View source for Thermodynamic temperature

{{Short description|Measure of temperature relative to absolute zero}}
{{Thermodynamics|cTopic=sysprop}}
{{more citations needed|date=May 2025}}
'''Thermodynamic temperature''', also known as '''absolute temperature''', is a [[physical quantity]] that measures [[temperature]] starting from [[absolute zero]], the point at which particles have minimal thermal motion.

Thermodynamic temperature is typically expressed using the [[Kelvin scale]], on which the unit of measurement is the ''kelvin'' (unit symbol: K). This unit is the same interval as the degree Celsius, used on the [[Celsius scale]] but the scales are offset so that 0&nbsp;K on the Kelvin scale corresponds to absolute zero. For comparison, a temperature of 295&nbsp;K corresponds to 21.85&nbsp;°C and 71.33&nbsp;°F. Another absolute scale of temperature is the [[Rankine scale]], which is based on the [[Fahrenheit]] degree interval.

Historically, thermodynamic temperature was defined by [[Lord Kelvin]] in terms of a relation between the macroscopic quantities [[Work (thermodynamics)|thermodynamic work]] and [[Heat|heat transfer]] as defined in [[thermodynamics]], but the kelvin was redefined by international agreement in 2019 in terms of phenomena that are now understood as manifestations of the kinetic energy of free motion of particles such as atoms, molecules, and electrons.{{citation needed|date=July 2025}}

== Overview ==
Thermodynamic temperature can be defined in purely thermodynamic terms using the [[Carnot cycle]]. Thermodynamic temperature was rigorously defined historically long before particles such as atoms, molecules, and electrons were fully understood.

The [[International System of Units]] (SI) specifies the absolute scale for measuring temperature, and the unit of measure ''[[kelvin]]'' (symbol: K) for specific values along the scale. A temperature interval of one [[degree Celsius]] is the same as one kelvin. Since the [[2019 revision of the SI#Kelvin|2019 revision of the SI]], the kelvin has been defined in relation to the ''physical property'' underlying thermodynamic temperature: the kinetic energy of atomic free particle motion. The revision fixed the [[Boltzmann constant]] at exactly {{physconst|k|after=.}}

The property that imbues material substances with a temperature can be readily understood by examining the [[ideal gas law]], which relates, through the Boltzmann constant, how [[heat|heat energy]] causes precisely defined changes in the [[pressure]] and temperature of certain gases. This is because [[monatomic gas]]es like [[helium]] and [[argon]] behave kinetically like freely moving perfectly elastic and spherical billiard balls that move only in a specific subset of the possible motions that can occur in matter: that comprising the three ''translational'' [[Degrees of freedom (physics and chemistry)|degrees of freedom]]. The translational degrees of freedom are the familiar billiard ball-like movements along the {{math|''x''}}-, {{math|''y''}}-, and {{math|''z''}}-axes of 3D space (see [[#Nature of kinetic energy, translational motion, and temperature|''Fig.&nbsp;1'']], below). This is why the noble gases all have the [[Table of specific heat capacities|same heat capacity per atom]] and why that value is lowest of all the gases.

[[Molecule]]s (two or more chemically bound atoms), however, have ''internal structure'' and therefore have additional ''internal'' degrees of freedom (see [[#Internal_motions_of_molecules_and_internal_energy|''Fig. 3'']], below), which has the effect that molecules absorb more heat energy for any given rise in temperature than do the monatomic gases. Heat energy is born in all available degrees of freedom; this is in accordance with the [[equipartition theorem]], so all available internal degrees of freedom have the same average energy as do their three external degrees of freedom. However, the property that gives all gases their [[pressure]], which is the net force per unit area on a container arising from gas particles recoiling off it, is a function of the kinetic energy borne in the freely moving atoms' and molecules' three translational degrees of freedom.<ref>Georgia State University, HyperPhysics Project, "[http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/eqpar.html Equipartition of Energy]"</ref>

Fixing the Boltzmann constant at a specific value had the effect of precisely establishing the magnitude of the kelvin in terms of the average kinetic behavior of the noble gases. Moreover, the ''starting point'' of the thermodynamic temperature scale, absolute zero, was reaffirmed as the point at which ''zero average kinetic energy'' remains in a sample; the only remaining particle motion being that comprising random vibrations due to zero-point energy.

== Absolute zero of temperature ==
{{Main|Absolute zero}}

Temperature scales are numerical. The numerical zero of a temperature scale is not bound to the absolute zero of temperature. Nevertheless, some temperature scales have their numerical zero coincident with the absolute zero of temperature. Examples are the [[Kelvin temperature scale]] and the [[Rankine scale|Rankine temperature scale]]. Other temperature scales have their numerical zero far from the absolute zero of temperature. Examples are the Celsius scale and the Fahrenheit scale.

At the zero point of thermodynamic temperature, [[absolute zero]], the particle constituents of matter have minimal motion and can become no colder.<ref>Rankine, W. J. M., "A manual of the steam engine and other prime movers", Richard Griffin and Co., London (1859), p. 306–307.</ref><ref>[[William Thomson, 1st Baron Kelvin]], "Heat", Adam and Charles Black, Edinburgh (1880), p. 39.</ref> Absolute zero, which is a temperature of zero kelvins (0&nbsp;K), precisely corresponds to −273.15&nbsp;°C and −459.67&nbsp;°F. Matter at absolute zero has no remaining transferable average kinetic energy and the only remaining particle motion is due to an ever-pervasive [[quantum mechanics|quantum mechanical]] phenomenon called ZPE ([[zero-point energy]]).<ref name="T0">[[Image:Zero-point energy v.s. motion.jpg|thumb|300px|Absolute zero's relationship to zero-point energy]] While scientists are achieving temperatures ever closer to [[absolute zero]], they can not fully achieve a state of ''zero'' temperature. However, even if scientists could remove ''all'' kinetic thermal energy from matter, [[quantum mechanics|quantum mechanical]] ''[[zero-point energy]]'' (ZPE) causes particle motion that can never be eliminated. Encyclopædia Britannica Online [http://britannica.com/eb/article-9078341 defines zero-point] 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 [[Casimir effect]] and other phenomena. See ''[http://calphysics.org/zpe.html Zero Point Energy and Zero Point Field]''. See also ''[http://www.phys.ualberta.ca/~therman/lowtemp/projects1.htm Solid Helium] {{Webarchive|url=https://web.archive.org/web/20080212140020/http://www.phys.ualberta.ca/~therman/lowtemp/projects1.htm |date=2008-02-12 }}'' by the University of Alberta's Department of Physics to learn more about ZPE's effect on [[Bose–Einstein condensate]]s of helium.

Although absolute zero ({{nowrap|1=''T'' = 0}}) is not a state of zero molecular motion, it ''is''&nbsp;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. {{Anchor|thought experiment}}To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two {{nowrap|1=''T'' = 0}} helium atoms in zero gravity are carefully positioned and observed to have an average separation of {{val|620|ul=pm}} between them (a gap of ten atomic diameters). It is an "average" separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83&nbsp;yoctokelvins (1&nbsp;yK = {{val|1|e=-24|u=K}}). This is done in a way that directs this atom's velocity vector at the other atom. With 83&nbsp;yK of kinetic energy between them, the 620 pm gap through their common [[Barycentric coordinates (astronomy)|barycenter]] would close at a rate of 719&nbsp;pm/s and they would collide after 0.862 second. This is the same speed as shown in the ''[[#Overview|Fig.&nbsp;1]] ''animation above. Before being given the kinetic kick, both {{nowrap|1=''T'' = 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 {{nowrap|1=''T'' = 0}}, no [[kinetic energy]] is available for transfer to other systems.

Note too that absolute zero serves as the baseline atop which [[thermodynamics]] and its [[thermodynamic equations|equations]] are founded because they deal with the exchange of thermal energy between "''systems''" (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPE-induced particle motion ''within'' 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, ''virtual'' photons are still emitted at {{nowrap|1=''T'' = 0}}. Such photons are called "virtual" because they can't be intercepted and observed. Furthermore, this ''zero-point radiation'' has a unique ''zero-point spectrum''. However, even though a {{nowrap|1=''T'' = 0}} system emits zero-point radiation, no net heat flow ''Q'' out of such a system can occur because if the surrounding environment is at a temperature greater than {{nowrap|1=''T'' = 0}}, heat will flow inward, and if the surrounding environment is at '{{nowrap|1=''T'' = 0}}, there will be an equal flux of ZP radiation both inward and outward. A similar ''Q ''equilibrium exists at {{nowrap|1=''T'' = 0}} with the ZPE-induced [[spontaneous emission]] of photons (which is more properly called a ''stimulated'' 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 ''zero-kelvin point''. [http://pra.aps.org/abstract/PRA/v42/i4/p1847_1 ''Derivation of the classical electromagnetic zero-point radiation spectrum via a classical thermodynamic operation involving van der Waals forces''], Daniel C. Cole, Physical Review A, '''42''' (1990) 1847.</ref> Though the atoms in, for instance, a container of liquid [[helium]] that was ''precisely'' at absolute zero would still jostle slightly due to zero-point energy, a [[Carnot cycle|theoretically perfect heat engine]] with such helium as one of its [[working fluid]]s could never transfer any net kinetic energy ([[Heat|heat energy]]) to the other working fluid and no [[Work (thermodynamics)|thermodynamic work]] could occur.

Temperature is generally expressed in absolute terms when scientifically examining temperature's interrelationships with certain other physical properties of matter such as its [[Volume (thermodynamics)|volume]] or [[pressure]] (see [[Gay-Lussac's law]]), or the wavelength of its emitted [[black-body radiation]]. Absolute temperature is also useful when calculating chemical reaction rates (see [[Arrhenius equation]]). Furthermore, absolute temperature is typically used in [[cryogenics]] and related phenomena like [[superconductivity]], as per the following example usage:
"Conveniently, tantalum's transition temperature (''T''{{sub|c}}) of 4.4924 kelvins is slightly above the 4.2221&nbsp;K boiling point of helium."

== Rankine scale ==
{{main|Rankine scale}}
Though there have been many [[Conversion of units of temperature|other temperature scales]] throughout history, there have been only two scales for measuring thermodynamic temperature which have absolute zero as their null point (0): The Kelvin scale and the Rankine scale.

Throughout the scientific world where modern measurements are nearly always made using the International System of Units, thermodynamic temperature is measured using the Kelvin scale. The Rankine scale is part of [[English Engineering Units|English engineering units]] and finds use in certain engineering fields, particularly in legacy reference works. The Rankine scale uses the ''degree Rankine'' (symbol: °R) as its unit, which is the same magnitude as the [[Fahrenheit|degree Fahrenheit]] (symbol: °F).

A unit increment of one kelvin is exactly 1.8 times one degree Rankine; thus, to convert a specific temperature on the Kelvin scale to the Rankine scale, {{nowrap|1='''{{math|''x''}} K = 1.8 {{math|''x''}} °R'''}}, and to convert from a temperature on the Rankine scale to the Kelvin scale, {{nowrap|1='''{{math|''x''}} °R = {{math|''x''}}/1.8 K'''}}. Consequently, absolute zero is "0" for both scales, but the melting point of water ice (0&nbsp;°C and 273.15&nbsp;K) is 491.67&nbsp;°R.

To convert temperature ''intervals'' (a span or difference between two temperatures), the formulas from the preceding paragraph are applicable; for instance, an interval of 5 kelvins is precisely equal to an interval of 9 degrees Rankine.

== Modern redefinition of the kelvin ==

For 65 years, between 1954 and the [[2019 revision of the SI]], a temperature interval of one kelvin was defined as {{sfrac|273.16}} of the temperature difference between the [[triple point]] of water and absolute zero. The 1954 resolution by the [[International Bureau of Weights and Measures]] (BIPM), plus later resolutions and publications, defined the triple point of water as precisely 273.16&nbsp;K and acknowledged that it was "common practice" to accept that due to previous conventions (namely, that 0&nbsp;°C had long been defined as the melting point of water and that the triple point of water had long been experimentally determined to be indistinguishably close to 0.01&nbsp;°C), the difference between the Celsius scale and Kelvin scale is accepted as 273.15&nbsp;K; which is to say, 0&nbsp;°C corresponds to 273.15&nbsp;K.<ref name="BIPMbrocure">{{cite web |url=http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html |title=SI brochure, section 2.1.1.5 |access-date=9 May 2008 |publisher=[[International Bureau of Weights and Measures]] |archive-url=https://web.archive.org/web/20070926215600/http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html |archive-date=26 September 2007}}</ref> The net effect of this as well as later resolutions was twofold: 1) they defined absolute zero as precisely 0&nbsp;K, and 2) they defined that the triple point of special isotopically controlled water called [[Vienna Standard Mean Ocean Water]] occurred at precisely 273.16&nbsp;K and 0.01&nbsp;°C. One effect of the aforementioned resolutions was that the melting point of water, while ''very'' close to 273.15&nbsp;K and 0&nbsp;°C, was not a defining value and was subject to refinement with more precise measurements.

The 1954 BIPM standard did a good job of establishing—within the uncertainties due to [[isotope|isotopic variations]] between water samples—temperatures around the freezing and triple points of water, but required that ''intermediate values'' between the triple point and absolute zero, as well as extrapolated values from room temperature and beyond, to be experimentally determined via apparatus and procedures in individual labs. This shortcoming was addressed by the [[International Temperature Scale of 1990]], or ITS{{nbh}}90, which defined 13 additional points, from 13.8033&nbsp;K, to 1,357.77&nbsp;K. While definitional, ITS{{nbh}}90 had—and still has—some challenges, partly because eight of its extrapolated values depend upon the melting or freezing points of metal samples, which must remain exceedingly pure lest their melting or freezing points be affected—usually depressed.

The 2019 revision of the SI was primarily for the purpose of decoupling much of the SI system's definitional underpinnings from the [[kilogram]], which was the last physical artifact defining an [[SI base unit]] (a platinum/iridium cylinder stored under three nested bell jars in a safe located in France) and which had highly questionable stability. The solution required that four physical constants, including the Boltzmann constant, be definitionally fixed.

Assigning the Boltzmann constant a precisely defined value had no practical effect on modern thermometry except for the most exquisitely precise measurements. Before the revision, the triple point of water was exactly 273.16&nbsp;K and 0.01&nbsp;°C and the Boltzmann constant was experimentally determined to be {{val|1.38064903|(51)|e=-23|u=J|up=K}}, where the "(51)" denotes the uncertainty in the two least significant digits (the 03) and equals a [[Standard deviation|relative standard uncertainty]] of 0.37&nbsp;ppm.<ref name=codata2017>
{{cite journal
 |title=The CODATA 2017 values of ''h'', ''e'', ''k'', and ''N''<sub>A</sub> for the revision of the SI
 |collaboration=Committee on Data for Science and Technology (CODATA) Task Group on Fundamental Constants
 |first1=D B |last1=Newell |first2=F |last2=Cabiati |first3=J |last3=Fischer |first4=K |last4=Fujii
 |first5=S G |last5=Karshenboim |first6=H S |last6=Margolis |first7=E |last7=de Mirandés
 |first8=P J |last8=Mohr |first9=F |last9=Nez |first10=K |last10=Pachucki |first11=T J |last11=Quinn
 |first12=B N |last12=Taylor |first13=M |last13=Wang |first14=B M |last14=Wood |first15=Z |last15=Zhang
 |journal=Metrologia |volume=55 |issue=1 |pages=L13–L16
 |date=29 January 2018
 |doi=10.1088/1681-7575/aa950a |bibcode=2018Metro..55L..13N
 |doi-access=free
}}</ref> Afterwards, by defining the Boltzmann constant as exactly {{val|1.380649|e=-23|u=J|up=K}}, the 0.37&nbsp;ppm uncertainty was transferred to the triple point of water, which became an experimentally determined value of {{val|273.1600|0.0001|u=K}} ({{val|0.0100|0.0001|u=°C}}). That the triple point of water ended up being exceedingly close to 273.16&nbsp;K after the SI revision was no accident; the final value of the Boltzmann constant was determined, in part, through clever experiments with [[argon]] and helium that used the triple point of water for their key reference temperature.<ref>{{cite web|url=https://www.nist.gov/programs-projects/acoustic-thermometry|title=SI Redefinition – Kelvin: Boltzmann Constant|work=NIST |access-date=13 Dec 2020|publisher=[[National Institute of Standards and Technology]]|archive-url=https://web.archive.org/web/20200701093442/https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant|archive-date=1 July 2020}}</ref><ref>{{cite web|url=https://www.nist.gov/programs-projects/acoustic-thermometry|title=Acoustic Thermometry|work=NIST |access-date=13 Dec 2020|publisher=[[National Institute of Standards and Technology]]|archive-url=https://web.archive.org/web/20200923025023/https://www.nist.gov/programs-projects/acoustic-thermometry|archive-date=23 September 2020}}</ref>

Notwithstanding the 2019 revision, water triple-point cells continue to serve in modern thermometry as exceedingly precise calibration references at 273.16&nbsp;K and 0.01&nbsp;°C. Moreover, the triple point of water remains one of the 14 calibration points comprising ITS{{nbhyph}}90, which spans from the triple point of hydrogen (13.8033&nbsp;K) to the freezing point of copper (1,357.77&nbsp;K), which is a nearly hundredfold range of thermodynamic temperature.

== Relationship of temperature, motions, conduction, and thermal energy ==

[[Image:Translational motion.gif|thumb|upright=1.4|'''Figure 1''' The ''translational motion'' of fundamental particles of nature such as atoms and molecules is directly related to temperature. Here, the size of [[helium]] atoms relative to their spacing is shown to scale under 1950 [[atmosphere (unit)|atmospheres]] of pressure. These room-temperature atoms have a certain average speed (slowed down here two trillion-fold). At any given instant however, a particular helium atom may be moving much faster than average while another may be nearly motionless. Five atoms are colored red to facilitate following their motions. This animation illustrates [[statistical mechanics]], which is the science of how the group behavior of a large collection of microscopic objects emerges from the kinetic properties of each individual object.]]

=== Nature of kinetic energy, translational motion, and temperature ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->

The thermodynamic temperature of any ''bulk quantity'' 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 ''translational motion''. These simple movements in the three {{math|''x''}}-, {{math|''y''}}-, and {{math|''z''}}-axes dimensions of space means the particles move in the three spatial ''[[degrees of freedom (physics and chemistry)|degrees of freedom]]''. This particular form of kinetic energy is sometimes referred to as ''kinetic temperature''. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the [[ideal gas law]]'s formula {{math|1=''pV'' = ''nRT''}} and is embodied in the [[gas laws]].

Though the kinetic energy borne exclusively in the three translational degrees of freedom comprise the thermodynamic temperature of a substance, molecules, as can be seen in [[#Internal motions of molecules and internal energy|''Fig.&nbsp;3'']], can have other degrees of freedom, all of which fall under three categories: bond length, bond angle, and rotational. All three additional categories are not necessarily available to all molecules, and even for molecules that ''can'' experience all three, some can be "frozen out" below a certain temperature. Nonetheless, all those degrees of freedom that are available to the molecules under a particular set of conditions contribute to the [[specific heat capacity]] of a substance; which is to say, they increase the amount of heat (kinetic energy) required to raise a given amount of the substance by one kelvin or one degree Celsius.

The relationship of kinetic energy, mass, and velocity is given by the formula {{math|1=''E''<sub>k</sub>&nbsp;=&nbsp;{{sfrac|2}}''mv''{{i sup|2}}}}.<ref>At non-[[Special relativity|relativistic]] temperatures of less than about 30&nbsp;GK, [[classical mechanics]] are sufficient to calculate the velocity of particles. At 30&nbsp;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 [[Lorentz factor]]). Thus, the classic Newtonian formula for kinetic energy is in error less than half a percent for temperatures less than 30&nbsp;GK.</ref> 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.

The [[Boltzmann constant]] relates the thermodynamic temperature of a gas to the mean kinetic energy of a particle's translational motion:
<math display="block">\tilde{E} = \frac{3}{2} k_\text{B} T</math>
where:
* <math display="inline"> \tilde{E}</math> is the mean kinetic energy for an individual particle
* {{math|''k''<sub>B</sub>}} is the Boltzmann constant
* {{mvar|T}} is the thermodynamic temperature of the bulk quantity of the substance

[[Image:Maxwell Dist-Inverse Speed.png|thumb|upright=1.4|'''Figure 2''' 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 ''[[#Diffusion of thermal energy: black-body radiation|Fig.&nbsp;5]]'' below.]]
While the Boltzmann constant is useful for finding the mean kinetic energy in a sample of particles, it is important to note that even when a substance is isolated and in [[thermodynamic equilibrium]] (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in ''[[#Nature of kinetic energy, translational motion, and temperature|Fig.&nbsp;1]]'' 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 [[Maxwell–Boltzmann distribution]]. The graph shown here in ''Fig.&nbsp;2'' shows the speed distribution of 5500&nbsp;K helium atoms. They have a ''most probable'' speed of 4.780&nbsp;km/s (0.2092&nbsp;s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the {{math|''x''}}-axis to the right). This graph uses ''inverse speed'' for its {{math|''x''}}-axis so the shape of the curve can easily be compared to the curves in ''[[#Diffusion of thermal energy: black-body radiation|Fig.&nbsp;5]]'' below. In both graphs, zero on the {{math|''x''}}-axis represents infinite temperature. Additionally, the {{math|''x''}}- and {{math|''y''}}-axes on both graphs are scaled proportionally.

==== High speeds of translational motion ====
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 [[fluid]] produces [[Brownian motion]] that can be seen with an ordinary microscope. The translational motions of elementary particles are ''very'' fast<ref>Even room–temperature air has an average molecular translational ''speed'' of 1822&nbsp;km/hour. Assumptions: Average molecular weight of wet air = 28.838&nbsp;g/mol and {{mvar|T}} = 296.15&nbsp;K. Assumption's primary variables: An altitude of 194&nbsp;m above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23&nbsp;°C, a dew point of 9&nbsp;°C (40.85% relative humidity), and {{cvt|lk=in|760|mmHg|kPa}} sea level–corrected barometric pressure.</ref> and temperatures close to [[absolute zero]] are required to directly observe them. For instance, when scientists at the [[National Institute of Standards and Technology|NIST]] achieved a record-setting low temperature of 700&nbsp;nK (billionths of a kelvin) in 1994, they used [[optical lattice]] laser equipment to [[Adiabatic process|adiabatically]] cool [[caesium|cesium]] atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7&nbsp;mm per second in order to calculate their temperature.<ref>{{cite journal |title=Adiabatic Cooling of Cesium to 700&nbsp;nK in an Optical Lattice |first=A. |last=Kastberg |display-authors=etal|journal=Physical Review Letters |volume=74 |issue=9 |date=27 February 1995 |pages=1542–1545 |doi=10.1103/PhysRevLett.74.1542 |pmid=10059055 |bibcode=1995PhRvL..74.1542K }} A record cold temperature of 450&nbsp;[[Kelvin#SI prefixes|pK]] in a Bose–Einstein condensate of sodium atoms (achieved by A. E. Leanhardt ''et al.'' of [[Massachusetts Institute of Technology|MIT]]){{cn|date=September 2024}} equates to an average vector-isolated atom velocity of 0.4&nbsp;mm/s and an average atom speed of 0.7&nbsp;mm/s.</ref> Formulas for calculating the velocity and speed of translational motion are given in the following footnote.<ref name="Boltzmann">The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature as follows:
<math display="block">\tilde{v}^2 = \frac {k_\text{B} T}{m}</math>
where
* <math display="inline">\tilde{v}</math> is the vector-isolated mean velocity of translational particle motion
* {{math|''k''<sub>B</sub>}} ([[Boltzmann constant]])
* {{mvar|T}} is the thermodynamic temperature
* {{mvar|m}} is the [[molecular mass]] of the substance

The mean ''speed'' (not vector-isolated velocity) of an atom or molecule along any arbitrary path is calculated as follows:
<math display="block">\tilde{s} = \tilde{v} \cdot \sqrt{3}</math>
where <math display="inline">\tilde{s}</math> is the mean speed of translational particle motion.

The mean energy of the translational motions of a substance's constituent particles correlates to their mean ''speed'', not velocity. Thus, substituting <math display="inline">\tilde{s}</math> for {{mvar|v}} in the classic formula for kinetic energy, {{math|1=''E''<sub>k</sub> = {{sfrac|2}}''mv''{{i sup|2}}}} produces precisely the same value as does {{math|1=''E''<sub>mean</sub> = 3/2''k''<sub>B</sub>''T''}} (as shown in {{section link|#Nature of kinetic energy, translational motion, and temperature}}). 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 ''[[#Notes|Note 1]]'' above).</ref>

[[File:Argon atom at 1E-12 K.gif|thumb|left|upright=1.4|'''Figure 2.5''' This simulation illustrates an argon atom as it would appear through a 400-power optical microscope featuring a reticle graduated with 50&nbsp;μm (0.05&nbsp;mm) tick marks. This atom is moving with a velocity of 14.43&nbsp;μm/s, which gives the atom a kinetic temperature of one-trillionth of a kelvin. The atom requires 13.9 seconds to travel 200&nbsp;μm (0.2&nbsp;mm). Though the atom is being invisibly jostled due to zero-point energy, its translational motion seen here comprises all its kinetic energy.]]It is neither difficult to imagine atomic motions due to kinetic temperature, nor distinguish between such motions and those due to zero-point energy. Consider the following hypothetical thought experiment, as illustrated in ''Fig.&nbsp;2.5'' at left, with an atom that is exceedingly close to absolute zero. Imagine peering through a common optical microscope set to 400 power, which is about the maximum practical magnification for optical microscopes. Such microscopes generally provide fields of view a bit over 0.4&nbsp;mm in diameter. At the center of the field of view is a single levitated argon atom (argon comprises about 0.93% of air) that is illuminated and glowing against a dark backdrop. If this argon atom was at a beyond-record-setting ''one-trillionth'' of a kelvin above absolute zero,<ref>One-trillionth of a kelvin is to one kelvin as the thickness of two sheets of kitchen aluminum foil (0.04&nbsp;mm) is to the distance around Earth at the equator.</ref> and was moving perpendicular to the field of view towards the right, it would require 13.9 seconds to move from the center of the image to the 200&nbsp;μm tick mark. As the argon atom slowly moved, the positional jitter due to zero-point energy would be much less than the 200&nbsp;nm (0.0002&nbsp;mm) resolution of an optical microscope. Importantly, the atom's translational velocity of 14.43&nbsp;μm/s constitutes all its retained kinetic energy due to not being precisely at absolute zero. Were the atom ''precisely'' at absolute zero, imperceptible jostling due to zero-point energy would cause it to very slightly wander, but the atom would perpetually be located, on average, at the same spot within the field of view. This is analogous to a boat that has had its motor turned off and is now bobbing slightly in relatively calm and windless ocean waters; even though the boat randomly drifts to and fro, it stays in the same spot in the long term and makes no headway through the water. Accordingly, an atom that was precisely at absolute zero would not be "motionless", and yet, a statistically significant collection of such atoms would have zero net kinetic energy available to transfer to any other collection of atoms. This is because regardless of the kinetic temperature of the second collection of atoms, they too experience the effects of zero-point energy. Such are the consequences of [[statistical mechanics]] and the nature of thermodynamics.

==== Internal motions of molecules and internal energy ====
[[Image:Thermally Agitated Molecule.gif|thumb|upright=1.1|'''Figure 3''' Molecules have internal structures because they are composed of atoms that have different ways of moving within molecules. Being able to store kinetic energy in these ''internal degrees of freedom'' contributes to a substance's ''[[specific heat capacity]]'', or internal energy, allowing it to contain more internal energy at the same temperature.]]
As mentioned above, there are other ways molecules can jiggle besides the three translational degrees of freedom that imbue substances with their kinetic temperature. As can be seen in the animation at right, [[molecule]]s are complex objects; they are a population of atoms and thermal agitation can strain their internal [[chemical bond]]s in three different ways: via rotation, bond length, and bond angle movements; these are all types of ''internal degrees of freedom''. This makes molecules distinct from ''[[monatomic]]'' substances (consisting of individual atoms) like the [[noble gas]]es [[helium]] and [[argon]], which have only the three translational degrees of freedom (the {{math|''x''}}-, {{math|''y''}}-, and {{math|''z''}}-axes). Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an ''internal temperature''. Even though these motions are called "internal", the external portions of molecules still move—rather like the jiggling of a stationary [[water balloon]]. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as internal energy is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the [[equipartition theorem]], which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-[[local thermodynamic equilibrium]] (LTE) phenomena such as [[combustion]], the [[sublimation (chemistry)|sublimation]] of solids, and the [[diffusion]] of hot gases in a partial vacuum.

The kinetic energy stored internally in molecules causes substances to contain more heat energy at any given temperature and to absorb additional internal energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions, is not contributing to the molecules' translational motions at that same instant.<ref>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 ''single'' 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 ''destructively'' interfering with translational motion during a collision with a container's walls or another molecule. Averaged across any bulk quantity of a gas, the internal thermal motions of molecules have zero net effect upon the temperature, pressure, or volume of a gas. Molecules' internal degrees of freedom simply provide additional locations where kinetic energy is stored. This is precisely why molecular-based gases have greater specific internal capacity than monatomic gases (where additional internal energy must be added to achieve a given temperature rise).</ref> This extra kinetic energy simply increases the amount of internal energy that substance absorbs for a given temperature rise. This property is known as a substance's [[specific heat capacity]].

Different molecules absorb different amounts of internal energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances' molecules possess more internal degrees of freedom than others do. For instance, room-temperature [[nitrogen]], which is a [[diatomic]] molecule, has ''five'' active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has five-thirds the specific heat capacity per [[mole (unit)|mole]] (a specific number of molecules) as do the monatomic gases.<ref>When measured at constant-volume since different amounts of work must be performed if measured at constant-pressure. Nitrogen's {{math|''C<sub>v</sub>H''}} (100&nbsp;kPa, 20&nbsp;°C) equals {{val|20.8|u=J⋅mol<sup>–1</sup>⋅K<sup>–1</sup>}} vs. the monatomic gases, which equal 12.4717&nbsp;J⋅mol<sup>–1</sup>⋅K<sup>–1</sup>. {{cite book |first=W. H. |last=Freeman |title=Physical Chemistry |url=http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf |chapter=Part 3: Change|archive-url=https://wayback.archive-it.org/all/20070927061428/http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf |archive-date=2007-09-27 |at=Exercise 21.20b, p.&nbsp;787}} See also {{cite web |first=R. |last=Nave |publisher=Georgia State University |url=http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html |title=Molar Specific Heats of Gases |website=HyperPhysics}}</ref> Another example is [[gasoline]] (see [[Specific heat capacity#Table of specific heat capacities|table]] showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.

=== Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->

[[Image:1D normal modes (280 kB).gif|thumb|upright=1.2|'''Figure 4''' The temperature-induced translational motion of particles in solids takes the form of ''[[phonon]]s''. Shown here are phonons with identical [[amplitude]]s but with [[wavelength]]s ranging from 2 to 12 average inter-molecule separations (''a'').]]

''[[Heat conduction]]'' is the diffusion of thermal energy from hot parts of a system to cold parts. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term ''bulk'' 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 [[entropy]] increases).

One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers [[momentum]] from particle to particle in collisions. In gases, these translational motions are of the nature shown above in ''[[#Relationship of temperature, motions, conduction, and thermal energy|Fig.&nbsp;1]]''. 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; [[Convection (heat transfer)|convection]] speeds this process even more.<ref>The ''speed'' 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 ''heat [[flux]]'' (the thermal power passing per area) through gases is comparatively low. This is why the dead-air spaces in [[insulated glazing|multi-pane windows]] have insulating qualities.</ref>

Translational motion in ''solids'', however, takes the form of ''[[phonon]]s'' (see ''Fig.&nbsp;4'' at right). Phonons are constrained, quantized wave packets that travel at the speed of sound of a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is ''usually'' inefficient<ref>[[Diamond]] is a notable exception – it also has exceptionally ''high'' [[thermal conductivity]].</ref> and such solids are considered ''thermal insulators'' (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.

[[Metal]]s 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 ''conduction [[electron]]s''. This is why there is a near-perfect correlation between metals' [[thermal conductivity]] and their [[electrical conductivity]].<ref>Correlation is 752&nbsp;(W⋅m<sup>−1</sup>⋅K<sup>−1</sup>)/(MS⋅cm), {{mvar|σ}}&nbsp;=&nbsp;81, through a 7:1 range in conductivity. Value and standard deviation based on data for Ag, Cu, Au, Al, Ca, Be, Mg, Rh, Ir, Zn, Co, Ni, Os, Fe, Pa, Pt, and Sn. Data from ''CRC Handbook of Chemistry and Physics'', 1st Student Edition.</ref> Conduction electrons imbue metals with their extraordinary conductivity because they are ''[[Delocalized electron|delocalized]]'' (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]]'' (for more on ZPE, see ''[[#Notes|Note 1]]'' below). Furthermore, electrons are relatively light with a rest mass only {{frac|1836}} that of a [[proton]]. As [[Isaac Newton]] wrote with his [[Newton's laws of motion#Newton's third law|third law of motion]],

{{Quote|Law #3: All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.}}

However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, thermal energy is readily borne by mobile conduction electrons. Additionally, because they are delocalized and ''very'' fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.

=== Diffusion of thermal energy: black-body radiation ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->

[[Image:Wiens law.svg|thumb|upright=1.4|'''Figure 5''' The spectrum of black-body radiation has the form of a Planck curve. A 5500&nbsp;K black-body has a peak emittance wavelength of 527&nbsp;nm. Compare the shape of this curve to that of a Maxwell distribution in ''[[#Nature of kinetic energy, translational motion, and temperature|Fig.&nbsp;2]]''&nbsp;above.]]
[[Thermal radiation]] is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal [[photon]]s (known as [[black-body radiation]]). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even ''individual molecules'' 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 [[wavelength]]s in a spectrum that has a bell curve-like shape called a [[Planck's law of black body radiation|Planck curve]] (see graph in ''Fig.&nbsp;5'' at right). The top of a Planck curve ([[Wien's displacement law|the peak emittance wavelength]]) is located in a particular part of the [[electromagnetic spectrum]] depending on the temperature of the black-body. Substances at extreme [[cryogenics|cryogenic]] temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short [[gamma ray]]s (see {{section link|#Table of thermodynamic temperatures}}).

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.

As established by the [[Stefan–Boltzmann law]], the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black-body at 824&nbsp;K (just short of glowing dull red) emits 60 times the radiant [[Power (physics)|power]] as it does at 296&nbsp;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 [[Incandescent light bulb|incandescent lamp]], black-body radiation can be the principal mechanism by which thermal energy escapes a system.

==== Table of thermodynamic temperatures ====
{{main article|Orders of magnitude (temperature)}}
The table below shows various points on the thermodynamic scale, in order of increasing temperature.

{| class="wikitable" style="text-align:center"
|-
!
!Temperature
!Peak emittance<br />[[wavelength]]<ref>The cited emission wavelengths are for true black bodies in equilibrium. In this table, only the sun so qualifies. [https://physics.nist.gov/cgi-bin/cuu/Value?bwien CODATA recommended value] of {{val|2.897771955|end=...|e=-3|u=m⋅K}} used for Wien displacement law constant ''b''.</ref> of<br />[[Wien's displacement law|black-body photons]]
|-
|style="text-align:right"|[[Absolute zero]]<br />(precisely by definition)
|0 K
|{{resize|140%|[[Infinity|∞]]}}<ref name="T0"/>
|-
|style="text-align:right"|Coldest measured<br />temperature<ref name="recordcold">A record cold temperature of 450&nbsp;±80&nbsp;pK in a Bose–Einstein condensate (BEC) of sodium (<sup>23</sup>Na) atoms was achieved in 2003 by researchers at [[Massachusetts Institute of Technology|MIT]]. {{cite journal |title=Cooling Bose–Einstein Condensates Below 500 Picokelvin |first=A. E. |last=Leanhardt |display-authors=etal |journal=Science |volume=301 |issue=5639 |date=12 September 2003 |page=1515|doi=10.1126/science.1088827 |pmid=12970559 |bibcode=2003Sci...301.1513L }} The thermal velocity of the atoms averaged about 0.4&nbsp;mm/s. This record's peak emittance black-body radiation wavelength of 6400&nbsp;km is roughly the radius of Earth.</ref>
|450 [[Orders of magnitude (temperature)#SI multiples|pK]]
|6,400 [[Kilometre|km]]
|-
|style="text-align:right"|One [[Orders of magnitude (temperature)#SI multiples|millikelvin]]<br />(precisely by definition)
|0.001 K
|2.897&nbsp;77 [[Metre|m]]<br /> (radio, [[FM broadcasting|FM band]])<ref>The peak emittance wavelength of 2.897&nbsp;77&nbsp;m is a frequency of 103.456&nbsp;MHz.</ref>
|-
|style="text-align:right"|[[Cosmic microwave background|cosmic microwave<br />background radiation]]
|2.725&nbsp;K
|1.063&nbsp;[[Metre|mm]] (peak wavelength)
|-
|style="text-align:right"|[[Vienna Standard Mean Ocean Water|Water]]'s [[triple point]]
|273.16 K
|10.6083 [[Metre#SI prefixed forms of metre|μm]]<br />(long wavelength [[Infrared|I.R.]])
|-
|style="text-align:right"|[[ISO 1]] standard temperature<br />for precision [[metrology]]<br />(precisely 20&nbsp;°C by definition)
|293.15 K
|{{val|9.88495}}&nbsp;μm<br />(long wavelength [[Infrared|I.R.]])
|-
|-
|style="text-align:right"|[[Incandescent light bulb|Incandescent lamp]]{{efn-ua|For a true black body (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.}}
|2500 K{{efn-ua|The 2500&nbsp;K value is approximate.}}
|1.16&nbsp;μm<br />(near [[infrared]]){{efn-ua|name=Photosphere|Effective photosphere temperature.}}<!--Should this be here?-->
|-
|[[Sun]]'s visible surface<ref>{{Cite web |year=2015 |title=Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties |url=https://iau.org/static/resolutions/IAU2015_English.pdf |access-date=2023-08-27 |archive-date=2016-01-28 |archive-url=https://web.archive.org/web/20160128180606/https://iau.org/static/resolutions/IAU2015_English.pdf |url-status=dead }}</ref><ref>{{Cite book |last1=Hertel |first1=Ingolf V. |url=https://books.google.com/books?id=vr0UBQAAQBAJ&dq=5772+K+sun&pg=PA35 |title=Atoms, Molecules and Optical Physics 1: Atoms and Spectroscopy |last2=Schulz |first2=Claus-Peter |date=2014-10-24 |publisher=Springer |isbn=978-3-642-54322-7 |page=35 |language=en}}</ref><ref>{{Cite book |last1=Vignola |first1=Frank |url=https://books.google.com/books?id=q9WlDwAAQBAJ&dq=5772+K+sun&pg=PP26 |title=Solar and Infrared Radiation Measurements |edition=2nd |last2=Michalsky |first2=Joseph |last3=Stoffel |first3=Thomas |date=2019-07-30 |publisher=CRC Press |isbn=978-1-351-60020-0 |pages=chapter 2.1, 2.2 |language=en}}</ref><ref>{{Cite web |title=Sun Fact Sheet |url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html |access-date=2023-08-27 |website=NASA Space Science Center Coordinated Archive |archive-date=1998-02-22 |archive-url=https://web.archive.org/web/19980222081658/https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html |url-status=dead }}</ref>
|5772 K
|502&nbsp;[[Metre#SI prefixed forms of metre|nm]]<br />([[Color#Spectral colors|green light]])
|-
|style="text-align:right"|[[Lightning|Lightning bolt's]]<br />channel
|{{val|28,000|u=K}}
|100&nbsp;nm<br />(far [[ultraviolet]] light)
|-
|style="text-align:right"|[[Sun#Core|Sun's core]]
|16 [[Orders of magnitude (temperature)#SI multiples|MK]]
|0.18&nbsp;nm ([[X-ray]]s)
|-
|style="text-align:right"|[[Thermonuclear explosion]]<br />(peak temperature)<ref>The 350&nbsp;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&nbsp;MK. {{cite web |title=Nuclear Weapons Frequently Asked Questions |at=3.2.5 Matter At High Temperatures |url=https://nuclearweaponarchive.org/Nwfaq/Nfaq3.html#nfaq3.2}}{{fcn|date=September 2024}} All referenced data was compiled from publicly available sources.</ref>
|align="center"|350 MK
|align="center"|8.3 × 10<sup>−3</sup> nm<br />([[gamma ray]]s)
|-
|style="text-align:right"|Sandia National Labs'<br />[[Z Pulsed Power Facility|Z machine]]{{efn-ua|For a true black body (which the plasma was not). The Z machine's dominant emission originated from 40&nbsp;MK electrons (soft x–ray emissions) within the plasma.}}<ref>Peak temperature for a bulk quantity of matter was achieved by a pulsed-power machine used in fusion physics experiments. The term "bulk quantity" draws a distinction from collisions in particle accelerators wherein high "temperature" applies only to the debris from two subatomic particles or nuclei at any given instant. The >2&nbsp;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 {{val|3.58|0.41|u=GK}} ({{val|309|35|u=keV}}) for 3&nbsp;ns (ns 112 through 115). {{cite journal |title=Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2&nbsp;×&nbsp;10<sup>9</sup> Kelvin |first=M. G. |last=Haines |display-authors=etal |journal=Physical Review Letters |volume=96 |issue=7 |id=No. 075003 |year=2006 |article-number=075003 |doi=10.1103/PhysRevLett.96.075003|pmid=16606100 |bibcode=2006PhRvL..96g5003H }} For a press summary of this article, see {{cite web |date=March 8, 2006 |title=Sandia's Z machine exceeds two billion degrees Kelvin |publisher=Sandia |url=http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html |archive-url=https://web.archive.org/web/20060702185740/http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html |archive-date=2006-07-02 }}</ref>
|2 [[Orders of magnitude (temperature)#SI multiples|GK]]
|1.4 × 10<sup>−3</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|Core of a [[Silicon-burning process|high-mass star on its last day]]<ref>Core temperature of a high–mass (>8–11 solar masses) star after it leaves the main sequence on the [[Hertzsprung–Russell diagram]] and begins the [[Alpha reactions|alpha process]] (which lasts one day) of [[Silicon burning process|fusing silicon–28]] 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&nbsp;II [[supernova]].</ref>
|align="center"|3 GK
|align="center"|1 × 10<sup>−3</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|Merging binary [[neutron star]] system<ref>Based on a computer model that predicted a peak internal temperature of 30 MeV (350&nbsp;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&nbsp;km in diameter, and were orbiting around their barycenter (common center of mass) at about 390&nbsp;Hz during the last several milliseconds before they completely merged. The 350&nbsp;GK portion was a small volume located at the pair's developing common core and varied from roughly 1 to 7&nbsp;km across over a time span of around 5&nbsp;ms. Imagine two city-sized objects of unimaginable density orbiting each other at the same frequency as the G4 musical note (the 28th white key on a piano). At 350&nbsp;GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass 5% greater than its rest mass. {{cite journal|arxiv=astro-ph/0507099v2|doi=10.1111/j.1365-2966.2006.10238.x|title=Torus formation in neutron star mergers and well-localized short gamma-ray bursts|journal=Monthly Notices of the Royal Astronomical Society|volume=368|issue=4|pages=1489–1499|year=2006|last1=Oechslin|first1=R.|last2=Janka|first2=H.-T.|doi-access=free |bibcode=2006MNRAS.368.1489O|s2cid=15036056}} For a summary, see {{cite web|url=https://www.mpa-garching.mpg.de/mpa/research/current_research/hl2005-10/hl2005-10-en.html |title=Short Gamma-Ray Bursts: Death Throes of Merging Neutron Stars |publisher=Max-Planck-Institut für Astrophysik |access-date=24 September 2024}}</ref>
|350 GK
|8 × 10<sup>−6</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|[[Gamma-ray burst progenitors]]<ref>{{cite magazine |magazine=New Scientist |url=https://www.newscientist.com/article/mg20928026.300-eight-extremes-the-hottest-thing-in-the-universe.html |title=Eight extremes: The hottest thing in the universe |first=Stephen |last=Battersby |date=2 March 2011 |quote=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.}}</ref>
|1 [[Orders of magnitude (temperature)#SI multiples|TK]]
|3 × 10<sup>−6</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|[[CERN]]'s proton vs. nucleus collisions<ref>{{cite web |url=http://public.web.cern.ch/public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html |title=How do physicists study particles? |archive-url=https://web.archive.org/web/20071011103924/http://public.web.cern.ch/Public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html |archive-date=2007-10-11 |publisher=CERN}}</ref>
|10 TK
|3 × 10<sup>−7</sup> nm<br />(gamma rays)
|-
|}
{{notelist-ua}}

=== Heat of phase changes ===
[[Image:IceBlockNearJoekullsarlon.jpg|thumb|left|upright=1.4|'''Figure 6''' Ice and water: two phases of the same substance]]
The kinetic energy of particle motion is just one contributor to the total thermal energy in a substance; another is ''[[phase transition]]s'', which are the [[potential energy]] of molecular bonds that can form in a substance as it cools (such as during [[condensation|condensing]] and [[freezing]]). The thermal energy required for a phase transition is called ''[[latent heat]]''. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to ''break'' [[chemical bonds]] (such as during [[evaporation]] and [[melting]]). Almost everyone is familiar with the effects of phase transitions; for instance, [[steam]] at 100&nbsp;°C can cause severe burns much faster than the 100&nbsp;°C air from a [[blowdryer|hair dryer]]. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.

Even though thermal energy is liberated or absorbed during phase transitions, pure [[chemical element]]s, [[chemical compound|compounds]], and [[eutectic point|eutectic]] [[alloy]]s exhibit no temperature change whatsoever while they undergo them (see ''Fig.&nbsp;7'', below right). Consider one particular type of phase transition: melting. When a solid is melting, [[Crystal structure|crystal lattice]] [[chemical bond]]s are being broken apart; the substance is transitioning from what is known as a ''more ordered state'' to a ''less ordered state''. In ''Fig.&nbsp;7'', the melting of ice is shown within the lower left box heading from blue to green.

[[File:Energy through phase changes.png|thumb|upright=1.8|'''Figure 7''' 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&nbsp;kJ.]]

At one specific thermodynamic point, the [[melting point]] (which is 0&nbsp;°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 [[joule]] of added thermal energy only breaks the bonds of a specific quantity of its atoms or molecules,<ref>Water's enthalpy of fusion (0&nbsp;°C, 101.325&nbsp;kPa) equates to {{val|0.062284|u=eV}}&nbsp;per molecule so adding one joule of thermal energy to 0&nbsp;°C water ice causes {{val|1.0021|e=20}} water molecules to break away from the crystal lattice and become liquid.</ref> 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 [[popcorn]]: at a certain temperature, additional thermal energy cannot make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), thermal energy must be removed from a substance.

As stated above, the thermal energy required for a phase transition is called ''latent heat''. In the specific cases of melting and freezing, it is called ''[[Standard enthalpy change of fusion|enthalpy of fusion]]'' or ''heat of fusion''. 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&nbsp;kJ per mole for water and most of the metallic elements.<ref>Water's enthalpy of fusion is {{val|6.0095|u=kJ⋅mol<sup>−1</sup>}} K<sup>−1</sup> (0&nbsp;°C, 101.325&nbsp;kPa). {{cite web |website=Water Structure and Science |title=Water Properties (including isotopologues) |first=Martin |last=Chaplin |publisher=London South Bank University |url=http://www.lsbu.ac.uk/water/data.html |archive-url=https://web.archive.org/web/20201121051504/http://www.lsbu.ac.uk/water/water_properties.html |archive-date=2020-11-21}} The only metals with enthalpies of fusion ''not'' in the range of 6–30&nbsp;J&nbsp;mol<sup>−1</sup>&nbsp;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.</ref> 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&nbsp;kJ per mole.<ref>For xenon,  available values range from 2.3 to 3.1&nbsp;kJ/mol. {{cite web |url=https://www.webelements.com/webelements/elements/text/Xe/heat.html |website=WebElements |title=Xenon – 54Xe: the essentials |access-date=24 September 2024}} Helium's heat of fusion of only 0.021&nbsp;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.</ref> Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0&nbsp;°C into water at 0&nbsp;°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.<ref>''CRC Handbook of Chemistry and Physics'', 1st Student Edition{{fcn|date=September 2024}}</ref> The phase transition of [[boiling]] is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as ''[[standard enthalpy change of vaporization|enthalpy of vaporization]]'') is roughly 540 times that required for a one-degree increase.<ref>H<sub>2</sub>O specific heat capacity, {{math|''C<sub>p</sub>''}}&nbsp;= {{val|0.075327|u=kJ⋅mol<sup>−1</sup>⋅K<sup>−1</sup>}} (25&nbsp;°C); enthalpy of fusion&nbsp;= 6.0095&nbsp;kJ/mol (0&nbsp;°C, 101.325&nbsp;kPa); enthalpy of vaporization (liquid)&nbsp;= 40.657&nbsp;kJ/mol (100&nbsp;°C). {{cite web |website=Water Structure and Science |title=Water Properties (including isotopologues) |first=Martin |last=Chaplin |publisher=London South Bank University |url=http://www.lsbu.ac.uk/water/data.html |archive-url=https://web.archive.org/web/20201121051504/http://www.lsbu.ac.uk/water/water_properties.html |archive-date=2020-11-21}}</ref>

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 ''Fig.&nbsp;7''&nbsp;above); water vapors (gas phase) are liquefied on the skin with releasing a large amount of energy (enthalpy) to the environment including the skin, resulting in skin damage. In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient [[wet-bulb temperature]] that is dependent on [[relative humidity]]); the water evaporation on the skin takes a large amount of energy from the environment including the skin, reducing the skin temperature. Water's highly energetic enthalpy of vaporization is also an important factor underlying why ''solar pool covers'' (floating, insulated blankets that cover [[swimming pool]]s when the pools are not in use) are so effective at reducing heating costs: they prevent evaporation. (In other words, taking energy from water when it is evaporated is limited.) For instance, the evaporation of just 20&nbsp;mm of water from a 1.29&nbsp;m-deep pool chills its water {{convert|8.4|C-change}}.

=== Internal energy ===

The total energy of all translational and internal particle motions, including that of conduction electrons, plus the potential energy of phase changes, plus [[zero-point energy]]<ref name="T0"/> of a substance comprise the ''[[internal energy]]'' of it.

[[Image:Close-packed spheres, with umbrella light & camerea.jpg|thumb|left|'''Figure 8''' When many of the chemical elements, such as the [[noble gas]]es and [[platinum group|platinum-group metals]], freeze to a solid — the most ordered state of matter — their [[crystal structures]] have a ''[[close-packing|close-packed arrangement]]''. This yields the greatest possible packing density and the lowest energy state.]]

=== Internal energy at absolute zero ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->
As a substance cools, different forms of internal energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions is liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic energy or temperature decreases); the internal motions of molecules diminish (their internal energy or temperature decreases); conduction electrons (if the substance is an electrical conductor) travel ''somewhat'' slower;<ref>Mobile conduction electrons are ''delocalized'', 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 ''Fermi velocity'' of about {{val|1.6|e=6|u=m/s}}. Kinetic thermal energy adds to this speed and also causes delocalized electrons to travel farther away from the nuclei.</ref> and black-body radiation's peak emittance wavelength increases (the photons' energy decreases). When particles of a substance are as close as possible to complete rest and retain only ZPE (zero-point energy)-induced quantum mechanical motion, the substance is at the temperature of absolute zero ({{mvar|T}}&nbsp;=&nbsp;0).

[[Image:Liquid helium superfluid phase.jpg|thumb|right|upright=1.1|'''Figure 9''' Due to the effects of zero-point energy, helium at ambient pressure remains a [[superfluid]] even when exceedingly close to absolute zero; it will not freeze unless under 25 bar of pressure (c.&nbsp;25 atmospheres).]]
Whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero internal energy; one must be very precise with what one means by ''internal energy''. Often, all the phase changes that ''can'' occur in a substance, ''will'' have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, {{mvar|T}}&nbsp;=&nbsp;0 [[helium]] remains liquid at room pressure (''Fig.&nbsp;9'' at right) and must be under a pressure of at least {{convert|25|bar|MPa|abbr=on|lk=on}} to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21&nbsp;joules&nbsp;per&nbsp;mole) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures.

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 ''solid–solid phase transitions'' wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one.

The above complexities make for rather cumbersome blanket statements regarding the internal energy in {{mvar|T}}&nbsp;=&nbsp;0 substances. Regardless of pressure though, what ''can'' be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a ''[[close-packing|closest-packed arrangement]]'' (see ''Fig.&nbsp;8'', above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy.<ref name="T0"/><ref>No other [[crystal structure]] can exceed the 74.048% packing density of a ''closest-packed arrangement''. The two regular crystal lattices found in nature that have this density are ''[[hexagonal crystal system|hexagonal close packed]]'' (HCP) and ''[[cubic crystal system|face-centered cubic]]'' (FCC). These regular lattices are at the lowest possible energy state. [[Diamond]] is a closest-packed structure with an FCC crystal lattice. Note too that suitable crystalline chemical ''compounds'', 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 [[mineral]] known as ''magnesium aluminum [[spinel]]'' (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.</ref> One can also say that for a given substance at constant pressure, absolute zero is the point of lowest ''[[enthalpy]]'' (a measure of work potential that takes internal energy, pressure, and volume into consideration).<ref>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 [[beryllium]], [[osmium]], [[neon]], and [[iridium]] (but excludes helium), and therefore have zero latent heat of phase transitions to contribute to internal energy (symbol: ''U''). In the calculation of enthalpy (formula: {{math|''H''{{=}}''U''&nbsp;+&nbsp;''pV''}}), internal energy may exclude different sources of thermal energy (particularly ZPE) depending on the nature of the analysis. Accordingly, all {{mvar|T}}&nbsp;=&nbsp;0 closest-packed matter under a perfect vacuum has either minimal or zero enthalpy, depending on the nature of the analysis. {{cite journal |url=http://iupac.org/publications/pac/2001/pdf/7308x1349.pdf |title=Use of Legendre Transforms In Chemical Thermodynamics |first=Robert A. |last=Alberty |journal=Pure and Applied Chemistry |volume=73 |issue=8 |year=2001 |page=1349|doi=10.1351/pac200173081349 }}</ref> Lastly, all {{mvar|T}}&nbsp;=&nbsp;0 substances contain zero kinetic thermal energy.<ref name="T0"/><ref name="Boltzmann"/>
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== Practical applications for thermodynamic temperature ==
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 [[Gay-Lussac's law]] of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a cold {{Not a typo|gage}}<ref>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 ''within'' 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 ''pressure gage''. For the same reason, many of the largest American manufacturers of pressure transducers and instrumentation use the spelling ''{{Not a typo|gage pressure}}'' (the convention used here) in their formal documentation to distinguish it from the instrument, which is spelled ''pressure gauge''.</ref> pressure of 200&nbsp;[[pascal (unit)|kPa]]<!--

NOTE TO EDITORS:

Regarding the spelling of "GAGE" pressure, this is one of two legitimate spellings. Please read the second footnote attached to this paragraph when not in "edit" view. Wikipedia's official policy ([[Wikipedia:Manual of Style]]) is that the spelling convention used by the first major contributors should be retained. This article uses American English spelling conventions.

-->, then its [[absolute pressure]] is 300&nbsp;kPa.<ref>Pressure also must be in absolute terms. The air still in a tire at a [[Pressure measurement#Gauge|{{Not a typo|gage pressure}}]] of 0&nbsp;kPa expands too as it gets hotter. It is not uncommon for engineers to overlook that one must work in terms of absolute pressure when compensating for temperature. For instance, a dominant manufacturer of aircraft tires published a document on temperature-compensating tire pressure, which used {{Not a typo|gage pressure}} in the formula. However, the high {{Not a typo|gage pressures}} involved (180&nbsp;psi; 12.4 bar; 1.24 MPa) means the error would be quite small. With low-pressure automobile tires, where {{Not a typo|gage pressures}} are typically around 2&nbsp;bar (200&nbsp;kPa), failing to adjust to absolute pressure results in a significant error. {{cite web |title=Aircraft tire ratings |publisher=Air Michelin |url=http://airmichelin.com/pdfs/05%20-%20Aircraft%20Tire%20Ratings.pdf |archive-url=https://web.archive.org/web/20100215151931/http://airmichelin.com/pdfs/05%20-%20Aircraft%20Tire%20Ratings.pdf |archive-date=2010-02-15}}{{bsn|reason=Source lacks metadata|date=September 2024}}</ref><ref>A difference of 100&nbsp;kPa is used here instead of the 101.325&nbsp;kPa value of one [[atmosphere (unit)|standard atmosphere]]. In 1982, the [[International Union of Pure and Applied Chemistry]] (IUPAC) recommended that for the purposes of specifying the physical properties of substances, ''the standard pressure'' (atmospheric pressure) should be defined as precisely 100&nbsp;kPa (≈&nbsp;750.062&nbsp;Torr). Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100&nbsp;kPa equates to the mean pressure at an altitude of about 112&nbsp;m, which is closer to the 194&nbsp;m, worldwide median altitude of human habitation. For especially low-pressure or high-accuracy work, true atmospheric pressure must be measured. {{cite book |publisher=International Union of Pure and Applied Chemistry |title=Compendium of Chemical Terminology |url=http://goldbook.iupac.org/S05921.html |chapter=Standard pressure |date=2014 |doi=10.1351/goldbook.S05921 |edition=online 3rd}}</ref> Room temperature ("cold" in tire terms) is 296&nbsp;K. If the tire temperature is 20&nbsp;°C hotter (20&nbsp;kelvins), the solution is calculated as {{sfrac|316 K|296 K}}&nbsp;= 6.8% greater thermodynamic temperature ''and'' absolute pressure; that is, an absolute pressure of 320&nbsp;kPa, which is a {{Not a typo|gage pressure}} of 220&nbsp;kPa.
{{clear}}

== Relationship to ideal gas law ==

The thermodynamic temperature is closely linked to the [[ideal gas law]] and its consequences. It can be linked also to the second law of thermodynamics. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the [[energy conversion efficiency|efficiency]] of idealized [[heat engine]]s. Thus the [[ratio]] {{math|''T''<sub>2</sub>/''T''<sub>1</sub>}} of two temperatures {{math|''T''<sub>1</sub>}} and {{math|''T''<sub>2</sub>}} is the same in all absolute scales.

Strictly speaking, the temperature of a system is well-defined only if it is at [[thermal equilibrium]]. 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.

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 "[[entropy]]". To better understand the relationship between temperature and entropy, consider the relationship between heat, [[mechanical work|work]] and temperature illustrated in the [[Carnot cycle|Carnot heat engine]]. The engine converts heat into work by directing a temperature gradient between a higher temperature heat source, {{math|''T''<sub>H</sub>}}, and a lower temperature heat sink, {{math|''T''<sub>C</sub>}}, through a gas filled piston. The work done per cycle is equal in magnitude to net heat taken up, which is sum of the heat {{math|''q''<sub>H</sub>}} taken up by the engine from the high-temperature source, plus the waste [[Heat|heat given off]] by the engine, {{math|''q''<sub>C</sub>}} < 0.<ref name="PlanckBook">{{cite book |last=Planck |first=M. |author-link=Max Planck |title=Treatise on Thermodynamics |at=§§90, 137, eqs. (39), (40), and (65) |publisher=Dover Publications |year=1945}}</ref> The ''efficiency'' of the engine is the work divided by the heat put into the system or
<math display="block">\begin{align}\textrm{Efficiency} &= \frac {|w_\text{cy}|}{q_\text{H}} = \frac{q_\text{H}+q_\text{C}}{q_\text{H}} = 1 + \frac{q_\text{C}}{q_\text{H}}= 1 - \frac{|q_\text{C}|}{|q_\text{H}|} & (1)\end{align}</math>
where <math>w_\text{cy}</math> is the work done per cycle. Thus the efficiency depends only on {{math|{{abs|''q''<sub>C</sub>}} / {{abs|''q''<sub>H</sub>}}}}.

[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures {{math|''T''<sub>1</sub>}} and {{math|''T''<sub>2</sub>}} must have the same efficiency, that is to say, the efficiency is the function of only temperatures
<math display="block">\begin{align}\frac{|q_\text{C}|}{|q_\text{H}|} &= f(T_\text{H},T_\text{C}). & (2)\end{align}</math>

In addition, a reversible heat engine operating between a pair of thermal reservoirs at temperatures {{math|''T''<sub>1</sub>}} and {{math|''T''<sub>3</sub>}} must have the same efficiency as one consisting of two cycles, one between {{math|''T''<sub>1</sub>}} and another (intermediate) temperature {{math|''T''<sub>2</sub>}}, and the second between {{math|''T''<sub>2</sub>}} and {{math|''T''<sub>3</sub>}}. If this were not the case, then energy (in the form of {{mvar|q}}) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles as an engine design choice, and any reversible engine between the same reservoir at {{math|''T''<sub>1</sub>}} and {{math|''T''<sub>3</sub>}} must be equally efficient regardless of the engine design.

If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as below.
<math display="block">\begin{align}
\eta _1 &= 1 - \frac{|q_3|}{|q_1|} &= 1 - f(T_1, T_3)\\
\eta _2 &= 1 - \frac{|q_2|}{|q_1|} &= 1 - f(T_1, T_2) \\
\eta _3 &= 1 - \frac{|q_3|}{|q_2|} &= 1 - f(T_2, T_3)
\end{align}</math>

Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at {{math|''T''<sub>2</sub>}}. We also have used the fact that the heat <math>q_2</math> passes through the intermediate thermal reservoir at <math>T_2</math> without losing its energy. (I.e., <math>q_2</math> is not lost during its passage through the reservoir at <math>T_2</math>.) This fact can be proved by the following.
<math display="block">\begin{align}
 { {\eta }_{2}}&=1-\frac{|{ {q}_{2} } |}{|{ {q}_{1} }|}\to |{ {w}_{2} }|=|{ {q}_{1} }|-|{ {q}_{2} }|,\\
 { {\eta }_{3}}&=1-\frac{|{ {q}_{3} }|}{|{ {q}_{2} }^{*}|}\to |{ {w}_{3} }|=|{ {q}_{2} }^{*}|-|{ {q}_{3} }|,\\
 |{ {w}_{2} }|+|{ {w}_{3} }|&=(|{ {q}_{1} }|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\
 { {\eta}_{1}}&=1-\frac{|{ {q}_{3} }|}{|{ {q}_{1} }|}=\frac{(|{ {w}_{2} }|+|{ {w}_{3} }|)}{|{ {q}_{1} }|}=\frac{(|{ {q}_{1} }|-|{ {q}_{2} }|)+(|{ {q}_{2}} ^{*}|-|{ {q}_{3} }|)}{|{ {q}_{1} }|}.\\
\end{align}</math>

In order to have the consistency in the last equation, the heat <math>q_2</math> flown from the engine 2 to the intermediate reservoir must be equal to the heat <math>q_2^*</math> flown out from the reservoir to the engine 3.

With this understanding of {{math|''q''<sub>1</sub>}}, {{math|''q''<sub>2</sub>}} and {{math|''q''<sub>3</sub>}}, mathematically,
<math display="block">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).</math>

But since the first function is ''not'' a function of {{math|''T''<sub>2</sub>}}, the product of the final two functions ''must'' result in the removal of {{math|''T''<sub>2</sub>}} as a variable. The only way is therefore to define the function {{mvar|f}} as follows:
<math display="block">f(T_1,T_2) = \frac{g(T_2)}{g(T_1)}</math>
and
<math display="block">f(T_2,T_3) = \frac{g(T_3)}{g(T_2)}</math>
so that
<math display="block">f(T_1,T_3) = \frac{g(T_3)}{g(T_1)} = \frac{|q_3|}{|q_1|}.</math>

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 <math>g(T)</math>;<ref>Here, need to add a reason of requiring the function {{math|''g''(''T'')}} to be a monotonic function. The Carnot efficiency (efficiency of all reversible engines) may be a reason.</ref> it is a matter of convenience and convention that we choose <math>g(T) = T</math>. Choosing then ''one'' fixed reference temperature (i.e. triple point of water), we establish the thermodynamic temperature scale.

Such a definition coincides with that of the ideal gas derivation; also it is this ''definition'' of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of {{math|''T''<sub>H</sub>}} and {{math|''T''<sub>C</sub>}}, and hence derive that the (complete) Carnot cycle is isentropic:
<math display="block">\frac{|q_\text{C}|}{|q_\text{H}|} = f(T_\text{H},T_\text{C}) = \frac{T_\text{C}}{T_\text{H}}.  \ \ \ \ \ \ \ \ \ \ \ (3)</math>

Substituting this back into our first formula for efficiency yields a relationship in terms of temperature:
<math display="block">\textrm{Efficiency} = 1 + \frac{q_\text{C}}{q_\text{H}}= 1 - \frac{|q_\text{C}|}{|q_\text{H}|} = 1 - \frac{T_\text{C}}{T_\text{H}}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)</math>

Note that for {{math|1=''T''<sub>C</sub> = 0}} the efficiency is 100% and that efficiency becomes greater than 100% for {{math|''T''<sub>C</sub> < 0}}, which is unrealistic. Subtracting 1 from the right hand side of the Equation (4) and the middle portion gives <math>\frac {q_\text{C}}{q_\text{H}} = - \frac{T_\text{C}}{T_\text{H}}</math> and thus <ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote= eq.(64) |publisher=Dover Publications |year=1956}}</ref><ref name="PlanckBook" />
<math display="block">\frac {q_\text{H}}{T_\text{H}} + \frac{q_\text{C}}{T_\text{C}} = 0.</math>

The generalization of this equation is the [[Clausius theorem]], which proposes the existence of a [[state function]] <math>S</math> (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
<math display="block">\begin{align} dS &= \frac {dq_\mathrm{rev}}{T} &(5) \end{align}</math>
where the subscript ''rev'' indicates heat transfer in a reversible process. The function <math>S</math> is the [[entropy]] of the system, mentioned previously, and the change of <math>S</math> around any cycle is zero (as is necessary for any state function). The Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid a logic loop, we should first define [[statistical entropy|entropy]] through statistical mechanics):
<math display="block">T = \frac{dq_\mathrm{rev}}{dS}.</math>

For a constant-volume system (so no mechanical work <math>W</math>) in which the entropy <math>S</math> is a function <math>S(E)</math> of its [[internal energy]] <math>E</math>, <math>dE=dq_{rev} </math> and the thermodynamic temperature <math>T</math> is therefore given by
<math display="block">\frac{1}{T} = \frac{dS}{dE},</math>
so that the reciprocal of the thermodynamic temperature is the rate of change of entropy with respect to the internal energy at the constant volume.

== History ==
[[File:Guillaume Amontons, wood cutting.png|thumb|left|upright=0.8|[[Guillaume Amontons]]]]
[[Guillaume Amontons]] (1663–1705) published two papers in 1702 and 1703 that may be used to credit him as being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. He made the discovery while endeavoring to improve upon the air thermometers in use at the time. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation led him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240&nbsp;°C—only 33.15 degrees short of the true value of −273.15&nbsp;°C. Amonton's discovery of a one-to-one relationship between absolute temperature and absolute pressure was rediscovered a century later and popularized within the scientific community by [[Joseph Louis Gay-Lussac]]. Today, this principle of thermodynamics is commonly known as ''[[Gay-Lussac's law]]'' but is also known as ''Amonton's law''.

[[Image:Headshot of Anders Celsius.jpg|thumb|upright=0.8|[[Anders Celsius]]]]
In 1742, [[Anders Celsius]] (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 ''Observations of two persistent degrees on a thermometer'', 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.

[[Image:Carl von Linné.jpg|thumb|left|upright=0.8|[[Carl Linnaeus]]]]
Coincident with the death of Anders Celsius in 1744, the botanist [[Carl Linnaeus]] (1707–1778) effectively reversed<ref>{{cite web | title=A Brief History of Temperature Measurement | website=Chemical Suppliers | date=2024-03-01 | url=https://www.chemicalsuppliers.com/brief-history-of-temperature-measurement/ | access-date=2024-09-25}}</ref><ref>{{cite web|url=http://www.linnaeus.uu.se/online/life/6_32.html|archive-url=https://web.archive.org/web/20121030164902/http://www.linnaeus.uu.se/online/life/6_32.html|archive-date=2012-10-30|title=Linnaeus Thermometer|access-date=2006-10-05|url-status=live}}</ref>{{fcn|date=September 2024}} 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 ''Linnaeus-thermometer'', 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 ''[[centigrade scale]]''. Temperatures on the centigrade scale were often reported simply as ''degrees'' or, when greater specificity was desired, ''[[degrees centigrade]]''. The symbol for temperature values on this scale was °C (in several formats over the years). Because the term ''centigrade'' 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 "[[centesimal degree]]" was used when very precise, unambiguous language was required by international standards bodies such as the [[International Bureau of Weights and Measures]] (BIPM). The 9th CGPM ([[General Conference on Weights and Measures]] and the CIPM ([[International Committee for Weights and Measures]] formally adopted<ref>{{cite web|url=http://www.bipm.org/en/committees/cipm/cipm-1948.html|archive-url=https://web.archive.org/web/20161014011349/http://www.bipm.org/en/committees/cipm/cipm-1948.html|archive-date=2016-10-14|title=Adoption of "degree Celsius"|access-date=2006-09-23|url-status=live}}</ref> ''[[degree Celsius]]'' (symbol: °C) in 1948.

[[File:JHLambert.jpg|thumb|upright=0.8|[[Johann Heinrich Lambert]]]]
In his book ''Pyrometrie'' (1777)<ref>{{cite book |first=Johann Heinrich |last=Lambert |title=Pyrometrie |url=http://www.spiess-verlage.de/html/haude___spener.html |location=Berlin |publisher=Haude & Spener |year=1779}}</ref> completed four months before his death, [[Johann Heinrich Lambert]] (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&nbsp;°C.

[[File:Jacques Charles.jpg|thumb|left|upright=0.8|[[Jacques Alexandre César Charles]]]]
Notwithstanding the work of Guillaume Amontons 85 years earlier, [[Jacques Charles|Jacques Alexandre César Charles]] (1746–1823) is often credited with discovering (circa 1787), but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was {{math|1=''V''<sub>1</sub>/''T''<sub>1</sub> = ''V''<sub>2</sub>/''T''<sub>2</sub>}}.

[[File:Gaylussac.jpg|thumb|upright=0.8|[[Joseph Louis Gay-Lussac]]]]
[[Joseph Louis Gay-Lussac]] (1778–1850) published work in 1802 (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called [[Charles's law]] and is one of the [[gas laws]]. His are the first known formulas to use the number 273 for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273&nbsp;°C).

[[Image:William Thomson 1st Baron Kelvin.jpg|thumb|left|upright=0.8|[[William Thomson, 1st Baron Kelvin|Lord Kelvin]]]]
[[William Thomson, 1st Baron Kelvin|William Thomson]] (1824–1907), also known as Lord Kelvin, wrote in his 1848 paper "On an Absolute Thermometric Scale"<ref>{{cite magazine |title=On an Absolute Thermometric Scale |url=https://zapatopi.net/kelvin/papers/on_an_absolute_thermometric_scale.html |first=William |last=Thomson |magazine=Philosophical Magazine |date=October 1848}} Also published in {{cite book |first=William |last=Thomson |title=Mathematical and Physical Papers |volume=1 |publisher=Cambridge University Press |year=1882 |pages=100–106}}</ref> of the need for a scale whereby ''infinite cold'' (absolute zero) was the scale's zero point, and which used the degree Celsius for its unit increment. Like Gay-Lussac, Thomson calculated that absolute zero was equivalent to −273&nbsp;°C on the air thermometers of the time. This absolute scale is known today as the Kelvin thermodynamic temperature scale. Thomson's value of −273 was derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of −0.00366 expressed to five significant digits is −273.22&nbsp;°C which is remarkably close to the true value of −273.15&nbsp;°C.

In the paper he proposed to define temperature using idealized heat engines. In detail, he proposed that, given three heat reservoirs at temperatures <math>T_A, T_B, T_C</math>, if two reversible heat engines ([[Carnot heat engine|Carnot engine]]), one working between <math>T_A, T_B</math> and another between <math> T_B, T_C</math>, can produce the same amount of mechanical work <math>W</math> by letting the same amount of heat <math>Q</math> pass through, then define <math>T_A - T_B = T_B - T_C</math>.

Note that like Carnot, Kelvin worked under the assumption that heat is conserved ("the conversion of heat (or caloric) into mechanical effect is probably impossible"), and if heat <math>Q</math> goes into the heat engine, then heat <math>Q</math> must come out.<ref>{{Cite book |last=Lemons |first=Don S. |title=Thermodynamic weirdness: from Fahrenheit to Clausius |date=2020 |isbn=978-0-262-53894-7 |edition=First MIT Press Paperback |chapter=Chapter 4: Absolute Temperature |publisher=MIT Press |oclc=1143850952}}</ref>

Kelvin, realizing after Joule's experiments that heat is not a conserved quantity but is convertible with mechanical work, modified his scale in the 1851 work ''An Account of Carnot's Theory of the Motive Power of Heat''. In this work, he defined as follows:<ref>{{Cite book |last=Lemons |first=Don S. |title=Thermodynamic weirdness: from Fahrenheit to Clausius |date=2020 |isbn=978-0-262-53894-7 |edition=1st paperback |publisher=MIT Press |location=Cambridge, Massachusetts |chapter=Chapter 8: Absolute Temperature—Again |oclc=1143850952}}</ref>

{{quote|Given two heat reservoirs <math>T_A, T_B</math>, and a reversible heat engine working between them, such that if during an engine cycle, heat <math>Q_A</math> moves into the engine, and heat <math>Q_B</math> comes out of the engine, then <math>\frac{T_A}{T_B} = \frac{Q_A}{Q_B}</math>.}}

The above definition fixes the ratios between absolute temperatures, but it does not fix a scale for absolute temperature. For the scale, Thomson proposed to use the Celsius degree, that is, <math>\frac{1}{100}</math> the interval between the freezing and the boiling point of water.

[[File:Rankine William signature.jpg|thumb|upright=0.8|[[Macquorn Rankine]]]]
In 1859 [[Macquorn Rankine]] (1820–1872) proposed a thermodynamic temperature scale similar to William Thomson's but which used the degree [[Fahrenheit]] for its unit increment, that is, <math>\frac{1}{180}</math> the interval between the freezing and the boiling point of water. This absolute scale is known today as the [[rankine scale|Rankine]] thermodynamic temperature scale.

[[Image:Boltzmann2.jpg|thumb|left|upright=0.8|[[Ludwig Boltzmann]]]]
[[Ludwig Boltzmann]] (1844–1906) made major contributions to thermodynamics between 1877 and 1884 through an understanding of the role that particle kinetics and black body radiation played. His name is now attached to several of the formulas used today in thermodynamics.

Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed in the 1930s that absolute zero was equivalent to −273.15&nbsp;°C.

Resolution 3<ref>{{cite web | url=http://www.bipm.fr/en/CGPM/db/9/3/ | title=Welcome }}</ref> of the 9th [[General Conference on Weights and Measures]] (CGPM) in 1948 fixed the triple point of water at precisely 0.01&nbsp;°C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared "will be fixed at a later date". The implication is that if the value of absolute zero measured in the 1930s was truly −273.15&nbsp;°C, then the triple point of water (0.01&nbsp;°C) was equivalent to 273.16&nbsp;K. Additionally, both the International Committee for Weights and Measures (CIPM) and the CGPM formally adopted<ref>{{Cite web |date= |title=BIPM - 1948 |url=http://www.bipm.org/en/committees/cipm/cipm-1948.html |archive-url=https://web.archive.org/web/20150213182030/http://www.bipm.org/en/committees/cipm/cipm-1948.html |archive-date=2015-02-13 |website=www.bipm.org}}</ref> the name ''Celsius'' for the ''degree Celsius'' and the ''Celsius temperature scale''.{{refn|name=°CName|According to ''The Oxford English Dictionary'' (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 {{not a typo|28° Celsius}}". However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards the terminology used throughout the history of science. According to several writings of Terry Quinn CBE FRS, Director of the BIPM (1988–2004), including ''Temperature Scales from the early days of thermometry to the 21st century''<ref>{{Cite web | url=https://www.imeko.org/publications/tc12-2004/IMEKO-TC12-2004-PL-001.pdf | title=Temperature Scales from the early days of thermometry to the 21st century | website=www.imeko.org}}</ref> as well as ''Temperature'',<ref>{{cite book |last=Quinn |first=Terry |title=Temperature |edition=2nd |year=1990 |publisher=Academic Press |isbn=0-12-569681-7}}</ref> the term ''Celsius'' in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM was not even aware that ''degree Celsius'' was in sporadic, non-scientific use before that time. The twelve-volume, 1933 edition of the OED did not even have a listing for the word ''Celsius'' (but did have listings for both ''centigrade'' and ''centesimal'' in the context of temperature measurement). The 1948 adoption of ''Celsius'' accomplished three objectives:
# All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.
# 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 ''centigrade'' could continue to be used and would simultaneously inherit an intuitive association with the new name.
# The new name eliminated the ambiguity of the term ''centigrade'', freeing it to refer exclusively to the French-language name for the unit of angular measurement.}}

Resolution 3<ref>{{cite web | url=http://www.bipm.fr/en/CGPM/db/10/3/ | title=Welcome }}</ref> of the 10th CGPM in 1954 gave the Kelvin scale its modern definition by choosing the triple point of water as its upper defining point (with no change to absolute zero being the null point) and assigning it a temperature of precisely 273.16&nbsp;kelvins (what was actually written 273.16 ''degrees Kelvin'' 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&nbsp;°C.

Resolution 3<ref>{{cite web | url=http://www.bipm.fr/en/CGPM/db/13/3/ | title=Welcome }}</ref> of the 13th CGPM in 1967/1968 renamed the unit increment of thermodynamic temperature ''kelvin'', symbol K, replacing ''degree absolute'', symbol {{not a typo|°K}}. Further, feeling it useful to more explicitly define the magnitude of the unit increment, the 13th CGPM also decided in Resolution 4<ref>{{cite web | url=http://www.bipm.fr/en/CGPM/db/13/4/ | title=Welcome }}</ref> that "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water".

The CIPM affirmed in 2005<ref>{{cite web | url=http://www.bipm.fr/en/si/si_brochure/chapter2/2-1/kelvin.html | title=Welcome }}</ref> 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 [[Vienna Standard Mean Ocean Water]].

In November 2018, the 26th General Conference on Weights and Measures (CGPM) changed the definition of the Kelvin by fixing the Boltzmann constant to {{val|1.380649|e=-23}} when expressed in the unit J/K. [[2019 revision of the SI|This change (and other changes in the definition of SI units)]] was made effective on the 144th anniversary of the Metre Convention, 20 May 2019.

== See also ==
{{div col|colwidth=22em}}
* [[:Category:Thermodynamics]]
* [[Absolute zero]]
* [[Hagedorn temperature]]
* [[Adiabatic process]]
* [[Boltzmann constant]]
* [[Carnot heat engine]]
* [[Conversion of scales of temperature]]
* [[Energy conversion efficiency]]
* [[Enthalpy]]
** [[Enthalpy of fusion]]
** [[Enthalpy of vaporization]]
* [[Entropy]]
* [[Equipartition theorem]]
* [[Fahrenheit]]
* [[First law of thermodynamics]]
* [[Freezing]]
* [[Gas laws]]
* [[International System of Quantities]]
* [[International Temperature Scale of 1990]] (ITS-90)
* [[Ideal gas law]]
* [[Kelvin]]
* [[Laws of thermodynamics]]
* [[Maxwell–Boltzmann distribution]]
* [[Orders of magnitude (temperature)]]
* [[Phase transition]]
* [[Planck's law of black body radiation]]
* [[Rankine scale]]
* [[Specific heat capacity]]
* [[Temperature]]
* [[Thermal radiation]]
* [[Thermodynamic beta]]
* [[Thermodynamic equations]]
* [[Thermodynamic equilibrium]]
* [[Thermodynamics]]
* [[Timeline of heat engine technology]]
* [[Timeline of temperature and pressure measurement technology]]
* [[Triple point]]
{{div col end}}

== Notes ==
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->
: ''In the following notes, wherever numeric equalities are shown in ''concise form'', such as {{val|1.85487|(14)|e=43}}, the two digits between the parentheses denotes the [[uncertainty]] at 1-σ (1 [[standard deviation]], 68% confidence level) in the two least significant digits of the [[significand]].''

== References ==
{{reflist}}

{{Meteorological variables}}
{{SI base quantities}}
{{Authority control}}

{{DEFAULTSORT:Thermodynamic Temperature}}
[[Category:Temperature]]
[[Category:SI base quantities]]
[[Category:State functions]]