What is Boltzmann's constant denoted? Boltzmann constant value

Among the fundamental constants, Boltzmann's constant k occupies a special place. Back in 1899, M. Planck proposed the following four numerical constants as fundamental for the construction of unified physics: the speed of light c, quantum of action h, gravitational constant G and Boltzmann constant k. Among these constants, k occupies a special place. It does not define elementary physical processes and is not included in the basic principles of dynamics, but it establishes a connection between microscopic dynamic phenomena and macroscopic characteristics of the state of particles. It is also included in the fundamental law of nature that relates the entropy of the system S with the thermodynamic probability of its state W:

S=klnW (Boltzmann formula)

and determining the direction of physical processes in nature. Particular attention should be paid to the fact that the appearance of the Boltzmann constant in one or another formula of classical physics each time clearly indicates the statistical nature of the phenomenon it describes. Understanding the physical essence of Boltzmann's constant requires uncovering enormous layers of physics - statistics and thermodynamics, the theory of evolution and cosmogony.

Research by L. Boltzmann

Since 1866, the works of the Austrian theorist L. Boltzmann have been published one after another. In them, the statistical theory receives such a solid foundation that it turns into a genuine science about the physical properties of groups of particles.

The distribution was obtained by Maxwell for the simplest case of a monatomic ideal gas. In 1868, Boltzmann showed that polyatomic gases in a state of equilibrium will also be described by the Maxwell distribution.

Boltzmann develops in the works of Clausius the idea that gas molecules cannot be considered as separate material points. Polyatomic molecules also have rotation of the molecule as a whole and vibrations of its constituent atoms. He introduces the number of degrees of freedom of molecules as the number of “variables required to determine the position of all the constituent parts of a molecule in space and their position relative to each other” and shows that from experimental data on the heat capacity of gases it follows that there is a uniform distribution of energy between the various degrees of freedom. Each degree of freedom accounts for the same energy

Boltzmann directly linked the characteristics of the microworld with the characteristics of the macroworld. Here is the key formula that establishes this relationship:

1/2 mv2 = kT

Where m And v- respectively, the mass and average speed of movement of gas molecules, T- gas temperature (on the absolute Kelvin scale), and k- Boltzmann constant. This equation bridges the gap between the two worlds, linking atomic level properties (on the left side) with bulk properties (on the right side) that can be measured using human instruments, in this case thermometers. This relationship is provided by Boltzmann's constant k, equal to 1.38 x 10-23 J/K.

Finishing the conversation about the Boltzmann constant, I would like to once again emphasize its fundamental importance in science. It contains enormous layers of physics - atomism and the molecular-kinetic theory of the structure of matter, statistical theory and the essence of thermal processes. The study of the irreversibility of thermal processes revealed the nature of physical evolution, concentrated in the Boltzmann formula S=klnW. It should be emphasized that the position according to which a closed system will sooner or later reach a state of thermodynamic equilibrium is valid only for isolated systems and systems under stationary external conditions. Processes are continuously occurring in our Universe, the result of which is a change in its spatial properties. The nonstationarity of the Universe inevitably leads to the absence of statistical equilibrium in it.

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 SI system is

J/.

The numbers in parentheses indicate the standard error in the last digits of the quantity value. In principle, Boltzmann's constant can be obtained from the definition of absolute temperature and other physical constants. However, calculating the Boltzmann 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 T, the energy per each translational degree of freedom is equal, as follows from the Maxwell distribution kT/ 2 . At room temperature (300 ) 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 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 already 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 different microstates Z, corresponding to a given macroscopic state (for example, a state with a given total energy).

S = k ln Z.

Proportionality factor k and is Boltzmann's constant. This is an expression that defines the relationship between microscopic ( Z) and macroscopic states ( S), expresses the central idea of statistical mechanics.

Big Encyclopedic Dictionary One of the fundamental physical constants; equal to the ratio of the gas constant R to the Avogadro constant NA, denoted by k; named after the Austrian physicist L. Boltzmann. The bp is included in a number of the most important relations of physics: in the equation... ...

Physical encyclopedia- (k) universal physical. constant equal to the ratio of the universal gas (see) to the Avogadro constant NA: k = R/Na = (1.380658 ± 000012)∙10 23 J/K ... Big Polytechnic Encyclopedia

Physical constant k, equal to the ratio of the universal gas constant R to the Avogadro number NA: k = R/NA = 1.3807·10 23 J/K. Named after L. Boltzmann. * * * BOLTZMANN’S CONSTANT BOLTZMANN’S CONSTANT, physical constant k, equal to... ... encyclopedic Dictionary

Phys. constant k, equal to the ratio of the universal. gas constant R to the Avogadro number NA: k = R/NA = 1.3807 x 10 23 J/K. Named after L. Boltzmann... Natural science. encyclopedic Dictionary

One of the basic physical constants (See Physical constants), equal to the ratio of the universal gas constant R to the Avogadro number NA. (number of molecules in 1 mole or 1 kmol of a substance): k = R/NA. Named after L. Boltzmann. B. p.... ... Great Soviet Encyclopedia

Boltzmann's constant builds a bridge from the macrocosm to the microcosm, connecting temperature with the kinetic energy of molecules.

Ludwig Boltzmann is one of the creators of the molecular kinetic theory of gases, on which the modern picture of the relationship between the movement of atoms and molecules, on the one hand, and the macroscopic properties of matter, such as temperature and pressure, on the other, is based. In this picture, gas pressure is determined by the elastic impacts of gas molecules on the walls of the vessel, and temperature is determined by the speed of movement of the molecules (or rather, their kinetic energy). The faster the molecules move, the higher the temperature.

Boltzmann's constant makes it possible to directly relate the characteristics of the microworld with the characteristics of the macroworld - in particular, with thermometer readings. Here is the key formula that establishes this relationship:

1/2 mv 2 = kT

Where m And v— respectively, the mass and average speed of gas molecules, T is the gas temperature (on the absolute Kelvin scale), and k — Boltzmann's constant. This equation bridges the gap between the two worlds, linking the characteristics of the atomic level (on the left side) with volumetric properties(on the right side), which can be measured using human instruments, in this case thermometers. This connection is provided by the Boltzmann constant k, equal to 1.38 x 10 -23 J/K.

The branch of physics that studies the connections between the phenomena of the microworld and the macroworld is called statistical mechanics. There is hardly an equation or formula in this section that does not include Boltzmann's constant. One of these relationships was derived by the Austrian himself, and it is simply called Boltzmann equation:

S = k log p + b

Where S— entropy of the system ( cm. Second law of thermodynamics) p- so-called statistical weight(a very important element of the statistical approach), and b- another constant.

Throughout his life, Ludwig Boltzmann was literally ahead of his time, developing the foundations of the modern atomic theory of the structure of matter, entering into fierce disputes with the overwhelming conservative majority of the scientific community of his day, who considered atoms only a convention, convenient for calculations, but not objects of the real world. When his statistical approach did not meet with the slightest understanding even after the advent of the special theory of relativity, Boltzmann committed suicide in a moment of deep depression. Boltzmann's equation is carved on his tombstone.

Boltzmann, 1844-1906

Austrian physicist. Born in Vienna into the family of a civil servant. Studied at the University of Vienna on the same course with Josef Stefan ( cm. Stefan-Boltzmann law). Having defended his degree in 1866, he continued his scientific career, holding at various times professorships in the departments of physics and mathematics at the universities of Graz, Vienna, Munich and Leipzig. Being one of the main proponents of the reality of the existence of atoms, he made a number of outstanding theoretical discoveries that shed light on how phenomena at the atomic level affect the physical properties and behavior of matter.

For a constant related to the energy of blackbody radiation, see Stefan-Boltzmann Constant

Constant value k	Dimension
1,380 6504(24) 10 −23
8,617 343(15) 10 −5
1,3807 10 −16
See also Values in various units below.

Boltzmann's constant (k or k B) is a physical constant that determines the relationship between the temperature of a substance and the energy of thermal motion of particles of this substance. 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 SI system is

In the table, the last numbers in parentheses indicate the standard error of the constant value. In principle, Boltzmann's constant can be obtained from the definition of absolute temperature and other physical constants. However, accurately calculating Boltzmann's constant using first principles is too complex and infeasible with the current state of knowledge.

Boltzmann's constant can be determined experimentally using Planck's law of thermal radiation, which describes the energy distribution in the spectrum of equilibrium radiation at a certain temperature of the emitting body, as well as other methods.

There is a relationship between the universal gas constant and Avogadro's number, from which the value of Boltzmann's constant follows:

The dimension of Boltzmann's constant is the same as that of entropy.

1. History
2 Ideal gas equation of state
3 Relationship between temperature and energy

3.1 Gas thermodynamics relations

4 Boltzmann multiplier
5 Role in the statistical determination of entropy
6 Role in semiconductor physics: thermal stress
7 Applications in other areas
8 Boltzmann's constant in Planck units
9 Boltzmann's constant in the theory of infinite nesting of matter
10 Values in different units
11 Links
12 See also

Story

In 1877, Boltzmann was the first to connect entropy and probability, but a fairly accurate value of the constant k as a coupling coefficient in the formula for entropy appeared only in the works of M. Planck. When deriving the law of black body radiation, Planck in 1900–1901. for the Boltzmann constant, he found a value of 1.346 10 −23 J/K, almost 2.5% less than the currently accepted value.

Before 1900, the relations that are now written with the Boltzmann constant were written using the gas constant R, and instead of the average energy per molecule, the total energy of the substance was used. Laconic formula of the form S = k log W on the bust of Boltzmann became such thanks to Planck. In his Nobel lecture in 1920, Planck wrote:

This constant is often called Boltzmann's constant, although, as far as I know, Boltzmann himself never introduced it - a strange state of affairs, despite the fact that Boltzmann's statements did not talk about the exact measurement of this constant.

This situation can be explained by the ongoing scientific debate at that time to clarify the essence of the atomic structure of matter. In the second half of the 19th century, there was considerable disagreement as to whether atoms and molecules were real or just a convenient way of describing phenomena. There was also no consensus as to whether the "chemical molecules" distinguished by their atomic mass were the same molecules as in the kinetic theory. Further in Planck's Nobel lecture one can find the following:

“Nothing can better demonstrate the positive and accelerating rate of progress than the art of experiment during the last twenty years, when many methods have been discovered at once for measuring the mass of molecules with almost the same accuracy as measuring the mass of a planet.”

Ideal gas equation of state

For an ideal gas, the unified gas law relating pressure is valid P, volume V, amount of substance n in moles, gas constant R and absolute temperature T:

In this equality, you can make a substitution. Then the gas law will be expressed in terms of the Boltzmann constant and the number of molecules N in gas volume V:

Relationship between temperature and energy

In a homogeneous ideal gas at absolute temperature T, the energy per each translational degree of freedom is equal, as follows from the Maxwell distribution, kT/ 2 . At room temperature (≈ 300 K) this energy is J, or 0.013 eV.

Gas thermodynamics relations

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 kT/ 2 . This agrees well with experimental data. 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.

Kinetic theory gives a formula for average pressure P ideal gas:

Considering that the average kinetic energy of rectilinear motion is equal to:

we find the equation of state of an ideal gas:

This relationship holds well for molecular gases; however, the dependence of the heat capacity changes, since the molecules can have additional internal degrees of freedom in relation to those degrees of freedom that are associated with the movement of molecules in space. For example, a diatomic gas already has approximately five degrees of freedom.

Boltzmann multiplier

In general, the system is in equilibrium with a thermal reservoir at a temperature T has a probability p occupy a state of energy E, which can be written using the corresponding exponential Boltzmann multiplier:

This expression involves the quantity kT with the dimension of energy.

Probability calculation is used not only for calculations in the kinetic theory of ideal gases, but also in other areas, for example in chemical kinetics in the Arrhenius equation.

Role in the statistical determination of entropy

Main article: Thermodynamic entropy

Entropy S of an isolated thermodynamic system in thermodynamic equilibrium is determined through the natural logarithm of the number of different microstates W, corresponding to a given macroscopic state (for example, a state with a given total energy E):

Proportionality factor k is Boltzmann's constant. This is an expression that defines the relationship between microscopic and macroscopic states (via W and entropy S accordingly), expresses the central idea of statistical mechanics and is the main discovery of Boltzmann.

Classical thermodynamics uses the Clausius expression for entropy:

Thus, the appearance of the Boltzmann constant k can be seen as a consequence of the connection between thermodynamic and statistical definitions of entropy.

Entropy can be expressed in units k, which gives the following:

In such units, entropy exactly corresponds to information entropy.

Characteristic energy kT equal to the amount of heat required to increase entropy S"for one nat.

Role in semiconductor physics: thermal stress

Unlike other substances, in semiconductors there is a strong dependence of electrical conductivity on temperature:

where the factor σ 0 depends rather weakly on temperature compared to the exponential, E A– conduction activation energy. The density of conduction electrons also depends exponentially on temperature. For the current through a semiconductor p-n junction, instead of the activation energy, consider the characteristic energy of a given p-n junction at temperature T as the characteristic energy of an electron in an electric field:

Where q- , A V T there is thermal stress depending on temperature.

This relationship is the basis for expressing the Boltzmann constant in units of eV∙K −1. At room temperature (≈ 300 K) the thermal voltage value is about 25.85 millivolts ≈ 26 mV.

In classical theory, a formula is often used, according to which the effective speed of charge carriers in a substance is equal to the product of the carrier mobility μ and the electric field strength. Another formula relates the carrier flux density to the diffusion coefficient D and with a carrier concentration gradient n :

According to the Einstein-Smoluchowski relation, the diffusion coefficient is related to mobility:

Boltzmann's constant k is also included in the Wiedemann-Franz law, according to which the ratio of the thermal conductivity coefficient to the electrical conductivity coefficient in metals is proportional to the temperature and the square of the ratio of the Boltzmann constant to the electric charge.

Applications in other areas

To delimit temperature regions in which the behavior of matter is described by quantum or classical methods, the Debye temperature is used:

Where - , is the limiting frequency of elastic vibrations of the crystal lattice, u– speed of sound in a solid, n– concentration of atoms.

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 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.