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<TITLE>Thermodynamic temperature - Wikipedia</TITLE>

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<H1 id="firstHeading" class="firstHeading" lang="en">Thermodynamic temperature</H1>			<DIV id="bodyContent" class="mw-body-content">
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<TH style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em;padding-bottom:0.3em;border-bottom:1px solid #aaa;"><A href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</A></TH>
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<TD style="padding:0.2em 0 0.4em;display:block;margin:0.3em 0 0.4em;"><A href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><IMG alt="Carnot heat engine 2.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" width="220" height="97" 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>
<DIV style="padding-top:0.2em;line-height:1.2em">The classical <A href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</A></DIV>
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<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>
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<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>
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<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>
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<TH style="padding:0.1em;background:#eaeaff;font-style:italic;"><A href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</A></TH>
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<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/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>
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<TH style="padding:0.1em;background:#eaeaff;font-style:italic;"><A href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</A></TH>
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<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" 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>
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<TH style="padding:0.1em;background:#eaeaff;font-style:italic;"><A href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</A></TH>
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<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>
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<DIV class="NavHead" style="font-size:105%;background:transparent;text-align:left;background:#ddf;text-align:center;"><A href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</A><A class="NavToggle" id="NavToggle4" href="#">[show]</A></DIV>
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<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>
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<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>
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<TH style="padding:0.1em;background:#eaeaff;font-style:italic;"><A href="/wiki/Process_function" title="Process function">Process functions</A></TH>
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<LI><A href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</A></LI>
<LI><A href="/wiki/Heat" title="Heat">Heat</A></LI>
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<TH style="padding:0.1em;background:#eaeaff;font-style:italic;"><A href="/wiki/State_function" title="State function">Functions of state</A></TH>
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<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>
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<TD style="vertical-align:middle; text-align:right"><A href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</A> </TD>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="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}">
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    <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="\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>
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        <mi>N</mi>
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  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="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}">
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    <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" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="\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 =-}">
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    <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" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="\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}">
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  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="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}">
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="\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}">
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        <mi>V</mi>
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    <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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="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}">
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        <mi mathvariant="normal">∂<!-- ∂ --></mi>
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    <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" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="\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 =}">
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        <mi>α<!-- α --></mi>
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    <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" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="\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}">
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        <mn>1</mn>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="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}">
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    <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="\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>
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        <mi>V</mi>
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    <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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="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}">
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        <mi mathvariant="normal">∂<!-- ∂ --></mi>
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    <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" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="\partial T" /></SPAN></TD>
</TR>
</TBODY></TABLE>
</TD>
</TR>
</TBODY></TABLE>
</DIV>
</DIV>
</DIV>
</TD>
</TR>
<TR>
<TD style="padding:0 0.1em 0.4em">
<DIV class="NavFrame collapsed" style="border:none;padding:0" id="NavFrame6">
<DIV class="NavHead" style="font-size:105%;background:transparent;text-align:left;background:#ddf;text-align:center;"><A href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</A><A class="NavToggle" id="NavToggle6" href="#">[show]</A></DIV>
<DIV class="NavContent" style="font-size: 105%; padding: 0.2em 0px 0.4em; text-align: center; display: none;">
<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 style="padding:0 0.1em 0.4em">
<DIV class="NavFrame collapsed" style="border:none;padding:0" id="NavFrame7">
<DIV class="NavHead" style="font-size:105%;background:transparent;text-align:left;background:#ddf;text-align:center;"><A href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</A><A class="NavToggle" id="NavToggle7" href="#">[show]</A></DIV>
<DIV class="NavContent" style="font-size: 105%; padding: 0.2em 0px 0.4em; text-align: center; display: none;">
<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>
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        <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" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="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" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="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>
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        <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" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="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" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="G(T,p)=H-TS" /></SPAN></LI>
</UL>
</DIV>
</DIV>
</DIV>
</TD>
</TR>
<TR>
<TD style="padding:0 0.1em 0.4em">
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<LI>History</LI>
<LI>Culture</LI>
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<A class="NavToggle" id="NavToggle8" href="#">[show]</A></DIV>
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<TABLE class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none">
<TBODY><TR>
<TH style="padding:0.1em;background:#eaeaff;font-style:italic;">History</TH>
</TR>
<TR>
<TD style="padding:0 0.1em 0.4em">
<DIV class="hlist">
<UL>
<LI><A href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</A></LI>
<LI><A href="/wiki/History_of_heat" title="History of heat">Heat</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 style="padding:0.1em;background:#eaeaff;font-style:italic;"><A href="/wiki/Philosophy_of_thermal_and_statistical_physics" title="Philosophy of thermal and statistical physics">Philosophy</A></TH>
</TR>
<TR>
<TD style="padding:0 0.1em 0.4em">
<DIV class="hlist">
<UL>
<LI><A href="/wiki/Entropy_(arrow_of_time)" 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 style="padding:0.1em;background:#eaeaff;font-style:italic;">Theories</TH>
</TR>
<TR>
<TD style="padding:0 0.1em 0.4em">
<DIV class="hlist">
<UL>
<LI><A href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</A></LI>
<LI><A href="/wiki/Theory_of_heat" title="Theory of heat">Theory of heat</A></LI>
</UL>
</DIV>
<UL>
<LI><A href="/wiki/Vis_viva" title="Vis viva"><I>Vis viva</I> <SPAN style="font-size:90%;">("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/Motive_power" title="Motive power">Motive power</A></LI>
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<DIV style="padding:0.2em 0.4em; line-height:1.2em;">"<A href="/wiki/An_Experimental_Enquiry_Concerning_the_Source_of_the_Heat_which_is_Excited_by_Friction" title="An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction">An Experimental Enquiry<BR />
Concerning ... Heat</A>"</DIV>
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<DIV style="padding:0.2em 0.4em; line-height:1.2em;">"<A href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<BR />
Heterogeneous Substances</A>"</DIV>
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<LI>
<DIV style="padding:0.2em 0.4em; line-height:1.2em;">"<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>"</DIV>
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<LI><A href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</A></LI>
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<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>
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<P><B>Thermodynamic temperature</B> is the absolute measure of <A href="/wiki/Temperature" title="Temperature">temperature</A> and is one of the principal parameters of <A href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</A>.</P>
<P>Thermodynamic temperature is defined by the <A href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">third law of thermodynamics</A> in which the theoretically lowest temperature is the null or zero point. At this point, <A href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</A>, the particle constituents of <A href="/wiki/Matter" title="Matter">matter</A> have minimal motion and can become no colder.<SUP id="cite_ref-1" class="reference"><A href="#cite_note-1">[1]</A></SUP><SUP id="cite_ref-2" class="reference"><A href="#cite_note-2">[2]</A></SUP> In the <A href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum-mechanical</A> description, matter at absolute zero is in its <A href="/wiki/Ground_state" title="Ground state">ground state</A>, which is its state of lowest <A href="/wiki/Energy" title="Energy">energy</A>. Thermodynamic temperature is often also called <B>absolute temperature</B>, for two reasons: one, proposed by Kelvin, that it does not depend on the properties of a particular material; two that it refers to an absolute zero according to the properties of the ideal gas.</P>
<P>The <A href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</A> specifies a particular scale for thermodynamic temperature. It uses the <A href="/wiki/Kelvin" title="Kelvin">kelvin</A> scale for measurement and selects the <A href="/wiki/Triple_point" title="Triple point">triple point</A> of water at 273.16 K as the fundamental fixing point. Other scales have been in use historically. The <A href="/wiki/Rankine_scale" title="Rankine scale">Rankine scale</A>, using the degree <A href="/wiki/Fahrenheit" title="Fahrenheit">Fahrenheit</A> as its unit interval, is still in use as part of the <A href="/wiki/English_Engineering_Units" class="mw-redirect" title="English Engineering Units">English Engineering Units</A> in the United States in some engineering fields. <A href="/wiki/ITS-90" class="mw-redirect" title="ITS-90">ITS-90</A> gives a practical means of estimating the thermodynamic temperature to a very high degree of accuracy.</P>
<P>Roughly, the temperature of a body at rest is a measure of the mean of the energy of the translational, vibrational and rotational motions of <A href="/wiki/Matter" title="Matter">matter</A>'s particle constituents, such as <A href="/wiki/Molecule" title="Molecule">molecules</A>, <A href="/wiki/Atom" title="Atom">atoms</A>, and <A href="/wiki/Subatomic_particle" title="Subatomic particle">subatomic particles</A>. The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, make up the total <A href="/wiki/Internal_energy" title="Internal energy">internal energy</A> of a substance. Internal energy is loosely called the <A href="/wiki/Heat" title="Heat">heat</A> energy or <A href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</A> in conditions when no <A href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</A> is done upon the substance by its surroundings, or by the substance upon the surroundings. Internal energy may be stored in a number of ways within a substance, each way constituting a "degree of freedom". At equilibrium, each degree of freedom will have on average the same energy: <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 k_{B}T/2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>k</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>B</mi>
          </mrow>
        </msub>
        <mi>T</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mo>/</mo>
        </mrow>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle k_{B}T/2}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8512b8d97f8f4061b3cb2cd612f033d0c7079b65" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.652ex; height:2.843ex;" alt="k_{B}T/2" /></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="{\displaystyle k_{B}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>k</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>B</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle k_{B}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.509ex;" alt="k_{B}" /></SPAN> is the <A href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</A>, unless that degree of freedom is in the quantum regime. The internal degrees of freedom (rotation, vibration, etc.) may be in the quantum regime at room temperature, but the translational degrees of freedom will be in the classical regime except at extremely low temperatures (fractions of kelvins) and it may be said that, for most situations, the thermodynamic temperature is specified by the average translational kinetic energy of the particles.</P>
<P></P>
<DIV id="toc" class="toc">
<DIV class="toctitle" lang="en" dir="ltr">
<H2>Contents</H2>
<SPAN class="toctoggle"> [<A role="button" tabindex="0" class="togglelink">hide</A>] </SPAN></DIV>
<UL>
<LI class="toclevel-1 tocsection-1"><A href="#Overview"><SPAN class="tocnumber">1</SPAN> <SPAN class="toctext">Overview</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-2"><A href="#Practical_realization"><SPAN class="tocnumber">1.1</SPAN> <SPAN class="toctext">Practical realization</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-3"><A href="#The_relationship_of_temperature,_motions,_conduction,_and_thermal_energy"><SPAN class="tocnumber">2</SPAN> <SPAN class="toctext">The relationship of temperature, motions, conduction, and thermal energy</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-4"><A href="#The_nature_of_kinetic_energy,_translational_motion,_and_temperature"><SPAN class="tocnumber">2.1</SPAN> <SPAN class="toctext">The nature of kinetic energy, translational motion, and temperature</SPAN></A></LI>
<LI class="toclevel-2 tocsection-5"><A href="#The_high_speeds_of_translational_motion"><SPAN class="tocnumber">2.2</SPAN> <SPAN class="toctext">The high speeds of translational motion</SPAN></A></LI>
<LI class="toclevel-2 tocsection-6"><A href="#The_internal_motions_of_molecules_and_specific_heat"><SPAN class="tocnumber">2.3</SPAN> <SPAN class="toctext">The internal motions of molecules and specific heat</SPAN></A></LI>
<LI class="toclevel-2 tocsection-7"><A href="#The_diffusion_of_thermal_energy:_Entropy,_phonons,_and_mobile_conduction_electrons"><SPAN class="tocnumber">2.4</SPAN> <SPAN class="toctext">The diffusion of thermal energy: Entropy, phonons, and mobile conduction electrons</SPAN></A></LI>
<LI class="toclevel-2 tocsection-8"><A href="#The_diffusion_of_thermal_energy:_Black-body_radiation"><SPAN class="tocnumber">2.5</SPAN> <SPAN class="toctext">The diffusion of thermal energy: Black-body radiation</SPAN></A>
<UL>
<LI class="toclevel-3 tocsection-9"><A href="#Table_of_thermodynamic_temperatures"><SPAN class="tocnumber">2.5.1</SPAN> <SPAN class="toctext">Table of thermodynamic temperatures</SPAN></A></LI>
<LI class="toclevel-3 tocsection-10"><A href="#The_heat_of_phase_changes"><SPAN class="tocnumber">2.5.2</SPAN> <SPAN class="toctext">The heat of phase changes</SPAN></A></LI>
<LI class="toclevel-3 tocsection-11"><A href="#Internal_energy"><SPAN class="tocnumber">2.5.3</SPAN> <SPAN class="toctext">Internal energy</SPAN></A></LI>
<LI class="toclevel-3 tocsection-12"><A href="#Internal_energy_at_absolute_zero"><SPAN class="tocnumber">2.5.4</SPAN> <SPAN class="toctext">Internal energy at absolute zero</SPAN></A></LI>
</UL>
</LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-13"><A href="#Practical_applications_for_thermodynamic_temperature"><SPAN class="tocnumber">3</SPAN> <SPAN class="toctext">Practical applications for thermodynamic temperature</SPAN></A></LI>
<LI class="toclevel-1 tocsection-14"><A href="#Definition_of_thermodynamic_temperature"><SPAN class="tocnumber">4</SPAN> <SPAN class="toctext">Definition of thermodynamic temperature</SPAN></A></LI>
<LI class="toclevel-1 tocsection-15"><A href="#History"><SPAN class="tocnumber">5</SPAN> <SPAN class="toctext">History</SPAN></A></LI>
<LI class="toclevel-1 tocsection-16"><A href="#See_also"><SPAN class="tocnumber">6</SPAN> <SPAN class="toctext">See also</SPAN></A></LI>
<LI class="toclevel-1 tocsection-17"><A href="#Notes"><SPAN class="tocnumber">7</SPAN> <SPAN class="toctext">Notes</SPAN></A></LI>
<LI class="toclevel-1 tocsection-18"><A href="#External_links"><SPAN class="tocnumber">8</SPAN> <SPAN class="toctext">External links</SPAN></A></LI>
</UL>
</DIV>
<P></P>
<H2><SPAN class="mw-headline" id="Overview">Overview</SPAN><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">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Temperature is a measure of the random submicroscopic motions and vibrations of the particle constituents of <A href="/wiki/Matter" title="Matter">matter</A>. These motions comprise the <A href="/wiki/Internal_energy" title="Internal energy">internal energy</A> of a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy per classical (i.e., non-quantum) degree of freedom of its constituent particles. "Translational motions" are almost always in the classical regime. Translational motions are ordinary, whole-body movements in <A href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</A> in which particles move about and exchange energy in collisions. <I><A href="#The_relationship_of_temperature.2C_motions.2C_conduction.2C_and_thermal_energy">Figure 1</A></I> below shows translational motion in gases; <I><A href="#The_diffusion_of_thermal_energy:_Entropy,_phonons,_and_mobile_conduction_electrons">Figure 4</A></I> below shows translational motion in solids. Thermodynamic temperature's <A href="/wiki/Null_(mathematics)" title="Null (mathematics)">null</A> point, absolute zero, is the temperature at which the particle constituents of matter are as close as possible to complete rest; that is, they have <A href="https://en.wiktionary.org/wiki/minimal#Adjective" class="extiw" title="wikt:minimal">minimal</A> motion, retaining only <A href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</A> motion.<SUP id="cite_ref-T0_3-0" class="reference"><A href="#cite_note-T0-3">[3]</A></SUP> Zero kinetic energy remains in a substance at absolute zero (see <I><A href="/wiki/Thermodynamic_temperature#Thermal_energy_at_absolute_zero" title="Thermodynamic temperature">Thermal energy at absolute zero</A></I>, below).</P>
<P>Throughout the scientific world where measurements are made in <A href="/wiki/International_System_of_Units" title="International System of Units">SI</A> units, thermodynamic temperature is measured in <A href="/wiki/Kelvin" title="Kelvin">kelvins</A> (symbol: K). Many engineering fields in the U.S. however, measure thermodynamic temperature using the <A href="/wiki/Rankine_scale" title="Rankine scale">Rankine scale</A>.</P>
<P>By <A rel="nofollow" class="external text" href="http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html">international agreement</A>, the unit <I>kelvin</I> and its scale are defined by two points: absolute zero, and the <A href="/wiki/Triple_point" title="Triple point">triple point</A> of <A href="/wiki/Vienna_Standard_Mean_Ocean_Water" title="Vienna Standard Mean Ocean Water">Vienna Standard Mean Ocean Water</A> (water with a specified blend of hydrogen and oxygen isotopes). Absolute zero, the lowest possible temperature, is defined as being precisely 0 K <I>and</I> −273.15 <A href="/wiki/Celsius" title="Celsius">°C</A>. The <A href="/wiki/Triple_point" title="Triple point">triple point</A> of water is defined as being precisely 273.16 K <I>and</I> 0.01 °C. This definition does three things:</P>
<OL>
<LI>It fixes the magnitude of the kelvin unit as being precisely 1 part in 273.16 parts the difference between absolute zero and the triple point of water;</LI>
<LI>It establishes that one kelvin has precisely the same magnitude as a one-degree increment on the <A href="/wiki/Celsius" title="Celsius">Celsius</A> scale; and</LI>
<LI>It establishes the difference between the two scales' null points as being precisely 273.15 kelvins (0 K = −273.15 °C and 273.16 K = 0.01 °C).</LI>
</OL>
<P>Temperatures expressed in kelvins (<I>T</I><SUB>K</SUB>) are converted to degrees Rankine (<I>T</I><SUB>°R</SUB>) simply by multiplying by 1.8 (<I>T</I><SUB>°R</SUB> = 1.8 × <I>T</I><SUB>K</SUB>). Temperatures expressed in degrees Rankine are converted to kelvins by <I>dividing</I> by 1.8 (<I>T</I><SUB>K</SUB> = <I>T</I><SUB>°R</SUB> ÷ 1.8).</P>
<H3><SPAN class="mw-headline" id="Practical_realization">Practical realization</SPAN><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: Practical realization">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV role="note" class="hatnote navigation-not-searchable">Main article: <A href="/wiki/ITS-90" class="mw-redirect" title="ITS-90">ITS-90</A></DIV>
<P>Although the kelvin and Celsius scales are defined using absolute zero (0 K) and the triple point of water (273.16 K and 0.01 °C), it is impractical to use this definition at temperatures that are very different from the triple point of water. <A href="/wiki/ITS-90" class="mw-redirect" title="ITS-90">ITS-90</A> is then designed to represent the thermodynamic temperature as closely as possible throughout its range. Many different thermometer designs are required to cover the entire range. These include helium vapor pressure thermometers, helium gas thermometers, <A href="/wiki/Resistance_thermometer" title="Resistance thermometer">standard platinum resistance thermometers</A> (known as SPRTs, PRTs or Platinum RTDs) and <A href="/wiki/Infrared_thermometer" title="Infrared thermometer">monochromatic radiation thermometers</A>.</P>
<P>For some types of thermometer the relationship between the property observed (e.g., length of a mercury column) and temperature, is close to linear, so for most purposes a linear scale is sufficient, without point-by-point calibration. For others a calibration curve or equation is required. The mercury thermometer, invented before the thermodynamic temperature was understood, originally <I>defined</I> the temperature scale; its linearity made readings correlate well with true temperature, i.e. the "mercury" temperature scale was a close fit to the true scale.</P>
<H2><SPAN id="The_relationship_of_temperature.2C_motions.2C_conduction.2C_and_thermal_energy"></SPAN><SPAN class="mw-headline" id="The_relationship_of_temperature,_motions,_conduction,_and_thermal_energy">The relationship of temperature, motions, conduction, and thermal energy</SPAN><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: The relationship of temperature, motions, conduction, and thermal energy">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:302px;"><A href="/wiki/File:Translational_motion.gif" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif" width="300" height="263" class="thumbimage" data-file-width="300" data-file-height="263" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Translational_motion.gif" class="internal" title="Enlarge"></A></DIV>
<B>Fig. 1</B> The <I>translational motion</I> of fundamental particles of nature such as atoms and molecules are 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)" 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.</DIV>
</DIV>
</DIV>
<H3><SPAN id="The_nature_of_kinetic_energy.2C_translational_motion.2C_and_temperature"></SPAN><SPAN class="mw-headline" id="The_nature_of_kinetic_energy,_translational_motion,_and_temperature">The nature of kinetic energy, translational motion, and temperature</SPAN><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: The nature of kinetic energy, translational motion, and temperature">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>The thermodynamic temperature is a measure of the average energy of the translational, vibrational, and rotational motions of <A href="/wiki/Matter" title="Matter">matter</A>'s particle constituents (<A href="/wiki/Molecule" title="Molecule">molecules</A>, <A href="/wiki/Atom" title="Atom">atoms</A>, and <A href="/wiki/Subatomic_particle" title="Subatomic particle">subatomic particles</A>). The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, contribute the total <A href="/wiki/Internal_energy" title="Internal energy">internal energy</A> (loosely, the <A href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</A>) of a substance. Thus, internal energy may be stored in a number of ways (degrees of freedom) within a substance. When the degrees of freedom are in the classical regime ("unfrozen") the temperature is very simply related to the average energy of those degrees of freedom at equilibrium. The three translational degrees of freedom are unfrozen except for the very lowest temperatures, and their kinetic energy is simply related to the thermodynamic temperature over the widest range. The <A href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</A>, which relates heat input and temperature change, is discussed below.</P>
<P>The relationship of kinetic energy, mass, and velocity is given by the formula <I>E<SUB>k</SUB></I> = <SPAN class="frac nowrap"><SUP>1</SUP>⁄<SUB>2</SUB></SPAN><I>mv</I><SUP>2</SUP>.<SUP id="cite_ref-4" class="reference"><A href="#cite_note-4">[4]</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>Except in the quantum regime at extremely low temperatures, 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 <I>x</I>, <I>y</I>, and <I>z</I>–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> The temperature derived from this translational kinetic energy is sometimes referred to as <I>kinetic temperature</I> and is equal to the thermodynamic temperature over a very wide range of temperatures. Since there are three <A href="/wiki/Translational_motion" class="mw-redirect" title="Translational motion">translational</A> degrees of freedom (e.g., motion along the x, y, and z axes), the translational kinetic energy is related to the kinetic temperature by:</P>
<DL>
<DD><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 {\bar {E}}\,=\,{\frac {3}{2}}k_{B}T_{k}}">
  <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>
        <mspace width="thinmathspace"></mspace>
        <mo>=</mo>
        <mspace width="thinmathspace"></mspace>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>3</mn>
            <mn>2</mn>
          </mfrac>
        </mrow>
        <msub>
          <mi>k</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>B</mi>
          </mrow>
        </msub>
        <msub>
          <mi>T</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>k</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\bar {E}}\,=\,{\frac {3}{2}}k_{B}T_{k}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c7137f4291f8f24e4ade0a72441c502f539bc3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:12.784ex; height:5.176ex;" alt="{\bar {E}}\,=\,{\frac {3}{2}}k_{B}T_{k}" /></SPAN></DD>
</DL>
<P>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="{\displaystyle \scriptstyle {\bar {E}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mstyle displaystyle="false" scriptlevel="1">
          <mrow class="MJX-TeXAtom-ORD">
            <mrow class="MJX-TeXAtom-ORD">
              <mover>
                <mi>E</mi>
                <mo stretchy="false">¯<!-- ¯ --></mo>
              </mover>
            </mrow>
          </mrow>
        </mstyle>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\bar {E}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad4184fc40f4ddaecbab9a33d081a32974bff86" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.26ex; height:2.009ex;" alt="\scriptstyle {\bar {E}}" /></SPAN> is the mean kinetic energy in <A href="/wiki/Joule" title="Joule">joules</A> (J) and is pronounced “E bar”</LI>
<LI><I>k<SUB>B</SUB></I> = <SPAN class="nowrap"><SPAN style="display:none" class="sortkey">6977138065039999999♠</SPAN>1.380<SPAN style="margin-left:.25em;">6504</SPAN>(24)<SPAN style="margin-left:0.25em;margin-right:0.15em;">×</SPAN>10<SUP>−23</SUP> J/K</SPAN> is the <A href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</A> and is pronounced “Kay sub bee”</LI>
<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="{\displaystyle T_{k}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>T</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>k</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle T_{k}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51cc852c6e446a4871f78e05492699a9525b9acb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.446ex; height:2.509ex;" alt="T_{k}" /></SPAN> is the kinetic temperature in kelvins (K) and is pronounced “Tee sub kay”</LI>
</UL>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:302px;"><A href="/wiki/File:Maxwell_Dist-Inverse_Speed.png" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/d0/Maxwell_Dist-Inverse_Speed.png/300px-Maxwell_Dist-Inverse_Speed.png" width="300" height="350" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/d0/Maxwell_Dist-Inverse_Speed.png/450px-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>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Maxwell_Dist-Inverse_Speed.png" class="internal" title="Enlarge"></A></DIV>
<B>Fig. 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="#The_diffusion_of_thermal_energy:_Black-body_radiation">Fig. 5 </A></I>below.</DIV>
</DIV>
</DIV>
<P>While the Boltzmann constant is useful for finding the mean kinetic energy of a particle, it's 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 occur across a wide range of speeds (see animation in <I><A href="#The_relationship_of_temperature.2C_motions.2C_conduction.2C_and_thermal_energy">Figure 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. 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="#The_diffusion_of_thermal_energy:_Black-body_radiation">Figure 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>–axis on both graphs are scaled proportionally.</P>
<H3><SPAN class="mw-headline" id="The_high_speeds_of_translational_motion">The high speeds of translational motion</SPAN><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: The high speeds of translational motion">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<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-5" class="reference"><A href="#cite_note-5">[5]</A></SUP> and temperatures close to absolute zero 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">caesium</A> atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second in order to calculate their temperature.<SUP id="cite_ref-6" class="reference"><A href="#cite_note-6">[6]</A></SUP>  Formulas for calculating the velocity and speed of translational motion are given in the following footnote.<SUP id="cite_ref-Boltzmann_7-0" class="reference"><A href="#cite_note-Boltzmann-7">[7]</A></SUP></P>
<H3><SPAN class="mw-headline" id="The_internal_motions_of_molecules_and_specific_heat">The internal motions of molecules and specific heat</SPAN><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: The internal motions of molecules and specific heat">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:268px;"><A href="/wiki/File:Thermally_Agitated_Molecule.gif" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Thermally_Agitated_Molecule.gif/266px-Thermally_Agitated_Molecule.gif" width="266" height="266" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/2/23/Thermally_Agitated_Molecule.gif 1.5x" data-file-width="280" data-file-height="280" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Thermally_Agitated_Molecule.gif" class="internal" title="Enlarge"></A></DIV>
<B>Fig. 3</B> Because of their internal structure and flexibility, molecules can store kinetic energy in <I>internal degrees of freedom</I> which contribute to the <A href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</A>.</DIV>
</DIV>
</DIV>
<P>There are other forms of internal energy besides the kinetic energy of translational motion. 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. Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an <I>internal temperature</I>. Even though these motions are called <I>internal</I>, 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 energy is removed from molecules, both their kinetic temperature (the temperature derived from 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 (i.e. unfrozen) 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 internal 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 at that same instant contributing to the molecules' translational motions.<SUP id="cite_ref-8" class="reference"><A href="#cite_note-8">[8]</A></SUP> This extra thermal energy simply increases the amount of energy a substance absorbs for a given temperature rise. This property is known as a substance's <A href="/wiki/Specific_heat_capacity" class="mw-redirect" title="Specific heat capacity">specific heat capacity</A>.</P>
<P>Different molecules absorb different amounts of thermal 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, <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 at room temperature: the three comprising translational motion plus two rotational degrees of freedom internally. Since the two internal degrees of freedom are essentially unfrozen, 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-9" class="reference"><A href="#cite_note-9">[9]</A></SUP> Another example is <A href="/wiki/Gasoline" title="Gasoline">gasoline</A> (see <A href="/wiki/Heat_capacity#Table_of_specific_heat_capacities" title="Heat capacity">table</A> showing its specific heat capacity). Gasoline can absorb a large amount of thermal 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>
<H3><SPAN id="The_diffusion_of_thermal_energy:_Entropy.2C_phonons.2C_and_mobile_conduction_electrons"></SPAN><SPAN class="mw-headline" id="The_diffusion_of_thermal_energy:_Entropy,_phonons,_and_mobile_conduction_electrons">The diffusion of thermal energy: Entropy, phonons, and mobile conduction electrons</SPAN><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: The diffusion of thermal energy: Entropy, phonons, and mobile conduction electrons">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
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<DIV class="thumbinner" style="width:277px;"><A href="/wiki/File:1D_normal_modes_(280_kB).gif" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/9/9b/1D_normal_modes_%28280_kB%29.gif" width="275" height="275" class="thumbimage" data-file-width="275" data-file-height="275" /></A>
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<B>Fig. 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 molecules.</DIV>
</DIV>
</DIV>
<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. 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="#Overview">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/Convective_heat_transfer" title="Convective heat transfer">convection</A> speeds this process even more.<SUP id="cite_ref-10" class="reference"><A href="#cite_note-10">[10]</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 a given substance's speed of sound. 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-11" class="reference"><A href="#cite_note-11">[11]</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" 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-12" class="reference"><A href="#cite_note-12">[12]</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 <SPAN class="frac nowrap"><SUP>1</SUP>⁄<SUB>1836</SUB></SPAN><SMALL>th</SMALL> 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>
<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're delocalized and <I>very</I> fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.</P>
<H3><SPAN class="mw-headline" id="The_diffusion_of_thermal_energy:_Black-body_radiation">The diffusion of thermal energy: Black-body radiation</SPAN><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: The diffusion of thermal energy: Black-body radiation">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
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<DIV class="thumbinner" style="width:302px;"><A href="/wiki/File:Wiens_law.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Wiens_law.svg/300px-Wiens_law.svg.png" width="300" height="250" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Wiens_law.svg/450px-Wiens_law.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Wiens_law.svg/600px-Wiens_law.svg.png 2x" data-file-width="720" data-file-height="600" /></A>
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<B>Fig. 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="#The_nature_of_kinetic_energy,_translational_motion,_and_temperature">Fig. 2 </A></I>above.</DIV>
</DIV>
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<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" class="mw-redirect" title="Black-body">black-body</A> radiation). 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 <I><A href="/wiki/Temperature#Table_of_common_temperatures" title="Temperature">Table of common temperatures</A></I>).</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-Boltzmann_law" class="mw-redirect" 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 <I>60 times</I> 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>
<H4><SPAN class="mw-headline" id="Table_of_thermodynamic_temperatures">Table of thermodynamic temperatures</SPAN><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: Table of thermodynamic temperatures">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<P>The full range of the thermodynamic temperature scale, from absolute zero to <A href="/wiki/Absolute_hot" title="Absolute hot">absolute hot</A>, and some notable points between them are shown in the table below.</P>
<TABLE border="1" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f5f5f5; border: 2px #525252 solid; border-collapse: collapse; font-size: 95%;">
<TBODY><TR>
<TD style="background:#d9d9d3" align="center"></TD>
<TD style="background:#d9d9d3" align="center"><B>kelvin</B></TD>
<TD style="background:#d9d9d3" align="center"><B>Peak emittance<BR />
<A href="/wiki/Wavelength" title="Wavelength">wavelength</A><SUP id="cite_ref-13" class="reference"><A href="#cite_note-13">[13]</A></SUP> of<BR />
<A href="/wiki/Wien%27s_displacement_law" title="Wien's displacement law">black-body photons</A></B></TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Absolute_zero" title="Absolute zero">Absolute zero</A><BR />
<P>(precisely by definition)</P>
</TD>
<TD align="center">0 K</TD>
<TD align="center">  <SPAN style="font-size:140%;"><A href="/wiki/Infinity" title="Infinity">∞</A></SPAN><SMALL> </SMALL><SUP id="cite_ref-T0_3-1" class="reference"><A href="#cite_note-T0-3">[3]</A></SUP></TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right">Coldest measured<BR />
temperature<SUP> </SUP><SUP id="cite_ref-recordcold_14-0" class="reference"><A href="#cite_note-recordcold-14">[14]</A></SUP></TD>
<TD align="center">450 <A href="/wiki/Kelvin#SI_prefixed_forms_of_kelvin" title="Kelvin">pK</A></TD>
<TD align="center">6,400 <A href="/wiki/Kilometre" title="Kilometre">kilometers</A></TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right">One <A href="/wiki/Kelvin#SI_prefixed_forms_of_kelvin" title="Kelvin">millikelvin</A><BR />
<P>(precisely by definition)</P>
</TD>
<TD align="center">0.001 K</TD>
<TD align="center">2.897 77 <A href="/wiki/Metre" title="Metre">meters</A><BR />
(Radio, <A href="/wiki/FM_broadcasting" title="FM broadcasting">FM band</A>)<SUP id="cite_ref-15" class="reference"><A href="#cite_note-15">[15]</A></SUP></TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Cosmic_Microwave_Background_Radiation" class="mw-redirect" title="Cosmic Microwave Background Radiation">Cosmic Microwave Background Radiation</A></TD>
<TD align="center">2.725 48(57) K</TD>
<TD align="center">1.063 <A href="/wiki/Metre" title="Metre">mm</A> (peak wavelength)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" 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><BR />
<P>(precisely by definition)</P>
</TD>
<TD align="center">273.16 K</TD>
<TD align="center">10,608.3 <A href="/wiki/Metre#SI_prefixed_forms_of_metre" title="Metre">nm</A><BR />
(Long wavelength <A href="/wiki/Infrared" title="Infrared">I.R.</A>)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Incandescent_light_bulb" title="Incandescent light bulb">Incandescent lamp</A><SUP>B</SUP></TD>
<TD align="center">2500 K</TD>
<TD align="center">1160 nm<BR />
(Near <A href="/wiki/Infrared" title="Infrared">infrared</A>)<SUP>C</SUP></TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Sun" title="Sun">Sun</A>’s visible surface<SUP>C</SUP><SUP id="cite_ref-16" class="reference"><A href="#cite_note-16">[16]</A></SUP></TD>
<TD align="center">5778 K</TD>
<TD align="center">501.5 nm<BR />
(<A href="/wiki/Color#Spectral_colors" title="Color">Green light</A>)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Lightning" title="Lightning">Lightning bolt’s</A><BR />
channel</TD>
<TD align="center">28,000 K</TD>
<TD align="center">100 nm<BR />
(Far <A href="/wiki/Ultraviolet" title="Ultraviolet">Ultraviolet</A> light)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Sun#Core" title="Sun">Sun’s core</A></TD>
<TD align="center">16 <A href="/wiki/Kelvin#SI_prefixed_forms_of_kelvin" title="Kelvin">MK</A></TD>
<TD align="center">0.18 nm (<A href="/wiki/X-ray" title="X-ray">X-rays</A>)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Thermonuclear_explosion" class="mw-redirect" title="Thermonuclear explosion">Thermonuclear explosion</A><BR />
(peak temperature)<SUP id="cite_ref-17" class="reference"><A href="#cite_note-17">[17]</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="background:#d9d9d3" align="right">Sandia National Labs’<BR />
<A href="/wiki/Z_Pulsed_Power_Facility" title="Z Pulsed Power Facility">Z machine</A> <SUP>D</SUP><SUP id="cite_ref-18" class="reference"><A href="#cite_note-18">[18]</A></SUP></TD>
<TD align="center">2 <A href="/wiki/Kelvin#SI_prefixed_forms_of_kelvin" title="Kelvin">GK</A></TD>
<TD align="center">1.4 × 10<SUP>−3</SUP> nm<BR />
(Gamma rays)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right">Core of a <A href="/wiki/Silicon_burning_process" class="mw-redirect" title="Silicon burning process">high–mass<BR />
star on its last day</A><SUP id="cite_ref-19" class="reference"><A href="#cite_note-19">[19]</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="background:#d9d9d3" align="right">Merging binary <A href="/wiki/Neutron_star" title="Neutron star">neutron<BR />
star</A> system <SUP id="cite_ref-20" class="reference"><A href="#cite_note-20">[20]</A></SUP></TD>
<TD align="center">350 GK</TD>
<TD align="center">8 × 10<SUP>−6</SUP> nm<BR />
(Gamma rays)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Gamma-ray_burst_progenitors" title="Gamma-ray burst progenitors">Gamma-ray burst<BR />
progenitors</A><SUP id="cite_ref-21" class="reference"><A href="#cite_note-21">[21]</A></SUP></TD>
<TD align="center">1 <A href="/wiki/Kelvin#SI_prefixed_forms_of_kelvin" title="Kelvin">TK</A></TD>
<TD align="center">3 × 10<SUP>−6</SUP> nm<BR />
(Gamma rays)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/Relativistic_Heavy_Ion_Collider" title="Relativistic Heavy Ion Collider">Relativistic Heavy<BR />
Ion Collider</A><SUP id="cite_ref-22" class="reference"><A href="#cite_note-22">[22]</A></SUP></TD>
<TD align="center">1 <A href="/wiki/Kelvin#SI_prefixed_forms_of_kelvin" title="Kelvin">TK</A></TD>
<TD align="center">3 × 10<SUP>−6</SUP> nm<BR />
(Gamma rays)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right"><A href="/wiki/CERN" title="CERN">CERN</A>’s proton vs.<BR />
nucleus collisions<SUP id="cite_ref-23" class="reference"><A href="#cite_note-23">[23]</A></SUP></TD>
<TD align="center">10 TK</TD>
<TD align="center">3 × 10<SUP>−7</SUP> nm<BR />
(Gamma rays)</TD>
</TR>
<TR>
<TD style="background:#d9d9d3" align="right">Universe <A href="/wiki/Planck_time" title="Planck time">5.391 × 10<SUP>−44</SUP> s</A><BR />
after the <A href="/wiki/Big_Bang" title="Big Bang">Big Bang</A></TD>
<TD align="center"><A href="/wiki/Planck_temperature" title="Planck temperature">1.417 × 10<SUP>32</SUP> K</A></TD>
<TD align="center"><A href="/wiki/Planck_length" title="Planck length">1.616 × 10<SUP>−26</SUP> nm</A><BR />
(Planck frequency)<SUP id="cite_ref-24" class="reference"><A href="#cite_note-24">[24]</A></SUP></TD>
</TR>
</TBODY></TABLE>
<P><SUP>A</SUP> The 2500 K value is approximate.<BR />
<SUP>B</SUP> For a true blackbody (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.<BR />
<SUP>C</SUP> Effective photosphere temperature.<BR />
<SUP>D</SUP> For a true blackbody (which the plasma was not). The Z machine's dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.</P>
<H4><SPAN class="mw-headline" id="The_heat_of_phase_changes">The heat of phase changes</SPAN><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: The heat of phase changes">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:302px;"><A href="/wiki/File:IceBlockNearJoekullsarlon.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/IceBlockNearJoekullsarlon.jpg/300px-IceBlockNearJoekullsarlon.jpg" width="300" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/IceBlockNearJoekullsarlon.jpg/450px-IceBlockNearJoekullsarlon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/IceBlockNearJoekullsarlon.jpg/600px-IceBlockNearJoekullsarlon.jpg 2x" data-file-width="2971" data-file-height="1984" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:IceBlockNearJoekullsarlon.jpg" class="internal" title="Enlarge"></A></DIV>
<B>Fig. 6</B>  Ice and water: two phases of the same substance</DIV>
</DIV>
</DIV>
<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> <I>exhibit no temperature change whatsoever</I> 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>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:402px;"><A href="/wiki/File:Energy_thru_phase_changes.png" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/1/18/Energy_thru_phase_changes.png/400px-Energy_thru_phase_changes.png" width="400" height="258" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/18/Energy_thru_phase_changes.png/600px-Energy_thru_phase_changes.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/18/Energy_thru_phase_changes.png/800px-Energy_thru_phase_changes.png 2x" data-file-width="960" data-file-height="620" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Energy_thru_phase_changes.png" class="internal" title="Enlarge"></A></DIV>
<B>Fig. 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.</DIV>
</DIV>
</DIV>
<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-25" class="reference"><A href="#cite_note-25">[25]</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 can't 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's 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-26" class="reference"><A href="#cite_note-26">[26]</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-27" class="reference"><A href="#cite_note-27">[27]</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-28" class="reference"><A href="#cite_note-28">[28]</A></SUP> And 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 <I>540 times</I> that required for a one-degree increase.<SUP id="cite_ref-29" class="reference"><A href="#cite_note-29">[29]</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). 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" title="Relative humidity">relative humidity</A>). 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 not in use) are so effective at reducing heating costs: they prevent evaporation. For instance, the evaporation of just 20 mm of water from a 1.29-meter-deep pool chills its water 8.4 degrees Celsius (15.1 °F).</P>
<H4><SPAN class="mw-headline" id="Internal_energy">Internal energy</SPAN><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: Internal energy">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<P>The total energy of all particle motion translational and internal, 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_3-2" class="reference"><A href="#cite_note-T0-3">[3]</A></SUP> comprise the <I><A href="/wiki/Internal_energy" title="Internal energy">internal energy</A></I> of a substance.</P>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:258px;"><A href="/wiki/File:Close-packed_spheres.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Close-packed_spheres.jpg/256px-Close-packed_spheres.jpg" width="256" height="253" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Close-packed_spheres.jpg/384px-Close-packed_spheres.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Close-packed_spheres.jpg/512px-Close-packed_spheres.jpg 2x" data-file-width="1024" data-file-height="1012" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Close-packed_spheres.jpg" class="internal" title="Enlarge"></A></DIV>
<B>Fig. 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">closest-packed arrangement</A></I>. This yields the greatest possible packing density and the lowest energy state.</DIV>
</DIV>
</DIV>
<H4><SPAN class="mw-headline" id="Internal_energy_at_absolute_zero">Internal energy at absolute zero</SPAN><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: Internal energy at absolute zero">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<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 temperature decreases); the internal motions of molecules diminish (their internal temperature decreases); conduction electrons (if the substance is an electrical conductor) travel <I>somewhat</I> slower;<SUP id="cite_ref-30" class="reference"><A href="#cite_note-30">[30]</A></SUP> and black-body radiation's peak emittance wavelength increases (the photons' energy decreases). When the particles of a substance are as close as possible to complete rest and retain only ZPE-induced quantum mechanical motion, the substance is at the temperature of absolute zero (<I>T</I>=0).</P>
<P>Note that 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 thermal 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, <I>T</I>=0 <A href="/wiki/Helium" title="Helium">helium</A> remains liquid at room pressure 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. Only if under at least 25 bar (2.5 MPa) of pressure will this latent thermal energy be liberated as helium freezes while approaching absolute zero. 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 <I>T</I>=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_3-3" class="reference"><A href="#cite_note-T0-3">[3]</A></SUP><SUP> </SUP><SUP id="cite_ref-31" class="reference"><A href="#cite_note-31">[31]</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-32" class="reference"><A href="#cite_note-32">[32]</A></SUP> Lastly, it is always true to say that all <I>T</I>=0 substances contain zero kinetic thermal energy.<SUP id="cite_ref-T0_3-4" class="reference"><A href="#cite_note-T0-3">[3]</A></SUP><SUP> </SUP><SUP id="cite_ref-Boltzmann_7-1" class="reference"><A href="#cite_note-Boltzmann-7">[7]</A></SUP></P>
<DIV style="clear:left;"></DIV>
<H2><SPAN class="mw-headline" id="Practical_applications_for_thermodynamic_temperature">Practical applications for thermodynamic temperature</SPAN><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: Practical applications for thermodynamic temperature">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:172px;"><A href="/wiki/File:Liquid_helium_superfluid_phase.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Liquid_helium_superfluid_phase.jpg/170px-Liquid_helium_superfluid_phase.jpg" width="170" height="116" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Liquid_helium_superfluid_phase.jpg/255px-Liquid_helium_superfluid_phase.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Liquid_helium_superfluid_phase.jpg/340px-Liquid_helium_superfluid_phase.jpg 2x" data-file-width="955" data-file-height="652" /></A>
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<DIV class="magnify"><A href="/wiki/File:Liquid_helium_superfluid_phase.jpg" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/Helium-4" title="Helium-4">Helium-4</A>, is a <A href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluid</A> at or below 2.17 kelvins, (2.17 Celsius degrees above <A href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</A>)</DIV>
</DIV>
</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 relatively cold pressure of 200 <A href="/wiki/Pascal_(unit)" title="Pascal (unit)">kPa</A>-gage , then in absolute terms (relative to a vacuum), its pressure is 300 kPa-absolute.<SUP id="cite_ref-33" class="reference"><A href="#cite_note-33">[33]</A></SUP><SUP id="cite_ref-34" class="reference"><A href="#cite_note-34">[34]</A></SUP><SUP id="cite_ref-35" class="reference"><A href="#cite_note-35">[35]</A></SUP> Room temperature ("cold" in tire terms) is 296 K. If the tire pressure is 20 °C hotter (20 kelvins), the solution is calculated as <SPAN class="frac nowrap"><SUP>316 K</SUP>⁄<SUB>296 K</SUB></SPAN> = 6.8% greater thermodynamic temperature <I>and</I> absolute pressure; that is, a pressure of 320 kPa-absolute, which is 220 kPa-gage.</P>
<DIV style="clear:both;"></DIV>
<H2><SPAN class="mw-headline" id="Definition_of_thermodynamic_temperature">Definition of thermodynamic temperature</SPAN><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: Definition of thermodynamic temperature">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>The thermodynamic temperature is defined by the <A href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</A> and its consequences. 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 <I><A href="/wiki/Ratio" title="Ratio">ratio</A></I> <I>T</I><SUB>2</SUB>/<I>T</I><SUB>1</SUB> of two temperatures<I>T</I><SUB>1</SUB> and<I>T</I><SUB>2</SUB> 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, <I>T</I><SUB>H</SUB>, and a lower temperature heat sync, <I>T</I><SUB>C</SUB>, through a gas filled piston. The work done per cycle is equal to the difference between the heat supplied to the engine by <I>T</I><SUB>H</SUB>, <I>q</I><SUB>H</SUB>, and the heat supplied to <I>T</I><SUB>C</SUB> by the engine, <I>q</I><SUB>C</SUB>. The <B>efficiency</B> of the engine is the work divided by the heat put into the system or</P>
<DL>
<DD><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 {\textrm {Efficiency}}={\frac {w_{cy}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}\qquad (1)}">
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    <annotation encoding="application/x-tex">{\displaystyle {\textrm {Efficiency}}={\frac {w_{cy}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}\qquad (1)}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd5176079506d9302abbac4b8e4c3838a146eed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:47.582ex; height:5.676ex;" alt="{\textrm {Efficiency}}={\frac {w_{cy}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}\qquad (1)" /></SPAN></DD>
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<P>where w<SUB>cy</SUB> is the work done per cycle. Thus the efficiency depends only on q<SUB>C</SUB>/q<SUB>H</SUB>.</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 <I>T</I><SUB>1</SUB> and <I>T</I><SUB>2</SUB> must have the same efficiency, that is to say, the efficiency is the function of only temperatures</P>
<DL>
<DD><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_{C}}{q_{H}}}=f(T_{H},T_{C})\qquad (2).}">
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})\qquad (2).}</annotation>
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<P>In addition, a reversible heat engine operating between temperatures <I>T</I><SUB>1</SUB> and <I>T</I><SUB>3</SUB> must have the same efficiency as one consisting of two cycles, one between <I>T</I><SUB>1</SUB> and another (intermediate) temperature <I>T</I><SUB>2</SUB>, and the second between <I>T</I><SUB>2</SUB> and<I>T</I><SUB>3</SUB>. If this were not the case, then energy (in the form of <I>Q</I>) 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.</P>
<P>With this understanding of <I>Q</I><SUB>1</SUB>, <I>Q</I><SUB>2</SUB> and <I>Q</I><SUB>3</SUB>, we note also that mathematically,</P>
<DL>
<DD><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 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}).}">
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    <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>
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<P>But the first function is <I>NOT</I> a function of <I>T</I><SUB>2</SUB>, therefore the product of the final two functions <I>MUST</I> result in the removal of <I>T</I><SUB>2</SUB> as a variable. The only way is therefore to define the function f as follows:</P>
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<DD><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 f(T_{1},T_{2})={\frac {g(T_{2})}{g(T_{1})}}.}">
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    <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{2})={\frac {g(T_{2})}{g(T_{1})}}.}</annotation>
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<P>and</P>
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<DD><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 f(T_{2},T_{3})={\frac {g(T_{3})}{g(T_{2})}}.}">
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    <annotation encoding="application/x-tex">{\displaystyle f(T_{2},T_{3})={\frac {g(T_{3})}{g(T_{2})}}.}</annotation>
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<P>so that</P>
<DL>
<DD><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 f(T_{1},T_{3})={\frac {g(T_{3})}{g(T_{1})}}={\frac {q_{3}}{q_{1}}}.}">
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    <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{3})={\frac {g(T_{3})}{g(T_{1})}}={\frac {q_{3}}{q_{1}}}.}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0461da30b05269396065f323052ec3461a5c63bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:24.89ex; height:6.509ex;" alt="f(T_{1},T_{3})={\frac {g(T_{3})}{g(T_{1})}}={\frac {q_{3}}{q_{1}}}." /></SPAN></DD>
</DL>
<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)}">
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        <mi>g</mi>
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    <annotation encoding="application/x-tex">{\displaystyle g(T)}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfafde1201bf1abd1bfdbaae832f38d3adab4027" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.562ex; height:2.843ex;" alt="g(T)" /></SPAN>; 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}">
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        <mi>g</mi>
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        <mi>T</mi>
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    <annotation encoding="application/x-tex">{\displaystyle g(T)=T}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb17ba66b1080d071d1856c110ebe0634351fed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.296ex; height:2.843ex;" alt="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>It is to be noted that 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 <I>T</I><SUB>H</SUB> and <I>T</I><SUB>C</SUB>, and hence derive that the (complete) Carnot cycle is isentropic:</P>
<DL>
<DD><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_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}}{T_{H}}}.\qquad (3).}">
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}}{T_{H}}}.\qquad (3).}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0ab53b6d4d00c5e75242e40436814d687773b3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:32.954ex; height:5.843ex;" alt="{\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}}{T_{H}}}.\qquad (3)." /></SPAN></DD>
</DL>
<P>Substituting this back into our first formula for efficiency yields a relationship in terms of temperature:</P>
<DL>
<DD><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 {\textrm {Efficiency}}=1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}}{T_{H}}}\qquad (4).}">
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    <annotation encoding="application/x-tex">{\displaystyle {\textrm {Efficiency}}=1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}}{T_{H}}}\qquad (4).}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/510268fcfb12d3bae0a349553d8966d21030dced" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:39.834ex; height:5.843ex;" alt="{\textrm {Efficiency}}=1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}}{T_{H}}}\qquad (4)." /></SPAN></DD>
</DL>
<P>Notice that for <I>T</I><SUB>C</SUB>=0 the efficiency is 100% and that efficiency becomes greater than 100% for <I>T</I><SUB>C</SUB><0, which cases are unrealistic. Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives</P>
<DL>
<DD><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_{H}}{T_{H}}}-{\frac {q_{C}}{T_{C}}}=0,}">
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              <mi>T</mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{H}}{T_{H}}}-{\frac {q_{C}}{T_{C}}}=0,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f045db4303eb5123561be131177ba7227f1cf831" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:15.308ex; height:5.343ex;" alt="{\frac {q_{H}}{T_{H}}}-{\frac {q_{C}}{T_{C}}}=0," /></SPAN></DD>
</DL>
<P>where the negative sign indicates heat ejected from the system. The generalization of this equation is <A href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</A>, which suggests the existence of a <A href="/wiki/State_function" title="State function">state function</A> <I>S</I> (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>
<DL>
<DD><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 dS={\frac {dq_{\mathrm {rev} }}{T}}\qquad (5),}">
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            <mi>T</mi>
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    <annotation encoding="application/x-tex">{\displaystyle dS={\frac {dq_{\mathrm {rev} }}{T}}\qquad (5),}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb00702ca591d38ddde4e9e26c91a50b1405599" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:19.641ex; height:5.343ex;" alt="dS={\frac {dq_{\mathrm {rev} }}{T}}\qquad (5)," /></SPAN></DD>
</DL>
<P>where the subscript indicates heat transfer in a reversible process. The function <I>S</I> corresponds to the <A href="/wiki/Entropy" title="Entropy">entropy</A> of the system, mentioned previously, and the change of <I>S</I> around any cycle is zero (as is necessary for any state function). Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid logic loop, we should first define <A href="/wiki/Statistical_entropy" class="mw-redirect" title="Statistical entropy">entropy</A> through statistical mechanics):</P>
<DL>
<DD><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={\frac {dq_{\mathrm {rev} }}{dS}}.}">
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        <mi>T</mi>
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            <mrow>
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    <annotation encoding="application/x-tex">{\displaystyle T={\frac {dq_{\mathrm {rev} }}{dS}}.}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25e229ed4361cfed3ecaf8c9e346cc7b368976cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:10.945ex; height:5.509ex;" alt="T={\frac {dq_{\mathrm {rev} }}{dS}}." /></SPAN></DD>
</DL>
<P>For a system in which the entropy <I>S</I> is a function <I>S</I>(<I>E</I>) of its energy <I>E</I>, the thermodynamic temperature <I>T</I> is therefore given by</P>
<DL>
<DD><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}{T}}={\frac {dS}{dE}},}">
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        <mo>=</mo>
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T}}={\frac {dS}{dE}},}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04adbdce39aa40002a5adca44cf8d1b8c094756a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:10.045ex; height:5.509ex;" alt="{\frac {1}{T}}={\frac {dS}{dE}}," /></SPAN></DD>
</DL>
<P>so that the reciprocal of the thermodynamic temperature is the rate of increase of entropy with energy.</P>
<H2><SPAN class="mw-headline" id="History">History</SPAN><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: History">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<UL>
<LI><B>Ca. 485 BC:</B>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:155px;"><A href="/wiki/File:Parmenides.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parmenides.jpg/153px-Parmenides.jpg" width="153" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parmenides.jpg/229px-Parmenides.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ed/Parmenides.jpg 2x" data-file-width="283" data-file-height="370" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Parmenides.jpg" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/Parmenides" title="Parmenides">Parmenides</A></DIV>
</DIV>
</DIV>
<A href="/wiki/Parmenides" title="Parmenides">Parmenides</A> in his treatise "On Nature" postulated the existente of <I>primum frigidum</I>, a hypothetical elementary substance source of all cooling or cold in the world.<SUP id="cite_ref-36" class="reference"><A href="#cite_note-36">[36]</A></SUP></LI>
<LI><B>1702–1703:</B>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:77px;"><A href="/wiki/File:Guillaume_Amontons.png" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/c/ca/Guillaume_Amontons.png" width="75" height="85" class="thumbimage" data-file-width="75" data-file-height="85" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Guillaume_Amontons.png" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/Guillaume_Amontons" title="Guillaume Amontons">Guillaume Amontons</A></DIV>
</DIV>
</DIV>
<A href="/wiki/Guillaume_Amontons" title="Guillaume Amontons">Guillaume Amontons</A> (1663–1705) published two papers 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.</LI>
<LI><B>1742:</B>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:178px;"><A href="/wiki/File:Headshot_of_Anders_Celsius.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Headshot_of_Anders_Celsius.jpg/176px-Headshot_of_Anders_Celsius.jpg" width="176" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Headshot_of_Anders_Celsius.jpg/264px-Headshot_of_Anders_Celsius.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Headshot_of_Anders_Celsius.jpg/352px-Headshot_of_Anders_Celsius.jpg 2x" data-file-width="923" data-file-height="1050" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Headshot_of_Anders_Celsius.jpg" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/Anders_Celsius" title="Anders Celsius">Anders Celsius</A></DIV>
</DIV>
</DIV>
<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.</LI>
<LI><B>1744:</B>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:167px;"><A href="/wiki/File:Carl_von_Linn%C3%A9.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Carl_von_Linn%C3%A9.jpg/165px-Carl_von_Linn%C3%A9.jpg" width="165" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Carl_von_Linn%C3%A9.jpg/248px-Carl_von_Linn%C3%A9.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Carl_von_Linn%C3%A9.jpg/331px-Carl_von_Linn%C3%A9.jpg 2x" data-file-width="1938" data-file-height="2342" /></A>
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<A href="/wiki/Carl_Linnaeus" title="Carl Linnaeus">Carl Linnaeus</A></DIV>
</DIV>
</DIV>
Coincident with the death of Anders Celsius, the famous botanist <A href="/wiki/Carl_Linnaeus" title="Carl Linnaeus">Carl Linnaeus</A> (1707–1778) effectively reversed<SUP id="cite_ref-37" class="reference"><A href="#cite_note-37">[37]</A></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> (<I>Bureau international des poids et mesures</I>) (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> (<I>Conférence générale des poids et mesures</I>) and the CIPM (<A href="/wiki/International_Committee_for_Weights_and_Measures" title="International Committee for Weights and Measures">International Committee for Weights and Measures</A> (<I>Comité international des poids et mesures</I>) formally adopted<SUP id="cite_ref-38" class="reference"><A href="#cite_note-38">[38]</A></SUP> <I><A href="/wiki/Degree_Celsius" class="mw-redirect" title="Degree Celsius">degree Celsius</A></I> (symbol: °C) in 1948.<SUP id="cite_ref-°CName_39-0" class="reference"><A href="#cite_note-°CName-39">[39]</A></SUP></LI>
<LI><B>1777:</B>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:154px;"><A href="/wiki/File:JHLambert.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/JHLambert.jpg/152px-JHLambert.jpg" width="152" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/JHLambert.jpg/228px-JHLambert.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/JHLambert.jpg/304px-JHLambert.jpg 2x" data-file-width="350" data-file-height="460" /></A>
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<A href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</A></DIV>
</DIV>
</DIV>
In his book <I>Pyrometrie</I> (Berlin: <A rel="nofollow" class="external text" href="http://www.spiess-verlage.de/html/haude___spener.html">Haude & Spener</A>, 1779) 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.</LI>
<LI><B>Circa 1787:</B>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:148px;"><A href="/wiki/File:Jacques_Alexandre_C%C3%A9sar_Charles.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Jacques_Alexandre_C%C3%A9sar_Charles.jpg/146px-Jacques_Alexandre_C%C3%A9sar_Charles.jpg" width="146" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Jacques_Alexandre_C%C3%A9sar_Charles.jpg/219px-Jacques_Alexandre_C%C3%A9sar_Charles.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Jacques_Alexandre_C%C3%A9sar_Charles.jpg/292px-Jacques_Alexandre_C%C3%A9sar_Charles.jpg 2x" data-file-width="586" data-file-height="802" /></A>
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<DIV class="magnify"><A href="/wiki/File:Jacques_Alexandre_C%C3%A9sar_Charles.jpg" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/Jacques_Alexandre_C%C3%A9sar_Charles" class="mw-redirect" title="Jacques Alexandre César Charles">Jacques Alexandre César Charles</A></DIV>
</DIV>
</DIV>
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, but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was <I>V</I><SUB>1</SUB>/<I>T</I><SUB>1</SUB> = <I>V</I><SUB>2</SUB>/<I>T</I><SUB>2</SUB>.</LI>
<LI><B>1802:</B>
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<DIV class="thumbinner" style="width:163px;"><A href="/wiki/File:Gaylussac.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Gaylussac.jpg/161px-Gaylussac.jpg" width="161" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Gaylussac.jpg/241px-Gaylussac.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Gaylussac.jpg/321px-Gaylussac.jpg 2x" data-file-width="1876" data-file-height="2335" /></A>
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<A href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</A></DIV>
</DIV>
</DIV>
<A href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</A> (1778–1850) published work (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 <I>273</I> for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C).</LI>
<LI><B>1848:</B>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:147px;"><A href="/wiki/File:William_Thomson_1st_Baron_Kelvin.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/William_Thomson_1st_Baron_Kelvin.jpg/145px-William_Thomson_1st_Baron_Kelvin.jpg" width="145" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/William_Thomson_1st_Baron_Kelvin.jpg/217px-William_Thomson_1st_Baron_Kelvin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/William_Thomson_1st_Baron_Kelvin.jpg/290px-William_Thomson_1st_Baron_Kelvin.jpg 2x" data-file-width="1128" data-file-height="1557" /></A>
<DIV class="thumbcaption">
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<A href="/wiki/William_Thomson,_1st_Baron_Kelvin" title="William Thomson, 1st Baron Kelvin">Lord Kelvin</A></DIV>
</DIV>
</DIV>
<A href="/wiki/William_Thomson,_1st_Baron_Kelvin" title="William Thomson, 1st Baron Kelvin">William Thomson</A>, (1824–1907) also known as Lord Kelvin, wrote in his paper, <I><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> of the need for a scale whereby <I>infinite cold</I> (absolute zero) was the scale's null 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. It's noteworthy that Thomson's value of <I>−273</I> was actually 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.</LI>
<LI><B>1859:</B>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:134px;"><A href="/wiki/File:Rankine_William_signature.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Rankine_William_signature.jpg/132px-Rankine_William_signature.jpg" width="132" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Rankine_William_signature.jpg/197px-Rankine_William_signature.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Rankine_William_signature.jpg/263px-Rankine_William_signature.jpg 2x" data-file-width="282" data-file-height="428" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Rankine_William_signature.jpg" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/William_John_Macquorn_Rankine" title="William John Macquorn Rankine">William John Macquorn Rankine</A></DIV>
</DIV>
</DIV>
<A href="/wiki/William_John_Macquorn_Rankine" title="William John Macquorn Rankine">William John 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. This absolute scale is known today as the <A href="/wiki/Rankine_scale" title="Rankine scale">Rankine</A> thermodynamic temperature scale.</LI>
<LI><B>1877–1884:</B>
<DIV class="thumb tleft">
<DIV class="thumbinner" style="width:165px;"><A href="/wiki/File:Boltzmann2.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/163px-Boltzmann2.jpg" width="163" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/245px-Boltzmann2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/327px-Boltzmann2.jpg 2x" data-file-width="490" data-file-height="600" /></A>
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<A href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</A></DIV>
</DIV>
</DIV>
<A href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</A> (1844–1906) made major contributions to thermodynamics 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.</LI>
<LI><B>Circa 1930s:</B> Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed that absolute zero was equivalent to −273.15 °C.</LI>
<LI><B>1948:</B> <A rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/9/3/">Resolution 3</A> of the 9th CGPM (Conférence Générale des Poids et Mesures, also known as the <A href="/wiki/General_Conference_on_Weights_and_Measures" title="General Conference on Weights and Measures">General Conference on Weights and Measures</A>) 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 <I>if</I> 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 CIPM (Comité international des poids et mesures, also known as the International Committee for Weights and Measures) and the CGPM <A rel="nofollow" class="external text" href="http://www.bipm.org/en/committees/cipm/cipm-1948.html">formally adopted</A> the name <I>Celsius</I> for the <I>degree Celsius</I> and the <I>Celsius temperature scale</I>.<SMALL> </SMALL><SUP id="cite_ref-°CName_39-1" class="reference"><A href="#cite_note-°CName-39">[39]</A></SUP></LI>
<LI><B>1954:</B> <A rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/10/3/">Resolution 3</A> of the 10th CGPM gave the kelvin scale its modern definition by choosing the triple point of water as its second defining point and assigned 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.</LI>
<LI><B>1967/1968:</B> <A rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/13/3/">Resolution 3</A> of the 13th CGPM 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 <A rel="nofollow" class="external text" href="http://www.bipm.fr/en/CGPM/db/13/4/">Resolution 4</A> that "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water".</LI>
<LI><B>2005:</B> The CIPM (Comité International des Poids et Mesures, also known as the <A href="/wiki/International_Committee_for_Weights_and_Measures" title="International Committee for Weights and Measures">International Committee for Weights and Measures</A>) <A rel="nofollow" class="external text" href="http://www.bipm.fr/en/si/si_brochure/chapter2/2-1/kelvin.html">affirmed</A> 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>.</LI>
</UL>
<H2><SPAN class="mw-headline" id="See_also">See also</SPAN><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: See also">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="div-col columns column-count column-count-3" style="-moz-column-count: 3; -webkit-column-count: 3; column-count: 3;">
<UL>
<LI><A href="/wiki/Absolute_hot" title="Absolute hot">Absolute hot</A></LI>
<LI><A href="/wiki/Absolute_zero" title="Absolute zero">Absolute zero</A></LI>
<LI><A href="/wiki/Planck_temperature" title="Planck temperature">Planck temperature</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/Black_body" title="Black body">Black-body</A></LI>
<LI><A href="/wiki/Boiling" title="Boiling">Boiling</A></LI>
<LI><A href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</A></LI>
<LI><A href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</A></LI>
<LI><A href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</A></LI>
<LI><A href="/wiki/Chemical_bond" title="Chemical bond">Chemical bond</A></LI>
<LI><A href="/wiki/Condensation" title="Condensation">Condensation</A></LI>
<LI><A href="/wiki/Convection" title="Convection">Convection</A></LI>
<LI><A href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">Degrees of freedom</A></LI>
<LI><A href="/wiki/Delocalized_electron" title="Delocalized electron">Delocalized electron</A></LI>
<LI><A href="/wiki/Diffusion" title="Diffusion">Diffusion</A></LI>
<LI><A href="/wiki/Elastic_collision" title="Elastic collision">Elastic collision</A></LI>
<LI><A href="/wiki/Electron" title="Electron">Electron</A></LI>
<LI><A href="/wiki/Energy" title="Energy">Energy</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></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/Evaporation" title="Evaporation">Evaporation</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/Heat" title="Heat">Heat</A></LI>
<LI><A href="/wiki/Heat_conduction" class="mw-redirect" title="Heat conduction">Heat conduction</A></LI>
<LI><A href="/wiki/Heat_engine" title="Heat engine">Heat engine</A></LI>
<LI><A href="/wiki/Heat_death_of_the_universe" title="Heat death of the universe">Heat death of the universe</A></LI>
<LI><A href="/wiki/Internal_energy" title="Internal energy">Internal energy</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">ITS-90</A></LI>
<LI><A href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</A></LI>
<LI><A href="/wiki/Joule" title="Joule">Joule</A></LI>
<LI><A href="/wiki/Kelvin" title="Kelvin">Kelvin</A></LI>
<LI><A href="/wiki/Kinetic_energy" title="Kinetic energy">Kinetic energy</A></LI>
<LI><A href="/wiki/Latent_heat" title="Latent heat">Latent heat</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/Melting" title="Melting">Melting</A></LI>
<LI><A href="/wiki/Mole_(unit)" title="Mole (unit)">Mole</A></LI>
<LI><A href="/wiki/Molecule" title="Molecule">Molecule</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/Phonon" title="Phonon">Phonon</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/Potential_energy" title="Potential energy">Potential energy</A></LI>
<LI>Quantum mechanics:
<UL>
<LI><A href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</A></LI>
<LI><A href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics (main article)</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Rankine_scale" title="Rankine scale">Rankine scale</A></LI>
<LI><A href="/wiki/Specific_heat_capacity" class="mw-redirect" title="Specific heat capacity">Specific heat capacity</A></LI>
<LI><A href="/wiki/Standard_enthalpy_change_of_fusion" class="mw-redirect" title="Standard enthalpy change of fusion">Standard enthalpy change of fusion</A></LI>
<LI><A href="/wiki/Standard_enthalpy_change_of_vaporization" class="mw-redirect" title="Standard enthalpy change of vaporization">Standard enthalpy change of vaporization</A></LI>
<LI><A href="/wiki/Stefan-Boltzmann_law" class="mw-redirect" title="Stefan-Boltzmann law">Stefan–Boltzmann law</A></LI>
<LI><A href="/wiki/Sublimation_(chemistry)" class="mw-redirect" title="Sublimation (chemistry)">Sublimation</A></LI>
<LI><A href="/wiki/Temperature" title="Temperature">Temperature</A></LI>
<LI><A href="/wiki/Temperature_conversion_formulas" class="mw-redirect" title="Temperature conversion formulas">Temperature conversion formulas</A></LI>
<LI><A href="/wiki/Thermal_conductivity" title="Thermal conductivity">Thermal conductivity</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/Category:Thermodynamics" title="Category:Thermodynamics">Thermodynamics Category (list of articles)</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>
<LI><A href="/wiki/Gas_constant" title="Gas constant">Universal gas constant</A></LI>
<LI><A href="/wiki/Vienna_Standard_Mean_Ocean_Water" title="Vienna Standard Mean Ocean Water">Vienna Standard Mean Ocean Water</A> (VSMOW)</LI>
<LI><A href="/wiki/Wien%27s_displacement_law" title="Wien's displacement law">Wien's displacement law</A></LI>
<LI><A href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">Work (Mechanical)</A></LI>
<LI><A href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work (thermodynamics)</A></LI>
<LI><A href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</A></LI>
</UL>
</DIV>
<H2><SPAN class="mw-headline" id="Notes">Notes</SPAN><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: Notes">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DL>
<DD><I>In the following notes, wherever numeric equalities are shown in</I> concise form<I>, such as <SPAN class="nowrap"><SPAN style="display:none" class="sortkey">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>
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<DIV class="mw-references-wrap mw-references-columns">
<OL class="references">
<LI id="cite_note-1"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-1"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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-2"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-2"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A href="/wiki/William_Thomson,_1st_Baron_Kelvin" 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-T0-3"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-T0_3-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-T0_3-1"><SUP><I><B>b</B></I></SUP></A> <A href="#cite_ref-T0_3-2"><SUP><I><B>c</B></I></SUP></A> <A href="#cite_ref-T0_3-3"><SUP><I><B>d</B></I></SUP></A> <A href="#cite_ref-T0_3-4"><SUP><I><B>e</B></I></SUP></A></SPAN>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:302px;"><SPAN class="reference-text"><A href="/wiki/File:Zero-point_energy_v.s._motion.jpg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Zero-point_energy_v.s._motion.jpg/300px-Zero-point_energy_v.s._motion.jpg" width="300" height="300" class="thumbimage" 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></SPAN>
<DIV class="thumbcaption">
<DIV class="magnify"><SPAN class="reference-text"><A href="/wiki/File:Zero-point_energy_v.s._motion.jpg" class="internal" title="Enlarge"></A></SPAN></DIV>
<SPAN class="reference-text">Absolute zero's relationship to zero-point energy</SPAN></DIV>
</DIV>
</DIV>
<SPAN class="reference-text">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></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.</SPAN>
<P><SPAN class="reference-text">Although absolute zero (<I>T</I>=0) 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 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 <I>T</I>=0 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's 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 style="display:none" class="sortkey">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 <I>T</I>=0 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 <I>T</I>=0, no kinetic energy is available for transfer to other systems. The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE 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 <I>T</I>>0 K gases. However, in <I>T</I>=0 <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 25 <A href="/wiki/Bar_(unit)" title="Bar (unit)">bar</A> or 2.5 <A href="/wiki/Megapascal" class="mw-redirect" title="Megapascal">MPa</A>), ZPE is very much a form of thermal energy and may properly be included when tallying a substance's internal energy.</SPAN></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 <I>T</I>=0. 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 <I>T</I>=0 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 <I>T</I>=0, heat will flow inward, and if the surrounding environment is at <I>T</I>=0, there will be an equal flux of ZP radiation both inward and outward. A similar <I>Q</I> equilibrium exists at <I>T</I>=0 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</I></A> <A href="/wiki/Thermodynamic_operation" title="Thermodynamic operation">thermodynamic operation</A> involving van der Waals forces, Daniel C. Cole, Physical Review A, <B>42</B> (1990) 1847.</P>
</LI>
<LI id="cite_note-4"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-4"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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-5"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-5"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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 style="display:none" class="sortkey">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 <I>T</I> = 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 dewpoint of 9 °C (40.85% relative humidity), and 760 <A href="/wiki/MmHg" class="mw-redirect" title="MmHg">mmHg</A> (101.325 kPa) sea level–corrected barometric pressure.</SPAN></LI>
<LI id="cite_note-6"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-6"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external text" href="http://www.science.uva.nl/research/aplp/eprints/KasPhiRol95.pdf"><I>Adiabatic Cooling of Cesium to 700 nK in an Optical Lattice</I></A>, A. Kastberg <I>et al.</I>, Physical Review Letters <B>74</B> (1995) 1542 <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.74.1542">10.1103/PhysRevLett.74.1542</A>. It's noteworthy that 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>) 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-7"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Boltzmann_7-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Boltzmann_7-1"><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>
<DL>
<DD><SPAN class="reference-text"><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 {\bar {v}}={\sqrt {\frac {k_{B}T}{m}}}}">
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        <mrow class="MJX-TeXAtom-ORD">
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            <mover>
              <mi>v</mi>
              <mo stretchy="false">¯<!-- ¯ --></mo>
            </mover>
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        </mrow>
        <mo>=</mo>
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          <msqrt>
            <mfrac>
              <mrow>
                <msub>
                  <mi>k</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>B</mi>
                  </mrow>
                </msub>
                <mi>T</mi>
              </mrow>
              <mi>m</mi>
            </mfrac>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\bar {v}}={\sqrt {\frac {k_{B}T}{m}}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b57160cad385df743a73427e2fb70c22d30c7f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:11.812ex; height:6.176ex;" alt="{\bar {v}}={\sqrt {\frac {k_{B}T}{m}}}" /></SPAN></SPAN></DD>
</DL>
<SPAN class="reference-text">where:</SPAN>
<UL>
<LI><SPAN class="reference-text"><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 {\bar {v}}}">
  <semantics>
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      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
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            <mover>
              <mi>v</mi>
              <mo stretchy="false">¯<!-- ¯ --></mo>
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          </mrow>
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      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\bar {v}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1d09340f8f6c1979330c2f23e514e38f243a3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.009ex;" alt="{\bar {v}}" /></SPAN> is the vector-isolated mean velocity of translational particle motion in m/s</SPAN></LI>
<LI><SPAN class="reference-text"><I>k<SUB>B</SUB></I> is the <A href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</A> = <SPAN class="nowrap"><SPAN style="display:none" class="sortkey">6977138065039999999♠</SPAN>1.380<SPAN style="margin-left:.25em;">6504</SPAN>(24)<SPAN style="margin-left:0.25em;margin-right:0.15em;">×</SPAN>10<SUP>−23</SUP> J/K</SPAN></SPAN></LI>
<LI><I>T</I> is the thermodynamic temperature in kelvins</LI>
<LI><I>m</I> is the molecular mass of substance in kilograms</LI>
</UL>
In the above formula, molecular mass, <I>m</I>, in kilograms per 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 style="display:none" class="sortkey">7026602214179000000♠</SPAN>6.022<SPAN style="margin-left:.25em;">141</SPAN><SPAN style="margin-left:.25em;">79</SPAN>(30)<SPAN style="margin-left:0.25em;margin-right:0.15em;">×</SPAN>10<SUP>26</SUP></SPAN> (which is the <A href="/wiki/Avogadro%27s_number" class="mw-redirect" title="Avogadro's number">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:
<DL>
<DD><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 {\bar {s}}={\bar {v}}{\sqrt {3}}}">
  <semantics>
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        <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>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mn>3</mn>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\bar {s}}={\bar {v}}{\sqrt {3}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae9ed7161f98a4805230fe423787f542b25c6d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.715ex; height:2.843ex;" alt="{\bar {s}}={\bar {v}}{\sqrt {3}}" /></SPAN></DD>
</DL>
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="{\displaystyle {\bar {s}}}">
  <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>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\bar {s}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6872cefff1312205ecbaf86ef1d2e6634d850b71" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.009ex;" alt="{\bar {s}}" /></SPAN> is the mean speed of translational particle motion in m/s</LI>
</UL>
Note that 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="{\displaystyle {\bar {s}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
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          <mrow class="MJX-TeXAtom-ORD">
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              <mi>s</mi>
              <mo stretchy="false">¯<!-- ¯ --></mo>
            </mover>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\bar {s}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6872cefff1312205ecbaf86ef1d2e6634d850b71" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.009ex;" alt="{\bar {s}}" /></SPAN> for <I>v</I> in the classic formula for kinetic energy, <I>E<SUB>k</SUB></I> = <SPAN class="frac nowrap"><SUP>1</SUP>⁄<SUB>2</SUB></SPAN><I>m</I> • <I>v</I><SUP> 2</SUP> produces precisely the same value as does <I>E<SUB>mean</SUB></I> = 3/2<I>k<SUB>B</SUB>T</I> (as shown in the section titled <I><A href="#The_nature_of_kinetic_energy,_translational_motion,_and_temperature">The nature of kinetic energy, translational motion, and temperature</A>)</I>.<SUB> </SUB> Note too that 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).</LI>
<LI id="cite_note-8"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-8"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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 internal energy is stored. This is precisely why molecular-based gases have greater specific heat capacity than monatomic gases (where additional thermal energy must be added to achieve a given temperature rise).</SPAN></LI>
<LI id="cite_note-9"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-9"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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 <I>C<SUB>v</SUB>H</I> (100 kPa, 20 °C) equals 20.8 J mol<SUP>−1</SUP> K<SUP>−1</SUP> vs. the monatomic gases, which equal 12.4717 J mol<SUP>−1</SUP> K<SUP>−1</SUP>. Citations: <A rel="nofollow" class="external text" href="http://www.whfreeman.com/">W.H. Freeman's</A> <I>Physical Chemistry</I>, Part 3: Change (<A rel="nofollow" class="external text" href="http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf">422 kB PDF, here</A>), Exercise 21.20b, p. 787. Also <A rel="nofollow" class="external text" href="http://www.gsu.edu/">Georgia State University's</A> <I><A rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html">Molar Specific Heats of Gases</A></I>.</SPAN></LI>
<LI id="cite_note-10"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-10"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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-11"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-11"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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" class="mw-redirect" title="Specific heat capacity">specific heat capacity</A>, it also has exceptionally <I>high</I> <A href="/wiki/Thermal_conductivity" title="Thermal conductivity">thermal conductivity</A>.</SPAN></LI>
<LI id="cite_note-12"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-12"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Correlation is 752 (W m<SUP>−1</SUP> K<SUP>−1</SUP>) /(MS·cm), σ = 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. Citation: Data from <I>CRC Handbook of Chemistry and Physics</I>, 1st Student Edition and <A rel="nofollow" class="external text" href="http://www.webelements.com/">this link</A> to Web Elements' home page.</SPAN></LI>
<LI id="cite_note-13"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-13"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The<SUP> </SUP>cited emission wavelengths are for true black bodies in equilibrium. In this table, only the sun so qualifies. CODATA 2006 recommended value of 2.897 7685(51) × 10<SUP>−3</SUP> m K used for Wien displacement law constant <I>b</I>.</SPAN></LI>
<LI id="cite_note-recordcold-14"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-recordcold_14-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">A<SUP> </SUP>record cold temperature of 450 ±80 pK in a Bose–Einstein condensate (BEC) of sodium atoms was achieved in 2003 by researchers at <A href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</A>. Citation: <I>Cooling Bose–Einstein Condensates Below 500 Picokelvin</I>, A. E. Leanhardt <I>et al</I>., Science <B>301</B>, 12 Sept. 2003, Pg. 1515. It’s noteworthy that this record’s peak emittance black-body wavelength of 6,400 kilometers is roughly the radius of Earth.</SPAN></LI>
<LI id="cite_note-15"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-15"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The<SUP> </SUP>peak emittance wavelength of 2.897 77 m is a frequency of 103.456 MHz</SPAN></LI>
<LI id="cite_note-16"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-16"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Measurement<SUP> </SUP>was made in 2002 and has an uncertainty of ±3 kelvins. A <A rel="nofollow" class="external text" href="http://www.kis.uni-freiburg.de/~hw/astroandsolartitles.html">1989 measurement</A> produced a value of 5777 ±2.5 K. Citation: <I>Overview of the Sun</I> (Chapter 1 lecture notes on Solar Physics by Division of Theoretical Physics, Dept. of Physical Sciences, University of Helsinki). <A rel="nofollow" class="external text" href="http://theory.physics.helsinki.fi/~sol_phys/Sol0601.pdf">Download paper (252 kB PDF</A>)</SPAN></LI>
<LI id="cite_note-17"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-17"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The<SUP> </SUP>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. Citation: <I>Nuclear Weapons Frequently Asked Questions, 3.2.5 Matter At High Temperatures.</I> <A rel="nofollow" class="external text" href="http://nuclearweaponarchive.org/Nwfaq/Nfaq3.html#nfaq3.2">Link to relevant Web page.</A> All referenced data was compiled from publicly available sources.</SPAN></LI>
<LI id="cite_note-18"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-18"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Peak<SUP> </SUP>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). Citation: <I>Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2 × 10<SUP>9</SUP> Kelvin</I>, M. G. Haines <I>et al.</I>, Physical Review Letters <B>96</B>, Issue 7, id. 075003. <A rel="nofollow" class="external text" href="http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html">Link to Sandia’s news release.</A></SPAN></LI>
<LI id="cite_note-19"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-19"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Core<SUP> </SUP>temperature of a high–mass (>8–11 solar masses) star after it leaves the <I>main sequence</I> on the <A href="/wiki/Hertzsprung-Russell_diagram" class="mw-redirect" title="Hertzsprung-Russell diagram">Hertzsprung–Russell diagram</A> and begins the <A href="/wiki/Alpha_reactions" class="mw-redirect" title="Alpha reactions"><I>alpha process</I></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>. Citation: <I>Stellar Evolution: The Life and Death of Our Luminous Neighbors</I> (by Arthur Holland and Mark Williams of the University of Michigan). <A rel="nofollow" class="external text" href="http://www.umich.edu/~gs265/star.htm">Link to Web site</A>. More informative links can be found <A rel="nofollow" class="external text" href="http://schools.qps.org/hermanga/images/Astronomy/chapter_21___stellar_explosions.htm">here</A>, and <A rel="nofollow" class="external text" href="http://cosserv3.fau.edu/~cis/AST2002/Lectures/C13/Trans/Trans.html">here</A>, and a concise treatise on stars by NASA is <A rel="nofollow" class="external text" href="http://www.nasa.gov/worldbook/star_worldbook.html">here</A>. <A rel="nofollow" class="external text" href="https://web.archive.org/web/20150720174212/http://www.umich.edu/~gs265/star.htm">Archived</A> July 20, 2015, at the <A href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</A>.</SPAN></LI>
<LI id="cite_note-20"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-20"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Based<SUP> </SUP>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). It’s also noteworthy that at 350 GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass (<I>m</I>) 5% greater than its rest mass (<I>m</I><SUB>0</SUB>).  Citation: <I>Torus Formation in Neutron Star Mergers and Well-Localized Short Gamma-Ray Bursts</I>, R. Oechslin <I>et al</I>. of <A rel="nofollow" class="external text" href="http://www.mpa-garching.mpg.de/">Max Planck Institute for Astrophysics.</A>, arXiv:astro-ph/0507099 v2, 22 Feb. 2006.  <A rel="nofollow" class="external text" href="https://arxiv.org/pdf/astro-ph/0507099.pdf">Download paper (725 kB PDF</A>) (from Cornell University Library’s arXiv.org server). To view a browser-based summary of the research, <A rel="nofollow" class="external text" href="http://www.mpa-garching.mpg.de/mpa/research/current_research/hl2005-10/hl2005-10-en.html">click here</A>.</SPAN></LI>
<LI id="cite_note-21"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-21"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">NewScientist: <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>, 07 March 2011, which stated “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[s]”</SPAN></LI>
<LI id="cite_note-22"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-22"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Results<SUP> </SUP>of research by Stefan Bathe using the <A rel="nofollow" class="external text" href="http://www.phenix.bnl.gov/">PHENIX</A> detector on the <A rel="nofollow" class="external text" href="http://www.bnl.gov/rhic/">Relativistic Heavy Ion Collider</A> at <A rel="nofollow" class="external text" href="http://www.bnl.gov/world/">Brookhaven National Laboratory</A> in Upton, New York, U.S.A.  Bathe has studied gold-gold, deuteron-gold, and proton-proton collisions to test the theory of quantum chromodynamics, the theory of the strong force that holds atomic nuclei together.  <A rel="nofollow" class="external text" href="http://www.bnl.gov/bnlweb/pubaf/pr/PR_display.asp?prID=06-56">Link to news release.</A></SPAN></LI>
<LI id="cite_note-23"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-23"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Citation:<SUP> </SUP><I><A rel="nofollow" class="external text" href="http://public.web.cern.ch/public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html">How do physicists study particles?</A></I> by <A rel="nofollow" class="external text" href="http://public.web.cern.ch/public/Welcome.html">CERN</A>.</SPAN></LI>
<LI id="cite_note-24"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-24"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The<SUP> </SUP>Planck frequency equals 1.854 87(14) × 10<SUP>43</SUP> Hz (which is the reciprocal of one Planck time). Photons at the Planck frequency have a wavelength of one Planck length. The Planck temperature of 1.416 79(11) × 10<SUP>32</SUP> K equates to a calculated <I>b </I>/<I>T</I> = λ<SUB><I>max</I></SUB> wavelength of 2.045 31(16) × 10<SUP>−26</SUP> nm. However, the actual peak emittance wavelength quantizes to the Planck length of 1.616 24(12) × 10<SUP>−26</SUP> nm.</SPAN></LI>
<LI id="cite_note-25"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-25"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Water's enthalpy of fusion (0 °C, 101.325 kPa) equates to <SPAN class="nowrap"><SPAN style="display:none" class="sortkey">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 style="display:none" class="sortkey">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-26"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-26"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Water's enthalpy of fusion is 6.0095 kJ mol<SUP>−1</SUP> K<SUP>−1</SUP> (0 °C, 101.325 kPa). Citation: <I>Water Structure and Science, Water Properties, Enthalpy of fusion, (0 °C, 101.325 kPa)</I> (by London South Bank University). <A rel="nofollow" class="external text" href="http://www.lsbu.ac.uk/water/data.html">Link to Web site.</A> 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. Citation: <A rel="nofollow" class="external text" href="http://www.webelements.com/">This link</A> to Web Elements' home page.</SPAN></LI>
<LI id="cite_note-27"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-27"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Xenon value citation: <A rel="nofollow" class="external text" href="http://www.webelements.com/webelements/elements/text/Xe/heat.html">This link</A> to WebElements' xenon data (available values range from 2.3 to 3.1 kJ/mol). It is also noteworthy that 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-28"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-28"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><I>CRC Handbook of Chemistry and Physics</I>, 1st Student Edition and <A rel="nofollow" class="external text" href="http://www.webelements.com/">Web Elements</A>.</SPAN></LI>
<LI id="cite_note-29"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-29"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">H<SUB>2</SUB>Ospecific heat capacity, <I>C<SUB>p</SUB></I> = 0.075327 kJ mol<SUP>−1</SUP> K<SUP>−1</SUP> (25 °C); Enthalpy of fusion = 6.0095 kJ/mol (0 °C, 101.325 kPa); Enthalpy of vaporization (liquid) = 40.657 kJ/mol (100 °C). Citation: <I>Water Structure and Science, Water Properties</I> (by London South Bank University). <A rel="nofollow" class="external text" href="http://www.lsbu.ac.uk/water/data.html">Link to Web site.</A></SPAN></LI>
<LI id="cite_note-30"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-30"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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 style="display:none" class="sortkey">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-31"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-31"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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-32"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-32"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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: <I><A href="/wiki/Enthalpy" title="Enthalpy">H<I> = </I>U<I> + </I>pV</A>)</I>, internal energy may exclude different sources of thermal energy (particularly ZPE) depending on the nature of the analysis. Accordingly, all <I>T</I>=0 closest-packed matter under a perfect vacuum has either minimal or zero enthalpy, depending on the nature of the analysis. <A rel="nofollow" class="external text" href="http://iupac.org/publications/pac/2001/pdf/7308x1349.pdf"><I>Use Of Legendre Transforms In Chemical Thermodynamics</I></A>, Robert A. Alberty, Pure Appl.Chem., <B>73</B> (2001) 1349.</SPAN></LI>
<LI id="cite_note-33"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-33"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Pressure also must be in absolute terms. The air still in a tire at 0 kPa-gage expands too as it gets hotter. It's 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. Referenced document: <I>Aircraft Tire Ratings</I> (<A rel="nofollow" class="external text" href="http://airmichelin.com/pdfs/05%20-%20Aircraft%20Tire%20Ratings.pdf">155 kB PDF, here</A>).</SPAN></LI>
<LI id="cite_note-34"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-34"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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>. (see <I>Honeywell-Sensotec's</I> <A rel="nofollow" class="external text" href="http://sensotec.com/pressurefaq.shtml">FAQ page</A> and Fluke Corporation's <A rel="nofollow" class="external text" href="http://us.fluke.com/usen/Home/Search.asp?txtSearchBox=%22gage+pressure%22&x=0&y=0">product search page</A>).</SPAN></LI>
<LI id="cite_note-35"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-35"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</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)" 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. Citation: IUPAC.org, Gold Book, <I><A rel="nofollow" class="external text" href="http://goldbook.iupac.org/S05921.html">Standard Pressure</A></I></SPAN></LI>
<LI id="cite_note-36"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-36"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><I>Absolute Zero and the Conquest of Cold</I> , Shachtman, Tom., Mariner Books, 1999.</SPAN></LI>
<LI id="cite_note-37"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-37"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external text" href="http://thermodynamics-information.net/"><I>A Brief History of Temperature Measurement</I></A> and; Uppsala University (Sweden), <A rel="nofollow" class="external text" href="http://www.linnaeus.uu.se/online/life/6_32.html"><I>Linnaeus' thermometer</I></A></SPAN></LI>
<LI id="cite_note-38"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-38"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external text" href="http://www.bipm.org/en/committees/cipm/cipm-1948.html">bipm.org</A></SPAN></LI>
<LI id="cite_note-°CName-39"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-°CName_39-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-°CName_39-1"><SUP><I><B>b</B></I></SUP></A></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 Dr. 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> (<A rel="nofollow" class="external text" href="http://www.imeko.org/publications/tc12-2004/PTC12-2004-PL-001.pdf">148 kB PDF, here</A>) as well as <I>Temperature</I> (2nd Edition / 1990 / Academic Press / 0125696817), 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 wasn't even aware that <I>degree Celsius</I> was in sporadic, non-scientific use before that time. It's also noteworthy that the twelve-volume, 1933 edition of 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:</SPAN>
<OL>
<LI><SPAN class="reference-text">All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.</SPAN></LI>
<LI><SPAN class="reference-text">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.</SPAN></LI>
<LI><SPAN class="reference-text">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.</SPAN></LI>
</OL>
</LI>
</OL>
</DIV>
</DIV>
<H2><SPAN class="mw-headline" id="External_links">External links</SPAN><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: External links">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<UL>
<LI><I><A rel="nofollow" class="external text" href="http://www.chm.davidson.edu/ChemistryApplets/KineticMolecularTheory/index.html">Kinetic Molecular Theory of Gases.</A></I> An explanation (with interactive animations) of the kinetic motion of molecules and how it affects matter. By David N. Blauch, <A rel="nofollow" class="external text" href="http://www.chm.davidson.edu/">Department of Chemistry</A>, <A rel="nofollow" class="external text" href="http://www2.davidson.edu/index.asp">Davidson College</A>.</LI>
<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. By Bernard Haisch, of <A rel="nofollow" class="external text" href="http://www.calphysics.org/index.html">Calphysics Institute</A>.</LI>
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<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 style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">Base quantity</TH>
<TD class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">
<TABLE class="wikitable sortable jquery-tablesorter" style="text-align:center; margin:0;">
<THEAD><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;" class="headerSort" tabindex="0" role="columnheader button" title="Sort ascending">Name</TH>
<TH style="text-align:center;" class="headerSort" tabindex="0" role="columnheader button" title="Sort ascending">Symbol</TH>
<TH style="text-align:center;" class="headerSort" tabindex="0" role="columnheader button" title="Sort ascending">Dimension<BR />
symbol</TH>
<TH class="unsortable"></TH>
<TH style="text-align:center;" class="headerSort" tabindex="0" role="columnheader button" title="Sort ascending">Unit name<BR />
(symbol)</TH>
<TH style="text-align:center;" class="headerSort" tabindex="0" role="columnheader button" title="Sort ascending">Example</TH>
</TR></THEAD><TBODY>

<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>L</TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Metre" title="Metre">metre</A> (m)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">r</VAR> = 10 m</SPAN></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>M</TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Kilogram" title="Kilogram">kilogram</A> (kg)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">m</VAR> = 10 kg</SPAN></TD>
</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>T</TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Second" title="Second">second</A> (s)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">t</VAR> = 10 s</SPAN></TD>
</TR>
<TR>
<TD style="text-align:left;"><A href="/wiki/Electric_current" title="Electric current">electric current</A></TD>
<TD><ABBR title="Uppercase letter i in italic"> <SPAN class="texhtml mvar" style="font-style:italic;">I</SPAN> </ABBR>, <SPAN class="texhtml mvar" style="font-style:italic;">i</SPAN></TD>
<TD><ABBR title="Uppercase letter i"> I </ABBR></TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Ampere" title="Ampere">ampere</A> (A)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">I</VAR> = 10 A</SPAN></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><ABBR title="Greek capital letter theta">Θ</ABBR></TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Kelvin" title="Kelvin">kelvin</A> (K)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">T</VAR> = 10 K</SPAN></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>N</TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Mole_(unit)" title="Mole (unit)">mole</A> (mol)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">n</VAR> = 10 mol</SPAN></TD>
</TR>
<TR>
<TD style="text-align:left;"><A href="/wiki/Luminous_intensity" title="Luminous intensity">luminous intensity</A></TD>
<TD><ABBR title="Uppercase letter i in italic, subscript lowercase letter v"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>I</I><SUB>v</SUB></SPAN></ABBR></TD>
<TD>J</TD>
<TD></TD>
<TD style="text-align:left;"><A href="/wiki/Candela" title="Candela">candela</A> (cd)</TD>
<TD style="text-align:left;"><SPAN class="texhtml texhtml-big" style="font-size:100%;"><VAR style="padding-right: 1px;">I</VAR><SUB>v</SUB> = 10 cd</SPAN></TD>
</TR>
</TBODY><TFOOT></TFOOT></TABLE>
</DIV>
</TD>
<TD class="navbox-image" rowspan="4" style="width:1px;padding:0px 0px 0px 2px">
<DIV><A href="/wiki/File:SI_base_unit.svg" class="image" title="The seven SI base quantities and their units"><IMG alt="The seven SI base quantities and their units" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/SI_base_unit.svg/125px-SI_base_unit.svg.png" width="125" height="125" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/SI_base_unit.svg/188px-SI_base_unit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/SI_base_unit.svg/250px-SI_base_unit.svg.png 2x" data-file-width="600" data-file-height="600" /></A></DIV>
</TD>
</TR>
<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%"><A href="/wiki/International_System_of_Units#Writing_unit_symbols_and_the_values_of_quantities" title="International System of Units">Specification</A></TH>
<TD class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">The quantity (not the unit) can have a specification: <SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>T</I><SUB>max</SUB> = 300 K</SPAN></DIV>
</TD>
</TR>
<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%"><A href="/wiki/SI_derived_unit" title="SI derived unit">Derived quantity</A></TH>
<TD class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em"></DIV>
<TABLE class="nowraplinks navbox-subgroup" style="border-spacing:0">
<TBODY><TR>
<TH scope="row" class="navbox-group" style="width:1%">Definition</TH>
<TD class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">A quantity <SPAN class="texhtml mvar" style="font-style:italic;">Q</SPAN> is expressed in the base quantities: <SPAN class="nowrap"><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=f\left({\mathit {l,m,t,I,T,n,I}}\mathrm {_{v}} \right)}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>Q</mi>
        <mo>=</mo>
        <mi>f</mi>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mrow class="MJX-TeXAtom-ORD">
              <mrow class="MJX-TeXAtom-ORD">
                <mi class="MJX-tex-mathit" mathvariant="italic">l</mi>
                <mo class="MJX-tex-mathit" mathvariant="italic">,</mo>
                <mi class="MJX-tex-mathit" mathvariant="italic">m</mi>
                <mo class="MJX-tex-mathit" mathvariant="italic">,</mo>
                <mi class="MJX-tex-mathit" mathvariant="italic">t</mi>
                <mo class="MJX-tex-mathit" mathvariant="italic">,</mo>
                <mi class="MJX-tex-mathit" mathvariant="italic">I</mi>
                <mo class="MJX-tex-mathit" mathvariant="italic">,</mo>
                <mi class="MJX-tex-mathit" mathvariant="italic">T</mi>
                <mo class="MJX-tex-mathit" mathvariant="italic">,</mo>
                <mi class="MJX-tex-mathit" mathvariant="italic">n</mi>
                <mo class="MJX-tex-mathit" mathvariant="italic">,</mo>
                <mi class="MJX-tex-mathit" mathvariant="italic">I</mi>
              </mrow>
            </mrow>
            <mrow class="MJX-TeXAtom-ORD">
              <msub>
                <mi></mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="normal">v</mi>
                </mrow>
              </msub>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle Q=f\left({\mathit {l,m,t,I,T,n,I}}\mathrm {_{v}} \right)}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3fdf26d2335b8b4dde70fbba15446ef8f26eafd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:24.155ex; height:2.843ex;" alt="Q=f\left({\mathit {l,m,t,I,T,n,I}}\mathrm {_{v}} \right)" /></SPAN></SPAN></DIV>
</TD>
</TR>
<TR>
<TH scope="row" class="navbox-group" style="width:1%"><A href="/wiki/Dimension_of_a_physical_quantity" class="mw-redirect" title="Dimension of a physical quantity">Derived dimension</A></TH>
<TD class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em"><SPAN class="nowrap">dim <SPAN class="texhtml mvar" style="font-style:italic;">Q</SPAN> = L<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>a</I></SPAN></SUP> <B>·</B> M<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>b</I></SPAN></SUP> <B>·</B> T<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>c</I></SPAN></SUP> <B>·</B> I<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>d</I></SPAN></SUP> <B>·</B> Θ<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>e</I></SPAN></SUP> <B>·</B> N<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>f</I></SPAN></SUP> <B>·</B> J<SUP><SPAN class="texhtml texhtml-big" style="font-size:100%;"><I>g</I></SPAN></SUP></SPAN> <SMALL>(Superscripts <SPAN class="texhtml mvar" style="font-style:italic;">a</SPAN>–<SPAN class="texhtml mvar" style="font-style:italic;">g</SPAN> are algebraic exponents, usually a positive, negative or zero integer.)</SMALL><BR /></DIV>
</TD>
</TR>
<TR>
<TH scope="row" class="navbox-group" style="width:1%">Example</TH>
<TD class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">Quantity <I>acceleration</I> = <SPAN class="texhtml"><I>l</I><SUP>1</SUP> <B>·</B> <I>t</I><SUP>−2</SUP></SPAN>, <SPAN class="nowrap">dim <I>acceleration</I> = L<SUP>1</SUP> <B>·</B> T<SUP>−2</SUP></SPAN><BR />
possible units: <SPAN class="nowrap">m<SUP>1</SUP> <B>·</B> s<SUP>−2</SUP></SPAN>, <SPAN class="nowrap">km<SUP>1</SUP> <B>·</B> Ms<SUP>−2</SUP></SPAN>, etc.</DIV>
</TD>
</TR>
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<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">See also</TH>
<TD class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">
<UL>
<LI><A href="/wiki/History_of_the_metric_system" title="History of the metric system">History of the metric system</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/Proposed_redefinition_of_SI_base_units" title="Proposed redefinition of SI base units">Proposed redefinitions</A></LI>
<LI><A href="/wiki/System_of_measurement" title="System of measurement">Systems of measurement</A></LI>
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<LI><B><A href="/wiki/Outline_of_the_metric_system" title="Outline of the metric system">Outline</A></B></LI>
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document_title: Thermodynamic temperature

token: Thermodynamic

token: temperature

token: wikipedia

token: Jump

token: to

token: navigation

token: search

token: <Table>

token: <Tr>

token: Thermodynamics

token: The

token: classical

token: Carnot

token: heat

token: engine

token: Branches

token: show

token: <Ul>

token: <Li>

token: Classical

token: Statistical

token: Chemical

token: Quantum

token: thermodynamics

token: Equilibrium

token: Non-equilibrium

token: Laws

token: Zeroth

token: First

token: Second

token: Third

token: Systems

token: State

token: Equation

token: of

token: state

token: Ideal

token: gas

token: Real

token: matter

token: Control

token: volume

token: Instruments

token: Processes

token: Isobaric

token: Isochoric

token: Isothermal

token: Adiabatic

token: Isentropic

question_text: molecular motion is independent of temperature and is unaffected by a change in temperature