How is the constant measured? Boltzmann's constant plays a major role in static mechanics
As an exact quantitative science, physics cannot do without a set of very important constants that are included as universal coefficients in equations that establish relationships between certain quantities. These are fundamental constants, thanks to which such relationships become invariant and are able to explain the behavior of physical systems at different scales.
Among such parameters that characterize the properties inherent in the matter of our Universe is the Boltzmann constant, a quantity included in a number of the most important equations. However, before turning to a consideration of its features and significance, one cannot help but say a few words about the scientist whose name it bears.
Ludwig Boltzmann: scientific achievements
One of the greatest scientists of the 19th century, the Austrian Ludwig Boltzmann (1844-1906) made a significant contribution to the development of molecular kinetic theory, becoming one of the creators of statistical mechanics. He was the author of the ergodic hypothesis, a statistical method in the description of an ideal gas, and the basic equation of physical kinetics. He worked a lot on issues of thermodynamics (Boltzmann's H-theorem, statistical principle for the second law of thermodynamics), radiation theory (Stefan-Boltzmann law). In his works he also touched upon some issues of electrodynamics, optics and other branches of physics. His name is immortalized in two physical constants, which will be discussed below.
Ludwig Boltzmann was a convinced and consistent supporter of the theory of the atomic-molecular structure of matter. For many years, he had to struggle with misunderstanding and rejection of these ideas in the scientific community of the time, when many physicists considered atoms and molecules to be an unnecessary abstraction, at best a conventional device for the convenience of calculations. A painful illness and attacks from conservative colleagues provoked Boltzmann into severe depression, which, unable to bear, led the outstanding scientist to commit suicide. On the grave monument, above the bust of Boltzmann, as a sign of recognition of his merits, the equation S = k∙logW is engraved - one of the results of his fruitful scientific work. The constant k in this equation is Boltzmann's constant.
Energy of molecules and temperature of matter
The concept of temperature serves to characterize the degree of heating of a particular body. In physics, an absolute temperature scale is used, which is based on the conclusion of the molecular kinetic theory about temperature as a measure reflecting the amount of energy of thermal motion of particles of a substance (meaning, of course, the average kinetic energy of a set of particles).
Both the SI joule and the erg used in the CGS system are too large units to express the energy of molecules, and in practice it was very difficult to measure temperature in this way. A convenient unit of temperature is the degree, and the measurement is carried out indirectly, through recording the changing macroscopic characteristics of a substance - for example, volume.
How do energy and temperature relate?
To calculate the states of real matter at temperatures and pressures close to normal, the model of an ideal gas is successfully used, that is, one whose molecular size is much smaller than the volume occupied by a certain amount of gas, and the distance between particles significantly exceeds the radius of their interaction. Based on the equations of kinetic theory, the average energy of such particles is determined as E av = 3/2∙kT, where E is the kinetic energy, T is the temperature, and 3/2∙k is the proportionality coefficient introduced by Boltzmann. The number 3 here characterizes the number of degrees of freedom of translational motion of molecules in three spatial dimensions.
The value k, which was later named the Boltzmann constant in honor of the Austrian physicist, shows how much of a joule or erg contains one degree. In other words, its value determines how much the energy of thermal chaotic motion of one particle of a monatomic ideal gas increases statistically, on average, with an increase in temperature by 1 degree.
How many times is a degree smaller than a joule?
The numerical value of this constant can be obtained in various ways, for example, by measuring absolute temperature and pressure, using the ideal gas equation, or using a Brownian motion model. Theoretical derivation of this value at the current level of knowledge is not possible.
Boltzmann's constant is equal to 1.38 × 10 -23 J/K (here K is kelvin, a degree on the absolute temperature scale). For a group of particles in 1 mole of an ideal gas (22.4 liters), the coefficient relating energy to temperature (universal gas constant) is obtained by multiplying Boltzmann’s constant by Avogadro’s number (the number of molecules in a mole): R = kN A, and is 8.31 J/(mol∙kelvin). However, unlike the latter, the Boltzmann constant is more universal in nature, since it is included in other important relationships, and also serves to determine another physical constant.
Statistical distribution of molecular energies
Since macroscopic states of matter are the result of the behavior of a large collection of particles, they are described using statistical methods. The latter also includes finding out how the energy parameters of gas molecules are distributed:
- Maxwellian distribution of kinetic energies (and velocities). It shows that in a gas in a state of equilibrium, most molecules have velocities close to some most probable speed v = √(2kT/m 0), where m 0 is the mass of the molecule.
- Boltzmann distribution of potential energies for gases located in the field of any forces, for example, Earth's gravity. It depends on the relationship between two factors: attraction to the Earth and the chaotic thermal movement of gas particles. As a result, the lower the potential energy of the molecules (closer to the surface of the planet), the higher their concentration.
Both statistical methods are combined into a Maxwell-Boltzmann distribution containing an exponential factor e - E/ kT, where E is the sum of kinetic and potential energies, and kT is the already known average energy of thermal motion, controlled by the Boltzmann constant.
Constant k and entropy
In a general sense, entropy can be characterized as a measure of the irreversibility of a thermodynamic process. This irreversibility is associated with the dissipation - dissipation - of energy. In the statistical approach proposed by Boltzmann, entropy is a function of the number of ways in which a physical system can be realized without changing its state: S = k∙lnW.
Here the constant k specifies the scale of entropy growth with an increase in this number (W) of system implementation options, or microstates. Max Planck, who brought this formula to its modern form, suggested giving the constant k the name Boltzmann.
Stefan-Boltzmann radiation law
The physical law that establishes how the energetic luminosity (radiation power per unit surface) of an absolutely black body depends on its temperature has the form j = σT 4, that is, the body emits proportional to the fourth power of its temperature. This law is used, for example, in astrophysics, since the radiation of stars is close in characteristics to blackbody radiation.
In this relationship there is another constant, which also controls the scale of the phenomenon. This is the Stefan-Boltzmann constant σ, which is approximately 5.67 × 10 -8 W/(m 2 ∙K 4). Its dimension includes kelvins - which means it is clear that the Boltzmann constant k is involved here too. Indeed, the value of σ is defined as (2π 2 ∙k 4)/(15c 2 h 3), where c is the speed of light and h is Planck’s constant. So the Boltzmann constant, combined with other world constants, forms a quantity that again connects energy (power) and temperature - in this case in relation to radiation.
The physical essence of the Boltzmann constant
It was already noted above that Boltzmann’s constant is one of the so-called fundamental constants. The point is not only that it allows us to establish a connection between the characteristics of microscopic phenomena at the molecular level and the parameters of processes observed in the macrocosm. And not only that this constant is included in a number of important equations.
It is currently unknown whether there is any physical principle on the basis of which it could be derived theoretically. In other words, it does not follow from anything that the value of a given constant should be exactly that. We could use other quantities and other units instead of degrees as a measure of compliance with the kinetic energy of particles, then the numerical value of the constant would be different, but it would remain a constant value. Along with other fundamental quantities of this kind - the limiting speed c, the Planck constant h, the elementary charge e, the gravitational constant G - science accepts the Boltzmann constant as a given of our world and uses it for a theoretical description of the physical processes occurring in it.
Boltzmann Ludwig (1844-1906)- great Austrian physicist, one of the founders of molecular kinetic theory. In the works of Boltzmann, the molecular kinetic theory first appeared as a logically coherent, consistent physical theory. Boltzmann gave a statistical interpretation of the second law of thermodynamics. He did a lot to develop and popularize Maxwell's theory of the electromagnetic field. A fighter by nature, Boltzmann passionately defended the need for a molecular interpretation of thermal phenomena and bore the brunt of the struggle against scientists who denied the existence of molecules.
Equation (4.5.3) includes the ratio of the universal gas constant R to Avogadro's constant N A . This ratio is the same for all substances. It is called the Boltzmann constant, in honor of L. Boltzmann, one of the founders of molecular kinetic theory.
Boltzmann's constant is:
(4.5.4)
Equation (4.5.3) taking into account the Boltzmann constant is written as follows:
(4.5.5)
Physical meaning of the Boltzmann constant
Historically, temperature was first introduced as a thermodynamic quantity, and its unit of measurement was established - degrees (see § 3.2). After establishing the connection between temperature and the average kinetic energy of molecules, it became obvious that temperature can be defined as the average kinetic energy of molecules and expressed in joules or ergs, i.e., instead of the quantity T enter value T* so that 
The temperature thus defined is related to the temperature expressed in degrees as follows:
Therefore, Boltzmann's constant can be considered as a quantity that relates temperature, expressed in energy units, to temperature, expressed in degrees.
Dependence of gas pressure on the concentration of its molecules and temperature
Having expressed E from relation (4.5.5) and substituting it into formula (4.4.10), we obtain an expression showing the dependence of gas pressure on the concentration of molecules and temperature:
(4.5.6)
From formula (4.5.6) it follows that at the same pressures and temperatures, the concentration of molecules in all gases is the same.
This implies Avogadro's law: equal volumes of gases at the same temperatures and pressures contain the same number of molecules.
The average kinetic energy of the translational motion of molecules is directly proportional to the absolute temperature. Proportionality factor- Boltzmann constantk = 10 -23 J/K - need to remember.
§ 4.6. Maxwell distribution
In a large number of cases, knowledge of average values of physical quantities alone is not enough. For example, knowing the average height of people does not allow us to plan the production of clothing in different sizes. You need to know the approximate number of people whose height lies in a certain interval. Likewise, it is important to know the numbers of molecules that have velocities different from the average value. Maxwell was the first to discover how these numbers could be determined.
Probability of a random event
In §4.1 we already mentioned that to describe the behavior of a large collection of molecules, J. Maxwell introduced the concept of probability.
As has been repeatedly emphasized, it is in principle impossible to trace the change in speed (or momentum) of one molecule over a large interval of time. It is also impossible to accurately determine the velocities of all gas molecules at a given time. From the macroscopic conditions in which a gas is located (a certain volume and temperature), certain values of molecular speeds do not necessarily follow. The speed of a molecule can be considered as a random variable, which under given macroscopic conditions can take on different values, just as when throwing a die you can get any number of points from 1 to 6 (the number of sides of the die is six). It is impossible to predict the number of points that will come up when throwing a dice. But the probability of rolling, say, five points is determinable.
What is the probability of a random event occurring? Let a very large number be produced N tests (N - number of dice throws). At the same time, in N" cases, there was a favorable outcome of the tests (i.e., dropping a five). Then the probability of a given event is equal to the ratio of the number of cases with a favorable outcome to the total number of trials, provided that this number is as large as desired:
(4.6.1)
For a symmetrical die, the probability of any chosen number of points from 1 to 6 is .
We see that against the background of many random events, a certain quantitative pattern is revealed, a number appears. This number - the probability - allows you to calculate averages. So, if you throw 300 dice, then the average number of fives, as follows from formula (4.6.1), will be equal to: 300 = 50, and it makes absolutely no difference whether you throw the same dice 300 times or 300 identical dice at the same time .
There is no doubt that the behavior of gas molecules in a vessel is much more complex than the movement of a thrown dice. But here, too, one can hope to discover certain quantitative patterns that make it possible to calculate statistical averages, if only the problem is posed in the same way as in game theory, and not as in classical mechanics. It is necessary to abandon the insoluble problem of determining the exact value of the speed of a molecule at a given moment and try to find the probability that the speed has a certain value.
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.
The defining 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 experimental value in the International System of Units (SI) is:
J/.The numbers in parentheses indicate the standard error in the last digits of the quantity value. Boltzmann's constant can be obtained from the definition of absolute temperature and other physical constants. However, calculating Boltzmann's constant using first principles is too complex and infeasible with the current state of knowledge. In the natural system of Planck units, the natural unit of temperature is given so that Boltzmann's constant is equal to unity.
Relationship between temperature and energy
In a homogeneous ideal gas at absolute temperature, the energy per each translational degree of freedom is, as follows from the Maxwell distribution, . At room temperature (300°C) this energy is 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 .
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 approximately five degrees of freedom.
Definition of entropy
The entropy of a thermodynamic system is defined as the natural logarithm of the number of distinct microstates corresponding to a given macroscopic state (for example, a state with a given total energy).
The proportionality coefficient is Boltzmann's constant. This expression, which defines the connection between microscopic () and macroscopic states (), expresses the central idea of statistical mechanics.
see also
Notes
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See what "Boltzmann's constant" is in other dictionaries:
- (symbol k), the ratio of the universal GAS constant to the AVOGADRO NUMBER, equal to 1.381.10 23 joules per degree Kelvin. It indicates the relationship between the kinetic energy of a gas particle (atom or molecule) and its absolute temperature.... ... Scientific and technical encyclopedic dictionary
Boltzmann constant- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN Boltzmann constant ... Technical Translator's Guide
Boltzmann's constant- Boltzmann Constant Boltzmann Constant 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 ... Explanatory English-Russian dictionary on nanotechnology. - M.
Boltzmann constant- Bolcmano konstanta statusas T sritis fizika atitikmenys: engl. Boltzmann constant vok. Boltzmann Constante, f; Boltzmannsche Konstante, f rus. Boltzmann constant, f pranc. constante de Boltzmann, f … Fizikos terminų žodynas
Relationship S k lnW between entropy S and thermodynamic probability W (k Boltzmann constant). The statistical interpretation of the second law of thermodynamics is based on the Boltzmann principle: natural processes tend to transform the thermodynamic... ...
- (Maxwell Boltzmann distribution) equilibrium distribution of ideal gas particles by energy (E) in an external force field (for example, in a gravitational field); is determined by the distribution function f e E/kT, where E is the sum of kinetic and potential energies... Big Encyclopedic Dictionary
Not to be confused with Boltzmann's constant. Stefan Boltzmann's constant (also Stefan's constant), a physical constant that is the constant of proportionality in Stefan Boltzmann's law: the total energy emitted per unit area... Wikipedia
Value of the constant Dimension 1.380 6504(24)×10−23 J K−1 8.617 343(15)×10−5 eV K−1 1.3807×10−16 erg K−1 Boltzmann constant (k or kb) a physical constant that defines the relationship between temperature and energy. Named after the Austrian... ... Wikipedia
Statistically equilibrium distribution function over the momenta and coordinates of particles of an ideal gas, molecules of which obey the classical. mechanics, in an external potential field: Here the Boltzmann constant (universal constant), absolute... ... Mathematical Encyclopedia
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- The Universe and physics without “dark energy” (discoveries, ideas, hypotheses). In 2 volumes. Volume 1, O. G. Smirnov. The books are devoted to problems of physics and astronomy that have existed in science for tens and hundreds of years from G. Galileo, I. Newton, A. Einstein to the present day. The smallest particles of matter and planets, stars and...
