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Boltzmann constant (k (\displaystyle k) or k B (\displaystyle k_(\rm (B)))) - a physical constant that defines the relationship between temperature and energy. Named after the Austrian physicist Ludwig Boltzmann, who made major contributions to statistical physics, in which this constant plays a key role. Its value in the International System of Units SI according to changes in the definitions of basic SI units (2018) is exactly equal to
k = 1.380 649 × 10 − 23 (\displaystyle k=1(,)380\,649\times 10^(-23)) J/.Relationship between temperature and energy
In a homogeneous ideal gas at absolute temperature T (\displaystyle T), the energy per each translational degree of freedom is equal, as follows from the Maxwell distribution, k T / 2 (\displaystyle kT/2). At room temperature (300 ) this energy is 2 , 07 × 10 − 21 (\displaystyle 2(,)07\times 10^(-21)) J, or 0.013 eV. In a monatomic ideal gas, each atom has three degrees of freedom corresponding to three spatial axes, which means that each atom has an energy of 3 2 k T (\displaystyle (\frac (3)(2))kT).
Knowing the thermal energy, we can calculate the root mean square velocity of the atoms, which is inversely proportional to the square root of the atomic mass. The root mean square velocity at room temperature varies from 1370 m/s for helium to 240 m/s for xenon. In the case of a molecular gas, the situation becomes more complicated, for example, a diatomic gas has 5 degrees of freedom - 3 translational and 2 rotational (at low temperatures, when vibrations of atoms in the molecule are not excited and additional degrees of freedom are not added).
Definition of entropy
The entropy of a thermodynamic system is defined as the natural logarithm of the number of different microstates Z (\displaystyle Z), corresponding to a given macroscopic state (for example, a state with a given total energy).
S = k ln Z . (\displaystyle S=k\ln Z.)Proportionality factor k (\displaystyle k) and is Boltzmann's constant. This is an expression that defines the relationship between microscopic ( Z (\displaystyle Z)) and macroscopic states ( S (\displaystyle S)), expresses the central idea of statistical mechanics.
According to the Stefan–Boltzmann law, the density of integral hemispherical radiation E 0 depends only on temperature and varies proportionally to the fourth power of absolute temperature T:
The Stefan–Boltzmann constant σ 0 is a physical constant included in the law that determines the volumetric density of the equilibrium thermal radiation of an absolutely black body:
Historically, the Stefan-Boltzmann law was formulated before Planck's radiation law, from which it follows as a consequence. Planck's law establishes the dependence of the spectral flux density of radiation E 0 on wavelength λ and temperature T:
where λ – wavelength, m; With=2.998 10 8 m/s – speed of light in vacuum; T– body temperature, K;
h= 6.625 ×10 -34 J×s – Planck’s constant.
Physical constant k, equal to the ratio of the universal gas constant R=8314J/(kg×K) to Avogadro’s number N.A.=6.022× 10 26 1/(kg×mol):
Number of different system configurations from N particles for a given set of numbers n i(number of particles in i-the state to which the energy e i corresponds) is proportional to the value:
Magnitude W there is a number of ways of distribution N particles by energy levels. If relation (6) is true, then it is considered that the original system obeys Boltzmann statistics. Set of numbers n i, at which the number W maximum, occurs most frequently and corresponds to the most probable distribution.
Physical kinetics– microscopic theory of processes in statistically nonequilibrium systems.
The description of a large number of particles can be successfully carried out using probabilistic methods. For a monatomic gas, the state of a set of molecules is determined by their coordinates and the values of velocity projections on the corresponding coordinate axes. Mathematically, this is described by the distribution function, which characterizes the probability of a particle being in a given state:
is the expected number of molecules in a volume d d whose coordinates are in the range from to +d, and whose velocities are in the range from to +d.
If the time-averaged potential energy of interaction of molecules can be neglected in comparison with their kinetic energy, then the gas is called ideal. An ideal gas is called a Boltzmann gas if the ratio of the path length of the molecules in this gas to the characteristic size of the flow L of course, i.e.
because the path length is inversely proportional nd 2(n is the numerical density 1/m 3, d is the diameter of the molecule, m).
Size
called H-Boltzmann function for a unit volume, which is associated with the probability of detecting a system of gas molecules in a given state. Each state corresponds to certain numbers of filling six-dimensional space-velocity cells into which the phase space of the molecules under consideration can be divided. Let's denote W the probability that there will be N 1 molecules in the first cell of the space under consideration, N 2 in the second, etc.
Up to a constant that determines the origin of the probability, the following relation is valid:
,
Where 
– H-function of a region of space A occupied by gas. From (9) it is clear that W And H interconnected, i.e. a change in the probability of a state leads to a corresponding evolution of the H function.
Boltzmann's principle establishes the connection between entropy S physical system and thermodynamic probability W her states:
(published according to the publication: Kogan M.N. Dynamics of a rarefied gas. - M.: Nauka, 1967.)
General view of the CUBE:
where is the mass force due to the presence of various fields (gravitational, electric, magnetic) acting on the molecule; J– collision integral. It is this term of the Boltzmann equation that takes into account the collisions of molecules with each other and the corresponding changes in the velocities of interacting particles. The collision integral is a five-dimensional integral and has the following structure:
Equation (12) with integral (13) was obtained for collisions of molecules in which no tangential forces arise, i.e. colliding particles are considered to be perfectly smooth.
During the interaction, the internal energy of the molecules does not change, i.e. these molecules are assumed to be perfectly elastic. We consider two groups of molecules that have velocities and before colliding with each other (collision) (Fig. 1), and after the collision, respectively, velocities and . The difference in speed is called relative speed, i.e. . It is clear that for a smooth elastic collision . Distribution functions f 1 ", f", f 1 , f describe the molecules of the corresponding groups after and before collisions, i.e. 
; 
; 
; 
.
Rice. 1. Collision of two molecules.
(13) includes two parameters characterizing the location of colliding molecules relative to each other: b and ε; b– aiming distance, i.e. the smallest distance that molecules would approach in the absence of interaction (Fig. 2); ε is called the collision angular parameter (Fig. 3). Integration over b from 0 to ¥ and from 0 to 2p (two external integrals in (12)) covers the entire plane of force interaction perpendicular to the vector
Rice. 2. The trajectory of the molecules.
Rice. 3. Consideration of the interaction of molecules in a cylindrical coordinate system: z, b, ε
The Boltzmann kinetic equation is derived under the following assumptions and assumptions.
1. It is believed that mainly collisions of two molecules occur, i.e. the role of collisions of three or more molecules simultaneously is insignificant. This assumption allows us to use a single-particle distribution function for analysis, which above is simply called the distribution function. Taking into account the collision of three molecules leads to the need to use a two-particle distribution function in the study. Accordingly, the analysis becomes significantly more complicated.
2. Assumption of molecular chaos. It is expressed in the fact that the probabilities of detecting particle 1 at the phase point and particle 2 at the phase point are independent of each other.
3. Collisions of molecules with any impact distance are equally probable, i.e. the distribution function does not change at the interaction diameter. It should be noted that the analyzed element must be small so that f within this element does not change, but at the same time so that the relative fluctuation ~ is not large. The interaction potentials used in calculating the collision integral are spherically symmetric, i.e. 
.
Maxwell-Boltzmann distribution
The equilibrium state of the gas is described by the absolute Maxwellian distribution, which is an exact solution of the Boltzmann kinetic equation:
where m is the mass of the molecule, kg.
The general local Maxwellian distribution, otherwise called the Maxwell-Boltzmann distribution:
in the case when the gas moves as a whole with speed and the variables n, T depend on the coordinate
and time t.
In the Earth's gravitational field, the exact solution of the Boltzmann equation shows:
Where n 0 = density at the Earth's surface, 1/m3; g– gravity acceleration, m/s 2 ; h– height, m. Formula (16) is an exact solution of the Boltzmann kinetic equation either in unlimited space or in the presence of boundaries that do not violate this distribution, while the temperature must also remain constant.
This page was designed by Puzina Yu.Yu. with the support of the Russian Foundation for Basic Research - project No. 08-08-00638.
Born in 1844 in Vienna. Boltzmann is a pioneer and pioneer in science. His works and research were often incomprehensible and rejected by society. However, with the further development of physics, his works were recognized and subsequently published.
The scientist's scientific interests covered such fundamental areas as physics and mathematics. Since 1867, he worked as a teacher in a number of higher educational institutions. In his research, he established that this is due to the chaotic impacts of molecules on the walls of the vessel in which they are located, while the temperature directly depends on the speed of movement of particles (molecules), in other words, on their Therefore, the higher the speed these particles move, the higher the temperature. Boltzmann's constant is named after the famous Austrian scientist. It was he who made an invaluable contribution to the development of static physics.
Physical meaning of this constant quantity
Boltzmann's constant defines the relationship between temperature and energy. In static mechanics it plays a major key role. Boltzmann's constant is equal to k=1.3806505(24)*10 -23 J/K. The numbers in parentheses indicate the permissible error of the value relative to the last digits. It is worth noting that Boltzmann's constant can also be derived from other physical constants. However, these calculations are quite complex and difficult to perform. They require deep knowledge not only in the field of physics, but also