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<span>Toggle Roles of the Boltzmann constant subsection</span> </button> <ul id="toc-Roles_of_the_Boltzmann_constant-sublist" class="vector-toc-list"> <li id="toc-Role_in_the_equipartition_of_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Role_in_the_equipartition_of_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Role in the equipartition of energy</span> </div> </a> <ul id="toc-Role_in_the_equipartition_of_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Role_in_Boltzmann_factors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Role_in_Boltzmann_factors"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Role in Boltzmann factors</span> </div> </a> <ul id="toc-Role_in_Boltzmann_factors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Role_in_the_statistical_definition_of_entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Role_in_the_statistical_definition_of_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Role in the statistical definition of entropy</span> </div> </a> <ul id="toc-Role_in_the_statistical_definition_of_entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermal_voltage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermal_voltage"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Thermal voltage</span> </div> </a> <ul id="toc-Thermal_voltage-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Value_in_different_units" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Value_in_different_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Value in different units</span> </div> </a> <button aria-controls="toc-Value_in_different_units-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Value in different units subsection</span> </button> <ul id="toc-Value_in_different_units-sublist" class="vector-toc-list"> <li id="toc-Natural_units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Natural units</span> </div> </a> <ul id="toc-Natural_units-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Boltzmann constant</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 71 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-71" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">71 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Konstante_van_Boltzmann" title="Konstante van Boltzmann – Afrikaans" lang="af" hreflang="af" data-title="Konstante van Boltzmann" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AB%D8%A7%D8%A8%D8%AA_%D8%A8%D9%88%D9%84%D8%AA%D8%B2%D9%85%D8%A7%D9%86" title="ثابت بولتزمان – Arabic" lang="ar" hreflang="ar" data-title="ثابت بولتزمان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann – Asturian" lang="ast" hreflang="ast" data-title="Constante de Boltzmann" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Boltsman_sabiti" title="Boltsman sabiti – Azerbaijani" lang="az" hreflang="az" data-title="Boltsman sabiti" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8B%E0%A6%B2%E0%A6%9F%E0%A7%8D%E0%A6%9C%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A8_%E0%A6%A7%E0%A7%8D%E0%A6%B0%E0%A7%81%E0%A6%AC%E0%A6%95" title="বোলট্জম্যান ধ্রুবক – Bangla" lang="bn" hreflang="bn" data-title="বোলট্জম্যান ধ্রুবক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Boltzmann_si%C3%B4ng-s%C3%B2%CD%98" title="Boltzmann siông-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Boltzmann siông-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D1%81%D1%82%D0%B0%D1%8F%D0%BD%D0%BD%D0%B0%D1%8F_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Пастаянная Больцмана – Belarusian" lang="be" hreflang="be" data-title="Пастаянная Больцмана" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B0_%D0%BD%D0%B0_%D0%91%D0%BE%D0%BB%D1%86%D0%BC%D0%B0%D0%BD" title="Константа на Болцман – Bulgarian" lang="bg" hreflang="bg" data-title="Константа на Болцман" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta – Bosnian" lang="bs" hreflang="bs" data-title="Boltzmannova konstanta" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Digemmenn_Boltzmann" title="Digemmenn Boltzmann – Breton" lang="br" hreflang="br" data-title="Digemmenn Boltzmann" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Constant_de_Boltzmann" title="Constant de Boltzmann – Catalan" lang="ca" hreflang="ca" data-title="Constant de Boltzmann" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD_%D0%BA%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D1%82%D0%B8" title="Больцман константти – Chuvash" lang="cv" hreflang="cv" data-title="Больцман константти" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta – Czech" lang="cs" hreflang="cs" data-title="Boltzmannova konstanta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Boltzmanns_konstant" title="Boltzmanns konstant – Danish" lang="da" hreflang="da" data-title="Boltzmanns konstant" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Boltzmann-Konstante" title="Boltzmann-Konstante – German" lang="de" hreflang="de" data-title="Boltzmann-Konstante" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Boltzmanni_konstant" title="Boltzmanni konstant – Estonian" lang="et" hreflang="et" data-title="Boltzmanni konstant" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CE%B1%CE%B8%CE%B5%CF%81%CE%AC_%CF%84%CE%BF%CF%85_%CE%9C%CF%80%CF%8C%CE%BB%CF%84%CF%83%CE%BC%CE%B1%CE%BD" title="Σταθερά του Μπόλτσμαν – Greek" lang="el" hreflang="el" data-title="Σταθερά του Μπόλτσμαν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann – Spanish" lang="es" hreflang="es" data-title="Constante de Boltzmann" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Konstanto_de_Boltzmann" title="Konstanto de Boltzmann – Esperanto" lang="eo" hreflang="eo" data-title="Konstanto de Boltzmann" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Boltzmannen_konstantea" title="Boltzmannen konstantea – Basque" lang="eu" hreflang="eu" data-title="Boltzmannen konstantea" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AB%D8%A7%D8%A8%D8%AA_%D8%A8%D9%88%D9%84%D8%AA%D8%B2%D9%85%D9%86" title="ثابت بولتزمن – Persian" lang="fa" hreflang="fa" data-title="ثابت بولتزمن" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann – French" lang="fr" hreflang="fr" data-title="Constante de Boltzmann" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tairiseach_Boltzmann" title="Tairiseach Boltzmann – Irish" lang="ga" hreflang="ga" data-title="Tairiseach Boltzmann" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann – Galician" lang="gl" hreflang="gl" data-title="Constante de Boltzmann" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AC%E0%AB%8B%E0%AA%B2%E0%AB%8D%E0%AA%9F%E0%AB%8D%E0%AA%9D%E0%AA%AE%E0%AB%87%E0%AA%A8_%E0%AA%85%E0%AA%9A%E0%AA%B3%E0%AA%BE%E0%AA%82%E0%AA%95" title="બોલ્ટ્ઝમેન અચળાંક – Gujarati" lang="gu" hreflang="gu" data-title="બોલ્ટ્ઝમેન અચળાંક" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%BC%EC%B8%A0%EB%A7%8C_%EC%83%81%EC%88%98" title="볼츠만 상수 – Korean" lang="ko" hreflang="ko" data-title="볼츠만 상수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B8%D5%AC%D6%81%D5%B4%D5%A1%D5%B6%D5%AB_%D5%B0%D5%A1%D5%BD%D5%BF%D5%A1%D5%BF%D5%B8%D6%82%D5%B6" title="Բոլցմանի հաստատուն – Armenian" lang="hy" hreflang="hy" data-title="Բոլցմանի հաստատուն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A5%8D%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%A4%E0%A4%BE%E0%A4%82%E0%A4%95" title="बोल्ट्समान नियतांक – Hindi" lang="hi" hreflang="hi" data-title="बोल्ट्समान नियतांक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta – Croatian" lang="hr" hreflang="hr" data-title="Boltzmannova konstanta" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Konstanta_Boltzmann" title="Konstanta Boltzmann – Indonesian" lang="id" hreflang="id" data-title="Konstanta Boltzmann" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Costante_di_Boltzmann" title="Costante di Boltzmann – Italian" lang="it" hreflang="it" data-title="Costante di Boltzmann" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%91%D7%95%D7%A2_%D7%91%D7%95%D7%9C%D7%A6%D7%9E%D7%9F" title="קבוע בולצמן – Hebrew" lang="he" hreflang="he" data-title="קבוע בולצמן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%91%E1%83%9D%E1%83%9A%E1%83%AA%E1%83%9B%E1%83%90%E1%83%9C%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%A3%E1%83%93%E1%83%9B%E1%83%98%E1%83%95%E1%83%90" title="ბოლცმანის მუდმივა – Georgian" lang="ka" hreflang="ka" data-title="ბოლცმანის მუდმივა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD_%D1%82%D2%B1%D1%80%D0%B0%D2%9B%D1%82%D1%8B%D1%81%D1%8B" title="Больцман тұрақтысы – Kazakh" lang="kk" hreflang="kk" data-title="Больцман тұрақтысы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Constans_Boltzmanniana" title="Constans Boltzmanniana – Latin" lang="la" hreflang="la" data-title="Constans Boltzmanniana" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bolcma%C5%86a_konstante" title="Bolcmaņa konstante – Latvian" lang="lv" hreflang="lv" data-title="Bolcmaņa konstante" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bolcmano_konstanta" title="Bolcmano konstanta – Lithuanian" lang="lt" hreflang="lt" data-title="Bolcmano konstanta" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Boltzmann-%C3%A1lland%C3%B3" title="Boltzmann-állandó – Hungarian" lang="hu" hreflang="hu" data-title="Boltzmann-állandó" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D1%86%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BA%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B0" title="Болцманова константа – Macedonian" lang="mk" hreflang="mk" data-title="Болцманова константа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8B%E0%B5%BE%E0%B4%9F%E0%B5%8D%E0%B4%B8%E0%B5%8D%E0%B4%AE%E0%B4%BE%E0%B5%BB_%E0%B4%B8%E0%B5%8D%E0%B4%A5%E0%B4%BF%E0%B4%B0%E0%B4%BE%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%82" title="ബോൾട്സ്മാൻ സ്ഥിരാങ്കം – Malayalam" lang="ml" hreflang="ml" data-title="ബോൾട്സ്മാൻ സ്ഥിരാങ്കം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%91%E1%83%9D%E1%83%9A%E1%83%AA%E1%83%9B%E1%83%90%E1%83%9C%E1%83%98%E1%83%A8_%E1%83%99%E1%83%9D%E1%83%9C%E1%83%A1%E1%83%A2%E1%83%90%E1%83%9C%E1%83%A2%E1%83%90" title="ბოლცმანიშ კონსტანტა – Mingrelian" lang="xmf" hreflang="xmf" data-title="ბოლცმანიშ კონსტანტა" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pemalar_Boltzmann" title="Pemalar Boltzmann – Malay" lang="ms" hreflang="ms" data-title="Pemalar Boltzmann" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D1%8B_%D1%82%D0%BE%D0%B3%D1%82%D0%BC%D0%BE%D0%BB" title="Больцманы тогтмол – Mongolian" lang="mn" hreflang="mn" data-title="Больцманы тогтмол" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Boltzmannconstante" title="Boltzmannconstante – Dutch" lang="nl" hreflang="nl" data-title="Boltzmannconstante" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%AB%E3%83%84%E3%83%9E%E3%83%B3%E5%AE%9A%E6%95%B0" title="ボルツマン定数 – Japanese" lang="ja" hreflang="ja" data-title="ボルツマン定数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Boltzmanns_konstant" title="Boltzmanns konstant – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Boltzmanns konstant" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Boltzmannkonstanten" title="Boltzmannkonstanten – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Boltzmannkonstanten" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Constanta_de_Boltzmann" title="Constanta de Boltzmann – Occitan" lang="oc" hreflang="oc" data-title="Constanta de Boltzmann" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Boltzmann_doimiysi" title="Boltzmann doimiysi – Uzbek" lang="uz" hreflang="uz" data-title="Boltzmann doimiysi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/Constante_d%27Boltzmann" title="Constante d'Boltzmann – Picard" lang="pcd" hreflang="pcd" data-title="Constante d'Boltzmann" data-language-autonym="Picard" data-language-local-name="Picard" class="interlanguage-link-target"><span>Picard</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sta%C5%82a_Boltzmanna" title="Stała Boltzmanna – Polish" lang="pl" hreflang="pl" data-title="Stała Boltzmanna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann – Portuguese" lang="pt" hreflang="pt" data-title="Constante de Boltzmann" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Constanta_Boltzmann" title="Constanta Boltzmann – Romanian" lang="ro" hreflang="ro" data-title="Constanta Boltzmann" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%BD%D0%B0%D1%8F_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Постоянная Больцмана – Russian" lang="ru" hreflang="ru" data-title="Постоянная Больцмана" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Boltzmann_constant" title="Boltzmann constant – Simple English" lang="en-simple" hreflang="en-simple" data-title="Boltzmann constant" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Boltzmannova_kon%C5%A1tanta" title="Boltzmannova konštanta – Slovak" lang="sk" hreflang="sk" data-title="Boltzmannova konštanta" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta – Slovenian" lang="sl" hreflang="sl" data-title="Boltzmannova konstanta" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D1%86%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BA%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B0" title="Болцманова константа – Serbian" lang="sr" hreflang="sr" data-title="Болцманова константа" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Boltzmannova konstanta" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Boltzmannin_vakio" title="Boltzmannin vakio – Finnish" lang="fi" hreflang="fi" data-title="Boltzmannin vakio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Boltzmanns_konstant" title="Boltzmanns konstant – Swedish" lang="sv" hreflang="sv" data-title="Boltzmanns konstant" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%8B%E0%AE%B2%E0%AF%8D%E0%AE%9F%E0%AF%8D%E0%AE%9A%E0%AF%81%E0%AE%AE%E0%AE%BE%E0%AE%A9%E0%AF%8D_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AE%BF%E0%AE%B2%E0%AE%BF" title="போல்ட்சுமான் மாறிலி – Tamil" lang="ta" hreflang="ta" data-title="போல்ட்சுமான் மாறிலி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/Boltsman_daimie" title="Boltsman daimie – Tatar" lang="tt" hreflang="tt" data-title="Boltsman daimie" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%84%E0%B8%87%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9A%E0%B9%87%E0%B8%AD%E0%B8%A5%E0%B8%97%E0%B8%8B%E0%B9%8C%E0%B8%A1%E0%B8%B1%E0%B8%99" title="ค่าคงตัวบ็อลทซ์มัน – Thai" lang="th" hreflang="th" data-title="ค่าคงตัวบ็อลทซ์มัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boltzmann_sabiti" title="Boltzmann sabiti – Turkish" lang="tr" hreflang="tr" data-title="Boltzmann sabiti" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D0%BB%D0%B0_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Стала Больцмана – Ukrainian" lang="uk" hreflang="uk" data-title="Стала Больцмана" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A8%D9%88%D9%84%D9%B9%D8%B2%D9%85%DB%8C%D9%86_%DA%A9%D8%A7_%D9%85%D8%B3%D8%AA%D9%82%D9%84" title="بولٹزمین کا مستقل – Urdu" lang="ur" hreflang="ur" data-title="بولٹزمین کا مستقل" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BA%B1ng_s%E1%BB%91_Boltzmann" title="Hằng số Boltzmann – Vietnamese" lang="vi" hreflang="vi" data-title="Hằng số Boltzmann" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E5%B8%B8%E6%95%B0" title="玻尔兹曼常数 – Wu" lang="wuu" hreflang="wuu" data-title="玻尔兹曼常数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B3%A2%E8%8C%B2%E6%9B%BC%E5%B8%B8%E6%95%B8" title="波茲曼常數 – Cantonese" lang="yue" hreflang="yue" data-title="波茲曼常數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B3%A2%E8%8C%B2%E6%9B%BC%E5%B8%B8%E6%95%B8" title="波茲曼常數 – Chinese" lang="zh" hreflang="zh" data-title="波茲曼常數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5962#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Boltzmann_constant" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Physical constant relating particle kinetic energy with temperature</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Stefan%E2%80%93Boltzmann_constant" class="mw-redirect" title="Stefan–Boltzmann constant">Stefan–Boltzmann constant</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Boltzmann constant</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Boltzmann2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/200px-Boltzmann2.jpg" decoding="async" width="200" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/300px-Boltzmann2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/400px-Boltzmann2.jpg 2x" data-file-width="490" data-file-height="600" /></a></span><div class="infobox-caption"><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>, the constant's namesake</div></td></tr><tr><th scope="row" class="infobox-label">Symbol:</th><td class="infobox-data"><var>k</var><sub>B</sub>, <var>k</var></td></tr><tr><th scope="row" class="infobox-label">Value in <a href="/wiki/Joule" title="Joule">joules</a> per <a href="/wiki/Kelvin" title="Kelvin">kelvin</a></th><td class="infobox-data"><span class="nowrap"><span data-sort-value="6977138064900000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J⋅K<sup>−1</sup></span><sup id="cite_ref-SI2019_1-0" class="reference"><a href="#cite_note-SI2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr><tr><th scope="row" class="infobox-label">Value in <a href="/wiki/Electronvolt" title="Electronvolt">electronvolts</a> per <a href="/wiki/Kelvin" title="Kelvin">kelvin</a></th><td class="infobox-data"><span class="nowrap"><span data-sort-value="6995861733326200000♠"></span>8.617<span style="margin-left:.25em;">333</span><span style="margin-left:.25em;">262</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup> eV⋅K<sup>−1</sup></span><sup id="cite_ref-SI2019_1-1" class="reference"><a href="#cite_note-SI2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr></tbody></table> <p>The <b>Boltzmann constant</b> (<span class="texhtml"><i>k</i><sub>B</sub></span> or <span class="texhtml mvar" style="font-style:italic;">k</span>) is the <a href="/wiki/Proportionality_factor" class="mw-redirect" title="Proportionality factor">proportionality factor</a> that relates the average relative <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a> of <a href="/wiki/Particle" title="Particle">particles</a> in a <a href="/wiki/Ideal_gas" title="Ideal gas">gas</a> with the <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a> of the gas.<sup id="cite_ref-Feynman1Ch39-10_2-0" class="reference"><a href="#cite_note-Feynman1Ch39-10-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It occurs in the definitions of the <a href="/wiki/Kelvin" title="Kelvin">kelvin</a> (K) and the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>, in <a href="/wiki/Planck%27s_law" title="Planck's law">Planck's law</a> of <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a> and <a href="/wiki/Boltzmann%27s_entropy_formula" title="Boltzmann's entropy formula">Boltzmann's entropy formula</a>, and is used in calculating <a href="/wiki/Johnson%E2%80%93Nyquist_noise" title="Johnson–Nyquist noise">thermal noise</a> in <a href="/wiki/Resistors" class="mw-redirect" title="Resistors">resistors</a>. The Boltzmann constant has <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensions</a> of <a href="/wiki/Energy" title="Energy">energy</a> divided by <a href="/wiki/Temperature" title="Temperature">temperature</a>, the same as <a href="/wiki/Entropy" title="Entropy">entropy</a> and <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a>. It is named after the Austrian scientist <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>. </p><p>As part of the <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision of the SI</a>, the Boltzmann constant is one of the seven "<a href="/wiki/Physical_constant" title="Physical constant">defining constants</a>" 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 <span class="nowrap"><span data-sort-value="6977138064900000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup></span> <a href="/wiki/Joules" class="mw-redirect" title="Joules">joules</a> per kelvin.<sup id="cite_ref-SI2019_1-2" class="reference"><a href="#cite_note-SI2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Roles_of_the_Boltzmann_constant">Roles of the Boltzmann constant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=1" title="Edit section: Roles of the Boltzmann constant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Ideal_gas_law_relationships.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Ideal_gas_law_relationships.svg/200px-Ideal_gas_law_relationships.svg.png" decoding="async" width="200" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Ideal_gas_law_relationships.svg/300px-Ideal_gas_law_relationships.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Ideal_gas_law_relationships.svg/400px-Ideal_gas_law_relationships.svg.png 2x" data-file-width="512" data-file-height="341" /></a><figcaption>Relationships between <a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's</a>, <a href="/wiki/Charles%27s_law" title="Charles's law">Charles's</a>, <a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's</a>, <a href="/wiki/Avogadro%27s_law" title="Avogadro's law">Avogadro's</a>, <a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">combined</a> and <span class="nowrap"><a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas laws</a></span>, with the <a class="mw-selflink selflink">Boltzmann constant</a> <span class="texhtml"><var style="padding-right: 1px;">k</var><sub><var style="padding-right: 1px;"></var></sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">R</var></span><span class="sr-only">/</span><span class="den"><var style="padding-right: 1px;">N</var><sub><var style="padding-right: 1px;">A</var></sub></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">n</var> <var style="padding-right: 1px;">R</var></span><span class="sr-only">/</span><span class="den"><var style="padding-right: 1px;">N</var></span></span>⁠</span></span> (in each law, <a href="/wiki/Physical_property" title="Physical property">properties</a> circled are variable and properties not circled are held constant)</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style><div class="quotebox pullquote floatright" style="width:35%; ;"> <div class="quotebox-title" style=""><a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">IUPAC</a> definition</div> <blockquote class="quotebox-quote left-aligned" style=""> <p><b>Boltzmann constant</b>: The Boltzmann constant, <i>k</i>, is one of seven fixed constants defining the International System of Units, the SI, with k = 1.380 649 x 10<sup>-23</sup> J K<sup>-1</sup>. The Boltzmann constant is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule). <sup id="cite_ref-Gold_Book_"Boltzmann_constant"_3-0" class="reference"><a href="#cite_note-Gold_Book_"Boltzmann_constant"-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> </blockquote> </div> <p>Macroscopically, the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> states that, for an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a>, the product of <a href="/wiki/Pressure" title="Pressure">pressure</a> <span class="texhtml mvar" style="font-style:italic;">p</span> and <a href="/wiki/Volume" title="Volume">volume</a> <span class="texhtml mvar" style="font-style:italic;">V</span> is proportional to the product of <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> <span class="texhtml mvar" style="font-style:italic;">n</span> and <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> <span class="texhtml mvar" style="font-style:italic;">T</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pV=nRT,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pV=nRT,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c5d5a7aae3733049c52bad27008bf7d60e17c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.586ex; height:2.509ex;" alt="{\displaystyle pV=nRT,}"></span> where <span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Molar_gas_constant" class="mw-redirect" title="Molar gas constant">molar gas constant</a> (<span class="nowrap"><span data-sort-value="7000831446261815324♠"></span>8.314<span style="margin-left:.25em;">462</span><span style="margin-left:.25em;">618</span><span style="margin-left:.25em;">153</span><span style="margin-left:.25em;">24</span> J⋅K<sup>−1</sup>⋅<a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a><sup>−1</sup></span>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Introducing the Boltzmann constant as the gas constant per molecule<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="texhtml"><var style="padding-right: 1px;">k</var> = <var style="padding-right: 1px;">R</var>/<var style="padding-right: 1px;">N</var><sub><var style="padding-right: 1px;">A</var></sub></span> (<i>N</i><sub>A</sub> being the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro constant</a>) transforms the ideal gas law into an alternative form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pV=NkT,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>V</mi> <mo>=</mo> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pV=NkT,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1efe84d5ec485bd2f904ff8acfeff93fd1184df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.702ex; height:2.509ex;" alt="{\displaystyle pV=NkT,}"></span> where <span class="texhtml mvar" style="font-style:italic;">N</span> is the <a href="/wiki/Number_of_particles" class="mw-redirect" title="Number of particles">number of molecules</a> of gas. </p> <div class="mw-heading mw-heading3"><h3 id="Role_in_the_equipartition_of_energy">Role in the equipartition of energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=2" title="Edit section: Role in the equipartition of energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Equipartition_of_energy" class="mw-redirect" title="Equipartition of energy">Equipartition of energy</a></div> <p>Given a <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamic</a> system at an <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">absolute temperature</a> <span class="texhtml mvar" style="font-style:italic;">T</span>, the average thermal energy carried by each microscopic degree of freedom in the system is <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <var style="padding-right: 1px;">k</var><var style="padding-right: 1px;">T</var></span> (i.e., about <span class="nowrap"><span data-sort-value="6979206999999999999♠"></span>2.07<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−21</sup> J</span>, or <span class="nowrap"><span data-sort-value="6979208282943310000♠"></span>0.013 <a href="/wiki/Electronvolt" title="Electronvolt">eV</a></span>, at room temperature). This is generally true only for classical systems with a <a href="/wiki/Thermodynamic_limit" title="Thermodynamic limit">large number of particles</a>, and in which quantum effects are negligible. </p><p>In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical</a> <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, this average is predicted to hold exactly for homogeneous <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gases</a>. Monatomic ideal gases (the six noble gases) possess three <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <var style="padding-right: 1px;">k</var><var style="padding-right: 1px;">T</var></span> per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the <a href="/wiki/Root-mean-square_speed" class="mw-redirect" title="Root-mean-square speed">root-mean-square speed</a> of the atoms, which turns out to be inversely proportional to the square root of the <a href="/wiki/Atomic_mass" title="Atomic mass">atomic mass</a>. The root mean square speeds found at room temperature accurately reflect this, ranging from <span class="nowrap"><span data-sort-value="7003137000000000000♠"></span>1370 m/s</span> for <a href="/wiki/Helium" title="Helium">helium</a>, down to <span class="nowrap"><span data-sort-value="7002240000000000000♠"></span>240 m/s</span> for <a href="/wiki/Xenon" title="Xenon">xenon</a>. </p><p><a href="/wiki/Kinetic_theory_of_gases#Pressure" title="Kinetic theory of gases">Kinetic theory</a> gives the average pressure <span class="texhtml mvar" style="font-style:italic;">p</span> for an ideal gas as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>V</mi> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8e12138aa2fb5c20185362a8ddf781c804bd59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:14.24ex; height:5.343ex;" alt="{\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.}"></span> </p><p>Combination with the ideal gas law <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pV=NkT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>V</mi> <mo>=</mo> <mi>N</mi> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pV=NkT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3da361ed1d92bf6756007a3e2226744eb5798d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.056ex; height:2.509ex;" alt="{\displaystyle pV=NkT}"></span> shows that the average translational kinetic energy is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>k</mi> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62595e846c20af33efebdf09248ab478785d707a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.246ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.}"></span> </p><p>Considering that the translational motion velocity vector <span class="texhtml"><b>v</b></span> 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. <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <var style="padding-right: 1px;">k</var><var style="padding-right: 1px;">T</var></span>. </p><p>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. </p> <div class="mw-heading mw-heading3"><h3 id="Role_in_Boltzmann_factors">Role in Boltzmann factors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=3" title="Edit section: Role in Boltzmann factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More generally, systems in equilibrium at temperature <span class="texhtml mvar" style="font-style:italic;">T</span> have probability <span class="texhtml"><var style="padding-right: 1px;">P</var><sub><var style="padding-right: 1px;">i</var></sub></span> of occupying a state <span class="texhtml mvar" style="font-style:italic;">i</span> with energy <span class="texhtml mvar" style="font-style:italic;">E</span> weighted by the corresponding <a href="/wiki/Boltzmann_factor" class="mw-redirect" title="Boltzmann factor">Boltzmann factor</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mi>Z</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693154403490b4a04b4a29983f850819a5dcb1bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.859ex; height:7.509ex;" alt="{\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},}"></span> where <span class="texhtml mvar" style="font-style:italic;">Z</span> is the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a>. Again, it is the energy-like quantity <span class="texhtml"><a href="/wiki/KT_(energy)" title="KT (energy)"><var style="padding-right: 1px;">k</var><var style="padding-right: 1px;">T</var></a></span> that takes central importance. </p><p>Consequences of this include (in addition to the results for ideal gases above) the <a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a> in <a href="/wiki/Chemical_kinetics" title="Chemical kinetics">chemical kinetics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Role_in_the_statistical_definition_of_entropy">Role in the statistical definition of entropy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=4" title="Edit section: Role in the statistical definition of entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Entropy_(statistical_thermodynamics)" title="Entropy (statistical thermodynamics)">Entropy (statistical thermodynamics)</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Zentralfriedhof_Vienna_-_Boltzmann.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Zentralfriedhof_Vienna_-_Boltzmann.JPG/220px-Zentralfriedhof_Vienna_-_Boltzmann.JPG" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Zentralfriedhof_Vienna_-_Boltzmann.JPG/330px-Zentralfriedhof_Vienna_-_Boltzmann.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Zentralfriedhof_Vienna_-_Boltzmann.JPG/440px-Zentralfriedhof_Vienna_-_Boltzmann.JPG 2x" data-file-width="1920" data-file-height="2560" /></a><figcaption>Boltzmann's grave in the <a href="/wiki/Zentralfriedhof" class="mw-redirect" title="Zentralfriedhof">Zentralfriedhof</a>, Vienna, with bust and entropy formula.</figcaption></figure> <p>In statistical mechanics, the <a href="/wiki/Entropy" title="Entropy">entropy</a> <span class="texhtml mvar" style="font-style:italic;">S</span> of an <a href="/wiki/Isolated_system" title="Isolated system">isolated system</a> at <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> is defined as the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> of <span class="texhtml mvar" style="font-style:italic;">W</span>, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy <span class="texhtml mvar" style="font-style:italic;">E</span>): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=k\,\ln W.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mi>W</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k\,\ln W.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf29694d93d7686ada0fbc1b0df588978bbcef8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.992ex; height:2.176ex;" alt="{\displaystyle S=k\,\ln W.}"></span> </p><p>This equation, which relates the microscopic details, or microstates, of the system (via <span class="texhtml mvar" style="font-style:italic;">W</span>) to its macroscopic state (via the entropy <span class="texhtml mvar" style="font-style:italic;">S</span>), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone. </p><p>The constant of proportionality <span class="texhtml mvar" style="font-style:italic;">k</span> serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of <a href="/wiki/Clausius" class="mw-redirect" title="Clausius">Clausius</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763f502a6b24c250c24db5ee31d00ba51ae5f060" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.728ex; height:5.843ex;" alt="{\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.}"></span> </p><p>One could choose instead a rescaled <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> entropy in microscopic terms such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>W</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>Q</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845bc1395be5677f059c36eb9e42f9e7f328f3d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.858ex; height:5.843ex;" alt="{\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.}"></span> </p><p>This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a>. </p><p>The characteristic energy <span class="texhtml mvar" style="font-style:italic;">kT</span> is thus the energy required to increase the rescaled entropy by one <a href="/wiki/Nat_(unit)" title="Nat (unit)">nat</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Thermal_voltage">Thermal voltage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=5" title="Edit section: Thermal voltage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Semiconductors" class="mw-redirect" title="Semiconductors">semiconductors</a>, the <a href="/wiki/Shockley_diode_equation" title="Shockley diode equation">Shockley diode equation</a>—the relationship between the flow of <a href="/wiki/Electric_current" title="Electric current">electric current</a> and the <a href="/wiki/Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">electrostatic potential</a> across a <a href="/wiki/P%E2%80%93n_junction" title="P–n junction">p–n junction</a>—depends on a characteristic voltage called the <i>thermal voltage</i>, denoted by <span class="texhtml"><var style="padding-right: 1px;">V</var><sub><var style="padding-right: 1px;">T</var></sub></span>. The thermal voltage depends on absolute temperature <span class="texhtml mvar" style="font-style:italic;">T</span> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {T} }={kT \over q}={RT \over F},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mi>q</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mi>F</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {T} }={kT \over q}={RT \over F},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ac3ed5b62b6f4cce6f364271d8147e97512140" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.538ex; height:5.843ex;" alt="{\displaystyle V_{\mathrm {T} }={kT \over q}={RT \over F},}"></span> where <span class="texhtml mvar" style="font-style:italic;">q</span> is the magnitude of the <a href="/wiki/Elementary_charge" title="Elementary charge">electrical charge on the electron</a> with a value <span class="nowrap"><span data-sort-value="6981160217663400000♠"></span>1.602<span style="margin-left:.25em;">176</span><span style="margin-left:.25em;">634</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−19</sup> C</span>.<sup id="cite_ref-physconst-e_6-0" class="reference"><a href="#cite_note-physconst-e-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.617333262\times 10^{-5}\ \mathrm {V/K} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>q</mi> </mfrac> </mrow> <mo>≈</mo> <mn>8.617333262</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">K</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.617333262\times 10^{-5}\ \mathrm {V/K} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9cd004c4ea2546ddfb0e440c3ca0b6f703afe1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.565ex; height:5.843ex;" alt="{\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.617333262\times 10^{-5}\ \mathrm {V/K} .}"></span> </p><p>At <a href="/wiki/Room_temperature" title="Room temperature">room temperature</a> 300 K (27 °C; 80 °F), <span class="texhtml"><var style="padding-right: 1px;">V</var><sub><var style="padding-right: 1px;">T</var></sub></span> is approximately <span class="nowrap"><span data-sort-value="6998258500000000000♠"></span>25.85 mV</span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> which can be derived by plugging in the values as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mi>q</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1.38</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>23</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <msup> <mi mathvariant="normal">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>×</mo> <mn>300</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> <mrow> <mn>1.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>19</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </mfrac> </mrow> <mo>≃</mo> <mn>25.85</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ac61442644781ccfcd5a069701d3ec996c861d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.88ex; height:6.176ex;" alt="{\displaystyle 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} }"></span> </p><p>At the <a href="/wiki/Standard_state" title="Standard state">standard state</a> temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately <span class="nowrap"><span data-sort-value="6998256900000000000♠"></span>25.69 mV</span>. The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the <a href="/wiki/Nernst_equation" title="Nernst equation">Nernst equation</a>); 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.<sup id="cite_ref-Kirby_9-0" class="reference"><a href="#cite_note-Kirby-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tabeling_10-0" class="reference"><a href="#cite_note-Tabeling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Boltzmann constant is named after its 19th century Austrian discoverer, <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>. Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> first introduced <span class="texhtml mvar" style="font-style:italic;">k</span>, and gave a more precise value for it (<span class="nowrap"><span data-sort-value="6977134600000000000♠"></span>1.346<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J/K</span>, about 2.5% lower than today's figure), in his derivation of the <a href="/wiki/Planck%27s_law" title="Planck's law">law of black-body radiation</a> in 1900–1901.<sup id="cite_ref-Planck01_11-0" class="reference"><a href="#cite_note-Planck01-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> 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 <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a> <span class="texhtml mvar" style="font-style:italic;">R</span>, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation <span class="texhtml"><i>S</i> = <i>k</i> ln <i>W</i></span> on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his <a href="/wiki/Planck_constant" title="Planck constant">eponymous <span class="texhtml mvar" style="font-style:italic;">h</span></a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>In 1920, Planck wrote in his <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a> lecture:<sup id="cite_ref-PlanckNobel_13-0" class="reference"><a href="#cite_note-PlanckNobel-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>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.</p></blockquote> <p>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 <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> tool for solving problems. There was no agreement whether <i>chemical</i> molecules, as measured by <a href="/wiki/Atomic_weight" class="mw-redirect" title="Atomic weight">atomic weights</a>, were the same as <i>physical</i> molecules, as measured by <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory</a>. Planck's 1920 lecture continued:<sup id="cite_ref-PlanckNobel_13-1" class="reference"><a href="#cite_note-PlanckNobel-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>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.</p></blockquote> <p>In versions of <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> prior to the <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision of the SI</a>, 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 <a href="/wiki/Kelvin#History" title="Kelvin">Kelvin § History</a>) and other SI base units (see <a href="/wiki/Joule#History" title="Joule">Joule § History</a>). </p><p>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.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> This decade-long effort was undertaken with different techniques by several laboratories;<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> it is one of the cornerstones of the 2019 revision of the SI. Based on these measurements, the <a href="/wiki/CODATA" class="mw-redirect" title="CODATA">CODATA</a> recommended <span class="nowrap"><span data-sort-value="6977138064900000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J/K</span> to be the final fixed value of the Boltzmann constant to be used for the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>As a precondition for redefining the Boltzmann constant, there must be one experimental value with a relative uncertainty below 1 <a href="/wiki/Parts_per_million" class="mw-redirect" title="Parts per million">ppm</a>, and at least one measurement from a second technique with a relative uncertainty below 3 ppm. The acoustic gas thermometry reached 0.2 ppm, and Johnson noise thermometry reached 2.8 ppm.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Value_in_different_units">Value in different units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=7" title="Edit section: Value in different units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Values of <span class="texhtml mvar" style="font-style:italic;">k</span> </th> <th>Comments </th></tr> <tr> <td><span class="nowrap"><span data-sort-value="6977138064900000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J⋅K<sup>−1</sup></span><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup id="cite_ref-physconst-k_20-0" class="reference"><a href="#cite_note-physconst-k-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></td> <td><a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> definition </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6995861733326200000♠"></span>8.617<span style="margin-left:.25em;">333</span><span style="margin-left:.25em;">262</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup> <a href="/wiki/Electronvolt" title="Electronvolt">eV</a>/K</span><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></td> <td> </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="7010208366191200000♠"></span>2.083<span style="margin-left:.25em;">661</span><span style="margin-left:.25em;">912</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>10</sup> <a href="/wiki/Hertz" title="Hertz">Hz</a>/K</span></td> <td>(<span class="texhtml"><i>k</i>/<i>h</i></span>) </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6984138064900000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−16</sup> <a href="/wiki/Erg" title="Erg">erg</a>/K</span></td> <td><a href="/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units" title="Centimetre–gram–second system of units">CGS</a>, 1 <a href="/wiki/Erg" title="Erg">erg</a> = <span class="nowrap"><span data-sort-value="6993100000000000000♠"></span>1<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup> J</span> </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6976329762348299999♠"></span>3.297<span style="margin-left:.25em;">623</span><span style="margin-left:.25em;">483</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−24</sup> <a href="/wiki/Calorie" title="Calorie">cal</a>/K</span></td> <td>1 <a href="/wiki/Calorie" title="Calorie">calorie</a> = <span class="nowrap"><span data-sort-value="7000418680000000000♠"></span>4.1868 J</span> </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6976183201304599999♠"></span>1.832<span style="margin-left:.25em;">013</span><span style="margin-left:.25em;">046</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−24</sup> cal/<a href="/wiki/Rankine_scale" title="Rankine scale">°R</a></span></td> <td> </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6976565730246599999♠"></span>5.657<span style="margin-left:.25em;">302</span><span style="margin-left:.25em;">466</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−24</sup> <a href="/wiki/Foot-pound_force" class="mw-redirect" title="Foot-pound force">ft lb</a>/°R</span></td> <td> </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6999695034800000000♠"></span>0.695<span style="margin-left:.25em;">034</span><span style="margin-left:.25em;">800</span>... <a href="/wiki/Wavenumber" title="Wavenumber">cm<sup>−1</sup></a>/K</span></td> <td>(<span class="texhtml"><i>k</i>/(<i>hc</i>)</span>) </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6994316681156300000♠"></span>3.166<span style="margin-left:.25em;">811</span><span style="margin-left:.25em;">563</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−6</sup> <a href="/wiki/Hartree" title="Hartree"><i>E</i><sub>h</sub></a>/K</span></td> <td> </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6997198720425900000♠"></span>1.987<span style="margin-left:.25em;">204</span><span style="margin-left:.25em;">259</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−3</sup> <a href="/wiki/Kcal" class="mw-redirect" title="Kcal">kcal</a>/(<a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a>⋅K)</span></td> <td>(<span class="texhtml"><i>kN</i><sub>A</sub></span>) </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6997831446261800000♠"></span>8.314<span style="margin-left:.25em;">462</span><span style="margin-left:.25em;">618</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−3</sup> kJ/(mol⋅K)</span></td> <td>(<span class="texhtml"><i>kN</i><sub>A</sub></span>) </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="2997771400832800000♠"></span>−228.599<span style="margin-left:.25em;">1672</span>... <a href="/wiki/Decibel" title="Decibel">dB</a>(W/K/Hz)</span></td> <td><span class="texhtml">10 log<sub>10</sub>(<i>k</i>/(1 W/K/Hz))</span>, used for <a href="/wiki/Thermal_noise" class="mw-redirect" title="Thermal noise">thermal noise</a> calculations </td></tr> <tr> <td><span class="nowrap"><span data-sort-value="6960153617918699999♠"></span>1.536<span style="margin-left:.25em;">179</span><span style="margin-left:.25em;">187</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−40</sup> kg/K</span><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></td> <td>(<span class="texhtml"><i>k</i>/<i>c</i><sup>2</sup></span>) </td></tr></tbody></table> <p>Since <span class="texhtml mvar" style="font-style:italic;">k</span> is a <a href="/wiki/Proportionality_factor" class="mw-redirect" title="Proportionality factor">proportionality factor</a> 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 <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> units means a change in temperature by <a href="/wiki/Kelvin" title="Kelvin">1 K</a> only changes a particle's energy by a small amount. A change of <span class="nowrap"><span data-sort-value="7002274149999999999♠"></span>1 <a href="/wiki/Celsius" title="Celsius">°C</a></span> is defined to be the same as a change of <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1 K</span>. The characteristic energy <span class="texhtml mvar" style="font-style:italic;">kT</span> is a term encountered in many physical relationships. </p><p>The Boltzmann constant sets up a relationship between wavelength and temperature (dividing <i>hc</i>/<i>k</i> by a wavelength gives a temperature) with one micrometer being related to <span class="nowrap"><span data-sort-value="7004143877770000000♠"></span>14<span style="margin-left:.25em;">387</span>.777 K</span>, and also a relationship between voltage and temperature (<i>kT</i> in units of eV corresponds to a voltage) with one volt being related to <span class="nowrap"><span data-sort-value="7004116045180000000♠"></span>11<span style="margin-left:.25em;">604</span>.518 K</span>. The ratio of these two temperatures, <span class="nowrap"><span data-sort-value="7004143877770000000♠"></span>14<span style="margin-left:.25em;">387</span>.777 K</span> / <span class="nowrap"><span data-sort-value="7004116045180000000♠"></span>11<span style="margin-left:.25em;">604</span>.518 K</span> ≈ 1.239842, is the numerical value of <i>hc</i> in units of eV⋅μm. </p> <div class="mw-heading mw-heading3"><h3 id="Natural_units">Natural units</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=8" title="Edit section: Natural units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Boltzmann constant provides a mapping from the characteristic microscopic energy <span class="texhtml mvar" style="font-style:italic;">E</span> to the macroscopic temperature scale <span class="texhtml"><var style="padding-right: 1px;">T</var> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">E</var></span><span class="sr-only">/</span><span class="den"><var style="padding-right: 1px;">k</var></span></span>⁠</span></span>. In fundamental physics, this mapping is often simplified by using the <a href="/wiki/Natural_units" title="Natural units">natural units</a> of setting <span class="texhtml mvar" style="font-style:italic;">k</span> to unity. This convention means that temperature and energy quantities have the same <a href="/wiki/Dimension_(physics)" class="mw-redirect" title="Dimension (physics)">dimensions</a>.<sup id="cite_ref-Kalinin_23-0" class="reference"><a href="#cite_note-Kalinin-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as <span class="texhtml">1 K = <span class="nowrap"><span data-sort-value="6977138064900000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J</span></span>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas.<sup id="cite_ref-Kalinin_23-1" class="reference"><a href="#cite_note-Kalinin-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998d98e22629427549d15750415f530b74f67244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.506ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}kT}"></span> above) becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af05e27c4ca927021993d2631f0be64a98d7ea5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.688ex; height:3.509ex;" alt="{\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T}"></span> </p><p>As another example, the definition of thermodynamic entropy coincides with the form of <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15740d52088aa5a731e31fa95173ab7d038fdb0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.479ex; height:5.509ex;" alt="{\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.}"></span> where <span class="texhtml"><var style="padding-right: 1px;">P</var><sub><var style="padding-right: 1px;">i</var></sub></span> is the probability of each <a href="/wiki/Microstate_(statistical_mechanics)" title="Microstate (statistical mechanics)">microstate</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Committee_on_Data_of_the_International_Science_Council" title="Committee on Data of the International Science Council">Committee on Data of the International Science Council</a></li> <li><a href="/wiki/Thermodynamic_beta" title="Thermodynamic beta">Thermodynamic beta</a></li> <li><a href="/wiki/List_of_scientists_whose_names_are_used_in_physical_constants" title="List of scientists whose names are used in physical constants">List of scientists whose names are used in physical constants</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Independent techniques exploited: acoustic gas thermometry, dielectric constant gas thermometry, <a href="/wiki/Johnson%E2%80%93Nyquist_noise" title="Johnson–Nyquist noise">Johnson noise thermometry</a>. Involved laboratories cited by CODATA in 2017: <a href="/wiki/Laboratoire_national_de_m%C3%A9trologie_et_d%27essais" title="Laboratoire national de métrologie et d'essais">LNE</a>-<a href="/wiki/Conservatoire_national_des_arts_et_m%C3%A9tiers" title="Conservatoire national des arts et métiers">Cnam</a> (France), <a href="/wiki/National_Physical_Laboratory_(United_Kingdom)" title="National Physical Laboratory (United Kingdom)">NPL</a> (UK), <a rel="nofollow" class="external text" href="https://www.inrim.it/">INRIM</a> (Italy), <a href="/wiki/Physikalisch-Technische_Bundesanstalt" title="Physikalisch-Technische Bundesanstalt">PTB</a> (Germany), <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a> (USA), <a rel="nofollow" class="external text" href="http://en.nim.ac.cn/">NIM</a> (China).</span> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-SI2019-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-SI2019_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SI2019_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-SI2019_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFNewellTiesinga2019" class="citation book cs1">Newell, David B.; Tiesinga, Eite (2019). <a rel="nofollow" class="external text" href="https://www.nist.gov/si-redefinition/meet-constants"><i>The International System of Units (SI)</i></a>. NIST Special Publication 330. Gaithersburg, Maryland: <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.6028%2Fnist.sp.330-2019">10.6028/nist.sp.330-2019</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:242934226">242934226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+International+System+of+Units+%28SI%29&rft.place=Gaithersburg%2C+Maryland&rft.series=NIST+Special+Publication+330&rft.pub=National+Institute+of+Standards+and+Technology&rft.date=2019&rft_id=info%3Adoi%2F10.6028%2Fnist.sp.330-2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A242934226%23id-name%3DS2CID&rft.aulast=Newell&rft.aufirst=David+B.&rft.au=Tiesinga%2C+Eite&rft_id=https%3A%2F%2Fwww.nist.gov%2Fsi-redefinition%2Fmeet-constants&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+constant" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|work=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-Feynman1Ch39-10-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Feynman1Ch39-10_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman1970" class="citation book cs1">Feynman, Richard (1970). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_39.html"><i>The Feynman Lectures on Physics Vol I</i></a>. 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Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-856864-3" title="Special:BookSources/978-0-19-856864-3"><bdi>978-0-19-856864-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Microfluidics&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=978-0-19-856864-3&rft.aulast=Tabeling&rft.aufirst=Patrick&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontomi0000tabe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+constant" class="Z3988"></span></span> </li> <li id="cite_note-Planck01-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Planck01_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlanck1901" class="citation journal cs1"><a href="/wiki/Max_Planck" title="Max Planck">Planck, Max</a> (1901). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19013090310">"Ueber das Gesetz der Energieverteilung im Normalspectrum"</a>. <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></i>. <b>309</b> (3): 553–63. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1901AnP...309..553P">1901AnP...309..553P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19013090310">10.1002/andp.19013090310</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Ueber+das+Gesetz+der+Energieverteilung+im+Normalspectrum&rft.volume=309&rft.issue=3&rft.pages=553-63&rft.date=1901&rft_id=info%3Adoi%2F10.1002%2Fandp.19013090310&rft_id=info%3Abibcode%2F1901AnP...309..553P&rft.aulast=Planck&rft.aufirst=Max&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Fandp.19013090310&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+constant" class="Z3988"></span>. English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081217042934/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html">"On the Law of Distribution of Energy in the Normal Spectrum"</a>. 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Freeman. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0716710889" title="Special:BookSources/0716710889"><bdi>0716710889</bdi></a>. <q>We prefer to use a more natural temperature scale ... the fundamental temperature has the units of energy.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermal+physics&rft.place=San+Francisco&rft.pages=41&rft.edition=2nd&rft.pub=W.+H.+Freeman&rft.date=1980&rft.isbn=0716710889&rft.aulast=Kittel&rft.aufirst=Charles&rft.au=Kroemer%2C+Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+constant" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMohrShirleyPhillipsTrott2022" class="citation journal cs1">Mohr, Peter J.; Shirley, Eric L.; Phillips, William D.; Trott, Michael (1 October 2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fac7bc2">"On the dimension of angles and their units"</a>. <i>Metrologia</i>. <b>59</b> (5): 053001. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2203.12392">2203.12392</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022Metro..59e3001M">2022Metro..59e3001M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fac7bc2">10.1088/1681-7575/ac7bc2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrologia&rft.atitle=On+the+dimension+of+angles+and+their+units&rft.volume=59&rft.issue=5&rft.pages=053001&rft.date=2022-10-01&rft_id=info%3Aarxiv%2F2203.12392&rft_id=info%3Adoi%2F10.1088%2F1681-7575%2Fac7bc2&rft_id=info%3Abibcode%2F2022Metro..59e3001M&rft.aulast=Mohr&rft.aufirst=Peter+J.&rft.au=Shirley%2C+Eric+L.&rft.au=Phillips%2C+William+D.&rft.au=Trott%2C+Michael&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1681-7575%252Fac7bc2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+constant" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_constant&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.bipm.org/utils/common/pdf/si_brochure_draft_ch2.pdf">Draft Chapter 2 for SI Brochure, following redefinitions of the base units</a> (prepared by the Consultative Committee for Units)</li> <li><a rel="nofollow" class="external text" href="https://www.sciencedaily.com/releases/2011/09/110920075520.htm">Big step towards redefining the kelvin: Scientists find new way to determine Boltzmann constant</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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