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 in. It can be determined using the maximum condition of the function:

or

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 of 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.

Maxwell and Boltzmann distributions. Transfer phenomena

Lecture plan:

Maxwell's law on the distribution of molecules over velocities. Characteristic velocities of molecules.

Boltzmann distribution.

Mean free path of molecules.

Transfer phenomena:

a) diffusion;

b). internal friction (viscosity);

c) thermal conductivity.

Maxwell's law on the distribution of molecules over velocities. Characteristic velocities of molecules.

Gas molecules move randomly and, as a result of collisions, their velocities change in magnitude and direction In a gas, there are molecules with both very high and very low velocities. One can raise the question of the number of molecules whose velocities lie in the range of and for a gas in a state of thermodynamic equilibrium in the absence of external force fields. In this case, some stationary velocity distribution of molecules does not change with time, which obeys the statistical law theoretically derived by Maxwell.

The greater the total number of molecules N, the greater the number of molecules N will have velocities in the range from and; the larger the interval of velocities , the greater the number of molecules will have the value of velocities in the specified interval.

We introduce the coefficient of proportionality f( .

, 

where f( is called the distribution function, which depends on the speed of molecules and characterizes the distribution of molecules over speeds.

If the form of the function is known, one can find the number of molecules whose velocities lie in the interval from to.

Using the methods of probability theory and the laws of statistics, Maxwell in 1860. theoretically obtained a formula that determines the number of molecules with velocities in the range from to.

, (2)

- The Maxwell distribution shows what proportion of the total number of molecules of a given gas has velocities in the range from to.

From the equations  and  follows the form of the function 

- (3)

velocity distribution function of ideal gas molecules.

From (3) it can be seen that the specific form of the function depends on the type of gas (on the mass of the molecule m 0 ) and temperature.

Most often, the law of distribution of molecules by velocities is written in the form:

The graph of the function is asymmetric (Fig. 1). The position of the maximum characterizes the most frequently encountered speed, which is called the most probable. Speeds in excess of  in, are more common than lower speeds.

is the fraction of the total number of molecules with velocities in this interval.

S total = 1.

With an increase in temperature, the maximum of the distribution shifts towards higher velocities, and the curve becomes flatter, but the area under the curve does not change, because S total = 1 .

The most probable speed is the one close to which the speeds of most of the molecules of a given gas turn out to be.

To determine it, we explore to the maximum.

4,

Previously, it was shown that

, ,

 .

In MKT, the concept of the arithmetic mean velocity of the translational motion of molecules of an ideal gas is also used.

- is equal to the ratio of the sum of the moduli of velocities of all molecules to

the number of molecules.

.

It can be seen from the comparison (Fig. 2) that the smallest is  in .

Boltzmann distribution.

Two factors - the thermal motion of molecules and the presence of the Earth's gravitational field bring the gas into a state in which its concentration and pressure decrease with height.

If there were no thermal motion of atmospheric air molecules, then all of them would be concentrated at the surface of the Earth. If there were no gravity, then the particles of the atmosphere would be scattered throughout the universe. Let's find the law of change of pressure with height.

The pressure of the gas column is determined by the formula.

Since the pressure decreases with increasing altitude,

where  gas density at altitude h.

Let's find p from the Mendeleev-Clapeyron equation

or.

Let us carry out the calculation for an isothermal atmosphere, assuming that T=const(does not depend on height).

.

at h=0 , , ,

, , ,

The barometric formula determines the gas pressure at any altitude.

We obtain an expression for the concentration of molecules at any height.

where is the potential energy of the molecule at a height h.

Boltzmann distribution in an external potential field.

Consequently, the distribution of molecules in height is their distribution in energy. Boltzmann proved that this distribution is valid not only in the case of the potential field of terrestrial gravitational forces, but also in any potential field of forces for a collection of any identical particles in a state of chaotic thermal motion.

It follows from the Boltzmann distribution that molecules are located with a higher concentration where their potential energy is lower.

Boltzmann distribution - distribution of particles in a potential force field.

Mean free path of molecules.

Due to the chaotic thermal motion of the gas molecules continuously collide with each other, go through a complex zigzag path. Between 2 collisions, the molecules move uniformly in a straight line.

M The minimum distance at which the centers of 2 molecules approach each other during a collision is called the effective diameter of the molecule. d(Fig. 4).

The quantity is called the effective cross section of the molecule.

Let us find the average number of collisions of a homogeneous gas molecule per unit time. A collision will occur if the centers of the molecules approach at a distance less than or equal to d. We assume that the molecule moves with speed , and the rest of the molecules are at rest. Then the number of collisions is determined by the number of molecules whose centers are located in a volume that is a cylinder with a base and a height equal to the path traveled by the molecule in 1s, i.e. .

AT In reality, all molecules move, and the possibility of a collision of 2 molecules determines their relative speed. It can be shown that if the Maxwell distribution is adopted for the velocities of molecules, .

.

For most gases under normal conditions

.

Average free path is the average distance traveled by a molecule between two successive collisions. It is equal to the ratio of the time passed  t way to the number of collisions during this time.

Maxwell and Boltzmann distributions. Transfer phenomena

Lecture plan:

1. Maxwell's law on the distribution of molecules over velocities. Characteristic velocities of molecules.

2. Boltzmann distribution.

3. Mean free path of molecules.

4. Transfer phenomena:

a) diffusion;

b). internal friction (viscosity);

c) thermal conductivity.

1. Maxwell's law on the distribution of molecules over velocities. Characteristic velocities of molecules.

Gas molecules move randomly and as a result of collisions their speeds change in magnitude and direction; in a gas there are molecules with both very high and very low velocities. One can raise the question of the number of molecules whose velocities lie in the interval from and for a gas in a state of thermodynamic equilibrium in the absence of external force fields. In this case, some stationary velocity distribution of molecules does not change with time, which obeys the statistical law theoretically derived by Maxwell.

The greater the total number of molecules N, the greater the number of molecules DN will have velocities in the interval o and; the larger the interval of velocities , the greater the number of molecules will have the velocities in the indicated interval.

We introduce the coefficient of proportionality f(u).

, (1)

where f(u) is called the distribution function, which depends on the speed of molecules and characterizes the distribution of molecules over speeds.

If the form of the function is known, one can find the number of molecules whose velocities lie in the interval from to.

Using the methods of probability theory and the laws of statistics, Maxwell in 1860. theoretically obtained a formula that determines the number of molecules with velocities in the range from to.

, (2)

- The Maxwell distribution shows what proportion of the total number of molecules of a given gas has velocities in the range from to.

Equations (1) and (2) imply the form of the function :

- (3)

velocity distribution function of ideal gas molecules.

From (3) it can be seen that the specific form of the function depends on the type of gas (on the mass of the molecule m0) and temperature.

Most often, the law of distribution of molecules according to speeds are written as:

The graph of the function is asymmetric (Fig. 1). The position of the maximum characterizes the most frequently encountered speed, which is called the most probable. Speeds in excess of u in, are more common than lower speeds.

is the fraction of the total number of molecules with velocities in this interval.

S total = 1.

As the temperature increases, the distribution maximum shifts towards higher velocities, and the curve becomes flatter, but the area under the curve does not change, because S total = 1.

The most probable speed is the one close to which the speeds of most of the molecules of a given gas turn out to be.

To determine it, we explore to the maximum.

4 ,

, .

Previously, it was shown that

, ,

=> .

In MKT, the concept of the arithmetic mean velocity of the translational motion of molecules of an ideal gas is also used.

- is equal to the ratio of the sum of the moduli of velocities of all molecules to

the number of molecules.

.

It can be seen from the comparison (Fig. 2) that the smallest is u in.

2. Boltzmann distribution.

Two factors - the thermal motion of molecules and the presence of the Earth's gravitational field bring the gas into a state in which its concentration and pressure decrease with height.

If there were no thermal motion of atmospheric air molecules, then all of them would be concentrated at the surface of the Earth. If there were no gravity, then the particles of the atmosphere would be scattered throughout the universe. Let's find the law of change of pressure with height.

The pressure of the gas column is determined by the formula.

Since the pressure decreases with increasing altitude,

where r gas density at altitude h.

Let's find p from the Mendeleev-Clapeyron equation

or.

Let us carry out the calculation for an isothermal atmosphere, assuming that T=const(does not depend on height).

.

at h=0 , , ,

, , ,

The barometric formula determines the gas pressure at any altitude.

We obtain an expression for the concentration of molecules at any height.

where is the potential energy of the molecule at a height h.

Boltzmann distribution in an external potential field.

Consequently, the distribution of molecules in height is their distribution in energy. Boltzmann proved that this distribution is valid not only in the case of the potential field of terrestrial gravitational forces, but also in any potential field of forces for a collection of any identical particles in a state of chaotic thermal motion.

It follows from the Boltzmann distribution that molecules are located with a higher concentration where their potential energy is lower.

Boltzmann distribution - distribution of particles in a potential force field.

3. Mean free path of molecules.

Due to the chaotic thermal motion of the gas molecules continuously collide with each other, go through a complex zigzag path. Between 2 collisions, the molecules move uniformly in a straight line.

M The minimum distance at which the centers of 2 molecules approach each other during a collision is called the effective diameter of the molecule. d(Fig. 4).

The quantity is called the effective cross section of the molecule.

Let us find the average number of collisions of a homogeneous gas molecule per unit time. A collision will occur if the centers of the molecules approach at a distance less than or equal to d. We assume that the molecule moves with speed , and the rest of the molecules are at rest. Then the number of collisions is determined by the number of molecules whose centers are located in a volume that is a cylinder with a base and a height equal to the path traveled by the molecule in 1s, i.e. .

In the statistical method, to determine the main characteristic (X is the set of coordinates and momenta of all particles of the system), one or another model of the structure of the body under consideration is used.

It turns out that it is possible to find general properties of general statistical patterns that do not depend on the structure of matter and are universal. The identification of such regularities is the main task of the thermodynamic method for describing thermal processes. All basic concepts and laws of thermodynamics can be revealed on the basis of statistical theory.

For an isolated (closed) system or a system in a constant external field, the state is called statistically equilibrium if the distribution function does not depend on time.

The specific form of the distribution function of the system under consideration depends both on the totality of external parameters and on the nature of the interaction with the surrounding bodies. Under the external parameters in this case we will understand the quantities determined by the position of the bodies not included in the system under consideration. These are, for example, the volume of the system V, the intensity of the force field, etc. Let's consider the two most important cases:

1) The system under consideration is energetically isolated. The total energy of particles E is constant. Wherein. E can be included in a, but highlighting it emphasizes the special role of E. The condition for the isolation of the system for given external parameters can be expressed by the equality:

2) The system is not closed - energy exchange is possible. In this case, it cannot be found; it will depend on the generalized coordinates and momenta of the particles of the surrounding bodies. This is possible if the interaction energy of the system under consideration with the surrounding bodies.

Under this condition, the distribution function of microstates depends on the average intensity of the thermal motion of the surrounding bodies, which is characterized by the temperature T of the surrounding bodies: .

Temperature also plays a special role. It has no (unlike a) analogue in mechanics: (does not depend on T).

In a state of statistical equilibrium does not depend on time, and all internal parameters are unchanged. In thermodynamics, this state is called the state of thermodynamic equilibrium. The concepts of statistical and thermodynamic equilibrium are equivalent.

Distribution function of a microscopic isolated system - Gibbs microcanonical distribution

The case of an energetically isolated system. Let us find the form of the distribution function for this case.

An essential role in finding the distribution function is played only by the integrals of motion - energy, - momentum of the system and - angular momentum. Only they are controlled.

The Hamiltonian plays a special role in mechanics, because it is the Hamiltonian function that determines the form of the particle motion equation. The conservation of the total momentum and angular momentum of the system in this case is a consequence of the equations of motion.

Therefore, it is precisely such solutions of the Liouville equation that are singled out when the dependence manifests itself only through the Hamiltonian:

Because, .

Of all possible values ​​of X (the set of coordinates and momenta of all particles of the system), those that are compatible with the condition are selected. The constant C can be found from the normalization condition:

where is the area of ​​the hypersurface in the phase space, distinguished by the condition of energy constancy.

Those. is the microcanonical Gibbs distribution.

In the quantum theory of the equilibrium state, there is also a microcanonical Gibbs distribution. Let's introduce the notation: - a complete set of quantum numbers characterizing the microstate of a system of particles, - the corresponding admissible energy values. They can be found by solving the stationary equation for the wave function of the system under consideration.

The distribution function of microstates in this case will be the probability for the system to be in a certain state: .

The quantum microcanonical Gibbs distribution can be written as:

where is the Kronecker symbol, - from the normalization: is the number of microstates with a given energy value (as well as). It's called statistical weight.

From the definition, all states that satisfy the condition have the same probability, equal. Thus, the quantum microcanonical Gibbs distribution is based on the principle of equal a priori probabilities.

The distribution function of the microstates of the system in the thermostat is the canonical Gibbs distribution.

Consider now a system exchanging energy with surrounding bodies. From a thermodynamic point of view, this approach corresponds to a system surrounded by a very large thermostat with temperature T. For a large system (our system + thermostat), the microcanonical distribution can be used, since such a system can be considered isolated. We will assume that the system under consideration is a small but macroscopic part of a larger system with temperature T and the number of particles in it. That is, equality (>>) is satisfied.

We will denote the variables of our system by X, and the thermostat variables by X1.

Then we write the microcanonical distribution for the entire system:

We will be interested in the probability of the state of a system of N particles for any possible states of the thermostat. This probability can be found by integrating this equation over the thermostat states

The Hamilton function of the system and thermostat can be represented as

We will neglect the energy of interaction between the system and the thermostat in comparison with both the energy of the system and the energy of the thermostat. This can be done because the interaction energy for a macrosystem is proportional to its surface area, while the energy of a system is proportional to its volume. However, neglecting the interaction energy compared to the energy of the system does not mean that it is equal to zero, otherwise the formulation of the problem loses its meaning.

Thus, the probability distribution for the system under consideration can be represented as

Let us turn to integration over the thermostat energy

Hence, using the -function property

In what follows, we will pass to the limiting case when the thermostat is very large. Let us consider a special case when the thermostat is an ideal gas with N1 particles with mass m each.

Let's find the value that represents the value

where is the volume of the phase space contained within the hypersurface. Then is the volume of the hyperspheric layer (compare with the expression for the three-dimensional space

For an ideal gas, the region of integration is given by the condition

As a result of integration within the specified boundaries, we obtain the volume of a 3N1-dimensional ball with a radius that will be equal to. Thus, we have

Where do we get

Thus, for the probability distribution we have

Let us now pass to the N1 limit, however, assuming that the ratio remains constant (the so-called thermodynamic limit). Then we get

Taking into account that

Then the distribution function of the system in the thermostat can be written as

where C is found from the normalization condition:

The function is called the classical statistical integral. Thus, the distribution function of the system in the thermostat can be represented as:

This is the canonical Gibbs distribution (1901).

In this distribution, T characterizes the average intensity of thermal motion - the absolute temperature of the particles of the environment.

Another form of writing the Gibbs distribution

When determining, microscopic states were considered different, differing only in the rearrangement of individual particles. This means that we are able to keep track of each particle. However, this assumption leads to a paradox.

The expression for the quantum canonical Gibbs distribution can be written by analogy with the classical one:

Statistical sum: .

It is a dimensionless analogue of the statistical integral. Then the free energy can be represented as:

Let us now consider a system located in a thermostat and capable of exchanging energy and particles with the environment. The derivation of the Gibbs distribution function for this case is in many ways similar to the derivation of the canonical distribution. For the quantum case, the distribution has the form:

This distribution is called the Gibbs grand canonical distribution. Here m is the chemical potential of the system, which characterizes the change in thermodynamic potentials when the number of particles in the system changes by one.

Z - from the normalization condition:

Here the summation goes not only over square numbers, but also over all possible values ​​of the number of particles.

Another form of writing: we introduce a function, but as previously obtained from thermodynamics, where is a large thermodynamic potential. As a result, we get

Here is the average value of the number of particles.

The classical distribution is similar.

Maxwell and Boltzmann distributions

The canonical Gibbs distribution establishes (for given) the explicit form of the distribution function for the values ​​of all coordinates and momenta of particles (6N-variables). But such a function is very complex. Often simpler functions are sufficient.

Maxwell distribution for an ideal monatomic gas. We can consider each gas molecule as a "system under consideration", belonging to a thermostat. Therefore, the probability of any molecule to have impulses in given intervals is given by the Gibbs canonical distribution: .

Replacing momenta with velocities and using the normalization conditions, we obtain

Maxwell's distribution function for velocity components. It is easy to get the distribution modulo as well.

In any system, the energy of which is equal to the sum of the energies of individual particles, there is an expression similar to Maxwell's. This is the Maxwell-Boltzmann distribution. Again, we will assume that the “system” is any one particle, while the rest play the role of a thermostat. Then the probability of the state of this chosen particle for any state of the others is given by the canonical distribution: , . For the rest of the quantities ... integrated

Maxwell and Boltzmann distributions

Maxwell distribution (velocity distribution of gas molecules). 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. Due to the huge number of 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 u x molecules take a normal distribution law. In this case, as shown by J.K. Maxwell, the probability density is written as follows:

similar for other axes

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):

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 - 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 u is modified (Fig. 2.6; T 1< Т 2 ). The Maxwell distribution allows one to calculate the number of molecules whose velocities lie within a certain interval Du. 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 du, to the total number N molecules:

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

If the speed interval du 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 du.

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 (du= 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 of 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 force 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 molecules throughout.

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.