View source for Boltzmann constant

{{distinguish|Stefan–Boltzmann constant}}
{{short description|Physical constant relating particle kinetic energy with temperature}}
{{Use dmy dates|date=June 2013}}

{|  class="wikitable" style="float: right;"
! Values of {{mvar|k}}<ref name="2014 CODATA">Barry N. Taylor of the Data Center in close collaboration with Peter J. Mohr of the Physical Measurement Laboratory's Atomic Physics Division, Termed the "[[Committee on Data for Science and Technology|2014 CODATA recommended values]]", they are generally recognized worldwide for use in all fields of science and technology. The values became available on 25 June 2015 and replaced the 2010 CODATA set. They are based on all of the data available through 31 December 2014. Available: [https://physics.nist.gov/cgi-bin/cuu/Value?k https://physics.nist.gov]</ref>
! Units

|-
| {{val|1.380649||e=-23}}<ref>{{Citation|url=https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf|title=The International System of Units (SI)|edition=9th |publisher=Bureau International des Poids et Mesures|year=2019|isbn=|location=|pages=129}}</ref>||J⋅K<sup>−1</sup>
|-
| {{val|8.617333262145|e=-5}} || [[electronvolt|eV]]⋅K<sup>−1</sup>
|-
| {{val|1.380649|e=-16}}  || [[erg]]⋅K<sup>−1</sup>
|-
| colspan=2 | <small>For details, see {{section link||Value in different units}} below.</small>
|}

The '''Boltzmann constant''' ({{math|''k''<sub>B</sub>}} or {{mvar|k}}), named after its discoverer, [[Ludwig Boltzmann]], is a [[physical constant]] that relates the average relative [[kinetic energy]] of [[particle]]s in a [[ideal gas|gas]] with the [[temperature]] of the gas. <ref name=Feynman1Ch39-10>{{cite book |author=Richard Feynman |title=The Feynman Lectures on Physics Vol I |publisher=Addison Wesley Longman |year=1970 |isbn=978-0-201-02115-8 |url=http://www.feynmanlectures.caltech.edu/I_39.html}}</ref> It occurs in the definitions of the [[kelvin]] and the [[gas constant]], and in [[Planck's law]] of [[black-body radiation]] and [[Boltzmann's entropy formula]]. The Boltzmann constant has the [[Dimensional analysis|dimension]] energy divided by temperature, the same as [[entropy]]. 

As part of the [[2019 redefinition of SI base units]], 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 {{val|1.380649|e=-23|u=J|up=K}}.<ref name=Minutes2014>{{cite conference
 |conference=SIM XXII General Assembly
 |location=Montevideo, Uruguay
 |url=http://www.sim-metrologia.org.br/docs/2016Presentations/BIPM%202016.pdf#page=10
 |conference-url=http://www.sim-metrologia.org.br/docs/2016Presentations/
 |page=10
 |title=Highlights in the work of the BIPM in 2016
 |first=Martin
 |last=Milton
 |date=14 November 2016
 |access-date=20 June 2017
 |archive-url=https://web.archive.org/web/20170901031605/http://www.sim-metrologia.org.br/docs/2016Presentations/BIPM%202016.pdf#page=10
 |archive-date=1 September 2017
 |url-status=dead
 }}</ref><ref>{{cite conference
 |title=Proceedings of the 106th meeting
 |conference=International Committee for Weights and Measures
 |date=16 October 2017
 |url=https://www.bipm.org/utils/en/pdf/CIPM/CIPM2017-EN.pdf?page=23
 |conference-url=https://www.bipm.org/en/committees/cipm/meeting/106.html
 |location=Sèvres
 |pages=17–23
}}
</ref>

This definition allows the [[kelvin]] to be defined in terms of the Boltzmann constant, the metre, the second, and the kilogram. Before 2019, its value in [[SI units]] was a measured quantity. Measurements of the Boltzmann constant depended on the definition of the [[kelvin]] in terms of the [[triple point of water]]. The measured values were used to determine the quantity that is used in the 2019 definition, to make the definition's value for the kelvin identical to the old value to within the limits of experimental accuracy at the time of the definition.

==Bridge from macroscopic to microscopic physics==
{{ideal_gas_law_relationships.svg}}
The Boltzmann constant, {{mvar|k}}, is a [[scale factor|scaling factor]] between macroscopic ([[thermodynamic temperature]]) and microscopic ([[thermal energy]]) physics. 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}} (in [[mole (unit)|moles]]) and [[absolute temperature]] {{mvar|T}}:
:<math>pV = nRT ,</math>
where {{mvar|R}} is the [[gas constant]] ({{val|8.31446261815324|u=J⋅K<sup>−1</sup>⋅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 transforms the ideal gas law into an alternative form:
:<math>p V = N k T ,</math>
where {{mvar|N}} is the [[Number of particles|number of molecules]] of gas. For {{nowrap|1={{mvar|n}} = 1 [[Mole (unit)|mol]]}}, {{mvar|N}} is equal to the number of particles in one mole ([[Avogadro's number]]).

===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}}''kT''}} (i.e., about {{val|2.07|e=−21|u=J}}, or {{val|0.013|ul=eV}}, at room temperature).

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, which means a thermal energy of {{math|{{sfrac|3|2}}''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> p = \frac{1}{3}\frac{N}{V} m \overline{v^2}.</math>
Combination with the ideal gas law
:<math>p V = N k T</math>
shows that the average translational kinetic energy is
:<math> \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}}''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 {{mvar|P<sub>i</sub>}} of occupying a state {{mvar|i}} with energy {{mvar|E}} weighted by the corresponding [[Boltzmann factor]]:
:<math>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 [[kT (energy)|{{mvar|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|200px|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>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>\Delta S = \int \frac{{\rm d}Q}{T}.</math>

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

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

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

===Role in semiconductor physics: the 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 {{math|''V''<sub>T</sub>}}.  The thermal voltage depends on absolute temperature {{mvar|T}} as
:<math> V_\mathrm{T}  =  { k T \over q },</math>
where {{mvar|q}} is the magnitude of the [[elementary charge|electrical charge on the electron]] with a value {{val|1.6021766208|(98)|e=−19|ul=C}}.<ref name="2014 CODATA" /> Equivalently,
:<math> { V_\mathrm{T} \over T } = { k \over q } \approx 8.61733034 \times 10^{-5}\ \mathrm{V/K}.</math>

At [[room temperature]] ({{val|300|u=K}}), {{math|''V''<sub>T</sub>}} 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=Third}}</ref><ref>{{cite arxiv |last1=Cataldo |first1=Enrico |last2=Lieto |first2=Alberto Di |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>  The thermal voltage is also important in plasmas and electrolyte solutions; 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=http://www.kirbyresearch.com/textbook |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}}</ref>

==History==
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 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">{{citation|first=Max |last=Planck |author-link=Max Planck |title=Ueber das Gesetz der Energieverteilung im Normalspectrum |url=http://www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf |journal=[[Annalen der Physik|Ann. Phys.]] |year=1901 |volume=309 |issue=3 |pages=553–63 |doi=10.1002/andp.19013090310 |bibcode=1901AnP...309..553P |archiveurl=https://web.archive.org/web/20120610124128/http://www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf |archivedate=10 June 2012 |url-status=dead |df=dmy }}. 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 |archiveurl=https://web.archive.org/web/20081217042934/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |archivedate=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 |first=Bertrand |last=Duplantier |title=Le mouvement brownien, 'divers et ondoyant' |language=French |trans-title=Brownian motion, 'diverse and undulating' |url=http://www.bourbaphy.fr/duplantier2.pdf |journal=Séminaire Poincaré 1 |pages=155–212 |year=2005}}</ref>

In 1920, Planck wrote in his [[Nobel Prize]] lecture:<ref name="PlanckNobel">{{citation | first = Max | last = Planck | author-link = Max Planck | title = The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture) | url = http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html | date = 2 June 1920}}</ref>
{{quotation|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" />
{{quotation|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 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}}</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|url=http://eprints.gla.ac.uk/142135/1/142135.pdf}}</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 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 [[2019 redefinition of SI base units]]. Based on these measurements, the [[CODATA]] recommended 1.380 649 × 10<sup>−23</sup> J⋅K<sup>−1</sup> to be the final fixed value of the Boltzmann constant to be used for the [[International System of Units]].<ref>{{Cite journal|last=Newell|first=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 A 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}}</ref>

==Value in different units==
{{See also|2019 redefinition of SI base units}}
{|  class="wikitable"
|-
! Values of {{mvar|k}}
! Units
! Comments
|-
| {{val|1.380649|e=−23}} || [[joule|J]]/[[kelvin|K]] || [[SI]] by definition, J/K = m<sup>2</sup>⋅kg/(s<sup>2</sup>⋅K) in SI base units
|-
| {{val|8.617333262145|e=−5}} || [[electronvolt|eV]]/K || [[electronvolt]] = {{val|1.602176634|e=-19|u=J}}
{{math|{{sfrac|1|''k''}}}} = {{val|11604.518121550|u=K/eV}}
|-
| {{val|2.0836612|(12)|e=10}} || [[Hertz|Hz]]/K || 2014 [[CODATA]] value<ref name="2014 CODATA" /><br /><!--
                                          -->{{val|1|u=Hz}}⋅[[Planck constant|{{mvar|h}}]] = {{val|6.626070040|(81)|e=−34|u=J}}<ref name="2014 CODATA" />
|-
| {{val|3.1668114|(29)|e=−6}} || [[Hartree|{{math|''E''<sub>H</sub>}}]]/K || {{math|''E''<sub>H</sub> {{=}} 2[[Rydberg constant|''R''<sub>∞</sub>]]''hc''}} = {{val|4.359744650|(54)|e=−18|u=J}}<ref name="2014 CODATA" /> = {{val|6.579683920729|(33)|u=Hz}}⋅''h''<ref name="2014 CODATA" />
|-
| {{val|1.0}} ||[[Hartree atomic units|Atomic units]]|| by definition
|-
| {{val|1.38064852|(79)|e=−16}}  || [[erg]]/K || [[Centimetre–gram–second system of units|CGS]] system, 1&nbsp;[[erg]] = {{val|1|e=−7|u=J}}
|-
| {{val|3.2976230|(30)|e=−24}} || [[Calorie|cal]]/K || 1&nbsp;[[steam table]]&nbsp;[[calorie]] = {{val|4.1868|u=J}}
|-
| {{val|1.8320128|(17)|e=−24}} || cal/[[Rankine scale|°R]] || 1&nbsp;[[Rankine scale|degree Rankine]] = {{sfrac|5|9}}&nbsp;K
|-
| {{val|5.6573016|(51)|e=−24}} || [[Foot-pound force|ft&thinsp;lb]]/°R || 1&nbsp;[[foot-pound force]] = {{val|1.3558179483314004|u=J}}
|-
| {{val|0.69503476|(63)}} || [[Wavenumber|cm<sup>−1</sup>]]/K || 2010 [[CODATA]] value<ref name="2014 CODATA" /><br /><!--
                                                     -->1&nbsp;cm<sup>−1&nbsp;</sup>⋅{{mvar|hc}} = {{val|1.986445683|(87)|e=−23|u=J}}
|-
| {{val|0.0019872041|(18)}} || kcal/([[mole (unit)|mol]]⋅K) || R noted kB, often used in statistical mechanics—using thermochemical calorie = 4.184 joule
|-
| {{val|0.0083144621|(75)}} || kJ/(mol⋅K) || R noted kB, often used in statistical mechanics
|-
| 4.10 || pN⋅nm || {{mvar|kT}} in piconewton nanometer at {{val|24|u=°C}}, used in biophysics
|-
| {{val|−228.5991678|(40)}} || dBW/(K⋅Hz) || in [[decibel watt]]s, used in telecommunications (see [[Johnson–Nyquist noise]])
|-
| 1.442 695 041... || Sh || in [[shannon (unit)|shannon]]s (logarithm base 2), used in [[information entropy]] (exact value {{sfrac|1|ln(2)}})
|-
| 1 || [[Nat (unit)|nat]] || in nats (logarithm base {{mvar|e}}), used in [[information entropy]] (''see {{section link||Planck units}}, below'')
|}

Since {{mvar|k}} is a [[physical constant]] of proportionality 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]] only changes a particle's energy by 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 {{mvar|kT}} 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 14&nbsp;387.770&nbsp;K, and also a relationship between voltage and temperature (multiplying the voltage by ''k'' in units of eV/K) with one volt being related to 11&nbsp;604.519&nbsp;K. The ratio of these two temperatures, 14&nbsp;387.770&nbsp;K/11&nbsp;604.519&nbsp;K&nbsp;≈&nbsp;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 {{mvar|E}} to the macroscopic temperature scale {{math|1=''T'' = {{sfrac|''E''|''k''}}}}. In physics research another definition is often encountered in setting {{mvar|k}} to unity, resulting in the [[Planck units]] or [[natural units]] for temperature and energy. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.<ref>{{citation | doi = 10.1007/s11018-005-0195-9 |author1=Kalinin, M |author2=Kononogov, S | title = Boltzmann's Constant, the Energy Meaning of Temperature, and Thermodynamic Irreversibility | journal = Measurement Techniques | pages = 632–36 | volume = 48 | issue = 7 | year = 2005}}</ref>

The equipartition formula for the energy associated with each classical degree of freedom then becomes
:<math>E_{\mathrm{dof}} = \tfrac{1}{2} T \ </math>
The use of natural units simplifies many physical relationships; in this form the definition of thermodynamic entropy coincides with the form of [[information entropy]]:
:<math> S = - \sum P_i \ln P_i.</math>
where {{mvar|P<sub>i</sub>}} is the probability of each [[Microstate (statistical mechanics)|microstate]].

The value chosen for a unit of the [[Planck temperature]] is that corresponding to the energy of the [[Planck mass]] or {{val|1.416808|(33)|e=32|u=K}}.<ref name="2014 CODATA" />

==See also==
* [[CODATA 2018]]
* [[Thermodynamic beta]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

==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}}

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