Boltzmann distribution - the energy distribution of particles (atoms, molecules) of an ideal gas under conditions of thermodynamic equilibrium, which was discovered in 1868-1871. Austrian physicist L. Boltzmann. According to him, the number of particles n i with total energy e i is equal to:

ni = Aω i exp (-e i /kT)

where ω i is the statistical weight (the number of possible states of a particle with energy e i). The constant A is found from the condition that the sum n i over all possible values ​​of i is equal to the given total number of particles N in the system (normalization condition): ∑n i = N. In the case when the motion of particles obeys classical mechanics, the energy e i can be considered to consist of kinetic energy e i, kin of a particle (molecule or atom), its internal energy e i, ext (for example, the excitation energy of electrons) and potential energy e i, sweat in an external field, depending on the position of the particle in space:

e i = e i, kin + e i, ext + e i, sweat

The particle velocity distribution (Maxwell distribution) is a special case of the Boltzmann distribution. It occurs when the internal excitation energy and the influence of external fields can be neglected. In accordance with it, the Boltzmann distribution formula can be represented as a product of three exponentials, each of which gives the distribution of particles over one type of energy.

In a constant gravitational field that creates an acceleration g, for particles of atmospheric gases near the surface of the Earth (or other planets), the potential energy is proportional to their mass m and height H above the surface, i.e. e i, sweat = mgH. After substituting this value into the Boltzmann distribution and summing it over all possible values ​​of the kinetic and internal energies of the particles, a barometric formula is obtained that expresses the law of decreasing the density of the atmosphere with height.

In astrophysics, especially in the theory of stellar spectra, the Boltzmann distribution is often used to determine the relative electron population of various energy levels of atoms.

The Boltzmann distribution was obtained in the framework of classical statistics. In 1924-1926. quantum statistics was created. It led to the discovery of the Bose-Einstein (for particles with integer spin) and Fermi-Dirac (for particles with half-integer spin) distributions. Both of these distributions go over to the Boltzmann distribution when the average number of quantum states available for the system significantly exceeds the number of particles in the system, that is, when there are many quantum states per particle, or, in other words, when the degree of filling of quantum states is small. The applicability condition for the Boltzmann distribution can be written as an inequality:

N/V .

where N is the number of particles, V is the volume of the system. This inequality is satisfied at high temperature and a small number of particles per unit volume (N/V). It follows from this that the larger the mass of particles, the wider the range of changes in T and N/V, the Boltzmann distribution is valid. For example, inside white dwarfs, the above inequality is violated for the electron gas, and therefore its properties should be described using the Fermi-Dirac distribution. However, it, and with it the Boltzmann distribution, remain valid for the ionic component of matter. In the case of a gas consisting of particles with zero rest mass (for example, a gas of photons), the inequality does not hold for any values ​​of T and N/V. Therefore, equilibrium radiation is described by Planck's radiation law, which is a special case of the Bose-Einstein distribution.

Due to the chaotic movement, changes in the position of each particle (molecule, atom, etc.) of a physical system (macroscopic body) are in the nature of a random process. Therefore, we can talk about the probability of finding a particle in a particular region of space.

It is known from kinematics that the position of a particle in space is characterized by its radius vector or coordinates.

Consider the probability dW() to detect a particle in a region of space defined by a small interval of values ​​of the radius-vector if the physical system is in thermodynamic equilibrium.

Vector spacing we will measure the volume dV=dxdydz.

Probability density (probability function of the distribution of the values ​​of the radius-vector )

The particle at a given moment of time is actually somewhere in the specified space, which means that the normalization condition must be satisfied:

Let us find the particle distribution probability function f() of a classical ideal gas. The gas occupies the entire volume V and is in a state of thermodynamic equilibrium with temperature T.

In the absence of an external force field, all positions of each particle are equally probable, i.e. gas occupies the entire volume with the same density. Therefore f() = const.

Using the normalization condition, we find that

T . e . f(r)=1/V.

If the number of gas particles is N, then the concentration n = N/V.

Therefore, f(r) =n/N .

Conclusion : in the absence of an external force field, the probability dW() to detect an ideal gas particle in a volume dV does not depend on the position of this volume in space, i.e. .

Let us place an ideal gas in an external force field.

As a result of the spatial redistribution of gas particles, the probability density f() ¹const.

The concentration of gas particles n and its pressure P will be different, i.e. within the limit Where D N is the average number of particles in the volume DV and pressure in the limit, Where D F is the absolute value of the average force acting normally on the site DS.

If the forces of the external field are potential and act in one direction (for example, the gravity of the Earth directed along the z axis), then the pressure forces acting on the upper dS 2 and lower dS 1 of the base of the volume dV will not be equal to each other (Fig. 2.2).

Rice. 2.2

In this case, the difference in pressure forces dF on the bases dS 1 and dS 2 must be compensated by the action of the forces of the external field .

Total pressure difference dF = nGdV,

where G is the force acting on one particle from the external field.

The difference in pressure forces (by definition of pressure) dF = dPdxdy. Therefore, dP = nGdz.

It is known from mechanics that the potential energy of a particle in an external force field is related to the strength of this field by the relation .

Then the pressure difference on the upper and lower bases of the selected volume dP = - n dW p .

In the state of thermodynamic equilibrium of a physical system, its temperature T within the volume dV is the same everywhere. Therefore, we use the ideal gas equation of state for pressure dP = kTdn.

Solving the last two equalities together, we get that

- ndW p = kTdn or .

After transformations, we find that

or

where ℓ nn o - constant of integration (n o - concentration of particles in the space where W p =0).

After potentiation, we get

Probability of finding an ideal gas particle in a volume dV located at a point determined by the radius vector , represent in the form

where P o \u003d n o kT.

Let us apply the Boltzmann distribution to atmospheric air in the Earth's gravitational field.

Part Earth's atmosphere includes gases: nitrogen - 78.1%; oxygen - 21%; argon-0.9%. Mass of the atmosphere -5.15× 10 18 kg. At an altitude of 20-25 km - an ozone layer.

Near the earth's surface, the potential energy of air particles at a height h W p =m o gh, Wherem o is the mass of the particle.

Potential energy at the level of the Earth (h=0) is equal to zero (W p =0).

If, in a state of thermodynamic equilibrium, the particles of the earth's atmosphere have a temperature T, then the change in atmospheric air pressure with height occurs according to the law

Formula (2.15) is called barometric formula ; applicable to rarefied gas mixtures.

Conclusion : for the earth's atmosphere the heavier the gas, the faster its pressure drops depending on the height, i.e. as the altitude increases, the atmosphere should become more and more enriched with light gases. Due to temperature changes, the atmosphere is not in equilibrium. Therefore, the barometric formula can be applied to small areas within which there is no change in temperature. In addition, the non-equilibrium of the earth's atmosphere is affected by the gravitational field of the earth, which cannot keep it close to the surface of the planet. There is a scattering of the atmosphere and the faster, the weaker the gravitational field. For example, the Earth's atmosphere dissipates rather slowly. During the existence of the Earth (~ 4-5 billion years), it lost a small part of its atmosphere (mainly light gases: hydrogen, helium, etc.).

The gravitational field of the Moon is weaker than the Earth's, so it has almost completely lost its atmosphere.

The non-equilibrium of the earth's atmosphere can be proved as follows. Let us assume that the Earth's atmosphere has come to a state of thermodynamic equilibrium and at any point in its space it has a constant temperature. We apply the Boltzmann formula (2.11), in which the role of potential energy is played by the potential energy of the Earth's gravitational field, i.e.

Where g- gravitational constant; M h - the mass of the Earth;m ois the mass of the air particle; ris the distance of the particle from the center of the Earth.= R h , where R h - radius of the earth, then

This means that n ¥ ¹ 0. But the number of particles in the Earth's atmosphere is finite. Therefore, such a number of particles cannot be distributed over an infinite volume.

Therefore, the earth's atmosphere cannot really be in an equilibrium state.

the law of change in pressure with height, assuming that the gravitational field is uniform, the temperature is constant, and the mass of all molecules is the same

Expression (45.2) is called barometric formula. It allows you to find the atmospheric pressure depending on the height or, by measuring the pressure, find the height: Since the heights are indicated relative to sea level, where the pressure is considered normal, the expression (45.2) can be written as

Where R - altitude pressure h.

The barometric formula (45.3) can be converted using the expression (42.6) p= nkT:

Where n is the concentration of molecules at a height h, n 0 - the same, on top h= 0. Since M = m 0 N A( N A is the Avogadro constant, T 0 – mass of one molecule), a R= kN A , That

Where m 0 gh\u003d P - potential energy of the molecule in the gravitational field, i.e.

Expression (45.5) is called Boltzmann distribution for an external potential field. It follows from the veto that at a constant temperature the density of a gas is greater where the potential energy of its molecules is lower.

If the particles have the same mass and are in a state of chaotic thermal motion, then the Boltzmann distribution (45.5) is valid in any external potential field, and not only in the field of gravity.

24. The law of uniform distribution of energy over degrees of freedom. Number of degrees of freedom. Average kinetic energy of thermal motion of molecules.

The average kinetic energy of a molecule having i-degrees of freedom accounts for this. This is Boltzmann's law on the uniform distribution of the average kinetic energy over the degrees of freedom. Molecules can be considered as systems of material points (atoms) performing both translational and rotational motions. When a point moves along a straight line, to estimate its position, it is necessary to know one coordinate, i.e. point has one degree of freedom. If the point of movement along the plane, its position is characterized by two coordinates; the point has two degrees of freedom. The position of a point in space is determined by 3 coordinates. The number of degrees of freedom is usually denoted by the letter i. Molecules that consist of an ordinary atom are considered material points and have three degrees of freedom (argon, helium). The average kinetic energy of gas molecules (per molecule) is determined by the expression The kinetic energy of the translational motion of atoms and molecules, averaged over a huge number of randomly moving particles, is a measure of what is called temperature. If the temperature T is measured in degrees Kelvin (K), then its relationship with Ek is given by the relation From equations (6) and (7) you can determine the value of the root-mean-square velocity of molecules thermal movement. From this follows Joule's law, confirmed by numerous experiments. The internal energy of an ideal gas depends only on its temperature and does not depend on volume. The molecular kinetic theory leads to the following expression for the internal energy of one mole of an ideal monatomic gas (helium, neon, etc.), whose molecules perform only translational motion: Since the potential energy of interaction molecules depends on the distance between them, in the general case, the internal energy U of the body depends, along with the temperature T, also on the volume V: U = U (T, V). It is customary to say that internal energy is a state function.

barometric formula. Consider a gas in equilibrium in a gravitational field. In this case, the sum of the forces acting on each element of the gas volume is zero. Let us allocate a small volume of gas at a height h(Fig. 2.7) and consider the forces acting on it:

The selected volume is affected by the force of gas pressure from below, the force of gas pressure from above and the force of gravity. Then the balance of forces will be written in the form

Where dm is the mass of the allocated volume. For this volume, we can write the Mendeleev-Clapeyron equation

Expressing magnitude dm, we can get the equation

Separating the variables, we get

We integrate the resulting equation, taking into account that the temperature is constant,

Let the surface pressure be p0, then the resulting equation can be easily transformed to the form

The resulting formula is called barometric and describes quite well the distribution of pressure along the height in the atmosphere of the Earth and other planets. It is important to remember that this formula was derived from the assumption of gas equilibrium, while the quantities g And T were assumed to be constant, which, of course, is not always true for the real atmosphere.

Boltzmann distribution. Let us write the barometric formula (2.24) in terms of the concentration of particles, using the fact that p = nkT:

Where m0 is the mass of a gas molecule.

The same conclusion can be drawn for any potential force (not necessarily for gravity). It can be seen from formula (2.25) that the numerator of the exponent is the potential energy of one molecule in the potential field. Then formula (2.25) can be written as

In this form, this formula is suitable for finding the concentration of molecules that are in equilibrium in the field of any potential force.

Find the number of gas particles whose coordinates are in the volume element dV = dxdydz

The total number of particles in the system can be written as

Here the integral is formally written over the entire space, but we must keep in mind that the volume of the system is finite, which will lead to the fact that the integration will be carried out over the entire volume of the system. Then the relation

just gives the probability that the particle will fall into the volume element dV. Then for this probability we write

where the value of the potential energy of the molecule will, generally speaking, depend on all three coordinates. Using the definition of the distribution function, we can write the distribution function of molecules in coordinates in the following form:

This is the Boltzmann distribution function with respect to particle coordinates (or potential energies, meaning that the potential energy depends on the coordinates). It is easy to show that the resulting function is normalized to one.

Relationship between Maxwell and Boltzmann distributions. The Maxwell and Boltzmann distributions are constituents of the Gibbs distribution. The temperature is determined by the average kinetic energy. Therefore, the question arises why the temperature is constant in a potential field, although, according to the law of conservation of energy, when the potential energy of particles changes, their kinetic energy should also change, and therefore, as it seems at first glance, their temperature. In other words, why in the gravitational field, when particles move upwards, the kinetic energy of all of them decreases, while the temperature remains constant, i.e. their average kinetic energy remains constant, and when particles move down, the energy of all particles increases, and the average energy remains constant?

This is explained by the fact that during the ascent, the slowest particles drop out of the flow, i.e. "coldest". Therefore, the energy calculation is based on a smaller number of particles, which at the initial height were, on average, “hotter”. In other words, if a certain number of particles arrived at altitude from zero altitude, then their average energy at altitude is equal to the average energy of all particles at zero altitude, some of which could not reach altitude due to low kinetic energy. However, if at zero height we calculate the average energy of particles that have reached a height , then it is greater than the average energy of all particles at zero height. Therefore, we can say that the average energy of the particles at a height really decreased and in this sense they "cooled" during the ascent. However, the average energy of all particles at zero height and height is the same, i.e. and the temperature is the same. On the other hand, a decrease in the density of particles with height is also a consequence of the removal of particles from the flow.

Therefore, the law of conservation of energy when particles rise to a height leads to a decrease in their kinetic energies and the elimination of particles from the flow. Due to this, on the one hand, the density of particles decreases with height, and on the other hand, their average kinetic energy is preserved, despite the fact that the kinetic energy of each of the particles decreases. This can be confirmed by direct calculation, which is recommended to be done as an exercise.

planetary atmosphere. The potential energy of a particle with a mass in the gravitational field of a spherical celestial body is equal to

where is the body mass; is the distance from the center of the body to the particle; is the gravitational constant. The atmosphere of the planets, including the Earth, is not in equilibrium. For example, due to the fact that the Earth's atmosphere is in a non-equilibrium state, its temperature is not constant, as it should be, but changes with altitude (decreases with altitude). Let us show that the equilibrium state of the planet's atmosphere is impossible in principle. If it were possible, then the density of the atmosphere would have to change with height according to formula (2.26), which takes the form

where the expression (2.28) for the potential energy is taken into account, is the radius of the planet. Formula (2.29) shows that as , the density tends to a finite limit

This means that if there are a finite number of molecules in the atmosphere, then they must be distributed over the entire infinite space, i.e. the atmosphere is scattered.

Since, in the end, all systems tend to an equilibrium state, the atmosphere of the planets gradually dissipates. Some of the celestial bodies, such as the Moon, have completely lost their atmosphere, while others, such as Mars, have a very rarefied atmosphere. Thus, the atmosphere of the Moon has reached an equilibrium state, and the atmosphere of Mars is already close to reaching an equilibrium state. Venus has a very dense atmosphere and, therefore, is at the beginning of the path to an equilibrium state.

For a quantitative consideration of the question of the loss of the atmosphere by the planets, it is necessary to take into account the distribution of molecules over velocities. The force of gravity can be overcome only by molecules whose speed exceeds the second cosmic one. These molecules are in the "tail" of the Maxwell distribution and their relative number is insignificant. However, over significant time intervals, the loss of molecules is sensitive. Since the escape velocity of heavy planets is greater than that of light planets, the intensity of atmospheric loss for massive celestial bodies is less than for light ones, i.e. lighter planets lose their atmosphere faster than heavy ones. The time of loss of the atmosphere also depends on the radius of the planet, the composition of the atmosphere, etc. A complete quantitative analysis of this issue is a challenging task.

Experimental verification of the Boltzmann distribution. When deriving the Boltzmann distribution, no restrictions were imposed on the mass of the particles. Therefore, in principle, it is also applicable to heavy particles. Let us take as these particles, for example, grains of sand. It is clear that they will be located in a certain layer near the vessel. Strictly speaking, this is a consequence of the Boltzmann distribution. For large masses of particles, the exponent changes so rapidly with height that it is zero everywhere outside the sand layer. As for the space inside the layer, the volume of the grains of sand must be taken into account there. This will be reduced to a purely mechanical problem of minimizing the potential energy for given constraints. Problems of this type are considered not in statistical physics, but in mechanics.

In order for heavy particles not to “sink to the bottom”, to be distributed in a sufficiently large layer at a height, it is necessary that their potential energy be sufficiently small. This can be achieved by placing the particles in a liquid whose density is only slightly less than the density of the material of the particles. Denoting the density and volume of particles and , and the density of the liquid - , we see that the force acting on the particle is equal to . Therefore, the potential energy of such a particle at a height from the bottom of the vessel is

Therefore, the height distribution of the concentrations of these particles is given by the formula

In order for the effect to be sufficiently noticeable, the particles must be sufficiently small. The number of such particles at different heights in the vessel is counted using a microscope. Experiments of this kind were first performed since 1906 by Zh.B. Perrin (1870-1942).

Having made measurements, one can first of all verify whether the particle concentration really changes according to an exponential law. Perrin proved that this is indeed the case, and hence the Boltzmann distribution is valid. Further, based on the validity of the distribution and by measuring the volumes and densities of particles by independent methods, it is possible to find the value of the Boltzmann constant from the results of the experiment, since all other quantities in (2.32) are known.

In this way, Perrin measured and obtained a result very close to modern. In another independent way, the value was obtained by Perrin from experiments with Brownian motion.

Subsequently, experiments of another type were also carried out, which fully confirmed the Boltzmann distribution. From experiments of another type, one can indicate, for example, the verification of the dependence of the polarization of polar dielectrics on temperature, which was considered above.

Example 2.2. Perrin used the distribution of gummigut grains in water to measure Avogadro's constant. The density of gum particles was r = 1.21×10 3 kg/m 3 , their volume t = 1.03×10 -19 m 3 . The temperature at which the experiment was carried out was . Find the height at which the density of distribution of gummigut grains has decreased by half.

Taking into account that, according to the condition of the problem, t (r - r 0) \u003d 0.22 × 10 -16 kg, we obtain on the basis of formula (2.32) h = kT ln2/ = 12.3×10 -6 m.

Example 2.3. Spherical particles with a radius of 10 -7 m are suspended in air at a temperature and pressure Pa. Find the mass of the suspended particle.

By formula (2.32) we find t(r - r 0) = kT ln2/ gh= 1.06 × 10 -23 kg.

Given that t \u003d 4.19 × 10 -21 m 3, we find (r - r 0) \u003d 2.53 × 10 -3 kg / m 3. Since r 0 \u003d 1.293 kg / m 3, we get r \u003d 1.296 kg / m 3 and, therefore, the mass of the particle

Boltzmann distribution

In the barometric formula in relation to M/R Divide both the numerator and denominator by Avogadro's number.

The mass of one molecule,

Boltzmann's constant.

Instead of R and substitute accordingly. (see lecture No. 7), where the density of molecules at a height h, the density of molecules at height .

From the barometric formula, as a result of substitutions and reductions, we obtain the distribution of the concentration of molecules in height in the Earth's gravity field.

It follows from this formula that as the temperature decreases, the number of particles at heights other than zero decreases (Fig. 8.10), turning to 0 at T=0 ( At absolute zero, all molecules would be located on the surface of the Earth). At high temperatures n decreases slightly with height, so

Hence, the distribution of molecules in height is also their distribution in terms of potential energy values.

where is the density of molecules in that place in space where the potential energy of the molecule has the value ; the density of molecules at the point where the potential energy is 0.

Boltzmann proved that the distribution (*) is true not only in the case of the potential field of terrestrial gravity forces, but also in any potential field of forces for a set of any identical particles in a state of chaotic thermal motion.

Thus, Boltzmann's law (*) gives the distribution of particles in a state of chaotic thermal motion according to the values ​​of potential energy. (Fig. 8.11)

4. Boltzmann distribution at discrete energy levels.

The distribution obtained by Boltzmann refers to the cases when the molecules are in an external field and their potential energy can be applied continuously. Boltzmann generalized his law to the case of a distribution that depends on the internal energy of the molecule.

It is known that the value of the internal energy of a molecule (or atom) E can take only a discrete set of allowed values. In this case, the Boltzmann distribution has the form:

where is the number of particles in a state with energy ;

The proportionality factor that satisfies the condition

Where N is the total number of particles in the system under consideration.

Then and as a result, for the case of discrete values ​​of energy, the Boltzmann distribution

But the state of the system in this case is thermodynamically nonequilibrium.

5. Maxwell-Boltzmann statistics

The Maxwell and Boltzmann distribution can be combined into one Maxwell-Boltzmann law, according to which the number of molecules whose velocity components lie in the range from to , and the coordinates in the range from x, y, z before x+dx, y+dy, z+dz, equals

where , the density of molecules in that place in space where ; ; ; total mechanical energy of the particle.

The Maxwell-Boltzmann distribution establishes the distribution of gas molecules in coordinates and velocities in the presence of an arbitrary potential force field.

Note: the Maxwell and Boltzmann distributions are components of a single distribution called the Gibbs distribution (this issue is discussed in detail in special courses on static physics, and we will limit ourselves to mentioning this fact only).

Questions for self-control.

1. Define probability.

2. What is the meaning of the distribution function?

3. What is the meaning of the normalization condition?

4. Write down the formula for determining the average value of the results of measuring x using the distribution function.

5. What is the Maxwell distribution?

6. What is the Maxwell distribution function? What is its physical meaning?

7. Plot a graph of the Maxwell distribution function and indicate the characteristic features of this function.

8. Indicate the most likely speed on the graph. Get an expression for . How does the graph change as the temperature rises?

9. Get the barometric formula. What does she define?

10. Get the dependence of the concentration of gas molecules in the gravity field on height.

11. Write down the Boltzmann distribution law a) for ideal gas molecules in the gravity field; b) for particles of mass m located in the rotor of a centrifuge rotating at an angular velocity .

12. Explain the physical meaning of the Maxwell-Boltzmann distribution.

Lecture #9

real gases

1. Forces of intermolecular interaction in gases. Van der Waals equation. Isotherms of real gases.

2. Metastable states. Critical condition.

3. Internal energy of a real gas.

4. Joule-Thomson effect. Liquefaction of gases and obtaining low temperatures.

1. Forces of intermolecular interaction in gases

Many real gases obey the laws of ideal gases. under normal conditions. Air can be considered ideal up to pressures ~ 10 atm. When the pressure rises deviations from ideality(deviation from the state described by the Mendeleev-Claperon equation) increase and at p=1000 atm reach more than 100%.

and attraction, A F - their resulting. The repulsive forces are considered positive, and the forces of mutual attraction are negative. The corresponding qualitative curve of the dependence of the interaction energy of molecules on the distance r between the centers of molecules is given on

rice. 9.1b). Molecules repel each other at short distances and attract each other at large distances. The rapidly increasing repulsive forces at small distances mean, roughly speaking, that molecules, as it were, occupy a certain volume, beyond which the gas cannot be compressed.