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class="vector-toc-numb">2</span> <span>Absolute zero of temperature</span> </div> </a> <ul id="toc-Absolute_zero_of_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Boltzmann_constant" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Boltzmann_constant"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Boltzmann constant</span> </div> </a> <ul id="toc-Boltzmann_constant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rankine_scale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rankine_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Rankine scale</span> </div> </a> <ul id="toc-Rankine_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_redefinition_of_the_kelvin" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modern_redefinition_of_the_kelvin"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Modern redefinition of the kelvin</span> </div> </a> <ul id="toc-Modern_redefinition_of_the_kelvin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_of_temperature,_motions,_conduction,_and_thermal_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_of_temperature,_motions,_conduction,_and_thermal_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Relationship of temperature, motions, conduction, and thermal energy</span> </div> </a> <button aria-controls="toc-Relationship_of_temperature,_motions,_conduction,_and_thermal_energy-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 Relationship of temperature, motions, conduction, and thermal energy subsection</span> </button> <ul id="toc-Relationship_of_temperature,_motions,_conduction,_and_thermal_energy-sublist" class="vector-toc-list"> <li id="toc-Nature_of_kinetic_energy,_translational_motion,_and_temperature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nature_of_kinetic_energy,_translational_motion,_and_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Nature of kinetic energy, translational motion, and temperature</span> </div> </a> <ul id="toc-Nature_of_kinetic_energy,_translational_motion,_and_temperature-sublist" class="vector-toc-list"> <li id="toc-High_speeds_of_translational_motion" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#High_speeds_of_translational_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>High speeds of translational motion</span> </div> </a> <ul id="toc-High_speeds_of_translational_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_motions_of_molecules_and_internal_energy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Internal_motions_of_molecules_and_internal_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Internal motions of molecules and internal energy</span> </div> </a> <ul id="toc-Internal_motions_of_molecules_and_internal_energy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Diffusion_of_thermal_energy:_entropy,_phonons,_and_mobile_conduction_electrons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_of_thermal_energy:_entropy,_phonons,_and_mobile_conduction_electrons"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons</span> </div> </a> <ul id="toc-Diffusion_of_thermal_energy:_entropy,_phonons,_and_mobile_conduction_electrons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffusion_of_thermal_energy:_black-body_radiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_of_thermal_energy:_black-body_radiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Diffusion of thermal energy: black-body radiation</span> </div> </a> <ul id="toc-Diffusion_of_thermal_energy:_black-body_radiation-sublist" class="vector-toc-list"> <li id="toc-Table_of_thermodynamic_temperatures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Table_of_thermodynamic_temperatures"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.1</span> <span>Table of thermodynamic temperatures</span> </div> </a> <ul id="toc-Table_of_thermodynamic_temperatures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Heat_of_phase_changes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_of_phase_changes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Heat of phase changes</span> </div> </a> <ul id="toc-Heat_of_phase_changes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internal_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Internal energy</span> </div> </a> <ul id="toc-Internal_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_energy_at_absolute_zero" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internal_energy_at_absolute_zero"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Internal energy at absolute zero</span> </div> </a> <ul id="toc-Internal_energy_at_absolute_zero-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Practical_applications_for_thermodynamic_temperature" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Practical_applications_for_thermodynamic_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Practical applications for thermodynamic temperature</span> </div> </a> <ul id="toc-Practical_applications_for_thermodynamic_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_ideal_gas_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_ideal_gas_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Relationship to ideal gas law</span> </div> </a> <ul id="toc-Relationship_to_ideal_gas_law-sublist" class="vector-toc-list"> </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">9</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </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">10</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">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">12</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 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Available in 43 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-43" 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">43 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%B1%D8%AC%D8%A9_%D8%AD%D8%B1%D8%A7%D8%B1%D8%A9_%D9%85%D8%B7%D9%84%D9%82%D8%A9" title="درجة حرارة مطلقة – Arabic" lang="ar" hreflang="ar" data-title="درجة حرارة مطلقة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%B0%E0%A6%AE_%E0%A6%A4%E0%A6%BE%E0%A6%AA%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BE" title="পরম তাপমাত্রা – Bangla" lang="bn" hreflang="bn" data-title="পরম তাপমাত্রা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Абсалютная тэрмадынамічная тэмпература – Belarusian" lang="be" hreflang="be" data-title="Абсалютная тэрмадынамічная тэмпература" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BC%D0%BF%D1%8D%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Абсалютная тэрмадынамічная тэмпэратура – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Абсалютная тэрмадынамічная тэмпэратура" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамична температура – Bulgarian" lang="bg" hreflang="bg" data-title="Термодинамична температура" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Temperatura_termodin%C3%A0mica" title="Temperatura termodinàmica – Catalan" lang="ca" hreflang="ca" data-title="Temperatura termodinàmica" 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%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамикăлла температура – Chuvash" lang="cv" hreflang="cv" data-title="Термодинамикăлла температура" 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/Termodynamick%C3%A1_teplota" title="Termodynamická teplota – Czech" lang="cs" hreflang="cs" data-title="Termodynamická teplota" 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/Termodynamisk_temperatur" title="Termodynamisk temperatur – Danish" lang="da" hreflang="da" data-title="Termodynamisk temperatur" 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/Thermodynamische_Temperatur" title="Thermodynamische Temperatur – German" lang="de" hreflang="de" data-title="Thermodynamische Temperatur" 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/Absoluutne_temperatuur" title="Absoluutne temperatuur – Estonian" lang="et" hreflang="et" data-title="Absoluutne temperatuur" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Temperatura_absoluta" title="Temperatura absoluta – Spanish" lang="es" hreflang="es" data-title="Temperatura absoluta" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%85%D8%A7%DB%8C_%D8%AA%D8%B1%D9%85%D9%88%D8%AF%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9%DB%8C" title="دمای ترمودینامیکی – Persian" lang="fa" hreflang="fa" data-title="دمای ترمودینامیکی" 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/Temp%C3%A9rature_thermodynamique" title="Température thermodynamique – French" lang="fr" hreflang="fr" data-title="Température thermodynamique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B4%EC%97%AD%ED%95%99_%EC%98%A8%EB%8F%84" title="열역학 온도 – Korean" lang="ko" hreflang="ko" data-title="열역학 온도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%A4%E0%A4%BE%E0%A4%AA%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="ऊष्मागतिक तापक्रम – Hindi" lang="hi" hreflang="hi" data-title="ऊष्मागतिक तापक्रम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Suhu_termodinamika" title="Suhu termodinamika – Indonesian" lang="id" hreflang="id" data-title="Suhu termodinamika" 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/Temperatura_assoluta" title="Temperatura assoluta – Italian" lang="it" hreflang="it" data-title="Temperatura assoluta" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%97%E1%83%94%E1%83%A0%E1%83%9B%E1%83%9D%E1%83%93%E1%83%98%E1%83%9C%E1%83%90%E1%83%9B%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A2%E1%83%94%E1%83%9B%E1%83%9E%E1%83%94%E1%83%A0%E1%83%90%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%90" title="თერმოდინამიკური ტემპერატურა – Georgian" lang="ka" hreflang="ka" data-title="თერმოდინამიკური ტემპერატურა" 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%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамикалық температура – Kazakh" lang="kk" hreflang="kk" data-title="Термодинамикалық температура" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Termodinamikai_h%C5%91m%C3%A9rs%C3%A9klet" title="Termodinamikai hőmérséklet – Hungarian" lang="hu" hreflang="hu" data-title="Termodinamikai hőmérséklet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Suhu_termodinamik" title="Suhu termodinamik – Malay" lang="ms" hreflang="ms" data-title="Suhu termodinamik" 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%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B8%D0%B9%D0%BD_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80" title="Термодинамикийн температур – Mongolian" lang="mn" hreflang="mn" data-title="Термодинамикийн температур" 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/Absolute_temperatuur" title="Absolute temperatuur – Dutch" lang="nl" hreflang="nl" data-title="Absolute temperatuur" 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/%E7%86%B1%E5%8A%9B%E5%AD%A6%E6%B8%A9%E5%BA%A6" title="熱力学温度 – Japanese" lang="ja" hreflang="ja" data-title="熱力学温度" 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/Termodynamisk_temperatur" title="Termodynamisk temperatur – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Termodynamisk temperatur" 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/Termodynamisk_temperatur" title="Termodynamisk temperatur – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Termodynamisk temperatur" 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-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kelviin" title="Kelviin – Oromo" lang="om" hreflang="om" data-title="Kelviin" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Termodinamik_harorat" title="Termodinamik harorat – Uzbek" lang="uz" hreflang="uz" data-title="Termodinamik harorat" 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-pt badge-Q70893996 mw-list-item" title=""><a href="https://pt.wikipedia.org/wiki/Temperatura_termodin%C3%A2mica" title="Temperatura termodinâmica – Portuguese" lang="pt" hreflang="pt" data-title="Temperatura termodinâmica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Absolyut_temperatura" title="Absolyut temperatura – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Absolyut temperatura" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамическая температура – Russian" lang="ru" hreflang="ru" data-title="Термодинамическая температура" 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/Absolute_temperature" title="Absolute temperature – Simple English" lang="en-simple" hreflang="en-simple" data-title="Absolute temperature" 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/Termodynamick%C3%A1_teplota" title="Termodynamická teplota – Slovak" lang="sk" hreflang="sk" data-title="Termodynamická teplota" 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/Absolutna_temperatura" title="Absolutna temperatura – Slovenian" lang="sl" hreflang="sl" data-title="Absolutna temperatura" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mutlak_s%C4%B1cakl%C4%B1k" title="Mutlak sıcaklık – Turkish" lang="tr" hreflang="tr" data-title="Mutlak sıcaklık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Absol%C3%BDut_temperatura" title="Absolýut temperatura – Turkmen" lang="tk" hreflang="tk" data-title="Absolýut temperatura" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамічна температура – Ukrainian" lang="uk" hreflang="uk" data-title="Термодинамічна температура" 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%AD%D8%B1%D8%AD%D8%B1%DA%A9%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AF%D8%B1%D8%AC%DB%81_%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA_%DB%8C%D8%A7_%D9%85%D8%B7%D9%84%D9%82_%D8%AF%D8%B1%D8%AC%DB%81_%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA" title="حرحرکیاتی درجہ حرارت یا مطلق درجہ حرارت – Urdu" lang="ur" hreflang="ur" data-title="حرحرکیاتی درجہ حرارت یا مطلق درجہ حرارت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Thermodynamische_temperateure" title="Thermodynamische temperateure – West Flemish" lang="vls" hreflang="vls" data-title="Thermodynamische temperateure" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%83%AD%E5%8A%9B%E5%AD%A6%E6%B8%A9%E5%BA%A6" title="热力学温度 – Wu" lang="wuu" hreflang="wuu" data-title="热力学温度" 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/%E7%B5%95%E5%B0%8D%E6%BA%AB%E5%BA%A6" title="絕對溫度 – Cantonese" lang="yue" hreflang="yue" data-title="絕對溫度" 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/%E7%83%AD%E5%8A%9B%E5%AD%A6%E6%B8%A9%E5%BA%A6" title="热力学温度 – Chinese" lang="zh" hreflang="zh" data-title="热力学温度" 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 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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">Measure of temperature relative to absolute zero</div> <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 ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a class="mw-selflink selflink">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><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 c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><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 \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><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 \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><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 \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><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 \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><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 U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><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 H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><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 A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><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 G(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Other</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Thermodynamic temperature</b> is a quantity defined in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> as distinct from <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory</a> or <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. </p><p>Historically, thermodynamic temperature was defined by <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a> in terms of a macroscopic relation between <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">thermodynamic work</a> and <a href="/wiki/Heat" title="Heat">heat transfer</a> as defined in thermodynamics, but the kelvin was redefined by international agreement in 2019 in terms of phenomena that are now understood as manifestations of the kinetic energy of free motion of microscopic particles such as atoms, molecules, and electrons. From the thermodynamic viewpoint, for historical reasons, because of how it is defined and measured, this microscopic kinetic definition is regarded as an "empirical" temperature. It was adopted because in practice it can generally be measured more precisely than can Kelvin's thermodynamic temperature. </p><p>A thermodynamic temperature of zero is of particular importance for the <a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">third law of thermodynamics</a>. By convention, it is reported on the <i><a href="/wiki/Kelvin" title="Kelvin">Kelvin scale</a></i> of <a href="/wiki/Temperature" title="Temperature">temperature</a> in which the <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit of measurement</a> is the <i>kelvin</i> (unit symbol: K). For comparison, a temperature of 295 K corresponds to 21.85 °C and 71.33 °F. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thermodynamic temperature, as distinct from SI temperature, is defined in terms of a macroscopic <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot cycle</a>. Thermodynamic temperature is of importance in thermodynamics because it is defined in purely thermodynamic terms. SI temperature is conceptually far different from thermodynamic temperature. Thermodynamic temperature was rigorously defined historically long before there was a fair knowledge of microscopic particles such as atoms, molecules, and electrons. </p><p>The <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI) specifies the international absolute scale for measuring temperature, and the unit of measure <i><a href="/wiki/Kelvin" title="Kelvin">kelvin</a></i> (unit symbol: K) for specific values along the scale. The kelvin is also used for denoting temperature <i>intervals</i> (a span or difference between two temperatures) as per the following example usage: "A 60/40 tin/lead solder is non-eutectic and is plastic through a range of 5 kelvins as it solidifies." A temperature interval of one degree Celsius is the same magnitude as one kelvin. </p><p>The magnitude of the kelvin was <a href="/wiki/2019_revision_of_the_SI#Kelvin" title="2019 revision of the SI">redefined in 2019</a> in relation to the <i>physical property</i> underlying thermodynamic temperature: the kinetic energy of atomic free particle motion. The revision fixed the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> at 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> joules per kelvin</span> (J/K).<sup id="cite_ref-Accessible_k_1-0" class="reference"><a href="#cite_note-Accessible_k-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The microscopic property that imbues material substances with a temperature can be readily understood by examining the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a>, which relates, per the Boltzmann constant, how <a href="/wiki/Heat" title="Heat">heat energy</a> causes precisely defined changes in the <a href="/wiki/Pressure" title="Pressure">pressure</a> and temperature of certain gases. This is because <a href="/wiki/Monatomic_gas" title="Monatomic gas">monatomic gases</a> like <a href="/wiki/Helium" title="Helium">helium</a> and <a href="/wiki/Argon" title="Argon">argon</a> behave kinetically like freely moving perfectly elastic and spherical billiard balls that move only in a specific subset of the possible motions that can occur in matter: that comprising the <i>three translational</i> <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a>. The translational degrees of freedom are the familiar billiard ball-like movements along the X, Y, and Z axes of 3D space (see <a href="#Nature_of_kinetic_energy,_translational_motion,_and_temperature"><i>Fig. 1</i></a>, below). This is why the noble gases all have the <a href="/wiki/Table_of_specific_heat_capacities" title="Table of specific heat capacities">same specific heat capacity per atom</a> and why that value is lowest of all the gases. </p><p><a href="/wiki/Molecule" title="Molecule">Molecules</a> (two or more chemically bound atoms), however, have <i>internal structure</i> and therefore have additional <i>internal</i> degrees of freedom (see <a href="#Internal_motions_of_molecules_and_internal_energy"><i>Fig. 3</i></a>, below), which makes molecules absorb more heat energy for any given amount of temperature rise than do the monatomic gases. Heat energy is born in all available degrees of freedom; this is in accordance with the <a href="/wiki/Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a>, so all available internal degrees of freedom have the same temperature as their three external degrees of freedom. However, the property that gives all gases their <a href="/wiki/Pressure" title="Pressure">pressure</a>, which is the net force per unit area on a container arising from gas particles recoiling off it, is a function of the kinetic energy borne in the freely moving atoms' and molecules' three translational degrees of freedom.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Fixing the Boltzmann constant at a specific value, along with other rule making, had the effect of precisely establishing the magnitude of the unit interval of SI temperature, the kelvin, in terms of the average kinetic behavior of the noble gases. Moreover, the <i>starting point</i> of the thermodynamic temperature scale, absolute zero, was reaffirmed as the point at which <i>zero average kinetic energy</i> remains in a sample; the only remaining particle motion being that comprising random vibrations due to zero-point energy. </p> <div class="mw-heading mw-heading2"><h2 id="Absolute_zero_of_temperature">Absolute zero of temperature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=2" title="Edit section: Absolute zero of temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Absolute_zero" title="Absolute zero">Absolute zero</a></div> <p>Temperature scales are numerical. The numerical zero of a temperature scale is not bound to the absolute zero of temperature. Nevertheless, some temperature scales have their numerical zero coincident with the absolute zero of temperature. Examples are the International SI temperature scale, the <a href="/wiki/Rankine_scale" title="Rankine scale">Rankine temperature scale</a>, and the thermodynamic temperature scale. Other temperature scales have their numerical zero far from the absolute zero of temperature. Examples are the Fahrenheit scale and the Celsius scale. </p><p>At the zero point of thermodynamic temperature, <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, the particle constituents of matter have minimal motion and can become no colder.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> Absolute zero, which is a temperature of zero kelvins (0 K), precisely corresponds to −273.15 °C and −459.67 °F. Matter at absolute zero has no remaining transferable average kinetic energy and the only remaining particle motion is due to an ever-pervasive <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> phenomenon called ZPE (<a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a>).<sup id="cite_ref-T0_5-0" class="reference"><a href="#cite_note-T0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Though the atoms in, for instance, a container of liquid <a href="/wiki/Helium" title="Helium">helium</a> that was <i>precisely</i> at absolute zero would still jostle slightly due to zero-point energy, a <a href="/wiki/Carnot_cycle" title="Carnot cycle">theoretically perfect heat engine</a> with such helium as one of its <a href="/wiki/Working_fluid" title="Working fluid">working fluids</a> could never transfer any net kinetic energy (<a href="/wiki/Heat" title="Heat">heat energy</a>) to the other working fluid and no <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">thermodynamic work</a> could occur. </p><p>Temperature is generally expressed in absolute terms when scientifically examining temperature's interrelationships with certain other physical properties of matter such as its <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">volume</a> or <a href="/wiki/Pressure" title="Pressure">pressure</a> (see <a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a>), or the wavelength of its emitted <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a>. Absolute temperature is also useful when calculating chemical reaction rates (see <a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a>). Furthermore, absolute temperature is typically used in <a href="/wiki/Cryogenics" title="Cryogenics">cryogenics</a> and related phenomena like <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>, as per the following example usage: "Conveniently, tantalum's transition temperature (<i>T</i><sub>c</sub>) of 4.4924 kelvin is slightly above the 4.2221 K boiling point of helium." </p> <div class="mw-heading mw-heading2"><h2 id="Boltzmann_constant">Boltzmann constant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=3" title="Edit section: Boltzmann constant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE (<a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a>) is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of <i><a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a></i>, i.e., unpredictability, to rebound kinetics; it is as likely that there will be <i>less</i> ZPE-induced particle motion after a given collision as <i>more</i>. This random nature of ZPE is why it has no net effect upon either the pressure or volume of any <i>bulk quantity</i> (a statistically significant quantity of particles) of gases. However, in temperature <span class="nowrap"><i>T</i> = 0</span> <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter</a>; e.g., solids and liquids, ZPE causes inter-atomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the real-world effects that ZPE has on substances can vary as one alters a thermodynamic system (for example, due to ZPE, helium won't freeze unless under a pressure of at least 2.5 <a href="/wiki/Megapascal" class="mw-redirect" title="Megapascal">MPa</a> (25 <a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a>)), ZPE is very much a form of thermal energy and may properly be included when tallying a substance's internal energy. </p> <div class="mw-heading mw-heading2"><h2 id="Rankine_scale">Rankine scale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=4" title="Edit section: Rankine scale"><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/Rankine_scale" title="Rankine scale">Rankine scale</a></div> <p>Though there have been many <a href="/wiki/Conversion_of_units_of_temperature" class="mw-redirect" title="Conversion of units of temperature">other temperature scales</a> throughout history, there have been only two scales for measuring thermodynamic temperature which have absolute zero as their null point (0): The Kelvin scale and the Rankine scale. </p><p>Throughout the scientific world where modern measurements are nearly always made using the International System of Units, thermodynamic temperature is measured using the Kelvin scale. The Rankine scale is part of <a href="/wiki/English_Engineering_Units" title="English Engineering Units">English engineering units</a> and finds use in certain engineering fields, particularly in legacy reference works. The Rankine scale uses the <i>degree Rankine</i> (symbol: °R) as its unit, which is the same magnitude as the <a href="/wiki/Fahrenheit" title="Fahrenheit">degree Fahrenheit</a> (symbol: °F). </p><p>A unit increment of one kelvin is exactly 1.8 times one degree Rankine; thus, to convert a specific temperature on the Kelvin scale to the Rankine scale, <span class="nowrap"><b><span class="texhtml"><i>x</i></span> K = 1.8 <span class="texhtml"><i>x</i></span> °R</b></span>, and to convert from a temperature on the Rankine scale to the Kelvin scale, <span class="nowrap"><b><span class="texhtml"><i>x</i></span> °R = <span class="texhtml"><i>x</i></span>/1.8 K</b></span>. Consequently, absolute zero is "0" for both scales, but the melting point of water ice (0 °C and 273.15 K) is 491.67 °R. </p><p>To convert temperature <i>intervals</i> (a span or difference between two temperatures), the formulas from the preceding paragraph are applicable; for instance, an interval of 5 kelvin is precisely equal to an interval of 9 degrees Rankine. </p> <div class="mw-heading mw-heading2"><h2 id="Modern_redefinition_of_the_kelvin">Modern redefinition of the kelvin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=5" title="Edit section: Modern redefinition of the kelvin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For 65 years, between 1954 and the <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision of the SI</a>, a temperature interval of one kelvin was defined as <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">1</span><span class="sr-only">/</span><span class="den">273.16</span></span>⁠</span> the difference between the <a href="/wiki/Triple_point" title="Triple point">triple point</a> of water and absolute zero. The 1954 resolution by the <a href="/wiki/International_Bureau_of_Weights_and_Measures" title="International Bureau of Weights and Measures">International Bureau of Weights and Measures</a> (known by the French-language acronym BIPM), plus later resolutions and publications, defined the triple point of water as precisely 273.16 K and acknowledged that it was "common practice" to accept that due to previous conventions (namely, that 0 °C had long been defined as the melting point of water and that the triple point of water had long been experimentally determined to be indistinguishably close to 0.01 °C), the difference between the Celsius scale and Kelvin scale is accepted as 273.15 kelvins; which is to say, 0 °C corresponds to 273.15 kelvins.<sup id="cite_ref-BIPMbrocure_6-0" class="reference"><a href="#cite_note-BIPMbrocure-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The net effect of this as well as later resolutions was twofold: 1) they defined absolute zero as precisely 0 K, and 2) they defined that the triple point of special isotopically controlled water called <a href="/wiki/Vienna_Standard_Mean_Ocean_Water" title="Vienna Standard Mean Ocean Water">Vienna Standard Mean Ocean Water</a> occurred at precisely 273.16 K and 0.01 °C. One effect of the aforementioned resolutions was that the melting point of water, while <i>very</i> close to 273.15 K and 0 °C, was not a defining value and was subject to refinement with more precise measurements. </p><p>The 1954 BIPM standard did a good job of establishing—within the uncertainties due to <a href="/wiki/Isotope" title="Isotope">isotopic variations</a> between water samples—temperatures around the freezing and triple points of water, but required that <i>intermediate values</i> between the triple point and absolute zero, as well as extrapolated values from room temperature and beyond, to be experimentally determined via apparatus and procedures in individual labs. This shortcoming was addressed by the <a href="/wiki/International_Temperature_Scale_of_1990" title="International Temperature Scale of 1990">International Temperature Scale of 1990</a>, or ITS‑90, which defined 13 additional points, from 13.8033 K, to 1,357.77 K. While definitional, ITS‑90 had—and still has—some challenges, partly because eight of its extrapolated values depend upon the melting or freezing points of metal samples, which must remain exceedingly pure lest their melting or freezing points be affected—usually depressed. </p><p>The 2019 revision of the SI was primarily for the purpose of decoupling much of the SI system's definitional underpinnings from the <a href="/wiki/Kilogram" title="Kilogram">kilogram</a>, which was the last physical artifact defining an <a href="/wiki/SI_base_unit" title="SI base unit">SI base unit</a> (a platinum/iridium cylinder stored under three nested bell jars in a safe located in France) and which had highly questionable stability. The solution required that four physical constants, including the Boltzmann constant, be definitionally fixed. </p><p>Assigning the Boltzmann constant a precisely defined value had no practical effect on modern thermometry except for the most exquisitely precise measurements. Before the revision, the triple point of water was exactly 273.16 K and 0.01 °C and the Boltzmann constant was experimentally determined to be <span class="nowrap"><span data-sort-value="6977138064903000000♠"></span>1.380<span style="margin-left:.25em;">649</span><span style="margin-left:.25em;">03</span>(51)<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J/K</span>, where the "(51)" denotes the uncertainty in the two least significant digits (the 03) and equals a <a href="/wiki/Standard_deviation" title="Standard deviation">relative standard uncertainty</a> of 0.37 ppm.<sup id="cite_ref-codata2017_7-0" class="reference"><a href="#cite_note-codata2017-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Afterwards, by defining the Boltzmann constant as 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> J/K</span>, the 0.37 ppm uncertainty was transferred to the triple point of water, which became an experimentally determined value of <span class="nowrap"><span data-sort-value="7002273160000000000♠"></span>273.1600<span style="margin-left:0.3em;margin-right:0.15em;">±</span>0.0001 K</span> (<span class="nowrap"><span data-sort-value="7002273159999999999♠"></span>0.0100<span style="margin-left:0.3em;margin-right:0.15em;">±</span>0.0001 °C</span>). That the triple point of water ended up being exceedingly close to 273.16 K after the SI revision was no accident; the final value of the Boltzmann constant was determined, in part, through clever experiments with <a href="/wiki/Argon" title="Argon">argon</a> and helium that used the triple point of water for their key reference temperature.<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><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Notwithstanding the 2019 revision, water triple-point cells continue to serve in modern thermometry as exceedingly precise calibration references at 273.16 K and 0.01 °C. Moreover, the triple point of water remains one of the 14 calibration points comprising ITS‑90, which spans from the triple point of hydrogen (13.8033 K) to the freezing point of copper (1,357.77 K), which is a nearly hundredfold range of thermodynamic temperature. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_of_temperature,_motions,_conduction,_and_thermal_energy"><span id="Relationship_of_temperature.2C_motions.2C_conduction.2C_and_thermal_energy"></span>Relationship of temperature, motions, conduction, and thermal energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=6" title="Edit section: Relationship of temperature, motions, conduction, and thermal energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Translational_motion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif" decoding="async" width="300" height="263" class="mw-file-element" data-file-width="300" data-file-height="263" /></a><figcaption><b>Figure 1</b> The <i>translational motion</i> of fundamental particles of nature such as atoms and molecules is directly related to temperature. Here, the size of <a href="/wiki/Helium" title="Helium">helium</a> atoms relative to their spacing is shown to scale under 1950 <a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atmospheres</a> of pressure. These room-temperature atoms have a certain average speed (slowed down here two trillion-fold). At any given instant however, a particular helium atom may be moving much faster than average while another may be nearly motionless. Five atoms are colored red to facilitate following their motions. This animation illustrates <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, which is the science of how the group behavior of a large collection of microscopic objects emerges from the kinetic properties of each individual object.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Nature_of_kinetic_energy,_translational_motion,_and_temperature"><span id="Nature_of_kinetic_energy.2C_translational_motion.2C_and_temperature"></span>Nature of kinetic energy, translational motion, and temperature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=7" title="Edit section: Nature of kinetic energy, translational motion, and temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The thermodynamic temperature of any <i>bulk quantity</i> of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as <i>translational motion</i>. These simple movements in the three X, Y, and Z–axis dimensions of space means the particles move in the three spatial <i><a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a></i>. This particular form of kinetic energy is sometimes referred to as <i>kinetic temperature</i>. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a>'s formula <span class="texhtml"><i>pV</i> = <i>nRT</i></span> and is embodied in the <a href="/wiki/Gas_laws" title="Gas laws">gas laws</a>. </p><p>Though the kinetic energy borne exclusively in the three translational degrees of freedom comprise the thermodynamic temperature of a substance, molecules, as can be seen in <a href="#Internal_motions_of_molecules_and_internal_energy"><i>Fig. 3</i></a>, can have other degrees of freedom, all of which fall under three categories: bond length, bond angle, and rotational. All three additional categories are not necessarily available to all molecules, and even for molecules that <i>can</i> experience all three, some can be "frozen out" below a certain temperature. Nonetheless, all those degrees of freedom that are available to the molecules under a particular set of conditions contribute to the <a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">specific heat capacity</a> of a substance; which is to say, they increase the amount of heat (kinetic energy) required to raise a given amount of the substance by one kelvin or one degree Celsius. </p><p>The relationship of kinetic energy, mass, and velocity is given by the formula <span class="texhtml"><i>E</i><sub>k</sub> = <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><i>mv</i><span style="padding-left:0.12em;"><sup>2</sup></span></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy, and precisely the same temperature, as those with four times the mass but half the velocity. </p><p>The extent to which the kinetic energy of translational motion in a statistically significant collection of atoms or molecules in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> (symbol: <span class="texhtml"><i>k</i><sub>B</sub></span>). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an <i>individual</i> particles' translational motion 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 {\tilde {E}}={\frac {3}{2}}k_{\text{B}}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}={\frac {3}{2}}k_{\text{B}}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd491f10198baf7e1e91b849a1371e9d5ce3c7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.116ex; height:5.176ex;" alt="{\displaystyle {\tilde {E}}={\frac {3}{2}}k_{\text{B}}T}"></span> where: </p> <ul><li><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="{\textstyle {\tilde {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tilde {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0ad40165ae88f337af34063c562ab599ae80d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.676ex;" alt="{\textstyle {\tilde {E}}}"></span> is the mean kinetic energy for an individual particle</li> <li><span class="texhtml"><i>k</i><sub>B</sub> = <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></span></li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the thermodynamic temperature of the bulk quantity of the substance</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Maxwell_Dist-Inverse_Speed.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/d/d0/Maxwell_Dist-Inverse_Speed.png/310px-Maxwell_Dist-Inverse_Speed.png" decoding="async" width="310" height="362" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/d0/Maxwell_Dist-Inverse_Speed.png/465px-Maxwell_Dist-Inverse_Speed.png 1.5x, //upload.wikimedia.org/wikipedia/en/d/d0/Maxwell_Dist-Inverse_Speed.png 2x" data-file-width="600" data-file-height="700" /></a><figcaption><b>Figure 2</b> The translational motions of helium atoms occur across a range of speeds. Compare the shape of this curve to that of a Planck curve in <i><a href="#Diffusion_of_thermal_energy:_black-body_radiation">Fig. 5</a></i> below.</figcaption></figure> <p>While the Boltzmann constant is useful for finding the mean kinetic energy in a sample of particles, it is important to note that even when a substance is isolated and in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in <i><a href="#Nature_of_kinetic_energy,_translational_motion,_and_temperature">Fig. 1</a></i> above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the <a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann distribution</a>. The graph shown here in <i>Fig. 2</i> shows the speed distribution of 5500 K helium atoms. They have a <i>most probable</i> speed of 4.780 km/s (0.2092 s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the <i>x</i>–axis to the right). This graph uses <i>inverse speed</i> for its <i>x</i>-axis so the shape of the curve can easily be compared to the curves in <i><a href="#Diffusion_of_thermal_energy:_black-body_radiation">Fig. 5</a></i> below. In both graphs, zero on the <i>x</i>-axis represents infinite temperature. Additionally, the <i>x</i>- and <i>y</i>-axes on both graphs are scaled proportionally. </p> <div class="mw-heading mw-heading4"><h4 id="High_speeds_of_translational_motion">High speeds of translational motion</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=8" title="Edit section: High speeds of translational motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a <a href="/wiki/Fluid" title="Fluid">fluid</a> produces <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> that can be seen with an ordinary microscope. The translational motions of elementary particles are <i>very</i> fast<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and temperatures close to <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a> are required to directly observe them. For instance, when scientists at the <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a> achieved a record-setting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used <a href="/wiki/Optical_lattice" title="Optical lattice">optical lattice</a> laser equipment to <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatically</a> cool <a href="/wiki/Caesium" title="Caesium">cesium</a> atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature.<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> Formulas for calculating the velocity and speed of translational motion are given in the following footnote.<sup id="cite_ref-Boltzmann_13-0" class="reference"><a href="#cite_note-Boltzmann-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Argon_atom_at_1E-12_K.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/fa/Argon_atom_at_1E-12_K.gif" decoding="async" width="300" height="300" class="mw-file-element" data-file-width="300" data-file-height="300" /></a><figcaption><b>Figure 2.5</b> This simulation illustrates an argon atom as it would appear through a 400-power optical microscope featuring a reticle graduated with 50-micron (0.05 mm) tick marks. This atom is moving with a velocity of 14.43 microns per second, which gives the atom a kinetic temperature of one-trillionth of a kelvin. The atom requires 13.9 seconds to travel 200 microns (0.2 mm). Though the atom is being invisibly jostled due to zero-point energy, its translational motion seen here comprises all its kinetic energy.</figcaption></figure><p>It is neither difficult to imagine atomic motions due to kinetic temperature, nor distinguish between such motions and those due to zero-point energy. Consider the following hypothetical thought experiment, as illustrated in <i>Fig. 2.5</i> at left, with an atom that is exceedingly close to absolute zero. Imagine peering through a common optical microscope set to 400 power, which is about the maximum practical magnification for optical microscopes. Such microscopes generally provide fields of view a bit over 0.4 mm in diameter. At the center of the field of view is a single levitated argon atom (argon comprises about 0.93% of air) that is illuminated and glowing against a dark backdrop. If this argon atom was at a beyond-record-setting <i>one-trillionth</i> of a kelvin above absolute zero,<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> and was moving perpendicular to the field of view towards the right, it would require 13.9 seconds to move from the center of the image to the 200-micron tick mark; this travel distance is about the same as the width of the period at the end of this sentence on modern computer monitors. As the argon atom slowly moved, the positional jitter due to zero-point energy would be much less than the 200-nanometer (0.0002 mm) resolution of an optical microscope. Importantly, the atom's translational velocity of 14.43 microns per second constitutes all its retained kinetic energy due to not being precisely at absolute zero. Were the atom <i>precisely</i> at absolute zero, imperceptible jostling due to zero-point energy would cause it to very slightly wander, but the atom would perpetually be located, on average, at the same spot within the field of view. This is analogous to a boat that has had its motor turned off and is now bobbing slightly in relatively calm and windless ocean waters; even though the boat randomly drifts to and fro, it stays in the same spot in the long term and makes no headway through the water. Accordingly, an atom that was precisely at absolute zero would not be "motionless", and yet, a statistically significant collection of such atoms would have zero net kinetic energy available to transfer to any other collection of atoms. This is because regardless of the kinetic temperature of the second collection of atoms, they too experience the effects of zero-point energy. Such are the consequences of <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> and the nature of thermodynamics. </p><div class="mw-heading mw-heading4"><h4 id="Internal_motions_of_molecules_and_internal_energy">Internal motions of molecules and internal energy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=9" title="Edit section: Internal motions of molecules and internal energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Thermally_Agitated_Molecule.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Thermally_Agitated_Molecule.gif/240px-Thermally_Agitated_Molecule.gif" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/23/Thermally_Agitated_Molecule.gif 1.5x" data-file-width="280" data-file-height="280" /></a><figcaption><b>Figure 3</b> Molecules have internal structures because they are composed of atoms that have different ways of moving within molecules. Being able to store kinetic energy in these <i>internal degrees of freedom</i> contributes to a substance's <i><a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">specific heat capacity</a></i>, or internal energy, allowing it to contain more internal energy at the same temperature.</figcaption></figure> <p>As mentioned above, there are other ways molecules can jiggle besides the three translational degrees of freedom that imbue substances with their kinetic temperature. As can be seen in the animation at right, <a href="/wiki/Molecule" title="Molecule">molecules</a> are complex objects; they are a population of atoms and thermal agitation can strain their internal <a href="/wiki/Chemical_bond" title="Chemical bond">chemical bonds</a> in three different ways: via rotation, bond length, and bond angle movements; these are all types of <i>internal degrees of freedom</i>. This makes molecules distinct from <i><a href="/wiki/Monatomic" class="mw-redirect" title="Monatomic">monatomic</a></i> substances (consisting of individual atoms) like the <a href="/wiki/Noble_gas" title="Noble gas">noble gases</a> <a href="/wiki/Helium" title="Helium">helium</a> and <a href="/wiki/Argon" title="Argon">argon</a>, which have only the three translational degrees of freedom (the X, Y, and Z axis). Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an <i>internal temperature</i>. Even though these motions are called "internal", the external portions of molecules still move—rather like the jiggling of a stationary <a href="/wiki/Water_balloon" title="Water balloon">water balloon</a>. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as internal energy is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the <a href="/wiki/Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a>, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-<a href="/wiki/Local_thermodynamic_equilibrium" class="mw-redirect" title="Local thermodynamic equilibrium">local thermodynamic equilibrium</a> (LTE) phenomena such as <a href="/wiki/Combustion" title="Combustion">combustion</a>, the <a href="/wiki/Sublimation_(chemistry)" class="mw-redirect" title="Sublimation (chemistry)">sublimation</a> of solids, and the <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> of hot gases in a partial vacuum. </p><p>The kinetic energy stored internally in molecules causes substances to contain more heat energy at any given temperature and to absorb additional internal energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions, is not contributing to the molecules' translational motions at that same instant.<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> This extra kinetic energy simply increases the amount of internal energy that substance absorbs for a given temperature rise. This property is known as a substance's <a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">specific heat capacity</a>. </p><p>Different molecules absorb different amounts of internal energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances' molecules possess more internal degrees of freedom than others do. For instance, room-temperature <a href="/wiki/Nitrogen" title="Nitrogen">nitrogen</a>, which is a <a href="/wiki/Diatomic" class="mw-redirect" title="Diatomic">diatomic</a> molecule, has <i>five</i> active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has five-thirds the specific heat capacity per <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> (a specific number of molecules) as do the monatomic gases.<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> Another example is <a href="/wiki/Gasoline" title="Gasoline">gasoline</a> (see <a href="/wiki/Specific_heat_capacity#Table_of_specific_heat_capacities" title="Specific heat capacity">table</a> showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom. </p> <div class="mw-heading mw-heading3"><h3 id="Diffusion_of_thermal_energy:_entropy,_phonons,_and_mobile_conduction_electrons"><span id="Diffusion_of_thermal_energy:_entropy.2C_phonons.2C_and_mobile_conduction_electrons"></span>Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=10" title="Edit section: Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:1D_normal_modes_(280_kB).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/1D_normal_modes_%28280_kB%29.gif/260px-1D_normal_modes_%28280_kB%29.gif" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/9b/1D_normal_modes_%28280_kB%29.gif 1.5x" data-file-width="275" data-file-height="275" /></a><figcaption><b>Figure 4</b> The temperature-induced translational motion of particles in solids takes the form of <i><a href="/wiki/Phonon" title="Phonon">phonons</a></i>. Shown here are phonons with identical <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a> but with <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a> ranging from 2 to 12 average inter-molecule separations (<i>a</i>).</figcaption></figure> <p><i><a href="/wiki/Heat_conduction" class="mw-redirect" title="Heat conduction">Heat conduction</a></i> is the diffusion of thermal energy from hot parts of a system to cold parts. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term <i>bulk</i> in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever thermal energy diffuses within an isolated system, temperature differences within the system decrease (and <a href="/wiki/Entropy" title="Entropy">entropy</a> increases). </p><p>One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers <a href="/wiki/Momentum" title="Momentum">momentum</a> from particle to particle in collisions. In gases, these translational motions are of the nature shown above in <i><a href="#Relationship_of_temperature,_motions,_conduction,_and_thermal_energy">Fig. 1</a></i>. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can move forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; <a href="/wiki/Convection_(heat_transfer)" title="Convection (heat transfer)">convection</a> speeds this process even more.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Translational motion in <i>solids</i>, however, takes the form of <i><a href="/wiki/Phonon" title="Phonon">phonons</a></i> (see <i>Fig. 4</i> at right). Phonons are constrained, quantized wave packets that travel at the speed of sound of a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is <i>usually</i> inefficient<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> and such solids are considered <i>thermal insulators</i> (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam. </p><p><a href="/wiki/Metal" title="Metal">Metals</a> however, are not restricted to only phonon-based heat conduction. Thermal energy conducts through metals extraordinarily quickly because instead of direct molecule-to-molecule collisions, the vast majority of thermal energy is mediated via very light, mobile <i>conduction <a href="/wiki/Electron" title="Electron">electrons</a></i>. This is why there is a near-perfect correlation between metals' <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> and their <a href="/wiki/Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">electrical conductivity</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Conduction electrons imbue metals with their extraordinary conductivity because they are <i><a href="/wiki/Delocalized_electron" title="Delocalized electron">delocalized</a></i> (i.e., not tied to a specific atom) and behave rather like a sort of quantum gas due to the effects of <i><a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a></i> (for more on ZPE, see <i><a href="#Notes">Note 1</a></i> below). Furthermore, electrons are relatively light with a rest mass only <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">1836</span></span> that of a <a href="/wiki/Proton" title="Proton">proton</a>. This is about the same ratio as a <a href="/wiki/.22_Short" title=".22 Short">.22 Short</a> bullet (29 <a href="/wiki/Grain_(measure)" class="mw-redirect" title="Grain (measure)">grains</a> or 1.88 <a href="/wiki/Gram" title="Gram">g</a>) compared to the rifle that shoots it. As <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> wrote with his <a href="/wiki/Newton%27s_laws_of_motion#Newton's_third_law" title="Newton's laws of motion">third law of motion</a>, </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>Law #3: All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.</p></blockquote> <p>However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, thermal energy is readily borne by mobile conduction electrons. Additionally, because they are delocalized and <i>very</i> fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons. </p> <div class="mw-heading mw-heading3"><h3 id="Diffusion_of_thermal_energy:_black-body_radiation">Diffusion of thermal energy: black-body radiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=11" title="Edit section: Diffusion of thermal energy: black-body radiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wiens_law.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Wiens_law.svg/310px-Wiens_law.svg.png" decoding="async" width="310" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Wiens_law.svg/465px-Wiens_law.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Wiens_law.svg/620px-Wiens_law.svg.png 2x" data-file-width="720" data-file-height="600" /></a><figcaption><b>Figure 5</b> The spectrum of black-body radiation has the form of a Planck curve. A 5500 K black-body has a peak emittance wavelength of 527 nm. Compare the shape of this curve to that of a Maxwell distribution in <i><a href="#Nature_of_kinetic_energy,_translational_motion,_and_temperature">Fig. 2</a></i> above.</figcaption></figure> <p><a href="/wiki/Thermal_radiation" title="Thermal radiation">Thermal radiation</a> is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal <a href="/wiki/Photon" title="Photon">photons</a> (known as <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a>). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even <i>individual molecules</i> with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. In any bulk quantity of a substance at equilibrium, black-body photons are emitted across a range of <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a> in a spectrum that has a bell curve-like shape called a <a href="/wiki/Planck%27s_law_of_black_body_radiation" class="mw-redirect" title="Planck's law of black body radiation">Planck curve</a> (see graph in <i>Fig. 5</i> at right). The top of a Planck curve (<a href="/wiki/Wien%27s_displacement_law" title="Wien's displacement law">the peak emittance wavelength</a>) is located in a particular part of the <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a> depending on the temperature of the black-body. Substances at extreme <a href="/wiki/Cryogenics" title="Cryogenics">cryogenic</a> temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short <a href="/wiki/Gamma_ray" title="Gamma ray">gamma rays</a> (see <a href="#Table_of_thermodynamic_temperatures">§ Table of thermodynamic temperatures</a>). </p><p>Black-body radiation diffuses thermal energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process. </p><p>As established by the <a href="/wiki/Stefan%E2%80%93Boltzmann_law" title="Stefan–Boltzmann law">Stefan–Boltzmann law</a>, the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black-body at 824 K (just short of glowing dull red) emits 60 times the radiant <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an <a href="/wiki/Incandescent_light_bulb" title="Incandescent light bulb">incandescent lamp</a>, black-body radiation can be the principal mechanism by which thermal energy escapes a system. </p> <div class="mw-heading mw-heading4"><h4 id="Table_of_thermodynamic_temperatures">Table of thermodynamic temperatures</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=12" title="Edit section: Table of thermodynamic temperatures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table below shows various points on the thermodynamic scale, in order of increasing temperature. </p> <table class="wikitable" style="text-align:center"> <tbody><tr> <th> </th> <th>Kelvin </th> <th>Peak emittance<br /><a href="/wiki/Wavelength" title="Wavelength">wavelength</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> of<br /><a href="/wiki/Wien%27s_displacement_law" title="Wien's displacement law">black-body photons</a> </th></tr> <tr> <td style="text-align:right"><a href="/wiki/Absolute_zero" title="Absolute zero">Absolute zero</a><br />(precisely by definition) </td> <td>0 K </td> <td><span style="font-size:140%;"><a href="/wiki/Infinity" title="Infinity">∞</a></span><sup id="cite_ref-T0_5-1" class="reference"><a href="#cite_note-T0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td style="text-align:right">Coldest measured<br />temperature<sup id="cite_ref-recordcold_21-0" class="reference"><a href="#cite_note-recordcold-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </td> <td>450 <a href="/wiki/Orders_of_magnitude_(temperature)#SI_multiples" title="Orders of magnitude (temperature)">pK</a> </td> <td>6,400 <a href="/wiki/Kilometre" title="Kilometre">km</a> </td></tr> <tr> <td style="text-align:right">One <a href="/wiki/Orders_of_magnitude_(temperature)#SI_multiples" title="Orders of magnitude (temperature)">millikelvin</a><br />(precisely by definition) </td> <td>0.001 K </td> <td>2.897 77 <a href="/wiki/Metre" title="Metre">m</a><br /> (radio, <a href="/wiki/FM_broadcasting" title="FM broadcasting">FM band</a>)<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">cosmic microwave<br />background radiation</a> </td> <td>2.725 K </td> <td>1.063 <a href="/wiki/Metre" title="Metre">mm</a> (peak wavelength) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Vienna_Standard_Mean_Ocean_Water" title="Vienna Standard Mean Ocean Water">Water</a>'s <a href="/wiki/Triple_point" title="Triple point">triple point</a> </td> <td>273.16 K </td> <td>10.6083 <a href="/wiki/Metre#SI_prefixed_forms_of_metre" title="Metre">μm</a><br />(long wavelength <a href="/wiki/Infrared" title="Infrared">I.R.</a>) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/ISO_1" title="ISO 1">ISO 1</a> standard temperature<br />for precision <a href="/wiki/Metrology" title="Metrology">metrology</a><br />(precisely 20 °C by definition) </td> <td>293.15 K </td> <td><span class="nowrap"><span data-sort-value="7000988495000000000♠"></span>9.884<span style="margin-left:.25em;">95</span></span> μm<br />(long wavelength <a href="/wiki/Infrared" title="Infrared">I.R.</a>) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Incandescent_light_bulb" title="Incandescent light bulb">Incandescent lamp</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>A<span class="cite-bracket">]</span></a></sup> </td> <td>2500 K<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>B<span class="cite-bracket">]</span></a></sup> </td> <td>1.16 μm<br />(near <a href="/wiki/Infrared" title="Infrared">infrared</a>)<sup id="cite_ref-Photosphere_25-0" class="reference"><a href="#cite_note-Photosphere-25"><span class="cite-bracket">[</span>C<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Sun" title="Sun">Sun</a>'s visible surface<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </td> <td>5772 K </td> <td>502 <a href="/wiki/Metre#SI_prefixed_forms_of_metre" title="Metre">nm</a><br />(<a href="/wiki/Color#Spectral_colors" title="Color">green light</a>) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Lightning" title="Lightning">Lightning bolt's</a><br />channel </td> <td>28,000 K </td> <td>100 nm<br />(far <a href="/wiki/Ultraviolet" title="Ultraviolet">ultraviolet</a> light) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Sun#Core" title="Sun">Sun's core</a> </td> <td>16 <a href="/wiki/Orders_of_magnitude_(temperature)#SI_multiples" title="Orders of magnitude (temperature)">MK</a> </td> <td>0.18 nm (<a href="/wiki/X-ray" title="X-ray">X-rays</a>) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Thermonuclear_explosion" class="mw-redirect" title="Thermonuclear explosion">Thermonuclear explosion</a><br />(peak temperature)<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </td> <td align="center">350 MK </td> <td align="center">8.3 × 10<sup>−3</sup> nm<br />(<a href="/wiki/Gamma_ray" title="Gamma ray">gamma rays</a>) </td></tr> <tr> <td style="text-align:right">Sandia National Labs'<br /><a href="/wiki/Z_Pulsed_Power_Facility" title="Z Pulsed Power Facility">Z machine</a><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>D<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </td> <td>2 <a href="/wiki/Orders_of_magnitude_(temperature)#SI_multiples" title="Orders of magnitude (temperature)">GK</a> </td> <td>1.4 × 10<sup>−3</sup> nm<br />(gamma rays) </td></tr> <tr> <td style="text-align:right">Core of a <a href="/wiki/Silicon-burning_process" title="Silicon-burning process">high-mass star on its last day</a><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </td> <td align="center">3 GK </td> <td align="center">1 × 10<sup>−3</sup> nm<br />(gamma rays) </td></tr> <tr> <td style="text-align:right">Merging binary <a href="/wiki/Neutron_star" title="Neutron star">neutron star</a> system<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </td> <td>350 GK </td> <td>8 × 10<sup>−6</sup> nm<br />(gamma rays) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/Gamma-ray_burst_progenitors" title="Gamma-ray burst progenitors">Gamma-ray burst progenitors</a><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </td> <td>1 <a href="/wiki/Orders_of_magnitude_(temperature)#SI_multiples" title="Orders of magnitude (temperature)">TK</a> </td> <td>3 × 10<sup>−6</sup> nm<br />(gamma rays) </td></tr> <tr> <td style="text-align:right"><a href="/wiki/CERN" title="CERN">CERN</a>'s proton vs. nucleus collisions<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </td> <td>10 TK </td> <td>3 × 10<sup>−7</sup> nm<br />(gamma rays) </td></tr> </tbody></table> <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-upper-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">For a true black body (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">The 2500 K value is approximate.</span> </li> <li id="cite_note-Photosphere-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Photosphere_25-0">^</a></b></span> <span class="reference-text">Effective photosphere temperature.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">For a true black body (which the plasma was not). The Z machine's dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Heat_of_phase_changes">Heat of phase changes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=13" title="Edit section: Heat of phase changes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:IceBlockNearJoekullsarlon.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/IceBlockNearJoekullsarlon.jpg/310px-IceBlockNearJoekullsarlon.jpg" decoding="async" width="310" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/IceBlockNearJoekullsarlon.jpg/465px-IceBlockNearJoekullsarlon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/IceBlockNearJoekullsarlon.jpg/620px-IceBlockNearJoekullsarlon.jpg 2x" data-file-width="2971" data-file-height="1984" /></a><figcaption><b>Figure 6</b> Ice and water: two phases of the same substance</figcaption></figure> <p>The kinetic energy of particle motion is just one contributor to the total thermal energy in a substance; another is <i><a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a></i>, which are the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> of molecular bonds that can form in a substance as it cools (such as during <a href="/wiki/Condensation" title="Condensation">condensing</a> and <a href="/wiki/Freezing" title="Freezing">freezing</a>). The thermal energy required for a phase transition is called <i><a href="/wiki/Latent_heat" title="Latent heat">latent heat</a></i>. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to <i>break</i> <a href="/wiki/Chemical_bonds" class="mw-redirect" title="Chemical bonds">chemical bonds</a> (such as during <a href="/wiki/Evaporation" title="Evaporation">evaporation</a> and <a href="/wiki/Melting" title="Melting">melting</a>). Almost everyone is familiar with the effects of phase transitions; for instance, <a href="/wiki/Steam" title="Steam">steam</a> at 100 °C can cause severe burns much faster than the 100 °C air from a <a href="/wiki/Blowdryer" class="mw-redirect" title="Blowdryer">hair dryer</a>. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin. </p><p>Even though thermal energy is liberated or absorbed during phase transitions, pure <a href="/wiki/Chemical_element" title="Chemical element">chemical elements</a>, <a href="/wiki/Chemical_compound" title="Chemical compound">compounds</a>, and <a href="/wiki/Eutectic_point" class="mw-redirect" title="Eutectic point">eutectic</a> <a href="/wiki/Alloy" title="Alloy">alloys</a> exhibit no temperature change whatsoever while they undergo them (see <i>Fig. 7</i>, below right). Consider one particular type of phase transition: melting. When a solid is melting, <a href="/wiki/Crystal_structure" title="Crystal structure">crystal lattice</a> <a href="/wiki/Chemical_bond" title="Chemical bond">chemical bonds</a> are being broken apart; the substance is transitioning from what is known as a <i>more ordered state</i> to a <i>less ordered state</i>. In <i>Fig. 7</i>, the melting of ice is shown within the lower left box heading from blue to green. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Energy_through_phase_changes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Energy_through_phase_changes.png/400px-Energy_through_phase_changes.png" decoding="async" width="400" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Energy_through_phase_changes.png/600px-Energy_through_phase_changes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Energy_through_phase_changes.png/800px-Energy_through_phase_changes.png 2x" data-file-width="960" data-file-height="620" /></a><figcaption><b>Figure 7</b> Water's temperature does not change during phase transitions as heat flows into or out of it. The total heat capacity of a mole of water in its liquid phase (the green line) is 7.5507 kJ.</figcaption></figure> <p>At one specific thermodynamic point, the <a href="/wiki/Melting_point" title="Melting point">melting point</a> (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are, on average, at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are all-or-nothing forces: they either hold fast, or break; there is no in-between state. Consequently, when a substance is at its melting point, every <a href="/wiki/Joule" title="Joule">joule</a> of added thermal energy only breaks the bonds of a specific quantity of its atoms or molecules,<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like <a href="/wiki/Popcorn" title="Popcorn">popcorn</a>: at a certain temperature, additional thermal energy cannot make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), thermal energy must be removed from a substance. </p><p>As stated above, the thermal energy required for a phase transition is called <i>latent heat</i>. In the specific cases of melting and freezing, it is called <i><a href="/wiki/Standard_enthalpy_change_of_fusion" class="mw-redirect" title="Standard enthalpy change of fusion">enthalpy of fusion</a></i> or <i>heat of fusion</i>. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> If the substance is one of the monatomic gases (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the thermal energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals' ratios are even greater, typically in the range of 400 to 1200 times.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The phase transition of <a href="/wiki/Boiling" title="Boiling">boiling</a> is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as <i><a href="/wiki/Standard_enthalpy_change_of_vaporization" class="mw-redirect" title="Standard enthalpy change of vaporization">enthalpy of vaporization</a></i>) is roughly 540 times that required for a one-degree increase.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>Water's sizable enthalpy of vaporization is why one's skin can be burned so quickly as steam condenses on it (heading from red to green in <i>Fig. 7</i> above); water vapors (gas phase) are liquefied on the skin with releasing a large amount of energy (enthalpy) to the environment including the skin, resulting in skin damage. In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient <a href="/wiki/Wet-bulb_temperature" title="Wet-bulb temperature">wet-bulb temperature</a> that is dependent on <a href="/wiki/Relative_humidity" class="mw-redirect" title="Relative humidity">relative humidity</a>); the water evaporation on the skin takes a large amount of energy from the environment including the skin, reducing the skin temperature. Water's highly energetic enthalpy of vaporization is also an important factor underlying why <i>solar pool covers</i> (floating, insulated blankets that cover <a href="/wiki/Swimming_pool" title="Swimming pool">swimming pools</a> when the pools are not in use) are so effective at reducing heating costs: they prevent evaporation. (In other words, taking energy from water when it is evaporated is limited.) For instance, the evaporation of just 20 mm of water from a 1.29-meter-deep pool chills its water 8.4 °C (15.1 °F). </p> <div class="mw-heading mw-heading3"><h3 id="Internal_energy">Internal energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=14" title="Edit section: Internal energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The total energy of all translational and internal particle motions, including that of conduction electrons, plus the potential energy of phase changes, plus <a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a><sup id="cite_ref-T0_5-2" class="reference"><a href="#cite_note-T0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> of a substance comprise the <i><a href="/wiki/Internal_energy" title="Internal energy">internal energy</a></i> of it. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Close-packed_spheres,_with_umbrella_light_%26_camerea.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Close-packed_spheres%2C_with_umbrella_light_%26_camerea.jpg/266px-Close-packed_spheres%2C_with_umbrella_light_%26_camerea.jpg" decoding="async" width="266" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Close-packed_spheres%2C_with_umbrella_light_%26_camerea.jpg/399px-Close-packed_spheres%2C_with_umbrella_light_%26_camerea.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Close-packed_spheres%2C_with_umbrella_light_%26_camerea.jpg/532px-Close-packed_spheres%2C_with_umbrella_light_%26_camerea.jpg 2x" data-file-width="1500" data-file-height="1443" /></a><figcaption><b>Figure 8</b> When many of the chemical elements, such as the <a href="/wiki/Noble_gas" title="Noble gas">noble gases</a> and <a href="/wiki/Platinum_group" title="Platinum group">platinum-group metals</a>, freeze to a solid — the most ordered state of matter — their <a href="/wiki/Crystal_structures" class="mw-redirect" title="Crystal structures">crystal structures</a> have a <i><a href="/wiki/Close-packing" class="mw-redirect" title="Close-packing">close-packed arrangement</a></i>. This yields the greatest possible packing density and the lowest energy state.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Internal_energy_at_absolute_zero">Internal energy at absolute zero</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=15" title="Edit section: Internal energy at absolute zero"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a substance cools, different forms of internal energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions is liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic energy or temperature decreases); the internal motions of molecules diminish (their internal energy or temperature decreases); conduction electrons (if the substance is an electrical conductor) travel <i>somewhat</i> slower;<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> and black-body radiation's peak emittance wavelength increases (the photons' energy decreases). When particles of a substance are as close as possible to complete rest and retain only ZPE (zero-point energy)-induced quantum mechanical motion, the substance is at the temperature of absolute zero (<span class="texhtml mvar" style="font-style:italic;">T</span> = 0). </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Liquid_helium_superfluid_phase.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Liquid_helium_superfluid_phase.jpg/240px-Liquid_helium_superfluid_phase.jpg" decoding="async" width="240" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Liquid_helium_superfluid_phase.jpg/360px-Liquid_helium_superfluid_phase.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Liquid_helium_superfluid_phase.jpg/480px-Liquid_helium_superfluid_phase.jpg 2x" data-file-width="864" data-file-height="648" /></a><figcaption><b>Figure 9</b> Due to the effects of zero-point energy, helium at ambient pressure remains a <a href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluid</a> even when exceedingly close to absolute zero; it will not freeze unless under 25 bar of pressure (c. 25 atmospheres).</figcaption></figure> <p>Whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero internal energy; one must be very precise with what one means by <i>internal energy</i>. Often, all the phase changes that <i>can</i> occur in a substance, <i>will</i> have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, <span class="texhtml mvar" style="font-style:italic;">T</span> = 0 <a href="/wiki/Helium" title="Helium">helium</a> remains liquid at room pressure (<i>Fig. 9</i> at right) and must be under a pressure of at least 25 <a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a> (2.5 <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">MPa</a>) to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21 joules per mole) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures. </p><p>A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars, or hundreds of gigapascals). These are known as <i>solid–solid phase transitions</i> wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one. </p><p>The above complexities make for rather cumbersome blanket statements regarding the internal energy in <span class="texhtml mvar" style="font-style:italic;">T</span> = 0 substances. Regardless of pressure though, what <i>can</i> be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a <i><a href="/wiki/Close-packing" class="mw-redirect" title="Close-packing">closest-packed arrangement</a></i> (see <i>Fig. 8</i>, above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy.<sup id="cite_ref-T0_5-3" class="reference"><a href="#cite_note-T0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> One can also say that for a given substance at constant pressure, absolute zero is the point of lowest <i><a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a></i> (a measure of work potential that takes internal energy, pressure, and volume into consideration).<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Lastly, all <span class="texhtml mvar" style="font-style:italic;">T</span> = 0 substances contain zero kinetic thermal energy.<sup id="cite_ref-T0_5-4" class="reference"><a href="#cite_note-T0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Boltzmann_13-1" class="reference"><a href="#cite_note-Boltzmann-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Practical_applications_for_thermodynamic_temperature">Practical applications for thermodynamic temperature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=16" title="Edit section: Practical applications for thermodynamic temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thermodynamic temperature is useful not only for scientists, it can also be useful for lay-people in many disciplines involving gases. By expressing variables in absolute terms and applying <a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a> of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a cold gage<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> pressure of 200 <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">kPa</a>, then its <a href="/wiki/Absolute_pressure" class="mw-redirect" title="Absolute pressure">absolute pressure</a> is 300 kPa.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Room temperature ("cold" in tire terms) is 296 K. If the tire temperature is 20 °C hotter (20 kelvins), the solution is calculated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">316 K</span><span class="sr-only">/</span><span class="den">296 K</span></span>⁠</span> = 6.8% greater thermodynamic temperature <i>and</i> absolute pressure; that is, an absolute pressure of 320 kPa, which is a gage pressure of 220 kPa. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_ideal_gas_law">Relationship to ideal gas law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermodynamic_temperature&action=edit&section=17" title="Edit section: Relationship to ideal gas law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The thermodynamic temperature is closely linked to the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> and its consequences. It can be linked also to the second law of thermodynamics. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the <a href="/wiki/Energy_conversion_efficiency" title="Energy conversion efficiency">efficiency</a> of idealized <a href="/wiki/Heat_engine" title="Heat engine">heat engines</a>. Thus the <a href="/wiki/Ratio" title="Ratio">ratio</a> <span class="texhtml"><i>T</i><sub>2</sub>/<i>T</i><sub>1</sub></span> of two temperatures <span class="texhtml"><i>T</i><sub>1</sub></span> and <span class="texhtml"><i>T</i><sub>2</sub></span> is the same in all absolute scales. </p><p>Strictly speaking, the temperature of a system is well-defined only if it is at <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a>. From a microscopic viewpoint, a material is at thermal equilibrium if the quantity of heat between its individual particles cancel out. There are many possible scales of temperature, derived from a variety of observations of physical phenomena. </p><p>Loosely stated, temperature differences dictate the direction of heat between two systems such that their combined energy is maximally distributed among their lowest possible states. We call this distribution "<a href="/wiki/Entropy" title="Entropy">entropy</a>". To better understand the relationship between temperature and entropy, consider the relationship between heat, <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> and temperature illustrated in the <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot heat engine</a>. The engine converts heat into work by directing a temperature gradient between a higher temperature heat source, <span class="texhtml"><i>T</i><sub>H</sub></span>, and a lower temperature heat sink, <span class="texhtml"><i>T</i><sub>C</sub></span>, through a gas filled piston. The work done per cycle is equal in magnitude to net heat taken up, which is sum of the heat <span class="texhtml"><i>q</i><sub>H</sub></span> taken up by the engine from the high-temperature source, plus the waste <a href="/wiki/Heat" title="Heat">heat given off</a> by the engine, <span class="texhtml"><i>q</i><sub>C</sub></span> < 0.<sup id="cite_ref-PlanckBook_48-0" class="reference"><a href="#cite_note-PlanckBook-48"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> The <i>efficiency</i> of the engine is the work divided by the heat put into the system or </p><p><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 {\begin{aligned}{\textrm {Efficiency}}&={\frac {|w_{\text{cy}}|}{q_{\text{H}}}}={\frac {q_{\text{H}}+q_{\text{C}}}{q_{\text{H}}}}=1+{\frac {q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}&(1)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Efficiency</mtext> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cy</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\textrm {Efficiency}}&={\frac {|w_{\text{cy}}|}{q_{\text{H}}}}={\frac {q_{\text{H}}+q_{\text{C}}}{q_{\text{H}}}}=1+{\frac {q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}&(1)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/131754203e6bfd56d279dd8409c5447c40056793" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.914ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\textrm {Efficiency}}&={\frac {|w_{\text{cy}}|}{q_{\text{H}}}}={\frac {q_{\text{H}}+q_{\text{C}}}{q_{\text{H}}}}=1+{\frac {q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}&(1)\end{aligned}}}"></span> </p><p>where <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 w_{\text{cy}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cy</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{\text{cy}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed5394caa3afcb00d063eaee51daaba02937251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.494ex; height:2.343ex;" alt="{\displaystyle w_{\text{cy}}}"></span> is the work done per cycle. Thus the efficiency depends only on <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>q</i><sub>C</sub></span>| / |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>q</i><sub>H</sub></span>|</span>. </p><p><a href="/wiki/Carnot_theorem_(thermodynamics)" class="mw-redirect" title="Carnot theorem (thermodynamics)">Carnot's theorem</a> states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures <span class="texhtml"><i>T</i><sub>1</sub></span> and <span class="texhtml"><i>T</i><sub>2</sub></span> must have the same efficiency, that is to say, the efficiency is the function of only temperatures <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 {\begin{aligned}{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}&=f(T_{\text{H}},T_{\text{C}}).&(2)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}&=f(T_{\text{H}},T_{\text{C}}).&(2)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2fd22950b8bf82a82b07a7f68c7fd4583731b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.466ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}&=f(T_{\text{H}},T_{\text{C}}).&(2)\end{aligned}}}"></span> </p><p>In addition, a reversible heat engine operating between a pair of thermal reservoirs at temperatures <span class="texhtml"><i>T</i><sub>1</sub></span> and <span class="texhtml"><i>T</i><sub>3</sub></span> must have the same efficiency as one consisting of two cycles, one between <span class="texhtml"><i>T</i><sub>1</sub></span> and another (intermediate) temperature <span class="texhtml"><i>T</i><sub>2</sub></span>, and the second between <span class="texhtml"><i>T</i><sub>2</sub></span> and <span class="texhtml"><i>T</i><sub>3</sub></span>. If this were not the case, then energy (in the form of <span class="texhtml mvar" style="font-style:italic;">q</span>) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles as an engine design choice, and any reversible engine between the same reservoir at <span class="texhtml"><i>T</i><sub>1</sub></span> and <span class="texhtml"><i>T</i><sub>3</sub></span> must be equally efficient regardless of the engine design. </p><p>If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as below. <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 {\begin{aligned}\eta _{1}&=1-{\frac {|q_{3}|}{|q_{1}|}}&=1-f(T_{1},T_{3})\\\eta _{2}&=1-{\frac {|q_{2}|}{|q_{1}|}}&=1-f(T_{1},T_{2})\\\eta _{3}&=1-{\frac {|q_{3}|}{|q_{2}|}}&=1-f(T_{2},T_{3})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\eta _{1}&=1-{\frac {|q_{3}|}{|q_{1}|}}&=1-f(T_{1},T_{3})\\\eta _{2}&=1-{\frac {|q_{2}|}{|q_{1}|}}&=1-f(T_{1},T_{2})\\\eta _{3}&=1-{\frac {|q_{3}|}{|q_{2}|}}&=1-f(T_{2},T_{3})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0087fc718f1763aea8380df71e94e1f119da3b86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:34.33ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\eta _{1}&=1-{\frac {|q_{3}|}{|q_{1}|}}&=1-f(T_{1},T_{3})\\\eta _{2}&=1-{\frac {|q_{2}|}{|q_{1}|}}&=1-f(T_{1},T_{2})\\\eta _{3}&=1-{\frac {|q_{3}|}{|q_{2}|}}&=1-f(T_{2},T_{3})\end{aligned}}}"></span> </p><p>Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at <span class="texhtml"><i>T</i><sub>2</sub></span>. We also have used the fact that the heat <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 q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> passes through the intermediate thermal reservoir at <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 T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> without losing its energy. (I.e., <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 q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> is not lost during its passage through the reservoir at <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 T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>.) This fact can be proved by the following. <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 {\begin{aligned}{{\eta }_{2}}&=1-{\frac {|{{q}_{2}}|}{|{{q}_{1}}|}}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\{{\eta }_{3}}&=1-{\frac {|{{q}_{3}}|}{|{{q}_{2}}^{*}|}}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\|{{w}_{2}}|+|{{w}_{3}}|&=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\{{\eta }_{1}}&=1-{\frac {|{{q}_{3}}|}{|{{q}_{1}}|}}={\frac {(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}}={\frac {(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}}.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{{\eta }_{2}}&=1-{\frac {|{{q}_{2}}|}{|{{q}_{1}}|}}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\{{\eta }_{3}}&=1-{\frac {|{{q}_{3}}|}{|{{q}_{2}}^{*}|}}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\|{{w}_{2}}|+|{{w}_{3}}|&=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\{{\eta }_{1}}&=1-{\frac {|{{q}_{3}}|}{|{{q}_{1}}|}}={\frac {(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}}={\frac {(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}}.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61fc326656ca8b33d19ffc5af890ca364149017" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:70.862ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}{{\eta }_{2}}&=1-{\frac {|{{q}_{2}}|}{|{{q}_{1}}|}}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\{{\eta }_{3}}&=1-{\frac {|{{q}_{3}}|}{|{{q}_{2}}^{*}|}}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\|{{w}_{2}}|+|{{w}_{3}}|&=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\{{\eta }_{1}}&=1-{\frac {|{{q}_{3}}|}{|{{q}_{1}}|}}={\frac {(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}}={\frac {(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}}.\\\end{aligned}}}"></span> </p><p>In order to have the consistency in the last equation, the heat <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 q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> flown from the engine 2 to the intermediate reservoir must be equal to the heat <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 q_{2}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97230842c1738b56bf51d10956cb0aee9e1ce62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.134ex; height:2.843ex;" alt="{\displaystyle q_{2}^{*}}"></span> flown out from the reservoir to the engine 3. </p><p>With this understanding of <span class="texhtml"><i>q</i><sub>1</sub></span>, <span class="texhtml"><i>q</i><sub>2</sub></span> and <span class="texhtml"><i>q</i><sub>3</sub></span>, mathematically, <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 f(T_{1},T_{3})={\frac {|q_{3}|}{|q_{1}|}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{3})={\frac {|q_{3}|}{|q_{1}|}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3629b1ae7b290d7bc2aaf2cfb64a1eb53d701ceb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.606ex; height:6.509ex;" alt="{\displaystyle f(T_{1},T_{3})={\frac {|q_{3}|}{|q_{1}|}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}"></span> </p><p>But since the first function is <i>not</i> a function of <span class="texhtml"><i>T</i><sub>2</sub></span>, the product of the final two functions <i>must</i> result in the removal of <span class="texhtml"><i>T</i><sub>2</sub></span> as a variable. The only way is therefore to define the function <span class="texhtml mvar" style="font-style:italic;">f</span> 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 f(T_{1},T_{2})={\frac {g(T_{2})}{g(T_{1})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{2})={\frac {g(T_{2})}{g(T_{1})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45d983f92be9e78067384a5da60cb3ece78b3ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.217ex; height:6.509ex;" alt="{\displaystyle f(T_{1},T_{2})={\frac {g(T_{2})}{g(T_{1})}}}"></span> and <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 f(T_{2},T_{3})={\frac {g(T_{3})}{g(T_{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{2},T_{3})={\frac {g(T_{3})}{g(T_{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbaea02441c5e3e66131df15c6272e982f801dd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.217ex; height:6.509ex;" alt="{\displaystyle f(T_{2},T_{3})={\frac {g(T_{3})}{g(T_{2})}}}"></span> so 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 f(T_{1},T_{3})={\frac {g(T_{3})}{g(T_{1})}}={\frac {|q_{3}|}{|q_{1}|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{3})={\frac {g(T_{3})}{g(T_{1})}}={\frac {|q_{3}|}{|q_{1}|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e795af9f32b6dcfa4fff449d541ba0616ee0ca80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.183ex; height:6.509ex;" alt="{\displaystyle f(T_{1},T_{3})={\frac {g(T_{3})}{g(T_{1})}}={\frac {|q_{3}|}{|q_{1}|}}.}"></span> </p><p>I.e. the ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our <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 g(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfafde1201bf1abd1bfdbaae832f38d3adab4027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.562ex; height:2.843ex;" alt="{\displaystyle g(T)}"></span>;<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> it is a matter of convenience and convention that we choose <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 g(T)=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(T)=T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb17ba66b1080d071d1856c110ebe0634351fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.296ex; height:2.843ex;" alt="{\displaystyle g(T)=T}"></span>. Choosing then <i>one</i> fixed reference temperature (i.e. triple point of water), we establish the thermodynamic temperature scale. </p><p>Such a definition coincides with that of the ideal gas derivation; also it is this <i>definition</i> of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of <span class="texhtml"><i>T</i><sub>H</sub></span> and <span class="texhtml"><i>T</i><sub>C</sub></span>, and hence derive that the (complete) Carnot cycle is isentropic: <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 {\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=f(T_{\text{H}},T_{\text{C}})={\frac {T_{\text{C}}}{T_{\text{H}}}}.\ \ \ \ \ \ \ \ \ \ \ (3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=f(T_{\text{H}},T_{\text{C}})={\frac {T_{\text{C}}}{T_{\text{H}}}}.\ \ \ \ \ \ \ \ \ \ \ (3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b5757dcb94d53aa7e5d9cf19e6d6fbb1f1f466" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.6ex; height:6.509ex;" alt="{\displaystyle {\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=f(T_{\text{H}},T_{\text{C}})={\frac {T_{\text{C}}}{T_{\text{H}}}}.\ \ \ \ \ \ \ \ \ \ \ (3)}"></span> </p><p>Substituting this back into our first formula for efficiency yields a relationship in terms of temperature: <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 {\textrm {Efficiency}}=1+{\frac {q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=1-{\frac {T_{\text{C}}}{T_{\text{H}}}}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Efficiency</mtext> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {Efficiency}}=1+{\frac {q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=1-{\frac {T_{\text{C}}}{T_{\text{H}}}}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d741a38f76cb5159f7b3878f6fc5fb61cc4c234" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.307ex; height:6.509ex;" alt="{\displaystyle {\textrm {Efficiency}}=1+{\frac {q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=1-{\frac {T_{\text{C}}}{T_{\text{H}}}}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)}"></span> </p><p>Note that for <span class="texhtml"><i>T</i><sub>C</sub> = 0</span> the efficiency is 100% and that efficiency becomes greater than 100% for <span class="texhtml"><i>T</i><sub>C</sub> < 0</span>, which is unrealistic. Subtracting 1 from the right hand side of the Equation (4) and the middle portion gives <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 {\frac {q_{\text{C}}}{q_{\text{H}}}}=-{\frac {T_{\text{C}}}{T_{\text{H}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{\text{C}}}{q_{\text{H}}}}=-{\frac {T_{\text{C}}}{T_{\text{H}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5b53a94714d79bbcdd46ae452355fc203a7a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.903ex; height:5.843ex;" alt="{\displaystyle {\frac {q_{\text{C}}}{q_{\text{H}}}}=-{\frac {T_{\text{C}}}{T_{\text{H}}}}}"></span> and thus <sup id="cite_ref-FermiBook_50-0" class="reference"><a href="#cite_note-FermiBook-50"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-PlanckBook_48-1" class="reference"><a href="#cite_note-PlanckBook-48"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> <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 {\frac {q_{\text{H}}}{T_{\text{H}}}}+{\frac {q_{\text{C}}}{T_{\text{C}}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{\text{H}}}{T_{\text{H}}}}+{\frac {q_{\text{C}}}{T_{\text{C}}}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae92f22aaf516574340f17b25b99c4c39097b6c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.019ex; height:5.343ex;" alt="{\displaystyle {\frac {q_{\text{H}}}{T_{\text{H}}}}+{\frac {q_{\text{C}}}{T_{\text{C}}}}=0.}"></span> </p><p>The generalization of this equation is the <a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a>, which proposes the existence of a <a href="/wiki/State_function" title="State function">state function</a> <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by </p><p><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 {\begin{aligned}dS&={\frac {dq_{\mathrm {rev} }}{T}}&(5)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mrow> <mi>T</mi> </mfrac> </mrow> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}dS&={\frac {dq_{\mathrm {rev} }}{T}}&(5)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32428ddb7c2713048749ecad9bdf6dba8e2921ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.746ex; height:5.509ex;" alt="{\displaystyle {\begin{aligned}dS&={\frac {dq_{\mathrm {rev} }}{T}}&(5)\end{aligned}}}"></span> </p><p>where the subscript <i>rev</i> indicates heat transfer in a reversible process. The function <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is the <a href="/wiki/Entropy" title="Entropy">entropy</a> of the system, mentioned previously, and the change of <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> around any cycle is zero (as is necessary for any state function). The Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid a logic loop, we should first define <a href="/wiki/Statistical_entropy" class="mw-redirect" title="Statistical entropy">entropy</a> through statistical mechanics): <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 T={\frac {dq_{\mathrm {rev} }}{dS}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {dq_{\mathrm {rev} }}{dS}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25e229ed4361cfed3ecaf8c9e346cc7b368976cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.945ex; height:5.509ex;" alt="{\displaystyle T={\frac {dq_{\mathrm {rev} }}{dS}}.}"></span> </p><p>For a constant-volume system (so no mechanical work <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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>) in which the entropy <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is a function <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 S(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1467f5ed55172dcd5d83f152603a6eb228912726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.084ex; height:2.843ex;" alt="{\displaystyle S(E)}"></span> of its <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, <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 dE=dq_{rev}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>E</mi> <mo>=</mo> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dE=dq_{rev}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d492714f1a41a0a1918ff16d5cf3fc2a3b54f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.88ex; height:2.509ex;" alt="{\displaystyle dE=dq_{rev}}"></span> and the thermodynamic temperature <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is therefore given by <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 {\frac {1}{T}}={\frac {dS}{dE}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mi>d</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T}}={\frac {dS}{dE}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04adbdce39aa40002a5adca44cf8d1b8c094756a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.045ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{T}}={\frac {dS}{dE}},}"></span> so that the reciprocal of the thermodynamic temperature is the rate of change of entropy with respect to the internal energy at the constant volume. </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=Thermodynamic_temperature&action=edit&section=18" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Guillaume_Amontons,_wood_cutting.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Guillaume_Amontons%2C_wood_cutting.png/180px-Guillaume_Amontons%2C_wood_cutting.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Guillaume_Amontons%2C_wood_cutting.png/270px-Guillaume_Amontons%2C_wood_cutting.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Guillaume_Amontons%2C_wood_cutting.png/360px-Guillaume_Amontons%2C_wood_cutting.png 2x" data-file-width="460" data-file-height="460" /></a><figcaption><a href="/wiki/Guillaume_Amontons" title="Guillaume Amontons">Guillaume Amontons</a></figcaption></figure> <p><a href="/wiki/Guillaume_Amontons" title="Guillaume Amontons">Guillaume Amontons</a> (1663–1705) published two papers in 1702 and 1703 that may be used to credit him as being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. He made the discovery while endeavoring to improve upon the air thermometers in use at the time. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation led him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240 °C—only 33.15 degrees short of the true value of −273.15 °C. Amonton's discovery of a one-to-one relationship between absolute temperature and absolute pressure was rediscovered a century later and popularized within the scientific community by <a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</a>. Today, this principle of thermodynamics is commonly known as <i><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></i> but is also known as <i>Amonton's law</i>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Headshot_of_Anders_Celsius.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Headshot_of_Anders_Celsius.jpg/180px-Headshot_of_Anders_Celsius.jpg" decoding="async" width="180" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Headshot_of_Anders_Celsius.jpg/270px-Headshot_of_Anders_Celsius.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Headshot_of_Anders_Celsius.jpg/360px-Headshot_of_Anders_Celsius.jpg 2x" data-file-width="923" data-file-height="1050" /></a><figcaption><a href="/wiki/Anders_Celsius" title="Anders Celsius">Anders Celsius</a></figcaption></figure> <p>In 1742, <a href="/wiki/Anders_Celsius" title="Anders Celsius">Anders Celsius</a> (1701–1744) created a "backwards" version of the modern Celsius temperature scale. In Celsius's original scale, zero represented the boiling point of water and 100 represented the melting point of ice. In his paper <i>Observations of two persistent degrees on a thermometer</i>, he recounted his experiments showing that ice's melting point was effectively unaffected by pressure. He also determined with remarkable precision how water's boiling point varied as a function of atmospheric pressure. He proposed that zero on his temperature scale (water's boiling point) would be calibrated at the mean barometric pressure at mean sea level. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Carl_von_Linn%C3%A9.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Carl_von_Linn%C3%A9.jpg/180px-Carl_von_Linn%C3%A9.jpg" decoding="async" width="180" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Carl_von_Linn%C3%A9.jpg/270px-Carl_von_Linn%C3%A9.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Carl_von_Linn%C3%A9.jpg/360px-Carl_von_Linn%C3%A9.jpg 2x" data-file-width="1938" data-file-height="2342" /></a><figcaption><a href="/wiki/Carl_Linnaeus" title="Carl Linnaeus">Carl Linnaeus</a></figcaption></figure> <p>Coincident with the death of Anders Celsius in 1744, the botanist <a href="/wiki/Carl_Linnaeus" title="Carl Linnaeus">Carl Linnaeus</a> (1707–1778) effectively reversed<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (September 2024)">full citation needed</span></a></i>]</sup> Celsius's scale upon receipt of his first thermometer featuring a scale where zero represented the melting point of ice and 100 represented water's boiling point. The custom-made <i>Linnaeus-thermometer</i>, for use in his greenhouses, was made by Daniel Ekström, Sweden's leading maker of scientific instruments at the time. For the next 204 years, the scientific and thermometry communities worldwide referred to this scale as the <i><a href="/wiki/Centigrade_scale" class="mw-redirect" title="Centigrade scale">centigrade scale</a></i>. Temperatures on the centigrade scale were often reported simply as <i>degrees</i> or, when greater specificity was desired, <i><a href="/wiki/Degrees_centigrade" class="mw-redirect" title="Degrees centigrade">degrees centigrade</a></i>. The symbol for temperature values on this scale was °C (in several formats over the years). Because the term <i>centigrade</i> was also the French-language name for a unit of angular measurement (one-hundredth of a right angle) and had a similar connotation in other languages, the term "<a href="/wiki/Centesimal_degree" class="mw-redirect" title="Centesimal degree">centesimal degree</a>" was used when very precise, unambiguous language was required by international standards bodies such as the <a href="/wiki/International_Bureau_of_Weights_and_Measures" title="International Bureau of Weights and Measures">International Bureau of Weights and Measures</a> (BIPM). The 9th CGPM (<a href="/wiki/General_Conference_on_Weights_and_Measures" title="General Conference on Weights and Measures">General Conference on Weights and Measures</a> and the CIPM (<a href="/wiki/International_Committee_for_Weights_and_Measures" class="mw-redirect" title="International Committee for Weights and Measures">International Committee for Weights and Measures</a> formally adopted<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> <i><a href="/wiki/Degree_Celsius" class="mw-redirect" title="Degree Celsius">degree Celsius</a></i> (symbol: °C) in 1948. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:JHLambert.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/JHLambert.jpg/180px-JHLambert.jpg" decoding="async" width="180" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/JHLambert.jpg/270px-JHLambert.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9b/JHLambert.jpg 2x" data-file-width="350" data-file-height="460" /></a><figcaption><a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a></figcaption></figure> <p>In his book <i>Pyrometrie</i> (1777)<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> completed four months before his death, <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> (1728–1777), sometimes incorrectly referred to as Joseph Lambert, proposed an absolute temperature scale based on the pressure/temperature relationship of a fixed volume of gas. This is distinct from the volume/temperature relationship of gas under constant pressure that Guillaume Amontons discovered 75 years earlier. Lambert stated that absolute zero was the point where a simple straight-line extrapolation reached zero gas pressure and was equal to −270 °C. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Jacques_Charles.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Jacques_Charles.jpg/180px-Jacques_Charles.jpg" decoding="async" width="180" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Jacques_Charles.jpg/270px-Jacques_Charles.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Jacques_Charles.jpg/360px-Jacques_Charles.jpg 2x" data-file-width="592" data-file-height="673" /></a><figcaption><a href="/wiki/Jacques_Alexandre_C%C3%A9sar_Charles" class="mw-redirect" title="Jacques Alexandre César Charles">Jacques Alexandre César Charles</a></figcaption></figure> <p>Notwithstanding the work of Guillaume Amontons 85 years earlier, <a href="/wiki/Jacques_Charles" title="Jacques Charles">Jacques Alexandre César Charles</a> (1746–1823) is often credited with discovering (circa 1787), but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was <span class="texhtml"><i>V</i><sub>1</sub>/<i>T</i><sub>1</sub> = <i>V</i><sub>2</sub>/<i>T</i><sub>2</sub></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gaylussac.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Gaylussac.jpg/180px-Gaylussac.jpg" decoding="async" width="180" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Gaylussac.jpg/270px-Gaylussac.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Gaylussac.jpg/360px-Gaylussac.jpg 2x" data-file-width="1876" data-file-height="2335" /></a><figcaption><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</a></figcaption></figure> <p><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</a> (1778–1850) published work in 1802 (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called <a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a> and is one of the <a href="/wiki/Gas_laws" title="Gas laws">gas laws</a>. His are the first known formulas to use the number 273 for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C). </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:William_Thomson_1st_Baron_Kelvin.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/William_Thomson_1st_Baron_Kelvin.jpg/180px-William_Thomson_1st_Baron_Kelvin.jpg" decoding="async" width="180" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/William_Thomson_1st_Baron_Kelvin.jpg/270px-William_Thomson_1st_Baron_Kelvin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/William_Thomson_1st_Baron_Kelvin.jpg/360px-William_Thomson_1st_Baron_Kelvin.jpg 2x" data-file-width="1128" data-file-height="1557" /></a><figcaption><a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">Lord Kelvin</a></figcaption></figure> <p><a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a> (1824–1907), also known as Lord Kelvin, wrote in his 1848 paper "On an Absolute Thermometric Scale"<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> of the need for a scale whereby <i>infinite cold</i> (absolute zero) was the scale's zero point, and which used the degree Celsius for its unit increment. Like Gay-Lussac, Thomson calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the kelvin thermodynamic temperature scale. Thomson's value of −273 was derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of −0.00366 expressed to five significant digits is −273.22 °C which is remarkably close to the true value of −273.15 °C. </p><p>In the paper he proposed to define temperature using idealized heat engines. In detail, he proposed that, given three heat reservoirs at temperatures <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 T_{A},T_{B},T_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A},T_{B},T_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae060d0ce1a984c720e7b96b262fa8dd793f212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.566ex; height:2.509ex;" alt="{\displaystyle T_{A},T_{B},T_{C}}"></span>, if two reversible heat engines (<a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot engine</a>), one working between <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 T_{A},T_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A},T_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd223f8fb3718ceb36c06c27b9880f0718d1a59b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.693ex; height:2.509ex;" alt="{\displaystyle T_{A},T_{B}}"></span> and another between <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 T_{B},T_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B},T_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f979aa9176c1f223cf7123e2f3d350cdb1b403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.509ex;" alt="{\displaystyle T_{B},T_{C}}"></span>, can produce the same amount of mechanical work <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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> by letting the same amount of heat <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> pass through, then define <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 T_{A}-T_{B}=T_{B}-T_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}-T_{B}=T_{B}-T_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f40c737372fa954ff193199e022df27302c8ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.115ex; height:2.509ex;" alt="{\displaystyle T_{A}-T_{B}=T_{B}-T_{C}}"></span>. </p><p>Note that like Carnot, Kelvin worked under the assumption that heat is conserved ("the conversion of heat (or caloric) into mechanical effect is probably impossible"), and if heat <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> goes into the heat engine, then heat <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> must come out.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>Kelvin, realizing after Joule's experiments that heat is not a conserved quantity but is convertible with mechanical work, modified his scale in the 1851 work <i>An Account of Carnot's Theory of the Motive Power of Heat</i>. In this work, he defined as follows:<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Given two heat reservoirs <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 T_{A},T_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A},T_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd223f8fb3718ceb36c06c27b9880f0718d1a59b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.693ex; height:2.509ex;" alt="{\displaystyle T_{A},T_{B}}"></span>, and a reversible heat engine working between them, such that if during an engine cycle, heat <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 Q_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227df710321dff901d1b8a6908f21b1b4c3762ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.303ex; height:2.509ex;" alt="{\displaystyle Q_{A}}"></span> moves into the engine, and heat <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 Q_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d47686ee57e8d2b64f2826b3a82b7a3beebd4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.318ex; height:2.509ex;" alt="{\displaystyle Q_{B}}"></span> comes out of the engine, then <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 {\frac {T_{A}}{T_{B}}}={\frac {Q_{A}}{Q_{B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {T_{A}}{T_{B}}}={\frac {Q_{A}}{Q_{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3ffa66e7e9d05fdd1af349a79e94267c564ada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.926ex; height:5.843ex;" alt="{\displaystyle {\frac {T_{A}}{T_{B}}}={\frac {Q_{A}}{Q_{B}}}}"></span>.</p></blockquote> <p>The above definition fixes the ratios between absolute temperatures, but it does not fix a scale for absolute temperature. For the scale, Thomson proposed to use the Celsius degree, that is, <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 {\frac {1}{100}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>100</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{100}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3620801d9068718ecf6e30de8efcb95026dd871d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.323ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{100}}}"></span> the interval between the freezing and the boiling point of water. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rankine_William_signature.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Rankine_William_signature.jpg/180px-Rankine_William_signature.jpg" decoding="async" width="180" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Rankine_William_signature.jpg/270px-Rankine_William_signature.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Rankine_William_signature.jpg/360px-Rankine_William_signature.jpg 2x" data-file-width="636" data-file-height="823" /></a><figcaption><a href="/wiki/Macquorn_Rankine" class="mw-redirect" title="Macquorn Rankine">Macquorn Rankine</a></figcaption></figure> <p>In 1859 <a href="/wiki/Macquorn_Rankine" class="mw-redirect" title="Macquorn Rankine">Macquorn Rankine</a> (1820–1872) proposed a thermodynamic temperature scale similar to William Thomson's but which used the degree <a href="/wiki/Fahrenheit" title="Fahrenheit">Fahrenheit</a> for its unit increment, that is, <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 {\frac {1}{180}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>180</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{180}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c49a468fbe28b80c01448e9c65397420e85b26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.323ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{180}}}"></span> the interval between the freezing and the boiling point of water. This absolute scale is known today as the <a href="/wiki/Rankine_scale" title="Rankine scale">Rankine</a> thermodynamic temperature scale. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Boltzmann2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/180px-Boltzmann2.jpg" decoding="async" width="180" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/270px-Boltzmann2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/360px-Boltzmann2.jpg 2x" data-file-width="490" data-file-height="600" /></a><figcaption><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a></figcaption></figure> <p><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> (1844–1906) made major contributions to thermodynamics between 1877 and 1884 through an understanding of the role that particle kinetics and black body radiation played. His name is now attached to several of the formulas used today in thermodynamics. </p><p>Gas thermometry experiments<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag may use weasel words or too-vague attribution. (September 2024)">by whom?</span></a></i>]</sup> carefully calibrated to the melting point of ice and boiling point of water showed in the 1930s that absolute zero was equivalent to −273.15 °C. </p><p>Resolution 3<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> of the 9th <a href="/wiki/General_Conference_on_Weights_and_Measures" title="General Conference on Weights and Measures">General Conference on Weights and Measures</a> (CGPM) in 1948 fixed the triple point of water at precisely 0.01 °C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared "will be fixed at a later date". The implication is that if the value of absolute zero measured in the 1930s was truly −273.15 °C, then the triple point of water (0.01 °C) was equivalent to 273.16 K. Additionally, both the International Committee for Weights and Measures (CIPM) and the CGPM formally adopted<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> the name <i>Celsius</i> for the <i>degree Celsius</i> and the <i>Celsius temperature scale</i>.<sup id="cite_ref-°CName_62-0" class="reference"><a href="#cite_note-°CName-62"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p><p>Resolution 3<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> of the 10th CGPM in 1954 gave the kelvin scale its modern definition by choosing the triple point of water as its upper defining point (with no change to absolute zero being the null point) and assigning it a temperature of precisely 273.16 kelvins (what was actually written 273.16 <i>degrees Kelvin</i> at the time). This, in combination with Resolution 3 of the 9th CGPM, had the effect of defining absolute zero as being precisely zero kelvins and −273.15 °C. </p><p>Resolution 3<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> of the 13th CGPM in 1967/1968 renamed the unit increment of thermodynamic temperature <i>kelvin</i>, symbol K, replacing <i>degree absolute</i>, symbol °K. Further, feeling it useful to more explicitly define the magnitude of the unit increment, the 13th CGPM also decided in Resolution 4<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> that "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water". </p><p>The CIPM affirmed in 2005<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin thermodynamic temperature scale would refer to water having an isotopic composition defined as being precisely equal to the nominal specification of <a href="/wiki/Vienna_Standard_Mean_Ocean_Water" title="Vienna Standard Mean Ocean Water">Vienna Standard Mean Ocean Water</a>. </p><p>In November 2018, the 26th General Conference on Weights and Measures (CGPM) changed the definition of the Kelvin by fixing the Boltzmann constant to <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> when expressed in the unit J/K. <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">This change (and other changes in the definition of SI units)</a> was made effective on the 144th anniversary of the Metre Convention, 20 May 2019. </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=Thermodynamic_temperature&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category:Thermodynamics</a></li> <li><a href="/wiki/Absolute_zero" title="Absolute zero">Absolute zero</a></li> <li><a href="/wiki/Hagedorn_temperature" title="Hagedorn temperature">Hagedorn temperature</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic process</a></li> <li><a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a></li> <li><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></li> <li><a href="/wiki/Conversion_of_scales_of_temperature" title="Conversion of scales of temperature">Conversion of scales of temperature</a></li> <li><a href="/wiki/Energy_conversion_efficiency" title="Energy conversion efficiency">Energy conversion efficiency</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a> <ul><li><a href="/wiki/Enthalpy_of_fusion" title="Enthalpy of fusion">Enthalpy of fusion</a></li> <li><a href="/wiki/Enthalpy_of_vaporization" title="Enthalpy of vaporization">Enthalpy of vaporization</a></li></ul></li> <li><a href="/wiki/Entropy" title="Entropy">Entropy</a></li> <li><a href="/wiki/Equipartition_theorem" title="Equipartition theorem">Equipartition theorem</a></li> <li><a href="/wiki/Fahrenheit" title="Fahrenheit">Fahrenheit</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First law of thermodynamics</a></li> <li><a href="/wiki/Freezing" title="Freezing">Freezing</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li> <li><a href="/wiki/International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a></li> <li><a href="/wiki/International_Temperature_Scale_of_1990" title="International Temperature Scale of 1990">International Temperature Scale of 1990</a> (ITS-90)</li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li> <li><a href="/wiki/Kelvin" title="Kelvin">Kelvin</a></li> <li><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws of thermodynamics</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann distribution</a></li> <li><a href="/wiki/Orders_of_magnitude_(temperature)" title="Orders of magnitude (temperature)">Orders of magnitude (temperature)</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Planck%27s_law_of_black_body_radiation" class="mw-redirect" title="Planck's law of black body radiation">Planck's law of black body radiation</a></li> <li><a href="/wiki/Rankine_scale" title="Rankine scale">Rankine scale</a></li> <li><a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">Specific heat capacity</a></li> <li><a href="/wiki/Temperature" title="Temperature">Temperature</a></li> <li><a href="/wiki/Thermal_radiation" title="Thermal radiation">Thermal radiation</a></li> <li><a href="/wiki/Thermodynamic_beta" title="Thermodynamic beta">Thermodynamic beta</a></li> <li><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Thermodynamic equations</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Timeline of heat engine technology</a></li> <li><a href="/wiki/Timeline_of_temperature_and_pressure_measurement_technology" title="Timeline of temperature and pressure measurement technology">Timeline of temperature and pressure measurement technology</a></li> <li><a href="/wiki/Triple_point" title="Triple point">Triple point</a></li></ul> </div> <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=Thermodynamic_temperature&action=edit&section=20" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>In the following notes, wherever numeric equalities are shown in </i>concise form<i>, such as <span class="nowrap"><span data-sort-value="7043185487000000000♠"></span>1.854<span style="margin-left:.25em;">87</span>(14)<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>43</sup></span>, the two digits between the parentheses denotes the <a href="/wiki/Uncertainty" title="Uncertainty">uncertainty</a> at 1-σ (1 <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>, 68% confidence level) in the two least significant digits of the <a href="/wiki/Significand" title="Significand">significand</a>.</i></dd></dl> <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-Accessible_k-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Accessible_k_1-0">^</a></b></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?k">CODATA Value: Boltzmann constant</a></i>. <i>The NIST Reference on Constants, Units, and Uncertainty</i>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Georgia State University, HyperPhysics Project, "<a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/eqpar.html">Equipartition of Energy</a>"</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Rankine, W. J. M., "A manual of the steam engine and other prime movers", Richard Griffin and Co., London (1859), p. 306–307.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson, 1st Baron Kelvin</a>, "Heat", Adam and Charles Black, Edinburgh (1880), p. 39.</span> </li> <li id="cite_note-T0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-T0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-T0_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-T0_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-T0_5-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-T0_5-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><figure typeof="mw:File/Thumb"><a href="/wiki/File:Zero-point_energy_v.s._motion.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Zero-point_energy_v.s._motion.jpg/300px-Zero-point_energy_v.s._motion.jpg" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Zero-point_energy_v.s._motion.jpg/450px-Zero-point_energy_v.s._motion.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/79/Zero-point_energy_v.s._motion.jpg 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Absolute zero's relationship to zero-point energy</figcaption></figure> While scientists are achieving temperatures ever closer to <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, they can not fully achieve a state of <i>zero</i> temperature. However, even if scientists could remove <i>all</i> kinetic thermal energy from matter, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> <i><a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a></i> (ZPE) causes particle motion that can never be eliminated. Encyclopædia Britannica Online <a rel="nofollow" class="external text" href="http://britannica.com/eb/article-9078341">defines zero-point</a> energy as the "vibrational energy that molecules retain even at the absolute zero of temperature". ZPE is the result of all-pervasive energy fields in the vacuum between the fundamental particles of nature; it is responsible for the <a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a> and other phenomena. See <i><a rel="nofollow" class="external text" href="http://calphysics.org/zpe.html">Zero Point Energy and Zero Point Field</a></i>. See also <i><a rel="nofollow" class="external text" href="http://www.phys.ualberta.ca/~therman/lowtemp/projects1.htm">Solid Helium</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080212140020/http://www.phys.ualberta.ca/~therman/lowtemp/projects1.htm">Archived</a> 2008-02-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i> by the University of Alberta's Department of Physics to learn more about ZPE's effect on <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensates</a> of helium. <p>Although absolute zero (<span class="nowrap"><i>T</i> = 0</span>) is not a state of zero molecular motion, it <i>is</i> the point of zero temperature and, in accordance with the Boltzmann constant, is also the point of zero particle kinetic energy and zero kinetic velocity. <span class="anchor" id="thought_experiment"></span>To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two <span class="nowrap"><i>T</i> = 0</span> helium atoms in zero gravity are carefully positioned and observed to have an average separation of 620 <a href="/wiki/Picometer" class="mw-redirect" title="Picometer">pm</a> between them (a gap of ten atomic diameters). It is an "average" separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83 yoctokelvins (1 yK = <span class="nowrap"><span data-sort-value="6976099999999999999♠"></span>1<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−24</sup> K</span>). This is done in a way that directs this atom's velocity vector at the other atom. With 83 yK of kinetic energy between them, the 620 pm gap through their common <a href="/wiki/Barycentric_coordinates_(astronomy)" class="mw-redirect" title="Barycentric coordinates (astronomy)">barycenter</a> would close at a rate of 719 pm/s and they would collide after 0.862 second. This is the same speed as shown in the <i><a href="#Overview">Fig. 1</a> </i>animation above. Before being given the kinetic kick, both <span class="nowrap"><i>T</i> = 0</span> atoms had zero kinetic energy and zero kinetic velocity because they could persist indefinitely in that state and relative orientation even though both were being jostled by ZPE. At <span class="nowrap"><i>T</i> = 0</span>, no <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> is available for transfer to other systems. </p><p>Note too that absolute zero serves as the baseline atop which <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> and its <a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">equations</a> are founded because they deal with the exchange of thermal energy between "<i>systems</i>" (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPE-induced particle motion <i>within</i> a system that is at absolute zero but there can never be a net outflow of thermal energy from such a system. Also, the peak emittance wavelength of black-body radiation shifts to infinity at absolute zero; indeed, a peak no longer exists and black-body photons can no longer escape. Because of ZPE, however, <i>virtual</i> photons are still emitted at <span class="nowrap"><i>T</i> = 0</span>. Such photons are called "virtual" because they can't be intercepted and observed. Furthermore, this <i>zero-point radiation</i> has a unique <i>zero-point spectrum</i>. However, even though a <span class="nowrap"><i>T</i> = 0</span> system emits zero-point radiation, no net heat flow <i>Q</i> out of such a system can occur because if the surrounding environment is at a temperature greater than <span class="nowrap"><i>T</i> = 0</span>, heat will flow inward, and if the surrounding environment is at '<span class="nowrap"><i>T</i> = 0</span>, there will be an equal flux of ZP radiation both inward and outward. A similar <i>Q </i>equilibrium exists at <span class="nowrap"><i>T</i> = 0</span> with the ZPE-induced <a href="/wiki/Spontaneous_emission" title="Spontaneous emission">spontaneous emission</a> of photons (which is more properly called a <i>stimulated</i> emission in this context). The graph at upper right illustrates the relationship of absolute zero to zero-point energy. The graph also helps in the understanding of how zero-point energy got its name: it is the vibrational energy matter retains at the <i>zero-kelvin point</i>. <a rel="nofollow" class="external text" href="http://pra.aps.org/abstract/PRA/v42/i4/p1847_1"><i>Derivation of the classical electromagnetic zero-point radiation spectrum via a classical thermodynamic operation involving van der Waals forces</i></a>, Daniel C. Cole, Physical Review A, <b>42</b> (1990) 1847. </p> </span></li> <li id="cite_note-BIPMbrocure-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-BIPMbrocure_6-0">^</a></b></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070926215600/http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html">"SI brochure, section 2.1.1.5"</a>. <a href="/wiki/International_Bureau_of_Weights_and_Measures" title="International Bureau of Weights and Measures">International Bureau of Weights and Measures</a>. Archived from <a rel="nofollow" class="external text" href="http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html">the original</a> on 26 September 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">9 May</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=SI+brochure%2C+section+2.1.1.5&rft.pub=International+Bureau+of+Weights+and+Measures&rft_id=http%3A%2F%2Fwww1.bipm.org%2Fen%2Fsi%2Fsi_brochure%2Fchapter2%2F2-1%2F2-1-1%2Fkelvin.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-codata2017-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-codata2017_7-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewellCabiatiFischerFujii2018" class="citation journal cs1">Newell, D B; Cabiati, F; Fischer, J; Fujii, K; Karshenboim, S G; Margolis, H S; de Mirandés, E; Mohr, P J; Nez, F; Pachucki, K; Quinn, T J; Taylor, B N; Wang, M; Wood, B M; Zhang, Z; et al. (Committee on Data for Science and Technology (CODATA) Task Group on Fundamental Constants) (29 January 2018). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Faa950a">"The CODATA 2017 values of <i>h</i>, <i>e</i>, <i>k</i>, and <i>N</i><sub>A</sub> for the revision of the SI"</a>. <i>Metrologia</i>. <b>55</b> (1): L13–L16. <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/2018Metro..55L..13N">2018Metro..55L..13N</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%2Faa950a">10.1088/1681-7575/aa950a</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=The+CODATA+2017+values+of+h%2C+e%2C+k%2C+and+N%3Csub%3EA%3C%2Fsub%3E+for+the+revision+of+the+SI&rft.volume=55&rft.issue=1&rft.pages=L13-L16&rft.date=2018-01-29&rft_id=info%3Adoi%2F10.1088%2F1681-7575%2Faa950a&rft_id=info%3Abibcode%2F2018Metro..55L..13N&rft.aulast=Newell&rft.aufirst=D+B&rft.au=Cabiati%2C+F&rft.au=Fischer%2C+J&rft.au=Fujii%2C+K&rft.au=Karshenboim%2C+S+G&rft.au=Margolis%2C+H+S&rft.au=de+Mirand%C3%A9s%2C+E&rft.au=Mohr%2C+P+J&rft.au=Nez%2C+F&rft.au=Pachucki%2C+K&rft.au=Quinn%2C+T+J&rft.au=Taylor%2C+B+N&rft.au=Wang%2C+M&rft.au=Wood%2C+B+M&rft.au=Zhang%2C+Z&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1681-7575%252Faa950a&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><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/20200701093442/https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant">"SI Redefinition – Kelvin: Boltzmann Constant"</a>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>. Archived from <a rel="nofollow" class="external text" href="https://www.nist.gov/programs-projects/acoustic-thermometry">the original</a> on 1 July 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">13 Dec</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=SI+Redefinition+%E2%80%93+Kelvin%3A+Boltzmann+Constant&rft.pub=National+Institute+of+Standards+and+Technology&rft_id=https%3A%2F%2Fwww.nist.gov%2Fprograms-projects%2Facoustic-thermometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><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/20200923025023/https://www.nist.gov/programs-projects/acoustic-thermometry">"Acoustic Thermometry"</a>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>. Archived from <a rel="nofollow" class="external text" href="https://www.nist.gov/programs-projects/acoustic-thermometry">the original</a> on 23 September 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">13 Dec</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Acoustic+Thermometry&rft.pub=National+Institute+of+Standards+and+Technology&rft_id=https%3A%2F%2Fwww.nist.gov%2Fprograms-projects%2Facoustic-thermometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">At non-<a href="/wiki/Special_relativity" title="Special relativity">relativistic</a> temperatures of less than about 30 GK, <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> are sufficient to calculate the velocity of particles. At 30 GK, individual neutrons (the constituent of neutron stars and one of the few materials in the universe with temperatures in this range) have a 1.0042 γ (gamma or <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>). Thus, the classic Newtonian formula for kinetic energy is in error less than half a percent for temperatures less than 30 GK.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Even room–temperature air has an average molecular translational <i>speed</i> (not vector-isolated velocity) of 1822 km/hour. This is relatively fast for something the size of a molecule considering there are roughly <span class="nowrap"><span data-sort-value="7016242000000000000♠"></span>2.42<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>16</sup></span> of them crowded into a single cubic millimeter. Assumptions: Average molecular weight of wet air = 28.838 g/mol and <span class="texhtml mvar" style="font-style:italic;">T</span> = 296.15 K. Assumption's primary variables: An altitude of 194 meters above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dew point of 9 °C (40.85% relative humidity), and 760 <a href="/wiki/Millimetre_of_mercury" title="Millimetre of mercury">mmHg</a> (101 kPa) sea level–corrected barometric pressure.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKastberg1995" class="citation journal cs1">Kastberg, A.; et al. (27 February 1995). "Adiabatic Cooling of Cesium to 700 nK in an Optical Lattice". <i>Physical Review Letters</i>. <b>74</b> (9): 1542–1545. <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/1995PhRvL..74.1542K">1995PhRvL..74.1542K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.74.1542">10.1103/PhysRevLett.74.1542</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10059055">10059055</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Adiabatic+Cooling+of+Cesium+to+700+nK+in+an+Optical+Lattice&rft.volume=74&rft.issue=9&rft.pages=1542-1545&rft.date=1995-02-27&rft_id=info%3Apmid%2F10059055&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.74.1542&rft_id=info%3Abibcode%2F1995PhRvL..74.1542K&rft.aulast=Kastberg&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> A record cold temperature of 450 <a href="/wiki/Kelvin#SI_prefixes" title="Kelvin">pK</a> in a Bose–Einstein condensate of sodium atoms (achieved by A. E. Leanhardt <i>et al.</i>. of <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</a>){{cn|{{subst:DATE}} equates to an average vector-isolated atom velocity of 0.4 mm/s and an average atom speed of 0.7 mm/s.</span> </li> <li id="cite_note-Boltzmann-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Boltzmann_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Boltzmann_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature 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 {\tilde {v}}={\sqrt {\frac {{\frac {k_{\text{B}}}{2}}\cdot T}{\frac {m}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>T</mi> </mrow> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {v}}={\sqrt {\frac {{\frac {k_{\text{B}}}{2}}\cdot T}{\frac {m}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f649b930b59edc41e4f35c00f0d4da2fb0fd6be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.732ex; height:8.676ex;" alt="{\displaystyle {\tilde {v}}={\sqrt {\frac {{\frac {k_{\text{B}}}{2}}\cdot T}{\frac {m}{2}}}}}"></span> where <ul><li><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="{\textstyle {\tilde {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tilde {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7012d40852be60aff6ca57e58341151d6f9a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\textstyle {\tilde {v}}}"></span> is the vector-isolated mean velocity of translational particle motion in m/s</li> <li><span class="texhtml"><i>k</i><sub>B</sub></span> (<a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>) = <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></li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the thermodynamic temperature in kelvins</li> <li><span class="texhtml mvar" style="font-style:italic;">m</span> is the molecular mass of substance in kg/particle</li></ul> In the above formula, molecular mass, <span class="texhtml mvar" style="font-style:italic;">m</span>, in kg/particle is the quotient of a substance's <a href="/wiki/Molar_mass" title="Molar mass">molar mass</a> (also known as <i>atomic weight</i>, <i><a href="/wiki/Atomic_mass" title="Atomic mass">atomic mass</a></i>, <i>relative atomic mass</i>, and <i><a href="/wiki/Atomic_mass_unit" class="mw-redirect" title="Atomic mass unit">unified atomic mass units</a></i>) in <a href="/wiki/Gram" title="Gram">g</a>/<a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a> or <a href="/wiki/Atomic_mass_unit" class="mw-redirect" title="Atomic mass unit">daltons</a> divided by <span class="nowrap"><span data-sort-value="7026602214076000000♠"></span>6.022<span style="margin-left:.25em;">140</span><span style="margin-left:.25em;">76</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>26</sup></span> (which is the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro constant</a> times one thousand). For <a href="/wiki/Diatomic" class="mw-redirect" title="Diatomic">diatomic</a> molecules such as <a href="/wiki/Hydrogen" title="Hydrogen">H<sub>2</sub></a>, <a href="/wiki/Nitrogen" title="Nitrogen">N<sub>2</sub></a>, and <a href="/wiki/Oxygen" title="Oxygen">O<sub>2</sub></a>, multiply atomic weight by two before plugging it into the above formula. The mean <i>speed</i> (not vector-isolated velocity) of an atom or molecule along any arbitrary path is calculated 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 {\tilde {s}}={\tilde {v}}\cdot {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {s}}={\tilde {v}}\cdot {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa3800a231b0d57ed0e0f9b0d0d3b6bcd8321f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.395ex; height:2.843ex;" alt="{\displaystyle {\tilde {s}}={\tilde {v}}\cdot {\sqrt {3}}}"></span> where <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="{\textstyle {\tilde {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tilde {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c3ffc0c541f91ea2f7e31860e17de0fafb1af3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\textstyle {\tilde {s}}}"></span> is the mean speed of translational particle motion in m/s. The mean energy of the translational motions of a substance's constituent particles correlates to their mean <i>speed</i>, not velocity. Thus, substituting <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="{\textstyle {\tilde {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tilde {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c3ffc0c541f91ea2f7e31860e17de0fafb1af3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\textstyle {\tilde {s}}}"></span> for <span class="texhtml mvar" style="font-style:italic;">v</span> in the classic formula for kinetic energy, <span class="texhtml"><i>E</i><sub>k</sub> = <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><i>mv</i><span style="padding-left:0.12em;"><sup>2</sup></span></span> produces precisely the same value as does <span class="texhtml"><i>E</i><sub>mean</sub> = 3/2<i>k</i><sub>B</sub><i>T</i></span> (as shown in <a href="#Nature_of_kinetic_energy,_translational_motion,_and_temperature">§ Nature of kinetic energy, translational motion, and temperature</a>). The Boltzmann constant and its related formulas establish that absolute zero is the point of both zero kinetic energy of particle motion and zero kinetic velocity (see also <i><a href="#Notes">Note 1</a></i> above).</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">One-trillionth of a kelvin is to one kelvin as the thickness of two sheets of kitchen aluminum foil (0.04 mm) is to the distance around Earth at the equator.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">The internal degrees of freedom of molecules cause their external surfaces to vibrate and can also produce overall spinning motions (what can be likened to the jiggling and spinning of an otherwise stationary water balloon). If one examines a <i>single</i> molecule as it impacts a containers' wall, some of the kinetic energy borne in the molecule's internal degrees of freedom can constructively add to its translational motion during the instant of the collision and extra kinetic energy will be transferred into the container's wall. This would induce an extra, localized, impulse-like contribution to the average pressure on the container. However, since the internal motions of molecules are random, they have an equal probability of <i>destructively</i> interfering with translational motion during a collision with a container's walls or another molecule. Averaged across any bulk quantity of a gas, the internal thermal motions of molecules have zero net effect upon the temperature, pressure, or volume of a gas. Molecules' internal degrees of freedom simply provide additional locations where kinetic energy is stored. This is precisely why molecular-based gases have greater specific internal capacity than monatomic gases (where additional internal energy must be added to achieve a given temperature rise).</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">When measured at constant-volume since different amounts of work must be performed if measured at constant-pressure. Nitrogen's <span class="texhtml"><i>C<sub>v</sub>H</i></span> (100 kPa, 20 °C) equals <span class="nowrap"><span data-sort-value="7001208000000000000♠"></span>20.8 J⋅mol<sup>–1</sup>⋅K<sup>–1</sup></span> vs. the monatomic gases, which equal 12.4717 J mol<sup>–1</sup> K<sup>–1</sup>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreeman" class="citation book cs1">Freeman, W. H. "Part 3: Change". <a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20070927061428/http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf"><i>Physical Chemistry</i></a> <span class="cs1-format">(PDF)</span>. Exercise 21.20b, p. 787. Archived from <a rel="nofollow" class="external text" href="http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2007-09-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Part+3%3A+Change&rft.btitle=Physical+Chemistry&rft.pages=Exercise+21.20b%2C+p.-787&rft.aulast=Freeman&rft.aufirst=W.+H.&rft_id=http%3A%2F%2Fwww.whfreeman.com%2Fcollege%2Fpdfs%2Fpchem8e%2FPC8eC21.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> See also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNave" class="citation web cs1">Nave, R. <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html">"Molar Specific Heats of Gases"</a>. <i>HyperPhysics</i>. Georgia State University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=HyperPhysics&rft.atitle=Molar+Specific+Heats+of+Gases&rft.aulast=Nave&rft.aufirst=R.&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Fkinetic%2Fshegas.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">The <i>speed</i> at which thermal energy equalizes throughout the volume of a gas is very rapid. However, since gases have extremely low density relative to solids, the <i>heat <a href="/wiki/Flux" title="Flux">flux</a></i> (the thermal power passing per area) through gases is comparatively low. This is why the dead-air spaces in <a href="/wiki/Insulated_glazing" title="Insulated glazing">multi-pane windows</a> have insulating qualities.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="/wiki/Diamond" title="Diamond">Diamond</a> is a notable exception. Highly quantized modes of phonon vibration occur in its rigid crystal lattice. Therefore, not only does diamond have exceptionally <i>poor</i> <a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">specific heat capacity</a>, it also has exceptionally <i>high</i> <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Correlation is 752 (W⋅m<sup>−1</sup>⋅K<sup>−1</sup>)/(MS⋅cm), <span class="texhtml mvar" style="font-style:italic;">σ</span> = 81, through a 7:1 range in conductivity. Value and standard deviation based on data for Ag, Cu, Au, Al, Ca, Be, Mg, Rh, Ir, Zn, Co, Ni, Os, Fe, Pa, Pt, and Sn. Data from <i>CRC Handbook of Chemistry and Physics</i>, 1st Student Edition.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">The cited emission wavelengths are for true black bodies in equilibrium. In this table, only the sun so qualifies. <a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?bwien">CODATA recommended value</a> of <span class="nowrap"><span data-sort-value="6997289777195500000♠"></span>2.897<span style="margin-left:.25em;">771</span><span style="margin-left:.25em;">955</span>...<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−3</sup> m⋅K</span> used for Wien displacement law constant <i>b</i>.</span> </li> <li id="cite_note-recordcold-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-recordcold_21-0">^</a></b></span> <span class="reference-text">A record cold temperature of 450 ±80 pK in a Bose–Einstein condensate (BEC) of sodium (<sup>23</sup>Na) atoms was achieved in 2003 by researchers at <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeanhardt2003" class="citation journal cs1">Leanhardt, A. E.; et al. (12 September 2003). "Cooling Bose–Einstein Condensates Below 500 Picokelvin". <i>Science</i>. <b>301</b> (5639): 1515. <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/2003Sci...301.1513L">2003Sci...301.1513L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1088827">10.1126/science.1088827</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12970559">12970559</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Cooling+Bose%E2%80%93Einstein+Condensates+Below+500+Picokelvin&rft.volume=301&rft.issue=5639&rft.pages=1515&rft.date=2003-09-12&rft_id=info%3Apmid%2F12970559&rft_id=info%3Adoi%2F10.1126%2Fscience.1088827&rft_id=info%3Abibcode%2F2003Sci...301.1513L&rft.aulast=Leanhardt&rft.aufirst=A.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> The thermal velocity of the atoms averaged about 0.4 mm/s. This record's peak emittance black-body radiation wavelength of 6,400 kilometers is roughly the radius of Earth.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">The peak emittance wavelength of 2.897 77 m is a frequency of 103.456 MHz.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://iau.org/static/resolutions/IAU2015_English.pdf">"Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties"</a> <span class="cs1-format">(PDF)</span>. 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Resolution+B3+on+recommended+nominal+conversion+constants+for+selected+solar+and+planetary+properties&rft.date=2015&rft_id=https%3A%2F%2Fiau.org%2Fstatic%2Fresolutions%2FIAU2015_English.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHertelSchulz2014" class="citation book cs1">Hertel, Ingolf V.; Schulz, Claus-Peter (2014-10-24). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vr0UBQAAQBAJ&dq=5772+K+sun&pg=PA35"><i>Atoms, Molecules and Optical Physics 1: Atoms and Spectroscopy</i></a>. Springer. p. 35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-54322-7" title="Special:BookSources/978-3-642-54322-7"><bdi>978-3-642-54322-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atoms%2C+Molecules+and+Optical+Physics+1%3A+Atoms+and+Spectroscopy&rft.pages=35&rft.pub=Springer&rft.date=2014-10-24&rft.isbn=978-3-642-54322-7&rft.aulast=Hertel&rft.aufirst=Ingolf+V.&rft.au=Schulz%2C+Claus-Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dvr0UBQAAQBAJ%26dq%3D5772%2BK%2Bsun%26pg%3DPA35&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVignolaMichalskyStoffel2019" class="citation book cs1">Vignola, Frank; Michalsky, Joseph; Stoffel, Thomas (2019-07-30). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q9WlDwAAQBAJ&dq=5772+K+sun&pg=PP26"><i>Solar and Infrared Radiation Measurements</i></a> (2nd ed.). CRC Press. pp. chapter 2.1, 2.2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-60020-0" title="Special:BookSources/978-1-351-60020-0"><bdi>978-1-351-60020-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solar+and+Infrared+Radiation+Measurements&rft.pages=chapter+2.1%2C+2.2&rft.edition=2nd&rft.pub=CRC+Press&rft.date=2019-07-30&rft.isbn=978-1-351-60020-0&rft.aulast=Vignola&rft.aufirst=Frank&rft.au=Michalsky%2C+Joseph&rft.au=Stoffel%2C+Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq9WlDwAAQBAJ%26dq%3D5772%2BK%2Bsun%26pg%3DPP26&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html">"Sun Fact Sheet"</a>. <i>NASA Space Science Center Coordinated Archive</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NASA+Space+Science+Center+Coordinated+Archive&rft.atitle=Sun+Fact+Sheet&rft_id=https%3A%2F%2Fnssdc.gsfc.nasa.gov%2Fplanetary%2Ffactsheet%2Fsunfact.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">The 350 MK value is the maximum peak fusion fuel temperature in a thermonuclear weapon of the Teller–Ulam configuration (commonly known as a "hydrogen bomb"). Peak temperatures in Gadget-style fission bomb cores (commonly known as an "atomic bomb") are in the range of 50 to 100 MK. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://nuclearweaponarchive.org/Nwfaq/Nfaq3.html#nfaq3.2">"Nuclear Weapons Frequently Asked Questions"</a>. 3.2.5 Matter At High Temperatures.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Nuclear+Weapons+Frequently+Asked+Questions&rft.pages=3.2.5+Matter+At+High+Temperatures&rft_id=http%3A%2F%2Fnuclearweaponarchive.org%2FNwfaq%2FNfaq3.html%23nfaq3.2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (September 2024)">full citation needed</span></a></i>]</sup> All referenced data was compiled from publicly available sources.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Peak temperature for a bulk quantity of matter was achieved by a pulsed-power machine used in fusion physics experiments. The term "bulk quantity" draws a distinction from collisions in particle accelerators wherein high "temperature" applies only to the debris from two subatomic particles or nuclei at any given instant. The >2 GK temperature was achieved over a period of about ten nanoseconds during "shot Z1137". In fact, the iron and manganese ions in the plasma averaged 3.58 ±0.41 GK (309 ±35 keV) for 3 ns (ns 112 through 115). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaines2006" class="citation journal cs1">Haines, M. G.; et al. (2006). "Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2 × 10<sup>9</sup> Kelvin". <i>Physical Review Letters</i>. <b>96</b> (7): 075003. <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/2006PhRvL..96g5003H">2006PhRvL..96g5003H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.96.075003">10.1103/PhysRevLett.96.075003</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16606100">16606100</a>. No. 075003.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Ion+Viscous+Heating+in+a+Magnetohydrodynamically+Unstable+Z+Pinch+at+Over+2+%C3%97+10%3Csup%3E9%3C%2Fsup%3E+Kelvin&rft.volume=96&rft.issue=7&rft.pages=075003&rft.date=2006&rft_id=info%3Apmid%2F16606100&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.96.075003&rft_id=info%3Abibcode%2F2006PhRvL..96g5003H&rft.aulast=Haines&rft.aufirst=M.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> For a press summary of this article, see <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/20060702185740/http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html">"Sandia's Z machine exceeds two billion degrees Kelvin"</a>. Sandia. March 8, 2006. Archived from <a rel="nofollow" class="external text" href="http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html">the original</a> on 2006-07-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sandia%27s+Z+machine+exceeds+two+billion+degrees+Kelvin&rft.pub=Sandia&rft.date=2006-03-08&rft_id=http%3A%2F%2Fwww.sandia.gov%2Fnews-center%2Fnews-releases%2F2006%2Fphysics-astron%2Fhottest-z-output.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Core temperature of a high–mass (>8–11 solar masses) star after it leaves the main sequence on the <a href="/wiki/Hertzsprung%E2%80%93Russell_diagram" title="Hertzsprung–Russell diagram">Hertzsprung–Russell diagram</a> and begins the <a href="/wiki/Alpha_reactions" class="mw-redirect" title="Alpha reactions">alpha process</a> (which lasts one day) of <a href="/wiki/Silicon_burning_process" class="mw-redirect" title="Silicon burning process">fusing silicon–28</a> into heavier elements in the following steps: sulfur–32 → argon–36 → calcium–40 → titanium–44 → chromium–48 → iron–52 → nickel–56. Within minutes of finishing the sequence, the star explodes as a Type II <a href="/wiki/Supernova" title="Supernova">supernova</a>.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Based on a computer model that predicted a peak internal temperature of 30 MeV (350 GK) during the merger of a binary neutron star system (which produces a gamma–ray burst). The neutron stars in the model were 1.2 and 1.6 solar masses respectively, were roughly 20 km in diameter, and were orbiting around their barycenter (common center of mass) at about 390 Hz during the last several milliseconds before they completely merged. The 350 GK portion was a small volume located at the pair's developing common core and varied from roughly 1 to 7 km across over a time span of around 5 ms. Imagine two city-sized objects of unimaginable density orbiting each other at the same frequency as the G4 musical note (the 28th white key on a piano). At 350 GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass 5% greater than its rest mass. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOechslinJanka2006" class="citation journal cs1">Oechslin, R.; Janka, H.-T. (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-2966.2006.10238.x">"Torus formation in neutron star mergers and well-localized short gamma-ray bursts"</a>. <i>Monthly Notices of the Royal Astronomical Society</i>. <b>368</b> (4): 1489–1499. <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/astro-ph/0507099v2">astro-ph/0507099v2</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/2006MNRAS.368.1489O">2006MNRAS.368.1489O</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/10.1111%2Fj.1365-2966.2006.10238.x">10.1111/j.1365-2966.2006.10238.x</a></span>. <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:15036056">15036056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=Torus+formation+in+neutron+star+mergers+and+well-localized+short+gamma-ray+bursts&rft.volume=368&rft.issue=4&rft.pages=1489-1499&rft.date=2006&rft_id=info%3Aarxiv%2Fastro-ph%2F0507099v2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15036056%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fj.1365-2966.2006.10238.x&rft_id=info%3Abibcode%2F2006MNRAS.368.1489O&rft.aulast=Oechslin&rft.aufirst=R.&rft.au=Janka%2C+H.-T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1365-2966.2006.10238.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> For a summary, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mpa-garching.mpg.de/mpa/research/current_research/hl2005-10/hl2005-10-en.html">"Short Gamma-Ray Bursts: Death Throes of Merging Neutron Stars"</a>. Max-Planck-Institut für Astrophysik<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Short+Gamma-Ray+Bursts%3A+Death+Throes+of+Merging+Neutron+Stars&rft.pub=Max-Planck-Institut+f%C3%BCr+Astrophysik&rft_id=http%3A%2F%2Fwww.mpa-garching.mpg.de%2Fmpa%2Fresearch%2Fcurrent_research%2Fhl2005-10%2Fhl2005-10-en.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBattersby2011" class="citation magazine cs1">Battersby, Stephen (2 March 2011). <a rel="nofollow" class="external text" href="https://www.newscientist.com/article/mg20928026.300-eight-extremes-the-hottest-thing-in-the-universe.html">"Eight extremes: The hottest thing in the universe"</a>. <i>New Scientist</i>. <q>While the details of this process are currently unknown, it must involve a fireball of relativistic particles heated to something in the region of a trillion kelvin.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Scientist&rft.atitle=Eight+extremes%3A+The+hottest+thing+in+the+universe&rft.date=2011-03-02&rft.aulast=Battersby&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fwww.newscientist.com%2Farticle%2Fmg20928026.300-eight-extremes-the-hottest-thing-in-the-universe.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><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/20071011103924/http://public.web.cern.ch/Public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html">"How do physicists study particles?"</a>. CERN. Archived from <a rel="nofollow" class="external text" href="http://public.web.cern.ch/public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html">the original</a> on 2007-10-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+do+physicists+study+particles%3F&rft.pub=CERN&rft_id=http%3A%2F%2Fpublic.web.cern.ch%2Fpublic%2FContent%2FChapters%2FAboutCERN%2FHowStudyPrtcles%2FHowSeePrtcles%2FHowSeePrtcles-en.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Water's enthalpy of fusion (0 °C, 101.325 kPa) equates to <span class="nowrap"><span data-sort-value="6979997899603163080♠"></span>0.062<span style="margin-left:.25em;">284</span> eV</span> per molecule so adding one joule of thermal energy to 0 °C water ice causes <span class="nowrap"><span data-sort-value="7020100210000000000♠"></span>1.0021<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>20</sup></span> water molecules to break away from the crystal lattice and become liquid.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Water's enthalpy of fusion is <span class="nowrap"><span data-sort-value="7000600950000000000♠"></span>6.0095 kJ⋅mol<sup>−1</sup></span> K<sup>−1</sup> (0 °C, 101.325 kPa). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaplin" class="citation web cs1">Chaplin, Martin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201121051504/http://www.lsbu.ac.uk/water/water_properties.html">"Water Properties (including isotopologues)"</a>. <i>Water Structure and Science</i>. London South Bank University. Archived from <a rel="nofollow" class="external text" href="http://www.lsbu.ac.uk/water/data.html">the original</a> on 2020-11-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Water+Structure+and+Science&rft.atitle=Water+Properties+%28including+isotopologues%29&rft.aulast=Chaplin&rft.aufirst=Martin&rft_id=http%3A%2F%2Fwww.lsbu.ac.uk%2Fwater%2Fdata.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> The only metals with enthalpies of fusion <i>not</i> in the range of 6–30 J mol<sup>−1</sup> K<sup>−1</sup> are (on the high side): Ta, W, and Re; and (on the low side) most of the group 1 (alkaline) metals plus Ga, In, Hg, Tl, Pb, and Np.</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">For xenon, available values range from 2.3 to 3.1 kJ/mol. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.webelements.com/webelements/elements/text/Xe/heat.html">"Xenon – 54Xe: the essentials"</a>. <i>WebElements</i><span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=WebElements&rft.atitle=Xenon+%E2%80%93+54Xe%3A+the+essentials&rft_id=http%3A%2F%2Fwww.webelements.com%2Fwebelements%2Felements%2Ftext%2FXe%2Fheat.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> Helium's heat of fusion of only 0.021 kJ/mol is so weak of a bonding force that zero-point energy prevents helium from freezing unless it is under a pressure of at least 25 atmospheres.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><i>CRC Handbook of Chemistry and Physics</i>, 1st Student Edition<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (September 2024)">full citation needed</span></a></i>]</sup></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">H<sub>2</sub>O specific heat capacity, <span class="texhtml"><i>C<sub>p</sub></i></span> = <span class="nowrap"><span data-sort-value="6998753270000000000♠"></span>0.075<span style="margin-left:.25em;">327</span> kJ⋅mol<sup>−1</sup>⋅K<sup>−1</sup></span> (25 °C); enthalpy of fusion = 6.0095 kJ/mol (0 °C, 101.325 kPa); enthalpy of vaporization (liquid) = 40.657 kJ/mol (100 °C). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaplin" class="citation web cs1">Chaplin, Martin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201121051504/http://www.lsbu.ac.uk/water/water_properties.html">"Water Properties (including isotopologues)"</a>. <i>Water Structure and Science</i>. London South Bank University. Archived from <a rel="nofollow" class="external text" href="http://www.lsbu.ac.uk/water/data.html">the original</a> on 2020-11-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Water+Structure+and+Science&rft.atitle=Water+Properties+%28including+isotopologues%29&rft.aulast=Chaplin&rft.aufirst=Martin&rft_id=http%3A%2F%2Fwww.lsbu.ac.uk%2Fwater%2Fdata.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">Mobile conduction electrons are <i>delocalized</i>, i.e. not tied to a specific atom, and behave rather like a sort of quantum gas due to the effects of zero-point energy. Consequently, even at absolute zero, conduction electrons still move between atoms at the <i>Fermi velocity</i> of about <span class="nowrap"><span data-sort-value="7006160000000000000♠"></span>1.6<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>6</sup> m/s</span>. Kinetic thermal energy adds to this speed and also causes delocalized electrons to travel farther away from the nuclei.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">No other <a href="/wiki/Crystal_structure" title="Crystal structure">crystal structure</a> can exceed the 74.048% packing density of a <i>closest-packed arrangement</i>. The two regular crystal lattices found in nature that have this density are <i><a href="/wiki/Hexagonal_crystal_system" class="mw-redirect" title="Hexagonal crystal system">hexagonal close packed</a></i> (HCP) and <i><a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">face-centered cubic</a></i> (FCC). These regular lattices are at the lowest possible energy state. <a href="/wiki/Diamond" title="Diamond">Diamond</a> is a closest-packed structure with an FCC crystal lattice. Note too that suitable crystalline chemical <i>compounds</i>, although usually composed of atoms of different sizes, can be considered as closest-packed structures when considered at the molecular level. One such compound is the common <a href="/wiki/Mineral" title="Mineral">mineral</a> known as <i>magnesium aluminum <a href="/wiki/Spinel" title="Spinel">spinel</a></i> (MgAl<sub>2</sub>O<sub>4</sub>). It has a face-centered cubic crystal lattice and no change in pressure can produce a lattice with a lower energy state.</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">Nearly half of the 92 naturally occurring chemical elements that can freeze under a vacuum also have a closest-packed crystal lattice. This set includes <a href="/wiki/Beryllium" title="Beryllium">beryllium</a>, <a href="/wiki/Osmium" title="Osmium">osmium</a>, <a href="/wiki/Neon" title="Neon">neon</a>, and <a href="/wiki/Iridium" title="Iridium">iridium</a> (but excludes helium), and therefore have zero latent heat of phase transitions to contribute to internal energy (symbol: <i>U</i>). In the calculation of enthalpy (formula: <span class="texhtml">{{{1}}}</span>), internal energy may exclude different sources of thermal energy (particularly ZPE) depending on the nature of the analysis. Accordingly, all <span class="texhtml mvar" style="font-style:italic;">T</span> = 0 closest-packed matter under a perfect vacuum has either minimal or zero enthalpy, depending on the nature of the analysis. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlberty2001" class="citation journal cs1">Alberty, Robert A. (2001). <a rel="nofollow" class="external text" href="http://iupac.org/publications/pac/2001/pdf/7308x1349.pdf">"Use of Legendre Transforms In Chemical Thermodynamics"</a> <span class="cs1-format">(PDF)</span>. <i>Pure and Applied Chemistry</i>. <b>73</b> (8): 1349. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fpac200173081349">10.1351/pac200173081349</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pure+and+Applied+Chemistry&rft.atitle=Use+of+Legendre+Transforms+In+Chemical+Thermodynamics&rft.volume=73&rft.issue=8&rft.pages=1349&rft.date=2001&rft_id=info%3Adoi%2F10.1351%2Fpac200173081349&rft.aulast=Alberty&rft.aufirst=Robert+A.&rft_id=http%3A%2F%2Fiupac.org%2Fpublications%2Fpac%2F2001%2Fpdf%2F7308x1349.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Regarding the spelling "gage" vs. "gauge" in the context of pressures measured relative to atmospheric pressure, the preferred spelling varies by country and even by industry. Further, both spellings are often used <i>within</i> a particular industry or country. Industries in British English-speaking countries typically use the spelling "gauge pressure" to distinguish it from the pressure-measuring instrument, which in the U.K., is spelled <i>pressure gage</i>. For the same reason, many of the largest American manufacturers of pressure transducers and instrumentation use the spelling <i>gage pressure</i> (the convention used here) in their formal documentation to distinguish it from the instrument, which is spelled <i>pressure gauge</i>.</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">Pressure also must be in absolute terms. The air still in a tire at a <a href="/wiki/Pressure_measurement#Gauge" title="Pressure measurement">gage pressure</a> of 0 kPa expands too as it gets hotter. It is not uncommon for engineers to overlook that one must work in terms of absolute pressure when compensating for temperature. For instance, a dominant manufacturer of aircraft tires published a document on temperature-compensating tire pressure, which used gage pressure in the formula. However, the high gage pressures involved (180 psi; 12.4 bar; 1.24 MPa) means the error would be quite small. With low-pressure automobile tires, where gage pressures are typically around 2 bar (200 kPa), failing to adjust to absolute pressure results in a significant error. <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/20100215151931/http://airmichelin.com/pdfs/05%20-%20Aircraft%20Tire%20Ratings.pdf">"Aircraft tire ratings"</a> <span class="cs1-format">(PDF)</span>. Air Michelin. Archived from <a rel="nofollow" class="external text" href="http://airmichelin.com/pdfs/05%20-%20Aircraft%20Tire%20Ratings.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-02-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Aircraft+tire+ratings&rft.pub=Air+Michelin&rft_id=http%3A%2F%2Fairmichelin.com%2Fpdfs%2F05%2520-%2520Aircraft%2520Tire%2520Ratings.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="Source lacks metadata (September 2024)">better source needed</span></a></i>]</sup></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text">A difference of 100 kPa is used here instead of the 101.325 kPa value of one <a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">standard atmosphere</a>. In 1982, the <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">International Union of Pure and Applied Chemistry</a> (IUPAC) recommended that for the purposes of specifying the physical properties of substances, <i>the standard pressure</i> (atmospheric pressure) should be defined as precisely 100 kPa (≈ 750.062 Torr). Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100 kPa equates to the mean pressure at an altitude of about 112 meters, which is closer to the 194–meter, worldwide median altitude of human habitation. For especially low-pressure or high-accuracy work, true atmospheric pressure must be measured. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">"Standard pressure". <a rel="nofollow" class="external text" href="http://goldbook.iupac.org/S05921.html"><i>Compendium of Chemical Terminology</i></a> (online 3rd ed.). International Union of Pure and Applied Chemistry. 2014. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fgoldbook.S05921">10.1351/goldbook.S05921</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Standard+pressure&rft.btitle=Compendium+of+Chemical+Terminology&rft.edition=online+3rd&rft.pub=International+Union+of+Pure+and+Applied+Chemistry&rft.date=2014&rft_id=info%3Adoi%2F10.1351%2Fgoldbook.S05921&rft_id=http%3A%2F%2Fgoldbook.iupac.org%2FS05921.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-PlanckBook-48"><span class="mw-cite-backlink">^ <a href="#cite_ref-PlanckBook_48-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-PlanckBook_48-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlanck1945" class="citation book cs1"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1945). <i>Treatise on Thermodynamics</i>. Dover Publications. §§90, 137, eqs. (39), (40), and (65).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Treatise+on+Thermodynamics&rft.pages=%C2%A7%C2%A790%2C+137%2C+eqs.+%2839%29%2C+%2840%29%2C+and+%2865%29&rft.pub=Dover+Publications&rft.date=1945&rft.aulast=Planck&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Here, need to add a reason of requiring the function <span class="texhtml"><i>g</i>(<i>T</i>)</span> to be a monotonic function. The Carnot efficiency (efficiency of all reversible engines) may be a reason.</span> </li> <li id="cite_note-FermiBook-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-FermiBook_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFermi1956" class="citation book cs1">Fermi, E. (1956). <i>Thermodynamics</i>. Dover Publications. p. 48. <q>eq.(64)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics&rft.pages=48&rft.pub=Dover+Publications&rft.date=1956&rft.aulast=Fermi&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.chemicalsuppliers.com/brief-history-of-temperature-measurement/">"A Brief History of Temperature Measurement"</a>. <i>Chemical Suppliers</i>. 2024-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-09-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Chemical+Suppliers&rft.atitle=A+Brief+History+of+Temperature+Measurement&rft.date=2024-03-01&rft_id=https%3A%2F%2Fwww.chemicalsuppliers.com%2Fbrief-history-of-temperature-measurement%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><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/20121030164902/www.linnaeus.uu.se/online/life/6_32.html">"Linnaeus Thermometer"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.linnaeus.uu.se/online/life/6_32.html">the original</a> on 2012-10-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Linnaeus+Thermometer&rft_id=http%3A%2F%2Fwww.linnaeus.uu.se%2Fonline%2Flife%2F6_32.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><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/20161014011349/www.bipm.org/en/committees/cipm/cipm-1948.html">"Adoption of "degree Celsius"<span class="cs1-kern-right"></span>"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.bipm.org/en/committees/cipm/cipm-1948.html">the original</a> on 2016-10-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Adoption+of+%22degree+Celsius%22&rft_id=http%3A%2F%2Fwww.bipm.org%2Fen%2Fcommittees%2Fcipm%2Fcipm-1948.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLambert1779" class="citation book cs1">Lambert, Johann Heinrich (1779). <a rel="nofollow" class="external text" href="http://www.spiess-verlage.de/html/haude___spener.html"><i>Pyrometrie</i></a>. Berlin: Haude & Spener.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pyrometrie&rft.place=Berlin&rft.pub=Haude+%26+Spener&rft.date=1779&rft.aulast=Lambert&rft.aufirst=Johann+Heinrich&rft_id=http%3A%2F%2Fwww.spiess-verlage.de%2Fhtml%2Fhaude___spener.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomson1848" class="citation magazine cs1">Thomson, William (October 1848). <a rel="nofollow" class="external text" href="http://zapatopi.net/kelvin/papers/on_an_absolute_thermometric_scale.html">"On an Absolute Thermometric Scale"</a>. <i>Philosophical Magazine</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+an+Absolute+Thermometric+Scale&rft.date=1848-10&rft.aulast=Thomson&rft.aufirst=William&rft_id=http%3A%2F%2Fzapatopi.net%2Fkelvin%2Fpapers%2Fon_an_absolute_thermometric_scale.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span> Also published in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomson1882" class="citation book cs1">Thomson, William (1882). <i>Mathematical and Physical Papers</i>. Vol. 1. Cambridge University Press. pp. 100–106.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+and+Physical+Papers&rft.pages=100-106&rft.pub=Cambridge+University+Press&rft.date=1882&rft.aulast=Thomson&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemons2020" class="citation book cs1">Lemons, Don S. (2020). "Chapter 4: Absolute Temperature". <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1143850952"><i>Thermodynamic weirdness: from Fahrenheit to Clausius</i></a> (First MIT Press Paperback ed.). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-53894-7" title="Special:BookSources/978-0-262-53894-7"><bdi>978-0-262-53894-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1143850952">1143850952</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+4%3A+Absolute+Temperature&rft.btitle=Thermodynamic+weirdness%3A+from+Fahrenheit+to+Clausius&rft.edition=First+MIT+Press+Paperback&rft.date=2020&rft_id=info%3Aoclcnum%2F1143850952&rft.isbn=978-0-262-53894-7&rft.aulast=Lemons&rft.aufirst=Don+S.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1143850952&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemons2020" class="citation book cs1">Lemons, Don S. (2020). "Chapter 8: Absolute Temperature—Again". <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1143850952"><i>Thermodynamic weirdness : from Fahrenheit to Clausius</i></a> (1st paperback ed.). Cambridge, Massachusetts: MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-53894-7" title="Special:BookSources/978-0-262-53894-7"><bdi>978-0-262-53894-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1143850952">1143850952</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+8%3A+Absolute+Temperature%E2%80%94Again&rft.btitle=Thermodynamic+weirdness+%3A+from+Fahrenheit+to+Clausius&rft.place=Cambridge%2C+Massachusetts&rft.edition=1st+paperback&rft.pub=MIT+Press&rft.date=2020&rft_id=info%3Aoclcnum%2F1143850952&rft.isbn=978-0-262-53894-7&rft.aulast=Lemons&rft.aufirst=Don+S.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1143850952&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/9/3/">"Welcome"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Welcome&rft_id=http%3A%2F%2Fwww.bipm.fr%2Fen%2FCGPM%2Fdb%2F9%2F3%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><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/20150213182030/http://www.bipm.org/en/committees/cipm/cipm-1948.html">"BIPM - 1948"</a>. <i>www.bipm.org</i>. Archived from <a rel="nofollow" class="external text" href="http://www.bipm.org/en/committees/cipm/cipm-1948.html">the original</a> on 2015-02-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.bipm.org&rft.atitle=BIPM+-+1948&rft_id=http%3A%2F%2Fwww.bipm.org%2Fen%2Fcommittees%2Fcipm%2Fcipm-1948.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.imeko.org/publications/tc12-2004/IMEKO-TC12-2004-PL-001.pdf">"Temperature Scales from the early days of thermometry to the 21st century"</a> <span class="cs1-format">(PDF)</span>. <i>www.imeko.org</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.imeko.org&rft.atitle=Temperature+Scales+from+the+early+days+of+thermometry+to+the+21st+century&rft_id=https%3A%2F%2Fwww.imeko.org%2Fpublications%2Ftc12-2004%2FIMEKO-TC12-2004-PL-001.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuinn1990" class="citation book cs1">Quinn, Terry (1990). <i>Temperature</i> (2nd ed.). Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0125696817" title="Special:BookSources/0125696817"><bdi>0125696817</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Temperature&rft.edition=2nd&rft.pub=Academic+Press&rft.date=1990&rft.isbn=0125696817&rft.aulast=Quinn&rft.aufirst=Terry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-°CName-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-°CName_62-0">^</a></b></span> <span class="reference-text">According to <i>The Oxford English Dictionary</i> (OED), the term "Celsius's thermometer" had been used at least as early as 1797. Further, the term "The Celsius or Centigrade thermometer" was again used in reference to a particular type of thermometer at least as early as 1850. The OED also cites this 1928 reporting of a temperature: "My altitude was about 5,800 metres, the temperature was 28° Celsius". However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards the terminology used throughout the history of science. According to several writings of Terry Quinn CBE FRS, Director of the BIPM (1988–2004), including <i>Temperature Scales from the early days of thermometry to the 21st century</i><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> as well as <i>Temperature</i>,<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> the term <i>Celsius</i> in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM was not even aware that <i>degree Celsius</i> was in sporadic, non-scientific use before that time. The twelve-volume, 1933 edition of the OED did not even have a listing for the word <i>Celsius</i> (but did have listings for both <i>centigrade</i> and <i>centesimal</i> in the context of temperature measurement). The 1948 adoption of <i>Celsius</i> accomplished three objectives: <ol><li>All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.</li> <li>Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius's name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name <i>centigrade</i> could continue to be used and would simultaneously inherit an intuitive association with the new name.</li> <li>The new name eliminated the ambiguity of the term <i>centigrade</i>, freeing it to refer exclusively to the French-language name for the unit of angular measurement.</li></ol> </span></li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/10/3/">"Welcome"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Welcome&rft_id=http%3A%2F%2Fwww.bipm.fr%2Fen%2FCGPM%2Fdb%2F10%2F3%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/13/3/">"Welcome"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Welcome&rft_id=http%3A%2F%2Fwww.bipm.fr%2Fen%2FCGPM%2Fdb%2F13%2F3%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/13/4/">"Welcome"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Welcome&rft_id=http%3A%2F%2Fwww.bipm.fr%2Fen%2FCGPM%2Fdb%2F13%2F4%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" class="Z3988"></span></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.bipm.fr/en/si/si_brochure/chapter2/2-1/kelvin.html">"Welcome"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Welcome&rft_id=http%3A%2F%2Fwww.bipm.fr%2Fen%2Fsi%2Fsi_brochure%2Fchapter2%2F2-1%2Fkelvin.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermodynamic+temperature" 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=Thermodynamic_temperature&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a rel="nofollow" class="external text" href="http://www.calphysics.org/zpe.html">Zero Point Energy and Zero Point Field.</a></i> A Web site with in-depth explanations of a variety of quantum effects. 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title="Template:SI base quantities"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:SI_base_quantities" title="Template talk:SI base quantities"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:SI_base_quantities" title="Special:EditPage/Template:SI base quantities"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="SI_base_quantities" style="font-size:114%;margin:0 4em"><a href="/wiki/SI_base_unit" title="SI base unit">SI base quantities</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Base quantities</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <table class="wikitable sortable" style="text-align:center; margin:0;"> <tbody><tr> <th colspan="3">Quantity </th> <th> </th> <th colspan="2"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> </th></tr> <tr> <th style="text-align:center;">Name </th> <th style="text-align:center;">Symbol </th> <th style="text-align:center;"><a href="/wiki/Dimensional_analysis" title="Dimensional analysis">Dimension<br />symbol</a> </th> <th class="unsortable"> </th> <th style="text-align:center;">Unit<br />name </th> <th style="text-align:center;">Unit<br />symbol </th></tr> <tr> <td style="text-align:left;"><a href="/wiki/Time" title="Time">time, duration</a> </td> <td><span class="texhtml mvar" style="font-style:italic;">t</span> </td> <td style="font-family:sans-serif;font-style:normal;">T </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Second" title="Second">second</a> </td> <td style="text-align:left;">s </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Length" title="Length">length</a> </td> <td><span class="texhtml mvar" style="font-style:italic;">l</span>, <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">r</span>, etc. </td> <td style="font-family:sans-serif;font-style:normal;">L </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Metre" title="Metre">metre</a> </td> <td style="text-align:left;">m </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Mass" title="Mass">mass</a> </td> <td><span class="texhtml mvar" style="font-style:italic;">m</span> </td> <td style="font-family:sans-serif;font-style:normal;">M </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Kilogram" title="Kilogram">kilogram</a> </td> <td style="text-align:left;">kg </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Electric_current" title="Electric current">electric current</a> </td> <td><abbr title="Uppercase italic letter 'i'"> <span class="texhtml mvar" style="font-style:italic;">I</span> </abbr>, <span class="texhtml mvar" style="font-style:italic;">i</span> </td> <td style="font-family:sans-serif;font-style:normal;"><abbr title="Uppercase sans-serif roman letter 'i'"> I </abbr> </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Ampere" title="Ampere">ampere</a> </td> <td style="text-align:left;">A </td></tr> <tr> <td style="text-align:left;"><a class="mw-selflink selflink">thermodynamic temperature</a> </td> <td><span class="texhtml mvar" style="font-style:italic;">T</span> </td> <td style="font-family:sans-serif;font-style:normal;"><abbr title="Greek capital letter theta">Θ</abbr> </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Kelvin" title="Kelvin">kelvin</a> </td> <td style="text-align:left;">K </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> </td> <td><span class="texhtml mvar" style="font-style:italic;">n</span> </td> <td style="font-family:sans-serif;font-style:normal;">N </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> </td> <td style="text-align:left;">mol </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Luminous_intensity" title="Luminous intensity">luminous intensity</a> </td> <td><span class="texhtml"><i>I</i><sub>v</sub></span> </td> <td style="font-family:sans-serif;font-style:normal;">J </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Candela" title="Candela">candela</a> </td> <td style="text-align:left;">cd </td></tr></tbody></table> </div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Unit_relations_in_the_new_SI.svg/200px-Unit_relations_in_the_new_SI.svg.png" decoding="async" width="200" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Unit_relations_in_the_new_SI.svg/300px-Unit_relations_in_the_new_SI.svg.png 1.5x, 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