What is the constant measured in? Boltzmann constant
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 value. Boltzmann's constant can be obtained from the definition absolute temperature and other physical constants. However, the calculation of the Boltzmann constant using basic principles is too complicated and impossible with the current level of knowledge. In Planck's natural system of units, the natural unit of temperature is given in such a way that Boltzmann's constant is equal to one.
Relationship between temperature and energy
In a homogeneous ideal gas at absolute temperature , the energy per translational degree of freedom is, as follows from the Maxwell distribution, . 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 energy in .
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 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 corresponding to a given macroscopic state (for example, a state with a given total energy).
The coefficient of proportionality is the Boltzmann constant. This expression defining the relation between microscopic () and macroscopic states () expresses the central idea of statistical mechanics.
see also
Notes
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See what the "Boltzmann constant" is in other dictionaries:
- (denotation 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's constant- - [A.S. Goldberg. English Russian Energy Dictionary. 2006] Topics energy in general EN Boltzmann constant … Technical Translator's Handbook
Boltzmann 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 a great contribution to statistical physics, in which this constant ... Explanatory English-Russian dictionary on nanotechnology. - M.
Boltzmann's constant- Bolcmano konstanta statusas T sritis fizika atitikmenys: engl. Boltzmann constant vok. Boltzmann Konstante, f; Boltzmannsche Konstante, f rus. Boltzmann's constant, fpranc. constante de Boltzmann, f … Fizikos terminų žodynas
The relation S k lnW between the entropy S and the thermodynamic probability W (k is Boltzmann's constant). The Boltzmann principle is based on the statistical interpretation of the second law of thermodynamics: natural processes tend to translate the thermodynamic ... ...
- (Maxwell Boltzmann distribution) equilibrium energy distribution of ideal gas particles (E) in an external force field (eg, in a gravitational field); is determined by the distribution function f e E/kT, where E is the sum of the kinetic and potential energies … Big Encyclopedic Dictionary
Not to be confused with the Boltzmann constant. Stefan Boltzmann's constant (also Stefan's constant), a physical constant that is a constant of proportionality in Stefan Boltzmann's law: total energy radiated per unit area ... Wikipedia
Constant value Unit 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 in terms of momenta and coordinates of particles of an ideal gas, molecules to which obey the classical. mechanics, in an external potential field: Here is the Boltzmann constant (universal constant), absolute ... ... Mathematical Encyclopedia
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- Universe and physics without "dark energy" (discoveries, ideas, hypotheses). In 2 volumes. Volume 1, O. G. Smirnov. The books are devoted to the problems of physics and astronomy that have existed in science for decades and hundreds of years from G. Galileo, I. Newton, A. Einstein to the present day. The smallest particles of matter and planets, stars and ...
The Boltzmann constant bridges the gap from the macrocosm to the microcosm, linking 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 painting 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. Within this picture, the gas pressure is due to elastic shocks gas molecules against the walls of the vessel, and the temperature is the speed of the molecules (or rather, their kinetic energy). The faster the molecules move, the higher the temperature.
The Boltzmann constant makes it possible to directly connect the characteristics of the microworld 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 - weight and average speed movement 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 violent disputes with the overwhelming conservative majority of the contemporary scientific community, who considered atoms only a convention convenient for calculations, but not objects. 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 in the family of a civil servant. Studied at 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 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 physical properties and behavior of matter.
According to the Stefan-Boltzmann law, the density of the integral hemispherical radiation E0 depends only on temperature and varies in proportion to the fourth power of absolute temperature T:
Stefan - Boltzmann constant σ 0 is a physical constant included in the law that determines the volumetric density of equilibrium thermal radiation of a black body:
Historically, the Stefan-Boltzmann law was formulated before Planck's radiation law, from which it follows as a corollary. Planck's law establishes the dependence of the spectral density of the radiation flux E 0 on wavelength λ and temperature T:
where λ is the wavelength, m; from\u003d 2.998 10 8 m / s - the speed of light in vacuum; T– body temperature, K;
h\u003d 6.625 × 10 -34 J × s - Planck's constant.
Physical constant k equal to the ratio of the universal gas constant R\u003d 8314J / (kg × K) to Avogadro's number NA\u003d 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:
Value W there are a number of ways to distribute N particles by energy levels. If relation (6) is valid, then it is considered that the original system obeys the Boltzmann statistics. Set of numbers n i, at which the number W maximum, occurs most often, 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 by probabilistic methods. For a monatomic gas, the state of a set of molecules is determined by their coordinates and by the values of the velocity projections on the corresponding coordinate axes. Mathematically, this is described by a distribution function that characterizes the probability of a particle being in a given state:
is the expected number of molecules in the 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 the 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 molecules in this gas to the characteristic size of the flow L of course, i.e.
because run length is inversely proportional nd 2(n is the number density 1/m3, d is the diameter of the molecule, m).
the value
called H- the Boltzmann function for a unit of volume, which is related to the probability of detecting a system of gas molecules in a given state. Each state corresponds to certain occupation numbers of six-dimensional space-velocity cells, into which the phase space of the considered molecules can be divided. Denote W the probability that in the first cell of the space under consideration there will be N 1 molecules, in the second cell N 2, etc.
Up to a constant that determines the origin of the probability, the following relation is valid:
,
where 
– H-function of the region of space BUT occupied by gas. From (9) it can be seen that W And H interrelated, i.e. a change in the probability of a state leads to a corresponding evolution of the H function.
The Boltzmann principle establishes a relationship between entropy S physical system and thermodynamic probability W her status:
(printed according to the publication: Kogan M.N. Dynamics of rarefied gas. - M .: Nauka, 1967.)
General view of the CUBE:
where is the body force due to the presence of various fields (gravitational, electric, magnetic) acting on the molecule; J is the 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 the interacting particles. The collision integral is a five-dimensional integral and has the following structure:
Equation (12) with integral (13) was obtained for the collision of molecules, in which there are no tangential forces, i.e. colliding particles are assumed to be perfectly smooth.
In the process of interaction, the internal energy of molecules does not change, i.e. it is assumed that these molecules are ideally elastic. Two groups of molecules are considered that have velocities and , respectively, before collision (collision) with each other (Fig. 1), and after collision, respectively, velocities and . The difference in speeds is called the 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.
Equation (13) includes two parameters that characterize the location of colliding molecules relative to each other: b and ε; b- aiming distance, i.e. the smallest distance that the molecules would approach in the absence of interaction (Fig. 2); ε is called the angular collision 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. Trajectory of the movement of 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 makes it possible to use the one-particle distribution function for analysis, which was simply called the distribution function above. 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 much 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. Equally probable collisions of molecules with any impact distance, i.e. the distribution function does not change on the interaction diameter. It should be noted that the analyzed element must be small in order to 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 the calculation of 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 the exact solution of the Boltzmann kinetic equation:
where m is the mass of the molecule, kg.
The general locally-Maxwellian distribution is otherwise called the Maxwell-Boltzmann distribution:
in the case when the gas moves as a whole with a 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 near the Earth's surface, 1/m 3 ; g- acceleration of gravity, m / s 2; h is the height, m. Formula (16) is the exact solution of the Boltzmann kinetic equation either in an infinite 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.
Physical meaning: Gas constant i is numerically equal to the work of expansion of one mole of an ideal gas in an isobaric process with an increase in temperature by 1 K
In the CGS system, the gas constant is:
The specific gas constant is:
In the formula we used:
Universal gas constant (Mendeleev's constant)
Boltzmann constant
Avogadro's number
Avogadro's Law - Equal volumes of different gases at constant temperature and pressure contain the same number of molecules.
There are 2 consequences of Avogadro's Law:
Corollary 1: One mole of any gas under the same conditions occupies the same volume
In particular, under normal conditions (T=0 °C (273K) and p=101.3 kPa), the volume of 1 mole of gas is 22.4 liters. This volume is called the molar volume of the gas Vm. You can recalculate this value to other temperatures and pressures using the Mendeleev-Clapeyron equation
1) Charles' law:
2) Gay-Lussac's law:
3) Law of Pain-Mariotte:
Consequence 2: The ratio of the masses of equal volumes of two gases is a constant value for these gases
This constant is called the relative density of gases and is denoted D. Since the molar volumes of all gases are the same (1st consequence of Avogadro's law), the ratio of the molar masses of any pair of gases is also equal to this constant:
In the formula we used:
Relative gas density
Pressure
Molar volume
Universal gas constant
Absolute temperature
Boyle Mariotte's law - At a constant temperature and mass of an ideal gas, the product of its pressure and volume is constant.
This means that as the pressure on the gas increases, its volume decreases, and vice versa. For a constant amount of gas, the Boyle-Mariotte law can also be interpreted as follows: at a constant temperature, the product of pressure and volume is a constant value. The Boyle-Mariotte law is fulfilled strictly for an ideal gas and is a consequence of Mendeleev's Clapeyron equation. For real gases, the Boyle-Mariotte law is fulfilled approximately. Almost all gases behave as ideal gases at not too high pressures and not too low temperatures.
To make it easier to understand Boyle's Law Mariotte Imagine that you are squeezing an inflated balloon. Since there is enough free space between the air molecules, you can easily compress the balloon by applying some force and doing some work, reducing the volume of gas inside it. This is one of the main differences between a gas and a liquid. In a ball of liquid water, for example, the molecules are tightly packed, as if the ball were filled with microscopic pellets. Therefore, water does not lend itself, unlike air, to elastic compression.
There is also:
Charles' Law:
Gay Lussac's law:
In the law we used:
Pressure in 1 vessel
Volume of 1 vessel
Pressure in 2nd vessel
Volume 2 vessels
Gay-Lussac's law - at constant pressure, the volume of a constant mass of gas is proportional to the absolute temperature
The volume V of a given mass of gas at constant gas pressure is directly proportional to the change in temperature
The Gay-Lussac law is valid only for ideal gases; real gases obey it at temperatures and pressures that are far from critical values. It is a special case of the Claiperon equation.
There is also:
Mendeleev's Clapeyron equation:
Charles' Law:
Boyle Mariotte's Law:
In the law we used:
Volume in 1 vessel
Temperature in 1 vessel
Volume in 1 vessel
Temperature in 1 vessel
Initial gas volume
Gas volume at temperature T
Thermal expansion coefficient of gases
Difference between initial and final temperatures
Henry's law - the law according to which, at a constant temperature, the solubility of a gas in a given liquid is directly proportional to the pressure of this gas over the solution. The law is suitable only for ideal solutions and low pressures.
Henry's law describes the process of dissolving a gas in a liquid. What is a liquid in which a gas is dissolved, we know from the example of carbonated drinks - non-alcoholic, low-alcohol, and on major holidays - champagne. All of these drinks contain carbon dioxide ( chemical formula CO2) is a harmless gas used in the food industry due to its good solubility in water, and all these drinks foam after opening a bottle or can, for the reason that the dissolved gas begins to be released from the liquid into the atmosphere, because after opening a sealed vessel, the pressure inside drops .
Actually, Henry's law states a fairly simple fact: the higher the pressure of a gas above the surface of a liquid, the more difficult it is for the gas dissolved in it to be released. And this is completely logical from the point of view of molecular kinetic theory, since in order to break free from the surface of a liquid, a gas molecule needs to overcome the energy of collisions with gas molecules above the surface, and the higher the pressure and, as a result, the number of molecules in the near-boundary region, the it is more difficult for a dissolved molecule to overcome this barrier.
In the formula we used:
Gas concentration in solution in fractions of a mole
Henry coefficient
Partial pressure of gas over solution
Kirchhoff's law of radiation - the ratio of emitting and absorbing abilities does not depend on the nature of the body, it is the same for all bodies.
By definition, a completely black body absorbs all the radiation falling on it, that is, for it (Absorption capacity of the body). Therefore, the function coincides with the emissivity
In the formula we used:
Emissivity of the body
Absorption capacity of the body
Kirchhoff function
Stefan-Boltzmann law - The energy luminosity of a black body is proportional to the fourth power of absolute temperature.
It can be seen from the formula that with an increase in temperature, the luminosity of a body does not just increase - it increases to a much greater extent. Double the temperature and the luminosity will increase 16 times!
Heated bodies radiate energy in the form electromagnetic waves various lengths. When we say that a body is "red-hot", it means that its temperature is high enough for thermal radiation to occur in the visible, light part of the spectrum. At the atomic level, radiation becomes a consequence of the emission of photons by excited atoms.
To understand how this law works, imagine an atom emitting light in the depths of the sun. Light is immediately absorbed by another atom, re-emitted by it - and thus transmitted along the chain from atom to atom, due to which the whole system is in a state energy balance. In an equilibrium state, light of a strictly defined frequency is absorbed by one atom in one place simultaneously with the emission of light of the same frequency by another atom in another place. As a result, the light intensity of each wavelength of the spectrum remains unchanged.
The temperature inside the Sun drops as you move away from its center. Therefore, as it moves towards the surface, the spectrum of light radiation is corresponding to higher temperatures than the temperature environment. As a result, upon repeated emission, according to Stefan-Boltzmann law, it will occur at lower energies and frequencies, but at the same time, due to the law of conservation of energy, a larger number of photons will be emitted. Thus, by the time it reaches the surface, the spectral distribution will correspond to the temperature of the surface of the Sun (about 5,800 K), and not to the temperature at the center of the Sun (about 15,000,000 K).
The energy that comes to the surface of the Sun (or to the surface of any hot object) leaves it in the form of radiation. The Stefan-Boltzmann law just tells us what is the radiated energy.
In the above wording Stefan-Boltzmann's law extends only to a completely black body, which absorbs all radiation falling on its surface. Real physical bodies absorb only part of the ray energy, and the rest is reflected by them, however, the pattern according to which the specific power of radiation from their surface is proportional to T in 4, as a rule, is also preserved in this case, however, in this case, the Boltzmann constant has to be replaced by another coefficient that will reflect the properties of a real physical body. Such constants are usually determined experimentally.
In the formula we used:
Energy luminosity of the body
Stefan-Boltzmann constant
Absolute temperature
Charles' law - the pressure of a given mass of ideal gas at constant volume is directly proportional to the absolute temperature
To make it easier to understand Charles' law, imagine the air inside a balloon. At a constant temperature, the air in the balloon will expand or contract until the pressure produced by its molecules reaches 101,325 pascals, equal to atmospheric pressure. In other words, until each impact of an air molecule from the outside, directed inside the ball, there will be a similar impact of an air molecule, directed from the inside of the ball to the outside.
If you lower the temperature of the air in the balloon (for example, by putting it in a large refrigerator), the molecules inside the balloon will move more slowly, hitting the walls of the balloon less vigorously from the inside. The molecules of the outside air will then put more pressure on the ball, compressing it, as a result, the volume of gas inside the ball will decrease. This will continue until the increase in gas density compensates for the decreased temperature, and then equilibrium is established again.
There is also:
Mendeleev's Clapeyron equation:
Gay Lussac's law:
Boyle Mariotte's Law:
In the law we used:
Pressure in 1 vessel
Temperature in 1 vessel
Pressure in 2 vessel
Temperature in 2 vessel
The first law of thermodynamics - The change in internal energy ΔU of a non-isolated thermodynamic system is equal to the difference between the amount of heat Q transferred to the system and the work A of external forces
Instead of the work A performed by external forces on a thermodynamic system, it is often more convenient to consider the work A' performed by a thermodynamic system on external bodies. Since these works are equal in absolute value, but opposite in sign:
Then after such a transformation first law of thermodynamics will look like:
First law of thermodynamics - In a non-isolated thermodynamic system, the change in internal energy is equal to the difference between the amount of heat Q received and the work A’ performed by this system
In simple terms first law of thermodynamics speaks of energy that cannot itself be created and disappear into nowhere, it is transferred from one system to another and turns from one form to another (mechanical to thermal).
An important consequence first law of thermodynamics is that it is impossible to create a machine (engine) that is capable of performing useful work without any external energy consumption. Such a hypothetical machine was called a perpetual motion machine of the first kind.
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 basic principles is too complicated and impossible with the current level of knowledge. IN 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.
