Boltzmann's constant meaning. Relationship between temperature and energy. The physical essence of the Boltzmann constant
Among the fundamental constants is the Boltzmann constant k occupies a special place. Back in 1899, M. Planck proposed the following four numerical constants as fundamental for building a unified physics: the speed of light c, action quantum h, the gravitational constant G and the 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 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, which 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. Special attention It should be noted that the appearance of the Boltzmann constant in one or another formula of classical physics every time quite clearly indicates the statistical nature of the phenomenon described by it. Understanding the physical essence of the Boltzmann constant requires the opening of huge layers of physics - statistics and thermodynamics, the theory of evolution and cosmogony.
Research by L. Boltzmann
Beginning in 1866, the works of the Austrian theoretician L. Boltzmann were published one after another. In them statistical theory receives such a solid justification that it turns into a true science of physical properties collectives of particles.
The distribution was obtained by Maxwell for the simplest case of a monatomic ideal gas. In 1868, Boltzmann shows that polyatomic gases in 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 constituent parts molecules in space and their positions relative to each other "and shows that from the experimental data on the heat capacity of gases follows a uniform distribution of energy between varying degrees freedom. Each degree of freedom has the same energy
Boltzmann directly connected the characteristics of the microcosm with the characteristics of the macrocosm. Here is the key formula that establishes this ratio:
1/2 mv2 = kT
where m and v- respectively, the mass and average speed of movement of gas molecules, T is the gas temperature (on the absolute Kelvin scale), and k is the Boltzmann constant. This equation bridges the two worlds by linking atomic level properties (on the left side) with bulk properties (on the right side) that can be measured with human instruments, in this case thermometers. This connection is provided by the Boltzmann constant k, equal to 1.38 x 10-23 J/K.
Finishing the conversation about the Boltzmann constant, I would like to emphasize once again its fundamental importance in science. It contains huge layers of physics - atomistics and 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 come to a state of thermodynamic equilibrium is valid only for isolated systems and systems that are in stationary external conditions. In our Universe, processes are continuously taking place, the result of which is a change in its spatial properties. The non-stationarity of the Universe inevitably leads to the absence of statistical equilibrium in it.
Boltzmann constant throws a bridge from the macrocosm to the microcosm, linking the 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. Within the framework of this picture, the gas pressure is due to the elastic impacts of gas molecules on the walls of the vessel, and the temperature is due to the speed 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 connect the characteristics of the microcosm with the characteristics of the macrocosm, in particular, with the readings of a thermometer. Here is the key formula that establishes this ratio:
1/2 mv 2 = kT
where m and v - respectively, the mass and average velocity of gas molecules, T is the gas temperature (on the absolute Kelvin scale), and k - Boltzmann's constant. This equation bridges the two worlds by linking the characteristics of the atomic level (on the left side) with bulk properties(on the right side) that can be measured with 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 microcosm and the macrocosm is called statistical mechanics. In this section, there is hardly an equation or formula in which the Boltzmann constant would not appear. One of these ratios was derived by the Austrian himself, and it is simply called Boltzmann equation:
S = k log p + b
where S- system entropy ( cm. second law of thermodynamics) p- so-called statistical weight(a very important element of the statistical approach), and b is 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 his time. scientific community, who considered atoms only a convention, convenient for calculations, but not objects of the real world. When his statistical approach did not meet the slightest understanding even after the advent of special theory 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 in the family of a civil servant. He studied at the University of Vienna on the same course with Josef Stefan ( cm. Stefan-Boltzmann law). Having defended his defense in 1866, he continued his scientific career, at various times holding professorships in the departments of physics and mathematics at the universities of Graz, Vienna, Munich and Leipzig. As one of the main supporters 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.
Boltzmann's constant (k (\displaystyle k) or k B (\displaystyle k_(\rm (B)))) is a physical constant that determines the relationship between temperature and energy. Named after the Austrian physicist Ludwig Boltzmann, who made a major contribution to statistical physics, in which this constant plays key role. Its value in the International System of Units SI according to the change in the definitions of the 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 translational degree of freedom is, as follows from the Maxwell distribution, kT / 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 energy in 3 2 k T (\displaystyle (\frac (3)(2))kT).
Knowing the thermal energy, one can calculate the rms atomic velocity, which is inversely proportional to square root 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 log Z . (\displaystyle S=k\ln Z.)Proportionality factor k (\displaystyle k) and is the Boltzmann 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.
Boltzmann's constant (k (\displaystyle k) or k B (\displaystyle k_(\rm (B)))) is a physical constant that determines 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 the change in the definitions of the 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 translational degree of freedom is, as follows from the Maxwell distribution, kT / 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 energy in 3 2 k T (\displaystyle (\frac (3)(2))kT).
Knowing the thermal energy, one can calculate the root-mean-square atomic velocity, 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 log Z . (\displaystyle S=k\ln Z.)Proportionality factor k (\displaystyle k) and is the Boltzmann 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.
Boltzmann constant (k or k b) is a physical constant that determines the relationship between and . Named after the Austrian physicist, who made a great contribution to, in which this constant plays a key role. Its experimental value in the system is
k = 1.380\;6505(24)\times 10^(-23) / .The numbers in parentheses indicate the standard error in the last digits of the value. In principle, the Boltzmann constant can be derived from the determination of absolute temperature and other physical constants. However, the calculation of the Boltzmann constant using the basic principles is too complicated and impossible for modern level knowledge. AT natural system Planck units The natural unit of temperature is given so that the Boltzmann constant is equal to one.
Relationship between temperature and energy.
Definition of entropy.
The 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 ZProportionality factor k and is the Boltzmann constant. This expression defining the relationship between microscopic (Z) and macroscopic states (S) expresses the central idea of statistical mechanics.
