• In a large number of cases, knowing the average values ​​of physical quantities alone is not enough. For example, knowing the average height of people does not allow planning the production of clothes of various sizes. You need to know the approximate number of people whose height lies in a certain interval.

    Similarly, it is important to know the numbers of molecules that have velocities other than the average. Maxwell was the first to find how these numbers can be determined.

Probability of a random event

In § 4.1 we have already mentioned that J. Maxwell introduced the concept of probability to describe the behavior of a large set of molecules.

As repeatedly emphasized, in principle it is impossible to follow the change in the speed (or momentum) of one molecule over a long time interval. It is also impossible to accurately determine the velocities of all gas molecules in this moment time. From the macroscopic conditions in which the gas is located (a certain volume and temperature), certain values ​​of the velocities of the molecules do not necessarily follow. The speed of a molecule can be considered as a random variable, which under given macroscopic conditions can take various meanings, just as when throwing a dice, any number of points from 1 to 6 (the number of faces of the die is six) can fall out. It is impossible to predict what number of points will fall out in a given throw of the die. But the probability of rolling, say, five points is defensible.

What is the probability of a random event occurring? Let a very large number N of trials be performed (N is the number of throws of the die). At the same time, in N "cases, a favorable outcome of the trials took place (i.e., the loss of five). Then the probability of this event is equal to the ratio of the number of cases with a favorable outcome to the total number of trials, provided that this number is arbitrarily large:

For a symmetric die, the probability of any chosen number of points from 1 to 6 is .

We see that against the background of many random events, a certain quantitative pattern is revealed, a number appears. This number - the probability - allows you to calculate averages. So, if you make 300 throws of a dice, then the average number of fives, as follows from formula (4.6.1), will be equal to 300 = 50, and it is completely indifferent to throw the same dice 300 times or simultaneously 300 identical dice.

Undoubtedly, the behavior of gas molecules in a vessel is much more complicated than the movement of a thrown dice. But even here one can hope to discover certain quantitative regularities that make it possible to calculate statistical averages, if only the problem is posed in the same way as in game theory, and not as in classical mechanics. We must abandon the unsolvable problem of determining the exact value of the speed of the molecule at a given moment and try to find the probability that the speed has a certain value.

Velocity distribution of molecules - Maxwell distribution

Maxwell assumed that in gases in a state of thermal equilibrium there is a certain distribution of velocities that does not change over time, in other words, the number of molecules with velocities in a given range of values ​​remains constant. And Maxwell found this distribution.

But Maxwell's main merit was not so much in the solution of this problem as in the very formulation of the new problem. He clearly realized that the random behavior of individual molecules under given macroscopic conditions is subject to a certain probabilistic, or statistical, law. This statistical law for the velocity distribution of gas molecules turned out to be relatively simple.

The velocity distribution of molecules can be visualized as follows. Let's choose rectangular system reference, on the axes of which we will plot the projections v x , v y , v z of particle velocities. As a result, a three-dimensional “velocity space” will be obtained, each point of which corresponds to a molecule with a strictly specified speed v, equal in absolute value to the length of the radius vector drawn from the origin of the reference system to this point (Fig. 4.7).

Rice. 4.7

A general idea of ​​the distribution of molecules over velocities will be obtained if the speed of each of the N molecules is represented by a point in this space of velocities (Fig. 4.8). The points will turn out to be located rather chaotically, but on average the density of points will decrease with distance from the origin (not all velocities of molecules are equally common).

Rice. 4.8

The picture of the distribution of points, of course, is not frozen. With the passage of time, the velocities of molecules change due to collisions and, consequently, the pattern of the distribution of points in the space of velocities changes. However, its change is such that the average density of points in any region of the velocity space will not change with time, it remains the same. This is what the existence of a certain statistical law means. The average density corresponds to the most probable velocity distribution.

The number of points AN in some small volume Δv x Δv y Δv z of the velocity space is obviously equal to this volume multiplied by the density of points inside it. (Similarly, the mass Δm of a certain volume ΔV is equal to the product of the density of matter ρ and this volume: Δm = ρΔV.) Denote by Nf(v x , v y , v z) the average density of points in the velocity space, i.e., the number of points per unit volume of space speeds (N is the total number of gas molecules). Then

In fact, ΔN is the number of molecules whose velocity projections lie in the ranges from v x to v x + Δv x , from v y to v y + Δv y and from v z to v z + Δv z (velocity radius vectors of these molecules end inside the velocity space volume Δv \u003d Δv x Δv y Δv z, having the shape of a cube (see Fig. 4.8).

The probability that the velocity projections of a molecule lie in a given velocity interval is equal to the ratio of the number of molecules with a given velocity value to the total number of molecules:

The function f(v x, v y, v z) is called the velocity distribution function of molecules and represents the probability density, i.e., the probability per unit volume of the velocity space.

In principle, the velocities of molecules at a given moment of time can turn out to be any. But the probability of different velocity distributions is not the same. Among all possible instantaneous distributions, there is one, the probability of which is greater than all others - the most probable distribution. Maxwell established that the distribution function f(v x , v y , v z), giving this most probable distribution of molecular velocities (Maxwell distribution), is determined by the ratio of the kinetic energy of the molecule

to the average energy of its thermal motion kT (k is the Boltzmann constant). This distribution has the form

Here, e ≈ 2.718 is the base of natural logarithms, and the value of A does not depend on speed.

Thus, according to Maxwell, the density of points representing molecules in the space of velocities is maximum near the origin (v = 0) and decreases with increasing v, and the faster, the lower the thermal motion energy kT. Figure 4.9 shows the dependence of the distribution function f on the projection v x, provided that the projections v y and v z are any. The distribution function has a characteristic bell-shaped shape, which is often found in statistical theories and is called the Gaussian curve.

Rice. 4.9

The constant A is found from the condition that the probability for the velocity of a molecule to have any value from zero to infinity must be equal to one. This condition is called the normalization condition. (Similarly, the probability of getting any number of points from 1 to 6 on a given throw of a die is equal to one.) The total probability is obtained by adding the probabilities of all possible mutually exclusive realizations of a random event.

Summing up the probabilities ΔW i of all possible values ​​of speed i , we obtain the equation

Having calculated the normalization constant A using equation (4.6.5), we can write the expression for the average number of particles with velocities in a given interval in the following form:

The speed of any molecule at a given time is random value. Therefore, the very distribution of molecules in terms of velocities at a given moment of time is random. But the average distribution, determined by the statistical law, is realized with necessity in certain macroscopic conditions and does not change with time. However, there are always deviations from the average - fluctuations. These deviations are equally likely to occur in either direction. That is why, on average, there is a certain distribution of molecules over velocities.

The distribution of molecules by Maxwell's velocities turns out to be valid not only for gases, but also for liquids and solids. Only in the case when it is impossible to apply to the description of the motion of particles classical mechanics, the Maxwell distribution ceases to be true.

Distribution of molecular velocity moduli

Let us find the average number of molecules whose absolute velocities lie in the interval from v to v + Δv.

The Maxwell distribution (4.6.4) determines the number of molecules whose velocity projections lie in the ranges from v x to v x + Δv x , from v y to v y + Δv y , from v z to v z + Δv z . The vectors of these velocities terminate inside the volume Δv x Δu y Δv z (see Fig. 4.8). Thus, the average number of molecules having a certain module and a certain direction of velocities is set, given by the position of the volume Δv x Δu y Δv z in the space of velocities.

All molecules whose velocity modules lie in the range from v to and + Δv are located in the velocity space inside a spherical layer with radius v and thickness Δv (Fig. 4.10). The volume of the spherical layer is equal to the product of the surface area of ​​the layer and its thickness: 4πv 2 Δv.

Rice. 4.10

The number of molecules inside this layer and, therefore, having given values ​​of the velocity modulus in the range from v to v + Δv, can be found from formula (4.6.2), if we replace the volume Δv x Δu y Δv z by the volume 4πv 2 Δv .

Thus, the desired average number of molecules is equal to

Since the probability of a certain value of the modulus of the velocity of a molecule is equal to the ratio , then for the probability density we obtain

A graph expressing the dependence of this function on speed is shown in Figure 4.11. We see that the function f(v) has a maximum not at zero, as the probability density f(v x , v y , v z). The reason for this is as follows. The density of points representing molecules in the velocity space will still be the highest near v = 0, but due to the growth of the volumes of spherical layers with an increase in the velocity modules (~ v 2), the function f(v) increases. In this case, the number of points inside the spherical layer grows faster than the function f(v x , v y , v z) decreases due to a decrease in the density of points.

Rice. 4.11

Can you explain what was said good example. Suppose an ordinary target with concentric circles is fired upon by a fairly accurate shooter. Bullet impacts are concentrated around the center of the target. The density of hits - the number of hits per unit area - will be maximum near the center of the target. We divide the target into separate narrow strips of width Δx (Fig. 4.12, a). Then the ratio of the number of hits on this strip to its width will be maximum near the center of the target.

Rice. 4.12

The dependence of the ratio of the number of hits in a given strip to its width has the form shown in Figure 4.12, b. Here again, a Gaussian curve is obtained, as for the distribution of f(v x) over the projections of velocities (see Fig. 4.9).

But a completely different result will be obtained if we count the number of hits in different target rings. In this case, the ratio of the number of hits in the ring with radius r to its width will be graphically characterized by the curve shown in Figure 4.12, c. Although the density of hits decreases with distance from the target center, the areas of the rings increase in proportion to r, which leads to a shift of the maximum of the curve from zero.

Most probable molecular speed

Knowing the formula (4.6.8) for the probability density of the molecular velocity moduli, one can find the value of the velocity corresponding to the maximum density of this probability (1). The speed (it is called the most probable) turns out to be equal to

Most molecules have velocities close to the most probable (see Fig. 4.11).

As the absolute temperature T increases, the most probable speed increases and, at the same time, the dependence curve To) becomes more and more smoothed (Fig. 4.13).

Rice. 4.13

The role of fast molecules

At any temperature, there is a certain number of molecules whose velocities, and hence the kinetic energies, are noticeably higher than the average.

It is known that many chemical reactions, for example, the combustion of conventional fuels (wood, coal, etc.) starts only at a certain, sufficient high temperature. The energy required to start the process of fuel oxidation, i.e. combustion (it is called the activation energy), is of the order of 10 -19 J. And at a temperature of 293 K (room temperature), the average kinetic energy of the thermal motion of molecules is approximately 5 10 -21 J Therefore, combustion does not occur. However, a temperature increase of only 2 times (up to 586 K) causes ignition. In this case, the average energy of molecules also increases by a factor of 2, but the number of molecules whose kinetic energy exceeds 10 -19 J increases by a factor of 108. This follows from the Maxwell distribution. Therefore, at a temperature of 293 K, you feel comfortable reading a book, and at 586 K, the book starts to burn.

The evaporation of a liquid is also determined by the fast molecules of the right tail of the Maxwellian distribution. The binding energy of water molecules at room temperature is much greater than kT. Nevertheless, evaporation occurs due to a small number of fast molecules whose kinetic energy exceeds kT.

Maxwell discovered new type physical law- statistical - and found the distribution of molecules by speed. He clearly understood the significance of his discovery. In a report to the Cambridge Philosophical Society, Maxwell said: "I believe that the most importance to develop our methods of thinking molecular theories have because they force a distinction to be made between two methods of cognition, which we may call dynamic and statistical.

(1) This is done according to the rules for finding the maximum of a known function. We need to calculate the derivative of this function with respect to speed and equate it to zero.

When gas molecules collide, they change their velocities. A 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

(14)

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:

(15)

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:

(16)

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 graph.

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:

(17)

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 velocity distribution of molecules established by Maxwell and the consequences arising 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:

. (18)

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

<v HF> = (19)

Distribution density function

Maxwell distribution is a probability distribution encountered in physics and chemistry. It underlies the kinetic theory of gases, which explains many of the fundamental properties of gases, including pressure and diffusion. The Maxwell distribution is also applicable to electronic transport processes and other phenomena. The Maxwell distribution applies to a variety of properties of individual molecules in a gas. It is usually thought of as the energy distribution of molecules in a gas, but it can also be applied to the distribution of velocities, momenta, and momentum modulus of molecules. It can also be expressed as a discrete distribution over a set of discrete energy levels, or as a continuous distribution over some energy continuum.

The Maxwell distribution can be obtained using statistical mechanics (see the origin of the partition function). As an energy distribution, it corresponds to the most probable energy distribution, in a collisionally dominated system consisting of a large number non-interacting particles, in which quantum effects are insignificant. Since the interaction between molecules in a gas is usually quite small, the Maxwell distribution gives a fairly good approximation of the situation that exists in a gas.

In many other cases, however, the condition of dominance of elastic collisions over all other processes is not even approximately satisfied. This is true, for example, in the physics of the ionosphere and space plasma, where the processes of recombination and collisional excitation (that is, radiative processes) have great importance, especially for electrons. The assumption of the applicability of the Maxwell distribution would in this case give not only quantitatively incorrect results, but would even prevent a correct understanding of the physics of processes on quality level. Also, in the case where the quantum de Broglie wavelength of the gas particles is not small compared to the distance between the particles, there will be deviations from the Maxwell distribution due to quantum effects.

The Maxwell energy distribution can be expressed as a discrete energy distribution:

,

where is the number of molecules having energy at the temperature of the system, is the total number of molecules in the system, and is Boltzmann's constant. (Note that sometimes the above equation is written with a factor denoting the degree of degeneracy energy levels. In this case, the sum will be over all energies, and not over all states of the system). Because speed is related to energy, Equation (1) can be used to derive the relationship between temperature and the velocities of molecules in a gas. The denominator in equation (1) is known as the canonical partition function.

Maxwell distribution

Momentum vector distribution

What follows is very different from the derivation proposed by James Clerk Maxwell and later described with less speculation by Ludwig Boltzmann.

When ideal gas, consisting of non-interacting atoms in the ground state, all the energy is in the form of kinetic energy. The kinetic energy is related to the momentum of the particle as follows

,

where is the square of the momentum vector.

We can therefore rewrite equation (1) as:

,

where is the partition function corresponding to the denominator in equation (1), is the molecular weight of the gas, is the thermodynamic temperature, and is the Boltzmann constant . This distribution is proportional to the probability density function of the molecule being in a state with these values ​​of the momentum components. Thus:

Normalization constant C, is determined from the condition according to which the probability that the molecules have any in general the momentum must be equal to one. Therefore, the integral of equation (4) over all values ​​of and must be equal to unity. It can be shown that:

.

Thus, for the integral in equation (4) to have a value of 1, it is necessary that

.

Substituting expression (6) into equation (4) and using the fact that , we get

.

Velocity vector distribution

Taking into account that the density of the velocity distribution is proportional to the momentum distribution density:

and using we get:

,

which is the Maxwell velocity distribution. The probability of finding a particle in an infinitesimal element near the velocity is

Distribution over the absolute value of the momentum

By integrating, we can find the distribution over the absolute magnitude of the momentum

Energy distribution

Finally, using the relations and , we obtain the kinetic energy distribution:

Speed ​​projection distribution

The Maxwell distribution for the velocity vector is the product of the distributions for each of the three directions:

,

where distribution in one direction:

This distribution has the form of a normal distribution. As one would expect for a gas at rest, the average velocity in any direction is zero.

Distribution modulo speeds

Usually, the distribution over the absolute value is more interesting than over the projections of the velocities of the molecules. speed module, v defined as:

so the velocity modulus will always be greater than or equal to zero. Since everything is normally distributed, there will be a chi-squared distribution with three degrees of freedom. If is a probability density function for the velocity modulus, then:

,

thus, the probability density function for the modulus of velocity is

Characteristic speed

Although Equation (11) gives the distribution of velocities, or in other words the proportion of molecules having a specific velocity, other quantities such as average particle velocities are often more interesting. In the following subsections, we define and obtain most likely speed, average speed And RMS speed.

Most likely speed

most likely speed, - the probability of possession of which by any molecule of the system is maximum, and which corresponds to the maximum value of . To find it, you need to calculate , equate it to zero and solve for :

average speed

RMS speed

Substituting and integrating, we get

Derivation of the Maxwell distribution

We now obtain the distribution formula in the same way as James Clerk Maxwell himself did.
Consider the space of velocity points (we represent each molecule as a point in the coordinate system ) in the stationary state of the gas. We choose an infinitesimal volume element . Since the gas is stationary, the number of velocity points in remains unchanged over time. The velocity space is isotropic, so the probability density functions for all directions are the same.

Maxwell suggested that the distributions of velocities in directions are statistically independent, that is, the velocity component of a molecule does not depend on and components.

- in fact, the probability of finding a high-speed point in the volume .

The right-hand side does not depend on and, hence the left-hand side does not depend on and either. However, and are equal, hence the left-hand side does not depend on . So this expression can only be equal to some constant.

Now you need to take a fundamental step - enter the temperature. Kinetic definition of temperature (as a measure of the average kinetic energy of the movement of molecules).

The molecules of any gas are in perpetual chaotic motion. The velocities of molecules can take on a variety of values. Molecules collide, as a result of collisions, the velocities of the molecules change. At any given moment in 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 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:

; . (10.10)

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 1). 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 . (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 gummigut balls 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 .

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

Then one 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 got out of addiction value of Avogadro's number. Exact value:

(10.17)

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 .

Statistical distributions

During thermal motion, the positions of particles, the magnitude and direction of their velocities change randomly. Due to the gigantic number of particles, the random nature of their motion manifests itself in the existence of certain statistical patterns in the distribution of particles of the system along coordinates, velocities, etc. Such distributions are characterized by corresponding distribution functions. The distribution function (probability density) characterizes the distribution of particles with respect to the corresponding variable (coordinates, velocities, etc.). Classical statistics are based on the following principles:

All particles of the classical system are distinguishable (i.e., they can be numbered and follow each particle);

All dynamic variables characterizing the state of the particle change continuously;

An unlimited number of particles can be in a given state.

In a state of thermal equilibrium, as if the velocities of molecules did not change during collisions, the root-mean-square velocity of molecules in a gas, at T=const, remains constant and equal to


This is explained by the fact that a certain stationary state is established in the gas. statistical distribution molecules according to their velocities, called the Maxwell distribution. The Maxwell distribution is described by some function f(u), called the velocity distribution function of molecules.

,

where N is the total number of molecules, dN(u) is the number of molecules whose velocities belong to the velocity interval from u to u + du.

Thus, the Maxwell function f(u) is equal to the probability that the velocity of a randomly selected molecule belongs to the unit velocity interval near the value of u. Or it is equal to the fraction of molecules whose velocities belong to the unit interval of velocities near the value of u.

The explicit form of the function f(u) was obtained theoretically by Maxwell:

.

The distribution function graph is shown in fig. 12. It follows from the graph that the distribution function tends to zero at u®0 and u®¥ and passes through a maximum at a certain speed u B, called most likely speed. This speed and close to it has the greatest number of molecules. The curve is asymmetric with respect to u B. The value of the most probable speed can be found using the condition for the maximum of the function f(u).

.

On fig. 13 shows the shift u B with temperature change, while the area under the graph remains constant and equal to 1, which follows from normalization conditions Maxwell functions

The normalization condition follows from the meaning of this integral - it determines the probability that the molecular velocity falls within the velocity range from 0 to ¥. This is a certain event, its probability, by definition, is taken equal to 1.