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.

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.

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

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.

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.

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.