Editing Boltzmann constant

{{Short description|Physical constant relating particle kinetic energy with temperature}}
{{Distinguish|Stefan–Boltzmann constant}}
{{Use dmy dates|date=December 2022}}

{{infobox
| above            = Boltzmann constant
| image1           = [[File:Boltzmann2.jpg|200px|upright=1]]
| caption1         = [[Ludwig Boltzmann]], the constant's namesake
| image2           = [[File:Max Planck by Hugo Erfurth 1938cr - restoration1.jpg|200px|upright=1]]
| caption2         = [[Max Planck]], who [[#History|discovered and named the constant]]
| label1           = Definitie average relative [[kinetic energy]] of [[particle]]s in a [[ideal gas|gas]] with the [[thermodynamic temperature]] of the gas
| label2           = Symbol:
| data2            = <var>k</var><sub>B</sub>, <var>k</var>
| label3           = Value in [[joule]]s per [[kelvin]]
| data3            = {{val|1.380649|e=-23|u=J.K-1}}<ref name="SI2019"/>
| label4           = Value in [[electronvolt]]s per [[kelvin]]
| data4            = {{val|8.617333262|e=-5|u=eV&sdot;K<sup>−1</sup>}}<ref name="SI2019"/>
}}
The '''Boltzmann constant''' ({{math|''k''<sub>B</sub>}} or {{mvar|k}}) is the [[proportionality factor]] that relates the average relative [[thermal energy]] of [[particle]]s in a [[ideal gas|gas]] with the [[thermodynamic temperature]] of the gas.<ref name="Feynman1Ch39-10">{{Cite book |last=Feynman |first=Richard |url=https://feynmanlectures.caltech.edu/I_39.html |title=The Feynman Lectures on Physics Vol I |publisher=Addison Wesley Longman |year=1970 |isbn=978-0-201-02115-8}}</ref> It occurs in the definitions of the [[kelvin]] (K) and the [[molar gas constant]], in [[Planck's law]] of [[black-body radiation]] and [[Boltzmann's entropy formula]], and is used in calculating [[Johnson–Nyquist noise|thermal noise]] in [[resistors]]. The Boltzmann constant has [[Dimensional analysis|dimensions]] of [[energy]] divided by [[temperature]], the same as [[entropy]] and [[heat capacity]]. It is named after the Austrian scientist [[Ludwig Boltzmann]].

As part of the [[2019 revision of the SI]], the Boltzmann constant is one of the seven "[[Physical constant|defining constants]]" that have been defined so as to have exact [[finite decimal]] values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly {{val|1.380649|e=-23}} [[joules]] per kelvin,<ref name="SI2019"/> with the effect of defining the SI unit kelvin.

== Roles of the Boltzmann constant ==
{{ideal_gas_law_relationships.svg}}
{{Quote box|width = 35%
 |title = [[International Union of Pure and Applied Chemistry|IUPAC]] definition
 |quote = '''Boltzmann constant''': The Boltzmann constant, ''k'', is one of seven fixed constants defining the International System of Units, the SI, with ''k'' = {{val|1.380649|e=−23|u=J K<sup>−1</sup>}}. The Boltzmann constant is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule). 
<ref name='Gold Book "Boltzmann constant"'>{{cite journal |title=Boltzmann constant |url=https://goldbook.iupac.org/terms/view/B00695 |website=Gold Book |date=2020 |publisher=IUPAC |access-date=1 April 2024 |doi=10.1351/goldbook.B00695|doi-access=free }}</ref>
}}
Macroscopically, the [[ideal gas law]] states that, for an [[ideal gas]], the product of [[pressure]] {{mvar|p}} and [[volume]] {{mvar|V}} is proportional to the product of [[amount of substance]] {{mvar|n}} and [[absolute temperature]] {{mvar|T}}:
<math display="block">pV = nRT ,</math>
where {{mvar|R}} is the [[molar gas constant]] ({{val|8.31446261815324|u=J⋅K<sup>−1</sup>⋅[[Mole (unit)|mol]]<sup>−1</sup>}}).<ref>{{Cite web|url=https://www.bipm.org/utils/en/pdf/CIPM/CIPM2017-EN.pdf?page=23|title=Proceedings of the 106th meeting|date=16–20 October 2017}}</ref> Introducing the Boltzmann constant as the gas constant per molecule<ref>{{cite book |last1=Petrucci |first1=Ralph H. |last2=Harwood |first2=William S. |last3=Herring |first3=F. Geoffrey |title=General Chemistry: Principles and Modern Applications |date=2002 |publisher=Prentice Hall |isbn=0-13-014329-4 |page=785 |edition=8th}}</ref> {{math|1=''k'' = ''R''/''N''{{sub|A}}}} ({{math|''N''<sub>A</sub>}} being the [[Avogadro constant]]) transforms the ideal gas law into an alternative form:
<math display="block">p V = N k T ,</math>
where {{mvar|N}} is the [[Number of particles|number of molecules]] of gas.

=== Role in the equipartition of energy ===
{{main|Equipartition of energy}}
Given a [[thermodynamics|thermodynamic]] system at an [[thermodynamic temperature|absolute temperature]] {{mvar|T}}, the average thermal energy carried by each microscopic degree of freedom in the system is {{math|{{sfrac|1|2}}&nbsp;''kT''}} (i.e., about {{val|2.07|e=−21|u=J}}, or {{val|0.013|ul=eV}}, at room temperature). This is generally true only for classical systems with a [[Thermodynamic limit|large number of particles]].

In [[classical mechanics|classical]] [[statistical mechanics]], this average is predicted to hold exactly for homogeneous [[ideal gas]]es. Monatomic ideal gases (the six noble gases) possess three [[degrees of freedom (physics and chemistry)|degrees of freedom]] per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of {{math|{{sfrac|3|2}}&nbsp;''kT''}} per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the [[root-mean-square speed]] 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 {{val|1370|u=m/s}} for [[helium]], down to {{val|240|u=m/s}} for [[xenon]].

[[Kinetic theory of gases#Pressure|Kinetic theory]] gives the average pressure {{mvar|p}} for an ideal gas as
<math display="block"> p = \frac{1}{3}\frac{N}{V} m \overline{v^2}.</math>

Combination with the ideal gas law
<math display="block">p V = N k T</math>
shows that the average translational kinetic energy is
<math display="block"> \tfrac{1}{2}m \overline{v^2} = \tfrac{3}{2} k T.</math>

Considering that the translational motion velocity vector {{math|'''v'''}} 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. {{math|{{sfrac|1|2}}&nbsp;''kT''}}.

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 {{mvar|T}} have probability {{math|''P''{{sub|''i''}}}} of occupying a state {{mvar|i}} with energy {{mvar|E}} weighted by the corresponding [[Boltzmann factor]]:
<math display="block">P_i \propto \frac{\exp\left(-\frac{E}{k T}\right)}{Z},</math>
where {{mvar|Z}} is the [[Partition function (statistical mechanics)|partition function]]. Again, it is the energy-like quantity {{math|[[kT (energy)|''kT'']]}} that takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the [[Arrhenius equation]] in [[chemical kinetics]].

=== Role in the statistical definition of entropy ===
<!-- [[Dimensionless entropy]] redirects here -->
{{Further|Entropy (statistical thermodynamics)}}
[[File:Zentralfriedhof Vienna - Boltzmann.JPG|thumb|right|Boltzmann's grave in the [[Zentralfriedhof]], Vienna, with bust and entropy formula.]]
In statistical mechanics, the [[entropy]] {{mvar|S}} of an [[isolated system]] at [[thermodynamic equilibrium]] is defined as the [[natural logarithm]] of {{mvar|W}}, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy {{mvar|E}}):
<math display="block">S = k \,\ln W.</math>

This equation, which relates the microscopic details, or microstates, of the system (via {{mvar|W}}) to its macroscopic state (via the entropy {{mvar|S}}), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.

The constant of proportionality {{mvar|k}} serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of [[Clausius]]:
<math display="block">\Delta S = \int \frac{{\rm d}Q}{T}.</math>

One could choose instead a rescaled [[dimensionless]] entropy in microscopic terms such that
<math display="block">{S' = \ln W}, \quad \Delta S' = \int \frac{\mathrm{d}Q}{k T}.</math>

This is a more natural form and this rescaled entropy corresponds exactly to Shannon's [[information entropy]].

The characteristic energy {{mvar|kT}} is thus the energy required to increase the rescaled entropy by one [[nat (unit)|nat]].

=== Thermal voltage ===
<!-- This section is linked from [[Bipolar junction transistor]] -->
In [[semiconductors]], the [[Shockley diode equation]]—the relationship between the flow of [[electric current]] and the [[electrostatic potential]] across a [[p–n junction]]—depends on a characteristic voltage called the ''thermal voltage'', denoted by {{math|''V''{{sub|T}}}}.  The thermal voltage depends on absolute temperature {{mvar|T}} as
<math display="block"> V_\mathrm{T}  =  { k T \over q } = { R T \over F },</math>
where {{mvar|q}} is the magnitude of the [[elementary charge|electrical charge on the electron]] with a value {{physconst|e|after=.}} Equivalently,
<math display="block"> { V_\mathrm{T} \over T } = { k \over q } \approx 8.617333262 \times 10^{-5}\ \mathrm{V/K}.</math>

At [[room temperature]] {{convert|300|K|C F}}, {{math|''V''{{sub|T}}}} is approximately {{val|25.85|u=mV}},<ref>{{cite book |last1=Rashid |first1=Muhammad H. |title=Microelectronic circuits: analysis and design |date=2016 |publisher=Cengage Learning |isbn=9781305635166 |pages=183&ndash;184 |edition=3rd}}</ref><ref>{{cite arXiv |last1=Cataldo |first1=Enrico |last2=Di Lieto |first2=Alberto |last3=Maccarrone |first3=Francesco |last4=Paffuti |first4=Giampiero |title=Measurements and analysis of current-voltage characteristic of a pn diode for an undergraduate physics laboratory |date=18 August 2016 |eprint=1608.05638v1 |class=physics.ed-ph}}</ref> which can be derived by plugging in the values as follows:
<math display="block">V_\mathrm{T}={kT \over q}
=\frac{1.38\times 10^{-23}\ \mathrm{J{\cdot}K^{-1}} \times 300\ \mathrm{K}}{1.6 \times 10^{-19}\ \mathrm{C}}
\simeq 25.85\ \mathrm{mV}</math>

At the [[standard state]] temperature of {{convert|298.15|K|C F}}, it is approximately {{val|25.69|u=mV}}. The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the [[Nernst equation]]); 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.<ref name="Kirby">{{cite book |last=Kirby |first=Brian J. |title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices |url=https://assets.cambridge.org/97805211/19030/frontmatter/9780521119030_frontmatter.pdf |year=2009 |publisher=Cambridge University Press |isbn=978-0-521-11903-0}}</ref><ref name="Tabeling">{{Cite book |last=Tabeling |first=Patrick |title=Introduction to Microfluidics |publisher=Oxford University Press |year=2006 |isbn=978-0-19-856864-3 |url-access=registration |url=https://archive.org/details/introductiontomi0000tabe }}</ref>

== History ==
The Boltzmann constant is named after its 19th century Austrian discoverer, [[Ludwig Boltzmann]]. Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until [[Max Planck]] first introduced {{mvar|k}}, and gave a more precise value for it ({{val|1.346|e=−23|u=J/K}}, about 2.5% lower than today's figure), in his derivation of the [[Planck's law|law of black-body radiation]] in 1900–1901.<ref name="Planck01">{{cite journal |first=Max |last=Planck |author-link=Max Planck |title=Ueber das Gesetz der Energieverteilung im Normalspectrum |journal=[[Annalen der Physik]] |year=1901 |volume=309 |issue=3 |pages=553–63 |doi=10.1002/andp.19013090310 |bibcode=1901AnP...309..553P |doi-access=free }}. English translation: {{cite web|url=http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |url-status=dead |title=On the Law of Distribution of Energy in the Normal Spectrum |archive-url=https://web.archive.org/web/20081217042934/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |archive-date=2008-12-17 }}</ref> 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]] {{mvar|R}}, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation {{math|1=''S'' = ''k'' ln ''W''}} on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his [[Planck constant|eponymous {{mvar|h}}]].<ref>{{Cite journal|last=Gearhart|first=Clayton A.|date=2002|title=Planck, the Quantum, and the Historians|url=http://link.springer.com/10.1007/s00016-002-8363-7|journal=Physics in Perspective|language=en|volume=4|issue=2|page=177|doi=10.1007/s00016-002-8363-7|bibcode=2002PhP.....4..170G |s2cid=26918826 |issn=1422-6944|url-access=subscription}}</ref>

In 1920, Planck wrote in his [[Nobel Prize]] lecture:<ref name="PlanckNobel">{{cite web| first = Max | last = Planck | author-link = Max Planck | title = The Genesis and Present State of Development of the Quantum Theory |work= Nobel Lectures, Physics 1901-1921 |publisher=Elsevier Publishing Company, Amsterdam |publication-date=1967 | url = http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html | date = 2 June 1920}}</ref>
{{blockquote|This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it—a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant.}}

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]] tool for solving problems. There was no agreement whether ''chemical'' molecules, as measured by [[atomic weight]]s, were the same as ''physical'' molecules, as measured by [[kinetic theory of gases|kinetic theory]]. Planck's 1920 lecture continued:<ref name="PlanckNobel" />
{{blockquote|Nothing can better illustrate the positive and hectic pace of progress which the art of experimenters has made over the past twenty years, than the fact that since that time, not only one, but a great number of methods have been discovered for measuring the mass of a molecule with practically the same accuracy as that attained for a planet.}}

In versions of [[International System of Units|SI]] prior to the [[2019 revision of the SI]], the Boltzmann constant was a measured quantity rather than having a fixed numerical value. Its exact definition also varied over the years due to redefinitions of the kelvin (see ''{{section link|Kelvin|History}}'') and other SI base units (see ''{{section link|Joule|History}}'').

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.<ref>{{cite journal |last1=Pitre |first1=L |last2=Sparasci |first2=F |last3=Risegari |first3=L |last4=Guianvarc'h |first4=C |last5=Martin |first5=C |last6=Himbert |first6=M E |last7=Plimmer |first7=M D |last8=Allard |first8=A |last9=Marty |first9=B |date=1 December 2017 |title=New measurement of the Boltzmann constant by acoustic thermometry of helium-4 gas |journal=Metrologia |volume=54 |issue=6 |pages=856–873 |doi=10.1088/1681-7575/aa7bf5 |last10=Giuliano Albo |first10=P A |last11=Gao |first11=B |last12=Moldover |first12=M R |last13=Mehl |first13=J B |bibcode=2017Metro..54..856P |hdl=11696/57295 |s2cid=53680647 |url=http://pdfs.semanticscholar.org/d37f/9e1d416196493f3d8a8c14290cdeb3b3ba43.pdf |archive-url=https://web.archive.org/web/20190305132022/http://pdfs.semanticscholar.org/d37f/9e1d416196493f3d8a8c14290cdeb3b3ba43.pdf |url-status=dead |archive-date=5 March 2019 }}</ref><ref>{{cite journal |last1=de Podesta |first1=Michael |last2=Mark |first2=Darren F |last3=Dymock |first3=Ross C |last4=Underwood |first4=Robin |last5=Bacquart |first5=Thomas |last6=Sutton |first6=Gavin |last7=Davidson |first7=Stuart |last8=Machin |first8=Graham |date=1 October 2017 |title=Re-estimation of argon isotope ratios leading to a revised estimate of the Boltzmann constant |journal=Metrologia |volume=54 |issue=5 |pages=683–692 |doi=10.1088/1681-7575/aa7880 |bibcode=2017Metro..54..683D |s2cid=125912713 |url=http://eprints.gla.ac.uk/142135/1/142135.pdf }}</ref><ref>{{cite journal |last1=Fischer |first1=J |last2=Fellmuth |first2=B |last3=Gaiser |first3=C |last4=Zandt |first4=T |last5=Pitre |first5=L |last6=Sparasci |first6=F |last7=Plimmer |first7=M D |last8=de Podesta |first8=M |last9=Underwood |first9=R |last10=Sutton |first10=G |last11=Machin |first11=G |last12=Gavioso |first12=R M |last13=Ripa |first13=D Madonna |last14=Steur |first14=P P M |last15=Qu |first15=J |date=2018 |title=The Boltzmann project |journal=Metrologia |volume=55 |issue=2 |pages=10.1088/1681–7575/aaa790 |doi=10.1088/1681-7575/aaa790 |issn=0026-1394 |pmc=6508687 |pmid=31080297 |bibcode=2018Metro..55R...1F }}</ref> This decade-long effort was undertaken with different techniques by several laboratories;{{efn|Independent techniques exploited: acoustic gas thermometry, dielectric constant gas thermometry, [[Johnson–Nyquist noise|Johnson noise thermometry]]. Involved laboratories cited by CODATA in 2017: [[Laboratoire national de métrologie et d'essais|LNE]]-[[Conservatoire national des arts et métiers|Cnam]] (France), [[National Physical Laboratory (United Kingdom)|NPL]] (UK), [https://www.inrim.it/ INRIM] (Italy), [[Physikalisch-Technische Bundesanstalt|PTB]] (Germany), [[National Institute of Standards and Technology|NIST]] (USA), [http://en.nim.ac.cn/ NIM] (China).}} it is one of the cornerstones of the revision of the SI. Based on these measurements, the value of {{val|1.380649|e=−23|u=J/K}} was recommended as the final fixed value of the Boltzmann constant to be used for the 2019 revision of the SI.<ref>{{cite journal |last1=Newell |first1=D. B. |last2=Cabiati |first2=F. |last3=Fischer |first3=J. |last4=Fujii |first4=K. |last5=Karshenboim |first5=S. G. |last6=Margolis |first6=H. S. |last7=Mirandés |first7=E. de |last8=Mohr |first8=P. J. |last9=Nez |first9=F. |date=2018 |title=The CODATA 2017 values of ''h'', ''e'', ''k'', and ''N''<sub>A</sub> for the revision of the SI |url=http://stacks.iop.org/0026-1394/55/i=1/a=L13 |journal=Metrologia |language=en |volume=55 |issue=1 |pages=L13 |doi=10.1088/1681-7575/aa950a |issn=0026-1394 |bibcode=2018Metro..55L..13N |doi-access=free }}</ref>

As a precondition for redefining the Boltzmann constant, there must be one experimental value with a relative uncertainty below 1 [[Parts per million|ppm]], and at least one measurement from a second technique with a relative uncertainty below 3&nbsp;ppm. The acoustic gas thermometry reached 0.2&nbsp;ppm, and Johnson noise thermometry reached 2.8&nbsp;ppm.<ref>{{cite journal |date=2017-06-29 |title=NIST 'Noise Thermometry' Yields Accurate New Measurements of Boltzmann Constant |url=https://www.nist.gov/news-events/news/2017/06/nist-noise-thermometry-yields-accurate-new-measurements-boltzmann-constant |journal=NIST |language=en}}</ref>

== Value in different units ==
{|  class="wikitable"
|-
! Values of {{mvar|k}}
! Comments
|-
| {{physconst|k}} || [[SI]] definition
|-
| {{val|8.617333262|end=...|e=−5|u=[[electronvolt|eV]]/K}}{{px2}}<ref>{{cite web | url=https://physics.nist.gov/cgi-bin/cuu/Value?Rkev | title=CODATA Value: kelvin-electron volt relationship }}</ref> || 
|-
| {{val|2.083661912|end=...|e=10|u=[[Hertz|Hz]]/K}} || ({{math|''k''/''h''}})
|-
| {{val|1.380649|e=-16|u=[[erg]]/K}} || [[Centimetre–gram–second system of units|CGS]], 1&nbsp;[[erg]] = {{val|1|e=−7|u=J}}
|-
| {{val|3.297623483|end=...|e=−24|u=[[Calorie|cal]]/K}} || 1&nbsp;[[calorie]] = {{val|4.1868|u=J}}
|-
| {{val|1.832013046|end=...|e=−24|u=cal/[[Rankine scale|°R]]}} ||
|-
| {{val|5.657302466|end=...|e=−24|u=[[Foot-pound force|ft&thinsp;lb]]/°R}} ||
|-
| {{val|0.695034800|end=...|u=[[Wavenumber|cm<sup>−1</sup>]]/K}} || ({{math|''k''/(''hc'')}})
|-
| {{val|3.166811563|e=−6|u=[[hartree|''E''<sub>h</sub>]]/K}} ||
|-
| {{val|1.987204259|end=...|e=−3|u=[[kcal]]/([[mole (unit)|mol]]⋅K)}} || ({{math|''kN''<sub>A</sub>}})
|-
| {{val|8.314462618|end=...|e=−3|u=kJ/(mol⋅K)}} || ({{math|''kN''<sub>A</sub>}})
|-
| {{val|−228.5991672|end=...|u=[[Decibel|dB]](W/K/Hz)}} || {{math|10 log<sub>10</sub>(''k''/(1 W/K/Hz))}}, used for [[thermal noise]] calculations
|-
| {{val|1.536179187|end=...|e=-40|u=kg/K}}{{px2}}<ref>{{cite web | url=https://physics.nist.gov/cgi-bin/cuu/Value?kkg | title=CODATA Value: kelvin-kilogram relationship }}</ref> || ({{math|''k''/''c''<sup>2</sup>}})
|}

Since {{mvar|k}} is a [[proportionality constant]] 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 [[SI]] units means a change in temperature by [[Kelvin|1&nbsp;K]] changes a particle's energy by only a small amount. A change of {{val|1|ul=°C}} is defined to be the same as a change of {{val|1|u=K}}.  The characteristic energy {{math|''kT''}} is a term encountered in many physical relationships.

The Boltzmann constant sets up a relationship between wavelength and temperature (dividing {{math|''hc''/''k''}} by a wavelength gives a temperature) with {{val|1000|ul=nm}} being related to {{val|14387.777|u=K}}, and also a relationship between voltage and temperature, with one volt corresponding to {{val|11604.518|u=K}}. The ratio of these two temperatures, {{val|14387.777|u=K}}&nbsp;/&nbsp;{{val|11604.518|u=K}}&nbsp;≈&nbsp;1.239842, is the numerical value of ''hc'' in units of eV⋅μm.

=== Natural units ===
The Boltzmann constant provides a mapping from the characteristic microscopic energy {{mvar|E}} to the macroscopic temperature scale {{math|1=''T'' = {{sfrac|''Ek''}}}}. In fundamental physics, this mapping is often simplified by using the [[natural units]] of setting {{mvar|k}} to unity. This convention means that temperature and energy quantities have the same [[Dimension (physics)|dimensions]].<ref name=Kalinin/><ref>{{cite book |last1=Kittel |first1=Charles |last2=Kroemer |first2=Herbert |title=Thermal physics |date=1980 |publisher=W. H. Freeman |location=San Francisco |isbn=0716710889 |pages=41 |edition=2nd |quote=We prefer to use a more natural temperature scale ... the fundamental temperature has the units of energy.}}</ref> In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as {{nowrap|1=1 K = {{val|1.380649|e=-23|u=J}}}}.<ref>{{cite journal |last1=Mohr |first1=Peter J. |last2=Shirley |first2=Eric L. |last3=Phillips |first3=William D. |last4=Trott |first4=Michael |title=On the dimension of angles and their units |journal=Metrologia |date=1 October 2022 |volume=59 |issue=5 |pages=053001 |doi=10.1088/1681-7575/ac7bc2|arxiv=2203.12392|bibcode=2022Metro..59e3001M |doi-access=free|quote=The scientific community could have decided to have a unit system in which temperature is measured in joules, but we find it to be more convenient to measure temperature in kelvins.}}</ref> With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas.<ref name=Kalinin>{{cite journal | doi = 10.1007/s11018-005-0195-9 |last1=Kalinin |first1=M. |last2=Kononogov |first2=S. | title = Boltzmann's Constant, the Energy Meaning of Temperature, and Thermodynamic Irreversibility | journal = Measurement Techniques | pages = 632–636 | volume = 48 | issue = 7 | year = 2005|bibcode=2005MeasT..48..632K |s2cid=118726162 }}</ref>

This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom becomes
<math display="block">E_{\mathrm{dof}} = \tfrac{1}{2} T </math>

As another example, the definition of thermodynamic entropy coincides with the form of [[information entropy]]:
<math display="block"> S = - \sum_i P_i \ln P_i.</math>
where {{math|''P''{{sub|''i''}}}} is the probability of each [[Microstate (statistical mechanics)|microstate]].

== See also ==
* [[Committee on Data of the International Science Council]]
* [[Thermodynamic beta]]
* [[List of scientists whose names are used in physical constants]]

== Notes ==
{{notelist}}
{{NoteFoot}}

== References ==
{{reflist | refs =

<ref name="SI2019">
{{cite book
 | last1 = Newell
 | first1 = David B.
 | last2 = Tiesinga
 | first2 = Eite
 | year = 2019
 | title = The International System of Units (SI)
 | work = NIST
 | series = NIST Special Publication 330
 | publisher = [[National Institute of Standards and Technology]]
 | location = Gaithersburg, Maryland
 | url = https://www.nist.gov/si-redefinition/meet-constants
 | doi = 10.6028/nist.sp.330-2019
 | s2cid = 242934226
}}</ref>

}}

== External links ==
* [http://www.bipm.org/utils/common/pdf/si_brochure_draft_ch2.pdf Draft Chapter 2 for SI Brochure, following redefinitions of the base units] (prepared by the Consultative Committee for Units)
* [https://www.sciencedaily.com/releases/2011/09/110920075520.htm Big step towards redefining the kelvin: Scientists find new way to determine Boltzmann constant]

{{Mole concepts}}
{{Authority control}}

{{DEFAULTSORT:Boltzmann Constant}}
[[Category:Ludwig Boltzmann|Constant]]
[[Category:Fundamental constants]]
[[Category:Statistical mechanics]]
[[Category:Thermodynamics]]