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The Boltzmann constant ( or ) is the
proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), constant; that is, when either their ratio or their product (mathem ...
that relates the average relative
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
of
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or chemical property, chemical p ...

particle
s in a
gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...
with the
thermodynamic temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. A thermodynamic temperature reading of zero is of particular importance for the third law of therm ...

thermodynamic temperature
of the gas. It occurs in the definitions of the
kelvin The kelvin is the base unit of temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal en ...

kelvin
and the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant The Boltzmann constant ( or ) is the proportionality fa ...
, and in
Planck's law Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of th ...
of
black-body radiation Black-body radiation is the within or surrounding a body in with its environment, emitted by a (an idealized opaque, non-reflective body). It has a specific spectrum of wavelengths, inversely related to intensity that depend only on the bod ...
and
Boltzmann's entropy formula In statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural ...
. The Boltzmann constant has
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
of energy divided by temperature, the same as
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

entropy
. It is named after the Austrian scientist
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...
. As part of the
2019 redefinition of SI base units Effective 20 May 2019, the 144th anniversary of the Metre Convention, the SI base units were redefined in agreement with the International System of Quantities. In the redefinition, four of the seven SI base units – the kilogram, ampere, ...
, the Boltzmann constant is one of the seven " defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly .


Roles of the Boltzmann constant

Macroscopically, the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

ideal gas law
states that, for an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, the product of
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

pressure
and
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
is proportional to the product of
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
(in moles) and
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...
: :pV = nRT , where is the molar
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant The Boltzmann constant ( or ) is the proportionality fa ...
(). Introducing the Boltzmann constant as the gas constant per molecule k = R/NA transforms the ideal gas law into an alternative form: :p V = N k T , where is the number of molecules of gas. For , is equal to the number of particles in one mole (the
Avogadro number The Avogadro constant (''N''A or ''L'') is the proportionality factor that relates the number of constituent particles (usually molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which ...
).


Role in the equipartition of energy

Given a
thermodynamic Thermodynamics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ent ...

thermodynamic
system at an
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...

absolute temperature
, the average thermal energy carried by each microscopic degree of freedom in the system is (i.e., about , or , at room temperature). In
classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, this average is predicted to hold exactly for homogeneous
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
es. Monatomic ideal gases (the six noble gases) possess three
degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
per atom, corresponding to the three spatial directions, which means a thermal energy of per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the
root-mean-square speed In mathematics and its applications, the root mean square (RMS or or rms) is defined as the square root of the mean square (the arithmetic mean of the square (algebra), squares of a Set (mathematics), set of numbers). The RMS is also known as the ...
of the atoms, which turns out to be inversely proportional to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from for
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...

helium
, down to for
xenon Xenon is a chemical element with the Symbol (chemistry), symbol Xe and atomic number 54. It is a colorless, dense, odorless noble gas found in Atmosphere of Earth, Earth's atmosphere in trace amounts. Although generally unreactive, x ...

xenon
.
Kinetic theory of the ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation o ...

Kinetic theory
gives the average pressure for an ideal gas as : p = \frac\frac m \overline. Combination with the ideal gas law :p V = N k T shows that the average translational kinetic energy is : \tfracm \overline = \tfrac k T. Considering that the translational motion velocity vector has three degrees of freedom (one for each dimension) gives the average energy per degree of freedom equal to one third of that, i.e. . The ideal gas equation is also obeyed closely by molecular gases; but the form for the heat capacity is more complicated, because the molecules possess additional internal degrees of freedom, as well as the three degrees of freedom for movement of the molecule as a whole. Diatomic gases, for example, possess a total of six degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and one vibrational). At lower temperatures, not all these degrees of freedom may fully participate in the gas heat capacity, due to quantum mechanical limits on the availability of excited states at the relevant thermal energy per molecule.


Role in Boltzmann factors

More generally, systems in equilibrium at temperature have probability of occupying a state with energy weighted by the corresponding
Boltzmann factor Factor, a Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republi ...
: :P_i \propto \frac, where is the partition function. Again, it is the energy-like quantity that takes central importance. Consequences of this include (in addition to the results for ideal gases above) the
Arrhenius equation In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 ...
in
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with thermodynamics, which deals with the direction in which a pro ...
.


Role in the statistical definition of entropy

In statistical mechanics, the
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

entropy
of an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
at
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic An axiom, postulate or assumption is a statement that is taken to be true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online ...
is defined as the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
of , the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy ): :S = k \,\ln W. This equation, which relates the microscopic details, or microstates, of the system (via ) to its macroscopic state (via the entropy ), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone. The constant of proportionality serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius: :\Delta S = \int \frac. One could choose instead a rescaled
dimensionless In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
entropy in microscopic terms such that :, \quad \Delta S' = \int \frac. This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent
information entropy In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent in the variable's possible outcomes. The concept of information entropy was introduced by Claude Shannon in hi ...
. The characteristic energy is thus the energy required to increase the rescaled entropy by one
nat Nat or NAT may refer to: Computing * Network address translation (NAT), in computer networking Organizations * National Actors Theatre, New York City, U.S. * National AIDS trust, a British charity * National Archives of Thailand * National Asse ...
.


The thermal voltage

In
semiconductors A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that quantifies how strongly it resists electric curren ...

semiconductors
, the
Shockley diode equation The ''Shockley diode equation'' or the ''diode law'', named after transistor file:MOSFET Structure.png, upright=1.4, Metal-oxide-semiconductor field-effect transistor (MOSFET), showing Metal gate, gate (G), body (B), source (S) and drain (D) term ...
—the relationship between the flow of
electric current An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
and the
electrostatic potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work (physics), work energy needed to move a unit of electric charge from a reference point to the sp ...
across a
p–n junction A p–n junction is a boundary or interface between two types of semiconductor material A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) i ...
—depends on a characteristic voltage called the ''thermal voltage'', denoted . The thermal voltage depends on absolute temperature as : V_\mathrm = , where is the magnitude of the electrical charge on the electron with a value Equivalently, : = \approx 8.61733034 \times 10^\ \mathrm. At
room temperature Colloquially, room temperature is the range of air temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat ...
, is approximately . and at the
standard state In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds composed of atoms, ...
temperature of , it is approximately . The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the
Nernst equation In electrochemistry, the Nernst equation is an equation that relates the reduction potential of an electrochemical reaction ( half-cell or electrochemical cell, full cell reaction) to the standard electrode potential, Thermodynamic temperature, t ...
); in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage.


History

The Boltzmann constant is named after its 19th century Austrian discoverer,
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...
. Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until
Max Planck Max Karl Ernst Ludwig Planck, (; ; 23 April 1858 – 4 October 1947) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citi ...

Max Planck
first introduced , and gave a more precise value for it (, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901.. English translation: Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and the Boltzmann constant, but rather using a form of the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant The Boltzmann constant ( or ) is the proportionality fa ...
, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his eponymous . In 1920, Planck wrote in his
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel Alfred Bernhard Nobel ( , ; 21 October 1833 – 10 December 1896) was a Swedish chemist, engineer, inventor, busines ...
lecture: This "peculiar state of affairs" is illustrated by reference to one of the great scientific debates of the time. There was considerable disagreement in the second half of the nineteenth century as to whether atoms and molecules were real or whether they were simply a
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be Mathematical optimisation, optimal, perfect, or Rationality, rational, but is nevertheless ...
tool for solving problems. There was no agreement whether ''chemical'' molecules, as measured by
atomic weight Relative atomic mass (symbol: ''A'') or atomic weight is a dimensionless physical quantity A physical quantity is a physical property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity ca ...
s, were the same as ''physical'' molecules, as measured by
kinetic theory of the ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation o ...

kinetic theory
. Planck's 1920 lecture continued: In versions of SI prior to the
2019 redefinition of the SI base units upright=1.35, The SI system after 1983, but before the 2019 redefinition: Dependence of base unit definitions on other base units (for example, the metre is defined as the distance travelled by light in a specific fraction of a second The ...
, the Boltzmann constant was a measured quantity rather than a fixed value. Its exact definition also varied over the years due to redefinitions of the kelvin (see ) and other SI base units (see ). In 2017, the most accurate measures of the Boltzmann constant were obtained by acoustic gas thermometry, which determines the speed of sound of a monatomic gas in a triaxial ellipsoid chamber using microwave and acoustic resonances. This decade-long effort was undertaken with different techniques by several laboratories; it is one of the cornerstones of the
2019 redefinition of SI base units Effective 20 May 2019, the 144th anniversary of the Metre Convention, the SI base units were redefined in agreement with the International System of Quantities. In the redefinition, four of the seven SI base units – the kilogram, ampere, ...
. Based on these measurements, the
CODATA The CODATA is the Committee on Data of the International Science Council The International Science Council (ISC) is an international non-governmental organization An international non-governmental organization (INGO) extends the concept of ...
recommended 1.380 649 × 10−23 J⋅K−1 to be the final fixed value of the Boltzmann constant to be used for the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
.


Value in different units

Since is a
proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), constant; that is, when either their ratio or their product (mathem ...
between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in units means a change in temperature by only changes a particle's energy by a small amount. A change of is defined to be the same as a change of . The characteristic energy is a term encountered in many physical relationships. The Boltzmann constant sets up a relationship between wavelength and temperature (dividing ''hc''/''k'' by a wavelength gives a temperature) with one micrometer being related to , and also a relationship between voltage and temperature (multiplying the voltage by ''k'' in units of eV/K) with one volt being related to . The ratio of these two temperatures,  /  ≈ 1.239842, is the numerical value of ''hc'' in units of eV⋅μm.


Planck units

The Boltzmann constant provides a mapping from this characteristic microscopic energy to the macroscopic temperature scale . In physics research another definition is often encountered in setting to unity, resulting in the
Planck units In and , Planck units are a set of defined exclusively in terms of four universal s, in such a manner that these physical constants take on the numerical value of when expressed in terms of these units. Originally proposed in 1899 by German p ...
or
natural units In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phy ...
for temperature and energy. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed. The equipartition formula for the energy associated with each classical degree of freedom then becomes :E_ = \tfrac T \ The use of natural units simplifies many physical relationships; in this form the definition of thermodynamic entropy coincides with the form of
information entropy In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent in the variable's possible outcomes. The concept of information entropy was introduced by Claude Shannon in hi ...
: : S = - \sum P_i \ln P_i. where is the probability of each
microstate upright=1.4, Map of the smallest states in the world by land area. Note many of these are not considered microstates A microstate or ministate is a sovereign state A sovereign state is a political entity that is represented by one centralize ...
. The value chosen for a unit of the Planck temperature is that corresponding to the energy of the Planck mass.


See also

* CODATA 2018 *
Thermodynamic beta In statistical thermodynamics In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (ph ...


Notes


References


External links


Draft Chapter 2 for SI Brochure, following redefinitions of the base units
(prepared by the Consultative Committee for Units)

{{DEFAULTSORT:Boltzmann Constant
Constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
Fundamental constants Statistical mechanics Thermodynamics