Boltzmann distribution is 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 it, 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). Constant A is found from the condition that the sum of 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 movement of particles obeys classical mechanics, the energy e i can be considered to consist of kinetic energy e i, the 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, pot in an external field, depending on the position of the particle in space:

e i = e i, kin + e i, vn + e i, sweat

The velocity distribution of particles (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 according to one type of energy.

In a constant gravitational field creating 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 over all possible values ​​of the kinetic and internal energies of particles, a barometric formula is obtained, expressing the law of decreasing atmospheric density with height.

In astrophysics, especially in the theory of stellar spectra, the Boltzmann distribution is often used to determine the relative electron occupancy of different atomic energy levels.

The Boltzmann distribution was obtained within 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 transform into the Boltzmann distribution when the average number of quantum states available to 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 occupation of quantum states is small. The condition for the applicability of the Boltzmann distribution can be written as the 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 it that the greater 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 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 the substance. 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 detecting a particle in a particular region of space.

From kinematics it is known that the position of a particle in space is characterized by its radius vector or coordinates.Consider the probability dW( ) detect a particle in a region of space defined by a small interval of radius vector values

, if the physical system is in a state of thermodynamic equilibrium. Vector interval

we will measure by volume dV=dxdydz. )

Probability density (probability function of distribution of radius vector values

The particle at a given moment in time is actually located somewhere in the specified space, which means the normalization condition must be satisfied:) 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(onst.

¹ c The concentration of gas particles n and its pressure P will be different, i.e. in the limit Where D N - average number of particles in volumeD V and pressure in the limit Where , Where F is the absolute value of the average force acting normally on the platformD

S. If the external field forces are potential and act in one direction (for example, the Earth's gravity

directed along the z axis), then the pressure forces acting on the upper dS 2 and lower dS 1 bases 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 external field forces

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 difference in pressure on the upper and lower bases of the allocated volume dP =

n dW p .

In a 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 ℓn

n o - constant of integration (n o - concentration of particles in that place in space where W p =0).

After potentiation, we get Probability of detecting an ideal gas particle in a volume dV located at a point defined by the radius vector

, let's represent it in the form

where P o = n o kT.

Let us apply the Boltzmann distribution to atmospheric air located in the Earth's gravitational field. Part gases included: nitrogen - 78.1%; oxygen - 21%; argon-0.9%. Atmospheric mass -5.15× 10 18 kg. At an altitude of 20-25 km there is 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 - particle mass.

Potential energy at the Earth level (h=0) is 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 for 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 increasingly enriched with light gases. Due to temperature changes, the atmosphere is not in an equilibrium state. Therefore, the barometric formula can be applied to small areas within which there is no change in temperature.

In addition, the disequilibrium of the Earth's atmosphere is affected by the Earth's gravitational field, which cannot keep it close to the surface of the planet.

The atmosphere dissipates faster, the weaker the gravitational field. For example, the earth's atmosphere dissipates quite 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 Moon's gravitational field is weaker than the Earth's, so it has almost completely lost its atmosphere.The non-equilibrium of the earth's atmosphere can be proven as follows. Let us assume that the Earth’s atmosphere has reached a state of thermodynamic equilibrium and at any point in its space it has a constant temperature. Let us 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.m oWhere g- gravitational constant; M s - mass of the Earth;- mass of air particle; r - the distance of the particle from the center of the Earth. r - = R

h ¥ ¹ , where R

radius of the Earth, then

Until now, we have considered the behavior of an ideal gas not affected by external force fields. It is well known from experience that under the action of external forces the uniform distribution of particles in space can be disrupted. So, under the influence of gravity, the molecules tend to sink to the bottom of the vessel. Intense thermal motion prevents sedimentation, and the molecules spread out so that their concentration gradually decreases as altitude increases.

Let us derive the law of pressure change with height, assuming that the gravitational field is uniform, the temperature is constant and the mass of all molecules is the same. If atmospheric pressure at altitude h is equal to p, then at altitude h+dh it is equal p+dp(at dh > 0, dp < 0, так как p decreases with increasing h).

Pressure difference at altitudes h And h+dh we can define it as the weight of air molecules enclosed in a volume with a base area of ​​1 and height dh.

Density at its best h, and since , then = const.

Then

From the Mendeleev-Clapeyron equation.

With a change in height from h 1 before h 2 pressure varies from p 1 before p2

Let's potentiate this expression (

The barometric formula shows how pressure changes with altitude

n concentration of molecules at height h,

n 0 concentration of molecules at height h =0.

Potential energy of molecules in a gravitational field

Boltzmann distribution in an external potential field. It follows from this that when T= const The gas density is greater where the potential energy of the molecules is less.

24.Real gas- a gas that is not described by the Clapeyron-Mendeleev equation of state for an ideal gas.

The relationships between its parameters show that molecules in a real gas interact with each other and occupy a certain volume. The state of a real gas is often described in practice by the generalized Mendeleev-Clapeyron equation:

where p is pressure; V - volume; T - temperature; Z r = Z r (p,T) - gas compressibility coefficient; m - mass; M - molar mass; R is the gas constant. There is also such a thing as critical temperature; if a gas is at a temperature above the critical temperature (individual for each gas, for example for carbon dioxide approximately 304 K), then it can no longer be turned into liquid, no matter what pressure is applied to it . This phenomenon occurs due to the fact that at a critical temperature the surface tension forces of the liquid are zero. If you continue to slowly compress a gas at a temperature above the critical temperature, then after it reaches a volume equal to approximately four of the intrinsic volumes of the molecules that make up the gas, the compressibility of the gas begins to drop sharply.

25. Phase transitions. Phase transitions of 1st and 2nd order. State diagrams of matter. Triple point. Classification of phase transitions During a first-order phase transition, the most important, primary extensive parameters change abruptly: specific volume (i.e. density), the amount of stored internal energy, concentration of components, etc. We emphasize: we mean an abrupt change in these quantities when changing temperature, pressure, etc., and not an abrupt change in time (for the latter, see the section Dynamics of phase transitions below). The most common examples of first-order phase transitions are: melting and solidification, boiling and condensation, sublimation and desublimation. During a second-order phase transition, the density and internal energy do not change, so such a phase transition may not be noticeable to the naked eye. The jump is experienced by their second derivatives with respect to temperature and pressure: heat capacity, coefficient of thermal expansion, various susceptibilities, etc. Second-order phase transitions occur in cases where the symmetry of the structure of a substance changes.

A triple point is a point on a phase diagram where three lines of phase transitions converge. The triple point is one of the characteristics of a chemical substance. Typically, the triple point is determined by the value of temperature and pressure at which a substance can be in equilibrium in three (hence the name) states of aggregation - solid, liquid and gaseous. At this point the lines of melting, boiling and sublimation converge.

STATE DIAGRAM (phase diagram) - a diagram depicting the dependence of the stable phase state of a single- or multicomponent substance on thermodynamics. parameters that determine this state (temperature T, pressure P, magnetic tension H or electric E fields, concentration With and etc.). Each point of D. s. (figurative point) indicates the phase composition of the substance at given thermodynamic values. parameters (coordinates of this point). Depending on the number of external parameters D. s. can be two-dimensional, three-dimensional and multidimensional. When studying phase equilibrium under AC conditions. pressure build isobaric. and isoconcentration sections and projections on the plane T-P or R-s. Naib. isobaric are fully studied. Shh sections T-R-s D. s., corresponding to atm. pressure.

26. Features of the surface layer of liquid. Surface tension coefficient.

The molecules of a substance in a liquid state are located almost close to each other. Unlike solid crystalline bodies, in which molecules form ordered structures throughout the entire volume of the crystal and can perform thermal vibrations around fixed centers, liquid molecules have greater freedom. Each molecule of a liquid, just like in a solid, is “sandwiched” on all sides by neighboring molecules and undergoes thermal vibrations around a certain equilibrium position. However, from time to time any molecule may move to a nearby vacant site. Such jumps in liquids occur quite often; therefore, molecules are not attached to specific centers, as in crystals (see §3.6), and can move throughout the entire volume of the liquid. This explains the fluidity of liquids. Due to the strong interaction between closely located molecules, they can form local (unstable) ordered groups containing several molecules. This phenomenon is called short-range order.

Surface tension- thermodynamic characteristic of the interface between two phases in equilibrium, determined by the work of reversible isothermokinetic formation of a unit area of ​​this interface, provided that the temperature, volume of the system and chemical potentials of all components in both phases remain constant. Surface tension has a double physical meaning - energetic (thermodynamic) and power (mechanical). Energy (thermodynamic) definition: surface tension is the specific work of increasing the surface when it is stretched, subject to constant temperature. Force (mechanical) definition: surface tension is the force acting per unit length of a line that bounds the surface of a liquid.

Boltzmann distribution

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

Mass of one molecule,

Boltzmann's constant.

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

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

From this formula it follows that with decreasing temperature, 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 by height is also their distribution by potential energy values.

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

Boltzmann proved that the distribution (*) is true 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.

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

Rice. 8.11

4. Boltzmann distribution at discrete energy levels.

The distribution obtained by Boltzmann applies to cases where the molecules are in an external field and their potential energy can be applied continuously. Boltzmann generalized the law he obtained to the case of a distribution depending 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 series of allowed values. In this case, the Boltzmann distribution has the form:

,

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

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 energy values, 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 range from x, y, z before x+dx, y+dy, z+dz, equals

Where , the density of molecules in the space where; ; ; total mechanical energy of a particle.

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

Note: 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 just mentioning this fact).

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 a formula to determine 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 probable speed on the graph. Get an expression for . How does the graph change as the temperature increases?

9. Obtain the barometric formula. What does it define?

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

11. Write down the Boltzmann distribution law a) for molecules of an ideal gas in a 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 No. 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 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 ideal gas laws under normal conditions. Air can be considered ideal up to pressures ~ 10 atm. When pressure increases deviations from ideality(deviation from the state described by the Mendeleev - Clayperon equation) increase and at p = 1000 atm reach more than 100%.

and attraction, A F – their resultant. 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 distance r between the centers of molecules is shown in

rice. 9.1b). At short distances molecules repel, at large distances they attract. The rapidly increasing repulsive forces at short distances mean, roughly speaking, that molecules seem to occupy a certain volume beyond which the gas cannot be compressed.