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

Let's select a separate molecule. The randomness of the movement allows, for example, for the projection of speed u x molecules accept the normal distribution law. In this case, as J. C. Maxwell showed, the probability density is written as follows:

similar for other axes

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

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

(2.36)

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

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

With increasing temperature, the maximum of the Maxwell curve shifts towards higher velocities and the distribution of molecules over u changes (Fig. 2.6; T 1< Т 2 ). The Maxwell distribution makes it possible to calculate the number of molecules whose velocities lie in a certain interval Du. We obtain the corresponding formula.

Since the total number N molecules in a gas are usually high, then the probability d P can be expressed as the ratio of the number d N molecules whose velocities lie in a certain range du to the total number N molecules:

or graphically calculate the area curved 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 the base du.

To the question of how many molecules have a speed equal to any specific value, a strange, at first glance, answer follows: if the speed is absolutely precisely specified, then the speed range 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 probability theory: for continuous random variable, what is the speed, it is impossible to “guess” absolutely exactly its value, which at least one molecule in the gas has.

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

The Maxwell distribution can be considered as the distribution of molecules not only by speed, but also by kinetic energy (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, that is, to establish the concentration of particles that have a certain value of potential energy.

Distribution of particles by potential energies in force fields— gravitational, electrical, etc.— called the Boltzmann distribution.

In relation to the gravitational field, this distribution can be written as a dependence of concentration P molecule from height h above the level of the Earth 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.

This distribution of molecules in the Earth’s gravitational field can be qualitatively explained, within the framework of molecular kinetic concepts, by the fact that molecules are influenced by two opposing factors: the gravitational field, under the influence of which all molecules are attracted to the Earth, and molecular chaotic motion , tending to scatter molecules evenly throughout the entire possible volume.

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

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

With the statistical method, to determine the main characteristic (X is the totality of coordinates and momenta of all particles of the system), certain models of the structure of the body in question are used.

It turns out to be possible to find general properties general statistical laws that do not depend on the structure of matter and are universal. Identifying such patterns is the main task of the thermodynamic method of 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 constant external field a 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 set of external parameters and on the nature of the interaction with surrounding bodies. In this case, by external parameters we mean quantities determined by the position of bodies not included in the system under consideration. This is, for example, the volume of the system V, the strength of the force field, etc. Let's consider the two most important cases:

1) The system under consideration is energetically isolated. The total particle energy E is constant. Wherein. E can be included in a, but highlighting it emphasizes the special role of E. The condition of 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 turns out to be possible if the energy of interaction of the system under consideration with surrounding bodies.

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

Temperature also plays a special role. It does not (unlike a) have an analogue in mechanics: (does not depend on T).

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

Distribution function of a microscopic isolated system - microcanonical Gibbs distribution

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

A significant 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 equation of particle motion. The conservation of total momentum and angular momentum of the system is a consequence of the equations of motion.

Therefore, it is precisely such solutions of the Liouville equation that are distinguished when the dependence is manifested only through the Hamiltonian:

Because, .

From all possible values of X (the set of coordinates and momenta of all particles in 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 phase space, allocated by the condition of constant energy.

Those. - microcanonical Gibbs distribution.

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

The microstate distribution function in this case will represent 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 normalization: - the number of microstates with a given energy value (as well as). It's called statistical weight.

From the definition, all states satisfying the condition have the same probability, equal. Thus, the quantum microcanonical Gibbs distribution is based on the principle of equal prior probabilities.

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

Let us now consider 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 constitutes a small but macroscopic part 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 for the entire system we write down the microcanonical distribution:

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

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 in comparison with 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's move on to integration over the thermostat energy

Hence, using the property of the -function

We will later move on to the limiting case when the thermostat is very large. Let's consider special case, when the thermostat is an ideal gas with N1 particles with mass m each.

Let us find the quantity that represents the quantity

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

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

As a result of integration within the indicated 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 it from?

Thus, for the probability distribution we have

Let us now move on 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 a 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 movement - absolute temperature environmental particles.

Another form of writing the Gibbs distribution

In the definition, microscopic states were considered different, differing only in the rearrangement of individual particles. This means that we are able to keep track of every particle. However, such an 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 Grand Canonical Gibbs 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 occurs not only over square numbers, but also over all possible values of the number of particles.

Another form of notation: let's introduce a function, but since it was previously obtained from thermodynamics, where is a large thermodynamic potential. As a result we get

Here is the average number of particles.

The classic distribution is similar.

Maxwell and Boltzmann distributions

The canonical Gibbs distribution establishes (given) an 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 having impulses in given intervals is given by the canonical Gibbs distribution: .

Replacing impulses with velocities and using normalization conditions, we obtain

Maxwell distribution function for velocity components. It is easy to obtain the distribution modulo.

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

Maxwell and Boltzmann distributions. Transference phenomena

Lecture outline:

1. Maxwell's law on the distribution of molecules by speed. Characteristic speeds of molecules.

2. Boltzmann distribution.

3. Average 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 by speed. Characteristic speeds of molecules.

Gas molecules move chaotically 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 about 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, a certain stationary velocity distribution of molecules that does not change with time is established, which obeys a 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 oty; the larger the interval of velocities , the greater the number of molecules will have velocities in the specified interval.

Let us introduce the proportionality coefficient f(u).

, (1)

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

If the form of the function is known, you 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)

- Maxwell's distribution shows what fraction total number molecules of a given gas have velocities in the range from to.

From equations (1) and (2) the form of the function follows:

- (3)

velocity distribution function of ideal gas molecules.

From (3) it is clear that the specific type of 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 according to speeds are written in the form:

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

- the proportion of the total number of molecules with velocities in this range.

S total = 1.

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

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

To determine it, we examine it to the maximum.

4 ,

, .

It was previously shown that

, ,

=> .

MKT also uses the concept of arithmetic mean speed forward motion ideal gas molecules.

- equal to the ratio of the sum of the velocity modules of all molecules to

number of molecules.

From the comparison it is clear (Fig. 2) that the smallest is u in.

2. Boltzmann distribution.

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

If there were no thermal movement of atmospheric air molecules, then they would all concentrate at the surface of the Earth. If there were no gravity, then atmospheric particles would be scattered throughout the Universe. Let's find the law of pressure change with height.

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

Since pressure decreases with increasing altitude,

Where r gas density at altitude h.

We'll find p from the Mendeleev-Clapeyron equation

or.

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

at h=0 , , ,

, , ,

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 height h.

Boltzmann distribution in an external potential field.

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

From the Boltzmann distribution it follows that molecules are located in greater concentration where their potential energy is lower.

Boltzmann distribution is the distribution of particles in a potential force field.

3. Average free path of molecules.

Due to the chaotic thermal motion, gas molecules continuously collide with each other and travel 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 upon 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 other 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 1 s, i.e. .

Maxwell and Boltzmann distributions. Transference phenomena

Lecture outline:

Maxwell's law on the distribution of molecules by speed. Characteristic speeds of molecules.

Boltzmann distribution.

Average free path of molecules.

Transference phenomena:

a).diffusion;

b).internal friction (viscosity);

c).thermal conductivity.

Maxwell's law on the distribution of molecules by speed. Characteristic speeds of molecules.

Gas molecules move chaotically and, as a result of collisions, their velocities change in magnitude and direction; gas contains molecules with both very high and very low velocities. One can raise the question about 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, a certain stationary velocity distribution of molecules that does not change with time is established, which obeys a 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 interval from and; the larger the interval of velocities, the greater the number of molecules will have velocities in the specified interval.

Let us introduce the proportionality coefficient f( .

, 

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

If the form of the function is known, you can find the number of molecules whose velocities lie in the range 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 equations  and  it follows the form of the function 

- (3)

velocity distribution function of ideal gas molecules.

From (3) it is clear that the specific type of function depends on the type of gas (on the mass of the molecule m 0 ) and temperature.

Most often, the law of molecular velocity distribution is written in the form:

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

- the proportion of the total number of molecules with velocities in this range.

S total = 1.

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

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

To determine it, we examine it to the maximum.

It was previously shown that

, ,

 .

MCT also uses the concept of the arithmetic mean velocity of translational motion of ideal gas molecules.

- equal to the ratio of the sum of the velocity modules of all molecules to

number of molecules.

From the comparison it is clear (Fig. 2) that the smallest is  V .

Boltzmann distribution.

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

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

Since pressure decreases with increasing altitude,

Where  gas density at altitude h.

We'll 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 height h.

Boltzmann distribution in an external potential field.

From the Boltzmann distribution it follows that molecules are located in greater concentration where their potential energy is lower.

Boltzmann distribution is the distribution of particles in a potential force field.

Average free path of molecules.

Due to the chaotic thermal motion, gas molecules continuously collide with each other and travel 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 upon collision is called the effective diameter of the molecule d(Fig. 4).

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

IN In reality, all molecules move, and the possibility of 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 a molecule travels between two successive collisions. It is equal to the ratio of distance traveled in time  t path to the number of collisions during this time.

Maxwell distribution (distribution of gas moleculesby speed). In an equilibrium state, gas parameters (pressure, volume and temperature) remain unchanged, but microstates - the relative position of molecules, their speeds - continuously change. Due to the huge number of molecules, it is practically impossible to determine the values of their speeds at any moment, but it is possible, considering the speed of molecules to be a continuous random variable, to indicate the distribution of molecules by speed.

Let's select a separate molecule. The randomness of the movement allows, for example, for the projection of speed  x molecules accept the normal distribution law. In this case, as J. C. Maxwell showed, the probability density is written as follows:

Where T 0 - the mass of the molecule, T- thermodynamic gas temperature, k - 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 a molecule has a velocity projection lying in the range from  x before  x + d X :

similar for other axes

Each of 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 using the probability multiplication theorem [see (2.6)]:

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

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

(2.33)

and the probability that the speed of a molecule is between  before  + d :

The graph of 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 function condition:

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

With increasing temperature, the maximum of the Maxwell curve shifts towards higher velocities and the distribution of molecules  changes (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 obtain the corresponding formula.

Since the total number N molecules in a gas are usually high, then the probability d P can be expressed as the ratio of the number d N molecules whose velocities lie within a certain range d , to the total number N molecules:

From (2.34) and (2.37) it follows that

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

or graphically calculate the area of a curved 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 the base d .

To the question of how many molecules have a speed equal to any specific value, a strange, at first glance, answer follows: if the speed is absolutely precisely specified, 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 probability theory: for a continuous random variable, such as speed, it is impossible to “guess” absolutely exactly its value, which at least one molecule in the gas has.

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

The Maxwell distribution can be considered as a distribution of molecules not only by speed, but also by kinetic energy (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, that is, to establish the concentration of particles with a certain specific value of potential energy.

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

In relation to the gravitational field, this distribution can be written as a dependence of concentration P molecules from height h above the level of the Earth 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.

This distribution of molecules in the Earth’s gravitational field can be qualitatively explained, within the framework of molecular kinetic concepts, by the fact that molecules are influenced by two opposing factors: the gravitational field, under the influence of which all molecules are attracted to the Earth, and molecular-chaotic motion, which tends to scatter the molecules evenly to the fullest 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 exponent, the ratio of the kinetic energy of the molecule to kT, in the second - the ratio of potential energy to kT.