies (since these concepts are interrelated).

Boltzmann distribution. If the molecules are in some external force field, for example, the gravitational field of the Earth, then it is possible to find the distribution over their potential energies, i.e., to establish the concentration of particles that have some specific value of potential energy.

Distribution of particles over potential energies in sifishing fields-gravitational, electrical, etc.-is called the Boltzmann distribution.

As applied to the gravitational field, this distribution can be written as a concentration dependence P molecules from height h above the ground level or from the potential energy of the molecule mgh:

Expression (2.40) is valid for ideal gas particles. Graphically, this exponential dependence is shown in fig. 2.8.

Such a distribution of molecules in the Earth's gravitational field can be qualitatively, within the framework of molecular-kinetic concepts, explained by the fact that molecules are influenced by two opposite factors: the gravitational field, under the influence of which all molecules are attracted to the Earth, and molecular-chaotic motion, which tends to uniformly scatter the molecules over the full extent possible.

In conclusion, it is useful to note some similarities between the exponential terms in the Maxwell and Boltzmann distributions:

In the first distribution, in the exponent, the ratio of the kinetic energy of the molecule to kT, in the second - the ratio of potential energy to kT.