View source for Temperature

{{Short description|Physical quantity of hot and cold}}
{{About|the physical quantity||Temperature (disambiguation)}}
{{pp-move|small=yes}}
{{pp-move}}
{{Infobox physical quantity
| name = Temperature
| image = Thermally Agitated Molecule.gif
| caption = Thermal vibration of a segment of protein's [[alpha helix]]. Its [[amplitude]] increases with temperature.
| unit = [[kelvin|K]]
| otherunits = [[Celsius|°C]], [[Fahrenheit|°F]], [[Rankine scale|°R]], [[Rømer scale|°Rø]], [[Réaumur scale|°Ré]], [[Newton scale|°N]], [[Delisle scale|°D]], [[Leiden scale|°L]], [[Wedgwood scale|°W]]
| symbols = {{mvar|T}}
| dimension = wikidata 
| intensive = Yes 
| derivations = <math>\frac{pV}{nR}</math>, <math>\frac{dq_\text{rev}}{dS}</math>
}}

'''Temperature''' is a [[physical quantity]] that quantitatively expresses the attribute of hotness or coldness. Temperature is [[measurement|measured]] with a [[thermometer]]. It reflects the average [[kinetic energy]] of the vibrating and colliding [[atom]]s making up a substance.

Thermometers are calibrated in various [[temperature scale]]s that historically have relied on various reference points and thermometric substances for definition. The most common scales are the [[Celsius]] scale with the unit symbol °C (formerly called ''centigrade''), the [[Fahrenheit]] scale (°F), and the [[Kelvin]] scale (K), the latter being used predominantly for scientific purposes. The kelvin is one of the seven base units in the [[International System of Units]] (SI).

[[Absolute zero]], i.e., zero kelvin or −273.15&nbsp;°C, is the lowest point in the [[thermodynamic temperature]] scale. Experimentally, it can be approached very closely but not actually reached, as recognized in the [[third law of thermodynamics]]. It would be impossible to extract energy as heat from a body at that temperature.

Temperature is important in all fields of [[natural science]], including [[physics]], [[chemistry]], [[Earth science]], [[astronomy]], [[medicine]], [[biology]], [[ecology]], [[Materials science|material science]], [[metallurgy]], [[mechanical engineering]] and [[geography]] as well as most aspects of daily life.

== Effects ==
[[File:Body Temp Variation.svg|thumb|Average daily variation in human body temperature]]Many physical processes are related to temperature; some of them are given below:
* the physical properties of materials including the [[Phases of matter|phase]] ([[solid]], [[liquid]], [[gas]]eous or [[Plasma (physics)|plasma]]), [[density]], [[solubility]], [[vapor pressure]], [[electrical conductivity]], [[hardness]], [[Wear|wear resistance]], [[thermal conductivity]], [[Corrosion|corrosion resistance]], strength
* the rate and extent to which [[chemical reaction]]s occur<ref>{{Cite book|url=https://books.google.com/books?id=UKkQAQAAMAAJ|title=Thermal discharges at nuclear power stations: their management and environmental impacts: a report prepared by a group of experts as the result of a panel meeting held in Vienna, 23–27 October 1972|last=Agency|first=International Atomic Energy|date=1974|publisher=International Atomic Energy Agency}}</ref>
* the amount and properties of [[thermal radiation]] emitted from the surface of an object
* [[air temperature]] affects all living organisms
* the [[speed of sound]], which in a gas is proportional to the square root of the absolute temperature<ref>{{Cite book|url=https://books.google.com/books?id=eVpITJfPxMEC&pg=PA34|title=The Art of Digital Audio|last=Watkinson|first=John|date=2001|publisher=Taylor & Francis|isbn=978-0-240-51587-8}}</ref>

== Scales ==
{{main|Scale of temperature}}
{{more citations needed section|date=January 2021}}
[[File:Thermometer CF.svg|thumb|Two thermometers showing temperature in Celsius and Fahrenheit]]
Temperature scales need two values for definition: the point chosen as zero degrees and the magnitudes of the incremental unit of temperature.

The [[Celsius]] scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that developed historically, which led to its zero point {{val|0|u=degC}} being defined as the freezing point of [[water]], and {{val|100|u=degC}} as the boiling point of water, both at [[atmospheric pressure]] at sea level. It was called a centigrade scale because of the 100-degree interval.<ref>Middleton, W.E.K. (1966), pp. 89–105.</ref> Since the standardization of the [[kelvin]] in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though numerically the scales differ by an exact offset of 273.15.

The [[Fahrenheit]] scale is in common use in the United States. Water freezes at {{val|32|u=degF}} and boils at {{val|212|u=degF}} at sea-level atmospheric pressure.

=== Absolute zero ===
At the [[absolute zero]] of temperature, no energy can be removed from matter as heat, a fact expressed in the [[third law of thermodynamics]]. At this temperature, matter contains no macroscopic thermal energy, but still has quantum-mechanical [[zero-point energy]] as predicted by the [[uncertainty principle]], although this does not enter into the definition of absolute temperature. Experimentally, absolute zero can be approached only very closely; it can never be reached (the lowest temperature attained by experiment is 38&nbsp;pK).<ref name="Lowest temperature">{{Cite web |author1=Joanna Thompson |date=2021-10-14 |title=Scientists just broke the record for the coldest temperature ever recorded in a lab |url=https://www.livescience.com/coldest-temperature-ever |access-date=2023-04-28 |website=Live Science |language=en}}</ref> Theoretically, in a body at a temperature of absolute zero, all classical motion of its particles has ceased and they are at complete rest in this classical sense. Absolute zero, defined as {{val|0|u=K}}, is exactly equal to {{val|−273.15|u=degC}}, or {{val|−459.67|u=degF}}.

=== Absolute scales ===
Referring to the [[Boltzmann constant]], to the [[Maxwell–Boltzmann distribution]], and to the Boltzmann [[Statistical mechanics|statistical mechanical definition]] of [[entropy]], as distinct from the Gibbs definition,<ref name="BoltzmannvsGibbs">Jaynes, E.T. (1965), pp. 391–398.</ref> for independently moving microscopic particles, disregarding interparticle potential energy, by international agreement, a temperature scale is defined and said to be absolute because it is independent of the characteristics of particular thermometric substances and [[thermometer]] mechanisms. Apart from absolute zero, it does not have a reference temperature. It is known as the [[Kelvin temperature scale|Kelvin scale]], widely used in science and technology. The kelvin (the unit name is spelled with a [[Letter case|lower-case]] 'k') is the unit of temperature in the [[International System of Units]] (SI). The temperature of a body in a state of thermodynamic equilibrium is always positive relative to absolute zero.

Besides the internationally agreed Kelvin scale, there is also a [[Thermodynamic temperature|thermodynamic temperature scale]], invented by [[Lord Kelvin]], also with its numerical zero at the absolute zero of temperature, but directly relating to purely macroscopic [[thermodynamics|thermodynamic]] concepts, including the macroscopic [[entropy]], though microscopically referable to the Gibbs statistical mechanical definition of entropy for the [[canonical ensemble]], that takes interparticle potential energy into account, as well as independent particle motion so that it can account for measurements of temperatures near absolute zero.<ref name="BoltzmannvsGibbs"/> This scale has a reference temperature at the [[triple point]] of water, the numerical value of which is defined by measurements using the aforementioned internationally agreed Kelvin scale.

=== Kelvin scale ===
Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the [[William Thomson, 1st Baron Kelvin|physicist who first defined it]]. It is an [[Absolute temperature|absolute]] scale. Its numerical zero point, {{val|0|u=K}}, is at the absolute zero of temperature. Since May 2019, the kelvin has been defined [[#Kinetic theory approach|through particle kinetic theory]], and statistical mechanics. In the [[International System of Units]] (SI), the magnitude of the kelvin is defined in terms of the [[Boltzmann constant]], the value of which is defined as fixed by international convention.<ref name="Boltzmann constant">[https://cryogenicsociety.org/36995/news/nist_explains_the_new_kelvin_definition/ Cryogenic Society] {{Webarchive|url=https://web.archive.org/web/20201107230319/https://cryogenicsociety.org/36995/news/nist_explains_the_new_kelvin_definition/ |date=2020-11-07 }} (2019).</ref><ref name="draft-resolution-A">{{citation|title=Draft Resolution A "On the revision of the International System of Units (SI)" to be submitted to the CGPM at its 26th meeting (2018)|url=https://www.bipm.org/utils/en/pdf/CGPM/Draft-Resolution-A-EN.pdf|access-date=2019-10-20|archive-date=2018-04-29|archive-url=https://web.archive.org/web/20180429025229/https://www.bipm.org/utils/en/pdf/CGPM/Draft-Resolution-A-EN.pdf|url-status=dead}}</ref>

=== Statistical mechanical ''versus'' thermodynamic temperature scales ===
Since May 2019, the magnitude of the kelvin is defined in relation to microscopic phenomena, characterized in terms of statistical mechanics. Previously, but since 1954, the International System of Units defined a scale and unit for the kelvin as a [[thermodynamic temperature]], by using the reliably reproducible temperature of the [[triple point]] of water as a second reference point, the first reference point being {{val|0|u=K}} at absolute zero. {{citation needed|date=January 2021}}

Historically, the temperature of the triple point of water was defined as exactly 273.16&nbsp;K. Today it is an empirically measured quantity. The freezing point of water at sea-level atmospheric pressure occurs at very close to {{val|273.15|u=K}} ({{val|0|u=degC}}).

== Classification of scales ==

There are various kinds of temperature scale. It may be convenient to classify them as empirically and theoretically based. Empirical temperature scales are historically older, while theoretically based scales arose in the middle of the nineteenth century.<ref name="Truesdell 1980">Truesdell, C.A. (1980), Sections 11 B, 11H, pp. 306–310, 320–332.</ref><ref>Quinn, T.&nbsp;J. (1983).</ref>

=== Empirical scales ===
Empirically based temperature scales rely directly on measurements of simple macroscopic physical properties of materials. For example, the length of a column of mercury, confined in a glass-walled capillary tube, is dependent largely on temperature and is the basis of the very useful mercury-in-glass thermometer. Such scales are valid only within convenient ranges of temperature. For example, above the boiling point of [[Mercury (element)|mercury]], a mercury-in-glass thermometer is impracticable. Most materials expand with temperature increase, but some materials, such as water, contract with temperature increase over some specific range, and then they are hardly useful as thermometric materials. A material is of no use as a thermometer near one of its phase-change temperatures, for example, its boiling-point.

In spite of these limitations, most generally used practical thermometers are of the empirically based kind. Especially, it was used for [[calorimetry]], which contributed greatly to the discovery of thermodynamics. Nevertheless, empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Empirically based thermometers, beyond their base as simple direct measurements of ordinary physical properties of thermometric materials, can be re-calibrated, by use of theoretical physical reasoning, and this can extend their range of adequacy.

=== Theoretical scales ===
Theoretically based temperature scales are based directly on theoretical arguments, especially those of kinetic theory and thermodynamics. They are more or less ideally realized in practically feasible physical devices and materials. Theoretically based temperature scales are used to provide calibrating standards for practical empirically based thermometers.

==== Microscopic statistical mechanical scale ====
In physics, the internationally agreed conventional temperature scale is called the Kelvin scale. It is calibrated through the internationally agreed and prescribed value of the Boltzmann constant,<ref name="Boltzmann constant"/><ref name="draft-resolution-A"/> referring to motions of microscopic particles, such as atoms, molecules, and electrons, constituent in the body whose temperature is to be measured. In contrast with the thermodynamic temperature scale invented by Kelvin, the presently conventional Kelvin temperature is not defined through comparison with the temperature of a reference state of a standard body, nor in terms of macroscopic thermodynamics.

Apart from the absolute zero of temperature, the Kelvin temperature of a body in a state of internal thermodynamic equilibrium is defined by measurements of suitably chosen of its physical properties, such as have precisely known theoretical explanations in terms of the [[Boltzmann constant]]. {{citation needed|date=January 2021}} That constant refers to chosen kinds of motion of microscopic particles in the constitution of the body. In those kinds of motion, the particles move individually, without mutual interaction. Such motions are typically interrupted by inter-particle collisions, but for temperature measurement, the motions are chosen so that, between collisions, the non-interactive segments of their trajectories are known to be accessible to accurate measurement. For this purpose, interparticle potential energy is disregarded.

In an [[ideal gas]], and in other theoretically understood bodies, the Kelvin temperature is defined to be proportional to the average kinetic energy of non-interactively moving microscopic particles, which can be measured by suitable techniques. The proportionality constant is a simple multiple of the Boltzmann constant. If molecules, atoms, or electrons<ref>Germer, L.H. (1925). 'The distribution of initial velocities among thermionic electrons', ''Phys. Rev.'', '''25''': 795–807. [http://journals.aps.org/pr/abstract/10.1103/PhysRev.25.795 here]</ref><ref>Turvey, K. (1990). 'Test of validity of Maxwellian statistics for electrons thermionically emitted from an oxide cathode', ''European Journal of Physics'', '''11'''(1): 51–59. [http://m.iopscience.iop.org/article/10.1088/0143-0807/11/1/010/meta here]</ref> are emitted from material and their velocities are measured, the spectrum of their velocities often nearly obeys a theoretical law called the [[Maxwell–Boltzmann distribution]], which gives a well-founded measurement of temperatures for which the law holds.<ref>Zeppenfeld, M., Englert, B.G.U., Glöckner, R., Prehn, A., Mielenz, M., Sommer, C., van Buuren, L.D., Motsch, M., Rempe, G. (2012).</ref> There have not yet been successful experiments of this same kind that directly use the [[Fermi–Dirac statistics|Fermi–Dirac distribution]] for thermometry, but perhaps that will be achieved in the future.<ref>Miller, J. (2013).</ref>

The speed of sound in a gas can be calculated theoretically from the gas's [[Molecule|molecular]] character, temperature, pressure, and the Boltzmann constant. For a gas of known molecular character and pressure, this provides a relation between temperature and the Boltzmann constant. Those quantities can be known or measured more precisely than can the thermodynamic variables that define the state of a sample of water at its triple point. Consequently, taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.<ref name="de Podesta">de Podesta, M., Underwood, R., Sutton, G., Morantz, P, Harris, P, Mark, D.F., Stuart, F.M., Vargha, G., Machin, M. (2013). A low-uncertainty measurement of the Boltzmann constant, ''Metrologia'', '''50''' (4): S213–S216,  BIPM & IOP Publishing Ltd</ref>

Measurement of the spectrum of electromagnetic radiation from an ideal three-dimensional [[black body]] can provide an accurate temperature measurement because the frequency of maximum spectral radiance of black-body radiation is directly proportional to the temperature of the black body; this is known as [[Wien's displacement law]] and has a theoretical explanation in [[Planck's law]] and the [[Bose–Einstein statistics|Bose–Einstein law]].

Measurement of the spectrum of noise-power produced by an electrical resistor can also provide accurate temperature measurement. The resistor has two terminals and is in effect a one-dimensional body. The Bose-Einstein law for this case indicates that the noise-power is directly proportional to the temperature of the resistor and to the value of its resistance and to the noise bandwidth. In a given frequency band, the noise-power has equal contributions from every frequency and is called [[Johnson noise]]. If the value of the resistance is known then the temperature can be found.<ref>Quinn, T.J. (1983), pp. 98–107.</ref><ref>Schooley, J.F. (1986), pp. 138–143.</ref>

==== Macroscopic thermodynamic scale ====
Historically, till May 2019, the definition of the Kelvin scale was that invented by Kelvin, based on a ratio of quantities of energy in processes in an ideal Carnot engine, entirely in terms of macroscopic thermodynamics. {{citation needed|date=January 2021}} That Carnot engine was to work between two temperatures, that of the body whose temperature was to be measured, and a reference, that of a body at the temperature of the triple point of water. Then the reference temperature, that of the triple point, was defined to be exactly {{val|273.16|u=K}}. Since May 2019, that value has not been fixed by definition but is to be measured through microscopic phenomena, involving the Boltzmann constant, as described above. The microscopic statistical mechanical definition does not have a reference temperature.

==== Ideal gas ====
A material on which a macroscopically defined temperature scale may be based is the [[ideal gas]]. The pressure exerted by a fixed volume and mass of an ideal gas is directly proportional to its temperature. Some natural gases show so nearly ideal properties over suitable temperature range that they can be used for thermometry; this was important during the development of thermodynamics and is still of practical importance today.<ref>Quinn, T.J. (1983), pp. 61–83.</ref><ref>Schooley, J.F. (1986), pp. 115–138.</ref> The ideal gas thermometer is, however, not theoretically perfect for thermodynamics. This is because the [[Ideal gas#Entropy|entropy of an ideal gas]] at its absolute zero of temperature is not a positive semi-definite quantity, which puts the gas in violation of the third law of thermodynamics. In contrast to real materials, the ideal gas does not liquefy or solidify, no matter how cold it is. Alternatively thinking, the ideal gas law, refers to the limit of infinitely high temperature and zero pressure; these conditions guarantee non-interactive motions of the constituent molecules.<ref>Adkins, C.J. (1968/1983), pp. 119–120.</ref><ref>Buchdahl, H.A. (1966), pp. 137–138.</ref><ref>Tschoegl, N.W. (2000), p.&nbsp;88.</ref>

== Kinetic theory approach ==
The magnitude of the kelvin is now defined in terms of kinetic theory, derived from the value of the [[Boltzmann constant]].

[[Kinetic theory of gases|Kinetic theory]] provides a microscopic account of temperature for some bodies of material, especially gases, based on macroscopic systems' being composed of many microscopic particles, such as molecules and [[ion]]s of various species, the particles of a species being all alike. It explains macroscopic phenomena through the [[classical mechanics]] of the microscopic particles. The [[equipartition theorem]] of kinetic theory asserts that each classical [[Degrees of freedom (physics and chemistry)|degree of freedom]] of a freely moving particle has an average kinetic energy of {{math|''k''<sub>B</sub>''T''/2}} where {{math|''k''<sub>B</sub>}} denotes the [[Boltzmann constant]]. {{citation needed|date=January 2021}} The translational motion of the particle has three degrees of freedom, so that, except at very low temperatures where quantum effects predominate, the average translational kinetic energy of a freely moving particle in a system with temperature {{math|''T''}} will be {{math|3''k''<sub>B</sub>''T''/2}}.

Molecules, such as oxygen (O<sub>2</sub>), have more [[Degrees of freedom (physics and chemistry)|degrees of freedom]] than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase of temperature due to an increase in the average translational kinetic energy of the molecules. Heating will also cause, through [[equipartition]]ing, the energy associated with vibrational and rotational modes to increase. Thus a [[diatomic]] gas will require more energy input to increase its temperature by a certain amount, i.e. it will have a greater [[heat capacity]] than a monatomic gas.

As noted above, the speed of sound in a gas can be calculated from the gas's molecular character, temperature, pressure, and the Boltzmann constant. Taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.<ref name="de Podesta"/>

It is possible to measure the average kinetic energy of constituent [[Microscopic scale|microscopic]] particles if they are allowed to escape from the bulk of the system, through a small hole in the containing wall. The spectrum of velocities has to be measured, and the average calculated from that. It is not necessarily the case that the particles that escape and are measured have the same velocity distribution as the particles that remain in the bulk of the system, but sometimes a good sample is possible.

== Thermodynamic approach ==
{{Thermodynamics|cTopic=[[List of thermodynamic properties|System properties]]}}
Temperature is one of the principal quantities in the study of [[thermodynamics]]. Formerly, the magnitude of the kelvin was defined in thermodynamic terms, but nowadays, as mentioned above, it is defined in terms of kinetic theory.

The thermodynamic temperature is said to be '''absolute''' for two reasons. One is that its formal character is independent of the properties of particular materials. The other reason is that its zero is, in a sense, absolute, in that it indicates absence of microscopic classical motion of the constituent particles of matter, so that they have a limiting specific heat of zero for zero temperature, according to the third law of thermodynamics. Nevertheless, a thermodynamic temperature does in fact have a definite numerical value that has been arbitrarily chosen by tradition and is dependent on the property of particular materials; it is simply less arbitrary than relative "degrees" scales such as [[Celsius scale|Celsius]] and [[Fahrenheit scale|Fahrenheit]].  Being an absolute scale with one fixed point (zero), there is only one degree of freedom left to arbitrary choice, rather than two as in relative scales. For the Kelvin scale since May 2019, by international convention, the choice has been made to use knowledge of modes of operation of various thermometric devices, relying on microscopic kinetic theories about molecular motion. The numerical scale is settled by a conventional definition of the value of the [[Boltzmann constant]], which relates macroscopic temperature to average microscopic kinetic energy of particles such as molecules. Its numerical value is arbitrary, and an alternate, less widely used absolute temperature scale exists called the [[Rankine scale]], made to be aligned with the Fahrenheit scale as [[Kelvin scale|Kelvin]] is with Celsius.

The thermodynamic definition of temperature is due to Kelvin. It is framed in terms of an idealized device called a [[Carnot engine]], imagined to run in a fictive continuous [[Carnot cycle|cycle of successive processes]] that traverse a cycle of states of its working body. The engine takes in a quantity of heat {{math|''Q''<sub>1</sub>}} from a hot reservoir and passes out a lesser quantity of waste heat {{math|''Q''<sub>2</sub> < 0}} to a cold reservoir. The net heat energy absorbed by the working body is passed, as thermodynamic work, to a work reservoir, and is considered to be the output of the engine. The cycle is imagined to run so slowly that at each point of the cycle the working body is in a state of thermodynamic equilibrium. The successive processes of the cycle are thus imagined to run reversibly with no [[entropy production]]. Then the quantity of entropy taken in from the hot reservoir when the working body is heated is equal to that passed to the cold reservoir when the working body is cooled. Then the absolute or thermodynamic temperatures, {{math|''T''<sub>1</sub>}} and {{math|''T''<sub>2</sub>}},  of the reservoirs are defined such that<ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote= eq.(64) |publisher=Dover Publications (still in print) |year=1956}}.</ref>
{{NumBlk|:|<math>\frac{T_1}{T_2} = -\frac{Q_1}{Q_2}.</math>|1}}

The zeroth law of thermodynamics allows this definition to be used to measure the absolute or thermodynamic temperature of an arbitrary body of interest, by making the other heat reservoir have the same temperature as the body of interest.

Kelvin's original work postulating absolute temperature was published in 1848. It was based on the work of Carnot, before the formulation of the first law of thermodynamics. Carnot had no sound understanding of heat and no specific concept of entropy. He wrote of 'caloric' and said that all the caloric that passed from the hot reservoir was passed into the cold reservoir. Kelvin wrote in his 1848 paper that his scale was absolute in the sense that it was defined "independently of the properties of any particular kind of matter". His definitive publication, which sets out the definition just stated, was printed in 1853, a paper read in 1851.<ref>[[William Thomson, 1st Baron Kelvin|Thomson, W. (Lord Kelvin)]] (1848).</ref><ref>[[William Thomson, 1st Baron Kelvin|Thomson, W. (Lord Kelvin)]] (1851).</ref><ref>[[J. R. Partington|Partington, J.R.]] (1949), pp. 175–177.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), pp. 321–322.</ref>

Numerical details were formerly settled by making one of the heat reservoirs a cell at the triple point of water, which was defined to have an absolute temperature of 273.16 K.<ref>Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}, pp. 160–162.</ref> Nowadays, the numerical value is instead obtained from measurement through the microscopic statistical mechanical international definition, as above.

=== Intensive variability ===
In thermodynamic terms, temperature is an [[Intensive and extensive properties|intensive variable]] because it is equal to a [[differential coefficient]] of one [[Intensive and extensive properties|extensive variable]] with respect to another, for a given body. It thus has the [[Dimensional analysis|dimensions]] of a [[ratio]] of two extensive variables. In thermodynamics, two bodies are often considered as connected by contact with a common wall, which has some specific permeability properties. Such specific permeability can be referred to a specific intensive variable. An example is a diathermic wall that is permeable only to heat; the intensive variable for this case is temperature. When the two bodies have been connected through the specifically permeable wall for a very long time, and have settled to a permanent steady state, the relevant intensive variables are equal in the two bodies; for a diathermal wall, this statement is sometimes called the zeroth law of thermodynamics.<ref>Tisza, L. (1966). ''Generalized Thermodynamics'', M.I.T. Press, Cambridge MA, pp. 47, 57.</ref><ref name="Munster 49 69">Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}, pp. 49, 69.</ref><ref name="Bailyn 14 214">Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, {{ISBN|0-88318-797-3}}, pp. 14–15, 214.</ref>

In particular, when the body is described by stating its [[internal energy]] {{math|''U''}}, an extensive variable, as a function of its [[entropy]] {{math|''S''}}, also an extensive variable, and other state variables {{math|''V'', ''N''}}, with {{math|''U'' {{=}} ''U'' (''S'', ''V'', ''N''}}), then the temperature is equal to the [[partial derivative]] of the internal energy with respect to the entropy:<ref name="Munster 49 69"/><ref name="Bailyn 14 214"/><ref name="Callen 146–148">[[Herbert Callen|Callen, H.B.]] (1960/1985), ''Thermodynamics and an Introduction to Thermostatistics'', (first edition 1960), second edition 1985, John Wiley & Sons, New York, {{ISBN|0-471-86256-8}}, pp. 146–148.</ref>

{{NumBlk|:|<math>T = \left ( \frac{\partial U}{\partial S} \right )_{V, N}.</math>|2}}

Likewise, when the body is described by stating its entropy {{math|''S''}} as a function of its internal energy {{math|''U''}}, and other state variables {{math|''V'', ''N''}}, with {{math|''S'' {{=}} ''S'' (''U'', ''V'', ''N'')}}, then the reciprocal of the temperature is equal to the partial derivative of the entropy with respect to the internal energy:<ref name="Munster 49 69"/><ref name="Callen 146–148"/><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics. From Heat Engines to Dissipative Structures'', John Wiley, Chichester, {{ISBN|0-471-97394-7}}, pp. 115–116.</ref>

{{NumBlk|:|<math>\frac{1}{T} = \left( \frac{\partial S}{\partial U} \right)_{V, N}.</math>|3}}

The above definition, equation (1), of the absolute temperature, is due to Kelvin. It refers to systems closed to the transfer of matter and has a special emphasis on directly experimental procedures. A presentation of thermodynamics by Gibbs starts at a more abstract level and deals with systems open to the transfer of matter; in this development of thermodynamics, the equations (2) and (3) above are actually alternative definitions of temperature.<ref>Tisza, L. (1966). ''Generalized Thermodynamics'', M.I.T. Press, Cambridge MA, p. 58.</ref>

=== Local thermodynamic equilibrium ===
Real-world bodies are often not in thermodynamic equilibrium and not homogeneous. For the study by methods of classical irreversible thermodynamics, a body is usually spatially and temporally divided conceptually into 'cells' of small size. If classical thermodynamic equilibrium conditions for matter are fulfilled to good approximation in such a 'cell', then it is homogeneous and a temperature exists for it. If this is so for every 'cell' of the body, then [[Thermodynamic equilibrium#Local and global equilibrium|local thermodynamic equilibrium]] is said to prevail throughout the body.<ref>[[Edward Arthur Milne|Milne, E.A.]] (1929). The effect of collisions on monochromatic radiative equilibrium, [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1928MNRAS..88..493M&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf ''Monthly Notices of the Royal Astronomical Society''], '''88''': 493–502.</ref><ref>Gyarmati, I. (1970). ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, Berlin, pp. 63–66.</ref><ref>Glansdorff, P., [[Ilya Prigogine|Prigogine, I.]], (1971). ''Thermodynamic Theory of Structure, Stability and Fluctuations'', Wiley, London, {{ISBN|0-471-30280-5}}, pp. 14–16.</ref><ref>Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, {{ISBN|0-88318-797-3}}, pp. 133–135.</ref><ref>[[Herbert Callen|Callen, H.B.]] (1960/1985), ''Thermodynamics and an Introduction to Thermostatistics'', (first edition 1960), second edition 1985, John Wiley & Sons, New York, {{ISBN|0-471-86256-8}}, pp. 309–310.</ref>

It makes good sense, for example, to say of the extensive variable {{math|''U''}}, or of the extensive variable {{math|''S''}}, that it has a density per unit volume or a quantity per unit mass of the system, but it makes no sense to speak of the density of temperature per unit volume or quantity of temperature per unit mass of the system. On the other hand, it makes no sense to speak of the internal energy at a point, while when local thermodynamic equilibrium prevails, it makes good sense to speak of the temperature at a point. Consequently, the temperature can vary from point to point in a medium that is not in global thermodynamic equilibrium, but in which there is local thermodynamic equilibrium.

Thus, when local thermodynamic equilibrium prevails in a body, the temperature can be regarded as a spatially varying local property in that body, and this is because the temperature is an intensive variable.

== Basic theory ==
{{Conjugate variables (thermodynamics)}}Temperature is a measure of a [[Categories (Aristotle)#The Praedicamenta|quality]] of a state of a material.<ref>Bryan, G.H. (1907). ''Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications'', B.G. Teubner, Leipzig, p. 3. {{cite web |url=http://www.e-booksdirectory.com/details.php?ebook=6455 |title=Thermodynamics by George Hartley Bryan  |access-date=2011-10-02 |url-status=live |archive-url=https://web.archive.org/web/20111118050819/http://www.e-booksdirectory.com/details.php?ebook=6455 |archive-date=2011-11-18 }}</ref>  The quality may be regarded as a more abstract entity than any particular temperature scale that measures it, and is called ''hotness'' by some writers.<ref>[[Brian Pippard|Pippard, A.B.]] (1957/1966), p. 18.</ref><ref>Adkins, C.J. (1968/1983), p. 20.</ref><ref>Bryan, G.H. (1907). ''Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications'', B.G. Teubner, Leipzig, p. 5: "... when a body is spoken of as growing hotter or colder an increase of temperature is always implied, for the hotness and coldness of a body are qualitative terms which can only refer to temperature." {{cite web |url=http://www.e-booksdirectory.com/details.php?ebook=6455 |title=Thermodynamics by George Hartley Bryan  |access-date=2011-10-02 |url-status=live |archive-url=https://web.archive.org/web/20111118050819/http://www.e-booksdirectory.com/details.php?ebook=6455 |archive-date=2011-11-18 }}</ref> The quality of hotness refers to the state of material only in a particular locality, and in general, apart from bodies held in a steady state of thermodynamic equilibrium, hotness varies from place to place. It is not necessarily the case that a material in a particular place is in a state that is steady and nearly homogeneous enough to allow it to have a well-defined hotness or temperature. Hotness may be represented abstractly as a one-dimensional [[manifold]]. Every valid temperature scale has its own one-to-one map into the hotness manifold.<ref name="Mach 1900">Mach, E. (1900). ''Die Principien der Wärmelehre. Historisch-kritisch entwickelt'', Johann Ambrosius Barth, Leipzig, section 22, pp. 56–57.</ref><ref name="Serrin 1986">Serrin, J. (1986). Chapter 1, 'An Outline of Thermodynamical Structure', pp. 3–32, especially p. 6, in ''New Perspectives in Thermodynamics'', edited by J. Serrin, Springer, Berlin, {{ISBN|3-540-15931-2}}.</ref>

When two systems in thermal contact are at the same temperature no heat transfers between them. When a temperature difference does exist heat flows spontaneously from the warmer system to the colder system until they are in [[thermal equilibrium]]. Such heat transfer occurs by conduction or by thermal radiation.<ref name="Maxwell 1872">Maxwell, J.C. (1872). ''Theory of Heat'', third edition, Longmans, Green, London, p. 32.</ref><ref name="Tait 1884 39-40">Tait, P.G. (1884). ''Heat'', Macmillan, London, Chapter VII, pp. 39–40.</ref><ref name="Planck 1897">Planck, M. (1897/1903). ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans, Green, London, pp. 1–2.</ref><ref>Planck, M. (1914), [https://openlibrary.org/books/OL7154661M/The_theory_of_heat_radiation ''The Theory of Heat Radiation''] {{webarchive|url=https://web.archive.org/web/20111118043153/https://openlibrary.org/books/OL7154661M/The_theory_of_heat_radiation |date=2011-11-18 }}, second edition, translated into English by M. Masius, Blakiston's Son & Co., Philadelphia, reprinted by Kessinger.</ref><ref name=dugdale>
{{cite book
 |author=J.S. Dugdale
 |title=Entropy and its Physical Interpretation
 |publisher=Taylor & Francis
 |year=1996
 |isbn=978-0-7484-0569-5
 |page=13
}}</ref><ref name=reif>
{{cite book
 |author=F. Reif
 |title=Fundamentals of Statistical and Thermal Physics
 |url=https://archive.org/details/fundamentalsofst00fred
 |url-access=registration
 |year=1965
 |publisher=McGraw-Hill
 |page=[https://archive.org/details/fundamentalsofst00fred/page/102 102]
 |isbn=9780070518001
}}</ref><ref name=Moran>
{{cite book
 |author1=M.J. Moran
 |author2=H.N. Shapiro
 |title=Fundamentals of Engineering Thermodynamics
 |edition=5
 |page=14
 |section=1.6.1
 |publisher=John Wiley & Sons, Ltd.
 |year=2006
 |isbn=978-0-470-03037-0
}}</ref><ref>
{{cite web
 |author      = T.W. Leland, Jr.
 |title       = Basic Principles of Classical and Statistical Thermodynamics
 |page        = 14
 |url         = http://www.uic.edu/labs/trl/1.OnlineMaterials/BasicPrinciplesByTWLeland.pdf
 |quote       = Consequently we identify temperature as a driving force which causes something called heat to be transferred.
 |url-status=live
 |archive-url  = https://web.archive.org/web/20110928205821/http://www.uic.edu/labs/trl/1.OnlineMaterials/BasicPrinciplesByTWLeland.pdf
 |archive-date = 2011-09-28
}}</ref>

Experimental physicists, for example [[Galileo Galilei#Engineering|Galileo]] and [[Newtons law of cooling|Newton]],<ref>Tait, P.G. (1884). ''Heat'', Macmillan, London, Chapter VII, pp. 42, 103–117.</ref> found that there are indefinitely many [[Scale of temperature|empirical temperature scales]]. Nevertheless, the [[zeroth law of thermodynamics]] says that they all measure the same quality. This means that for a body in its own state of internal thermodynamic equilibrium, every correctly calibrated thermometer, of whatever kind, that measures the temperature of the body, records one and the same temperature. For a body that is not in its own state of internal thermodynamic equilibrium, different thermometers can record different temperatures, depending respectively on the mechanisms of operation of the thermometers.

=== Bodies in thermodynamic equilibrium ===
For experimental physics, hotness means that, when comparing any two given bodies in their respective separate [[Thermodynamic equilibrium|thermodynamic equilibria]], any two suitably given empirical thermometers with numerical scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature.<ref>Beattie, J.A., Oppenheim, I. (1979). ''Principles of Thermodynamics'', Elsevier Scientific Publishing Company, Amsterdam, {{ISBN|978-0-444-41806-7}}, p. 29.</ref> This does not require the two thermometers to have a linear relation between their numerical scale readings, but it does require that the relation between their numerical readings shall be [[monotonic|strictly monotonic]].<ref>Landsberg, P.T. (1961). ''Thermodynamics with Quantum Statistical Illustrations'', Interscience Publishers, New York, p. 17.</ref><ref>{{cite journal | last1 = Thomsen | first1 = J.S. | year = 1962 | title = A restatement of the zeroth law of thermodynamics | journal = Am. J. Phys. | volume = 30 | issue = 4| pages = 294–296 |bibcode = 1962AmJPh..30..294T |doi = 10.1119/1.1941991 | doi-access = free }}</ref> A definite sense of greater hotness can be had, independently of [[calorimetry]], of thermodynamics, and of properties of particular materials, from [[Wien's displacement law#Frequency-dependent formulation|Wien's displacement law]] of [[thermal radiation]]: the temperature of a bath of [[thermal radiation]] is [[Proportionality (mathematics)|proportional]], by a universal constant, to the frequency of the maximum of its [[Frequency spectrum#Light|frequency spectrum]]; this frequency is always positive, but can have values that [[Third law of thermodynamics|tend to zero]]. Thermal radiation is initially defined for a cavity in thermodynamic equilibrium. These physical facts justify a mathematical statement that hotness exists on an ordered one-dimensional [[manifold]]. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.<ref name="Truesdell 1980"/><ref name="Mach 1900"/><ref name="Serrin 1986"/><ref>Maxwell, J.C. (1872). ''Theory of Heat'', third edition, Longman's, Green & Co, London, p. 45.</ref><ref name="Pitteri 1984">Pitteri, M. (1984). On the axiomatic foundations of temperature, Appendix G6 on pp. 522–544 of ''Rational Thermodynamics'', C. Truesdell, second edition, Springer, New York, {{ISBN|0-387-90874-9}}.</ref>

Except for a system undergoing a [[order parameter|first-order]] [[phase transition|phase change]] such as the melting of ice, as a closed system receives heat, without a change in its volume and without a change in external force fields acting on it, its temperature rises. For a system undergoing such a phase change so slowly that departure from thermodynamic equilibrium can be neglected, its temperature remains constant as the system is supplied with [[latent heat]]. Conversely, a loss of heat from a closed system, without phase change, without change of volume, and without a change in external force fields acting on it, decreases its temperature.<ref>Truesdell, C., Bharatha, S. (1977). ''The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech'', Springer, New York, {{ISBN|0-387-07971-8}}, p. 20.</ref>

=== Bodies in a steady state but not in thermodynamic equilibrium ===
While for bodies in their own thermodynamic equilibrium states, the notion of temperature requires that all empirical thermometers must agree as to which of two bodies is the hotter or that they are at the same temperature, this requirement is not safe for bodies that are in steady states though not in thermodynamic equilibrium. It can then well be that different empirical thermometers disagree about which is hotter, and if this is so, then at least one of the bodies does not have a well-defined absolute thermodynamic temperature. Nevertheless, any one given body and any one suitable empirical thermometer can still support notions of empirical, non-absolute, hotness, and temperature, for a suitable range of processes. This is a matter for study in [[non-equilibrium thermodynamics]]. {{citation needed|date=January 2021}}

=== Bodies not in a steady state ===
When a body is not in a steady-state, then the notion of temperature becomes even less safe than for a body in a steady state not in thermodynamic equilibrium. This is also a matter for study in [[non-equilibrium thermodynamics]].

=== Thermodynamic equilibrium axiomatics ===
For the axiomatic treatment of thermodynamic equilibrium, since the 1930s, it has become customary to refer to a [[zeroth law of thermodynamics]]. The customarily stated minimalist version of such a law postulates only that all bodies, which when thermally connected would be in thermal equilibrium, should be said to have the same temperature by definition, but by itself does not establish temperature as a quantity expressed as a real number on a scale. A more physically informative version of such a law views empirical temperature as a chart on a hotness manifold.<ref name="Mach 1900"/><ref name="Pitteri 1984"/><ref name="Serrin 1978">Serrin, J. (1978). The concepts of thermodynamics, in ''Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, August 1977'', edited by G.M. de La Penha, L.A.J. Medeiros, North-Holland, Amsterdam, {{ISBN|0-444-85166-6}}, pp. 411–451.</ref> While the zeroth law permits the definitions of many different empirical scales of temperature, the [[second law of thermodynamics]] selects the definition of a single preferred, [[absolute temperature]], unique up to an arbitrary scale factor, whence called the [[thermodynamic temperature]].<ref name="Truesdell 1980"/><ref name="Mach 1900"/><ref name="Maxwell 1872 155-158">Maxwell, J.C. (1872). ''Theory of Heat'', third edition, Longmans, Green, London, pp. 155–158.</ref><ref name="Tait 1884 68-69">Tait, P.G. (1884). ''Heat'', Macmillan, London, Chapter VII, Section 95, pp. 68–69.</ref><ref>Buchdahl, H.A. (1966), p. 73.</ref><ref>Kondepudi, D. (2008). ''Introduction to Modern Thermodynamics'', Wiley, Chichester, {{ISBN|978-0-470-01598-8}}, Section 32., pp. 106–108.</ref> If [[internal energy]] is considered as a function of the volume and entropy of a homogeneous system in thermodynamic equilibrium, thermodynamic absolute temperature appears as the partial derivative of [[internal energy]] with respect the [[entropy]] at constant volume. Its natural, intrinsic origin or null point is [[absolute zero]] at which the entropy of any system is at a minimum. Although this is the lowest absolute temperature described by the model, the [[third law of thermodynamics]] postulates that absolute zero cannot be attained by any physical system.

== Heat capacity ==
{{see also|Heat capacity|Calorimetry}}
When an energy transfer to or from a body is only as heat, the state of the body changes. Depending on the surroundings and the walls separating them from the body, various changes are possible in the body. They include chemical reactions, increase of pressure, increase of temperature and phase change. For each kind of change under specified conditions, the heat capacity is the ratio of the quantity of heat transferred to the magnitude of the change.<ref>{{cite book |last1=Green |first1=Don |last2=Perry |first2=Robert H. |title=Perry's Chemical Engineers' Handbook, Eighth Edition |year=2008 |publisher=McGraw-Hill Education |isbn=978-0071422949 |page=660 |edition=8th}}</ref>

For example, if the change is an increase in temperature at constant volume, with no phase change and no chemical change, then the temperature of the body rises and its pressure increases. The quantity of heat transferred, {{math|Δ''Q''}}, divided by the observed temperature change, {{math|Δ''T''}}, is the body's [[heat capacity]] at constant volume:
: <math qid=Q112187521> C_V = \frac{\Delta Q}{\Delta T}. </math>
If heat capacity is measured for a well-defined [[amount of substance]], the [[specific heat]] is the measure of the heat required to increase the temperature of such a unit quantity by one unit of temperature. For example, raising the temperature of water by one kelvin (equal to one degree Celsius) requires 4186 [[joules]] per [[kilogram]] (J/kg).

== Measurement ==
[[File:Pakkanen.jpg|thumb|A typical Celsius thermometer measures a winter day temperature of {{val|−17|u=degC}}.|597x597px]]
{{See also|Timeline of temperature and pressure measurement technology|International Temperature Scale of 1990|Comparison of temperature scales}}
[[Temperature measurement]] using modern scientific [[thermometer]]s and temperature scales goes back at least as far as the early 18th century, when [[Daniel Gabriel Fahrenheit]] adapted a thermometer (switching to [[mercury (element)|mercury]]) and a scale both developed by [[Ole Rømer|Ole Christensen Rømer]]. Fahrenheit's scale is still in use in the United States for non-scientific applications.

Temperature is measured with [[thermometers]] that may be [[calibration|calibrated]] to a variety of [[Temperature conversion formulas|temperature scales]]. In most of the world (except for [[Belize]], [[Myanmar]], [[Liberia]] and the [[United States]]), the Celsius scale is used for most temperature measuring purposes. Most scientists measure temperature using the Celsius scale and thermodynamic temperature using the [[Kelvin]] scale, which is the Celsius scale offset so that its null point is {{val|0|u=K}} = {{val|−273.15|u=degC}}, or [[absolute zero]]. Many engineering fields in the US, notably high-tech and US federal specifications (civil and military), also use the Kelvin and Celsius scales. Other engineering fields in the US also rely upon the [[Rankine scale]] (a shifted Fahrenheit scale) when working in thermodynamic-related disciplines such as [[combustion]].

=== Units ===
The basic unit of temperature in the [[International System of Units]] (SI) is the [[kelvin]]. It has the symbol K.

For everyday applications, it is often convenient to use the Celsius scale, in which {{val|0|u=degC}} corresponds very closely to the [[freezing point]] of water and {{val|100|u=degC}} is its [[boiling point]] at sea level. Because liquid droplets commonly exist in clouds at sub-zero temperatures, {{val|0|u=degC}} is better defined as the melting point of ice. In this scale, a temperature difference of 1 degree Celsius is the same as a {{gaps|1|kelvin}} increment, but the scale is offset by the temperature at which ice melts ({{val|273.15|u=K}}).

By international agreement,<ref>[http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html The kelvin in the SI Brochure] {{webarchive|url=https://web.archive.org/web/20070926215600/http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html |date=2007-09-26 }}</ref> until May 2019, the Kelvin and Celsius scales were defined by two fixing points: [[absolute zero]] and the [[triple point]] of [[Vienna Standard Mean Ocean Water]], which is water specially prepared with a specified blend of hydrogen and oxygen isotopes. Absolute zero was defined as precisely {{val|0|u=K}} and {{val|−273.15|u=degC}}. It is the temperature at which all classical translational motion of the particles comprising matter ceases and they are at complete rest in the classical model. Quantum-mechanically, however, zero-point motion remains and has an associated energy, the [[zero-point energy]].  Matter is in its [[ground state]],<ref>
{{cite web
 |url         = http://www.calphad.com/absolute_zero.html
 |title       = Absolute Zero
 |publisher   = Calphad.com
 |access-date  = 2010-09-16
 |url-status=live
 |archive-url  = https://web.archive.org/web/20110708112153/http://www.calphad.com/absolute_zero.html
 |archive-date = 2011-07-08
}}</ref> and contains no [[thermal energy]]. The temperatures {{val|273.16|u=K}} and {{val|0.01|u=degC}} were defined as those of the triple point of water. This definition served the following purposes: it fixed the magnitude of the kelvin as being precisely 1 part in 273.16 parts of the difference between absolute zero and the triple point of water; it established that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it established the difference between the null points of these scales as being {{val|273.15|u=K}} ({{val|0|u=K}} = {{val|−273.15|u=degC}} and {{val|273.16|u=K}} = {{val|0.01|u=degC}}). Since 2019, there has been a new definition based on the Boltzmann constant,<ref>[https://www.bipm.org/metrology/thermometry/units.html Definition agreed by the 26th General Conference on Weights and Measures (CGPM)] {{Webarchive|url=https://web.archive.org/web/20201009075414/https://www.bipm.org/metrology/thermometry/units.html |date=2020-10-09 }} in November 2018, implemented 20 May 2019</ref> but the scales are scarcely changed.

In the United States, the Fahrenheit scale is the most widely used. On this scale the freezing point of water corresponds to {{val|32|u=degF}} and the boiling point to {{val|212|u=degF}}. The Rankine scale, still used in fields of chemical engineering in the US, is an absolute scale based on the Fahrenheit increment.

==== Historical scales ====
{{see also|Conversion of scales of temperature}}
The following temperature scales are in use or have historically been used for measuring temperature:
* [[Kelvin scale]]
* [[Celsius scale]]
* [[Fahrenheit scale]]
* [[Rankine scale]]
* [[Delisle scale]]
* [[Newton scale]]
* [[Réaumur scale]]
* [[Rømer scale]]

==== Plasma physics ====
The field of [[plasma physics]] deals with phenomena of [[electromagnetic radiation|electromagnetic]] nature that involve very high temperatures. It is customary to express temperature as energy in a unit related to the [[electronvolt]] or kiloelectronvolt ([[Electronvolt#Temperature|eV/''k''<sub>B</sub> or keV/''k''<sub>B</sub>]]). The corresponding energy, which is [[dimensional analysis|dimensionally distinct]] from temperature, is then calculated as the product of the [[Boltzmann constant]] and temperature, <math>E = k_\text{B} T</math>. Then, 1{{nbsp}}eV/''k''<sub>B</sub> is {{val|11605|u=K}}. In the study of [[QCD matter]] one routinely encounters temperatures of the order of a few hundred MeV/''k''<sub>B</sub>, equivalent to about {{val|e=12|u=K}}.

=== Continuous or discrete ===
When one measures the variation of temperature across a region of space or time, do the temperature measurements turn out to be continuous or discrete? There is a widely held misconception that such temperature measurements must always be continuous.<ref name=":0">{{Cite journal |last1=Jha |first1=Aditya |last2=Campbell |first2=Douglas |last3=Montelle |first3=Clemency |last4=Wilson |first4=Phillip L. |date=2023-07-30 |title=On the Continuum Fallacy: Is Temperature a Continuous Function? |journal=Foundations of Physics |language=en |volume=53 |issue=4 |pages=69 |doi=10.1007/s10701-023-00713-x |bibcode=2023FoPh...53...69J |issn=1572-9516|doi-access=free }}</ref> This misconception partly originates from the historical view associated with the continuity of classical [[Physical quantity|physical quantities]], which states that physical quantities must assume every intermediate value between a starting value and a final value.<ref name=":0" /><ref>{{Cite journal |last=van Strien |first=Marij |date=2015-10-01 |title=Continuity in nature and in mathematics: Boltzmann and Poincaré |url=https://doi.org/10.1007/s11229-015-0701-9 |journal=Synthese |language=en |volume=192 |issue=10 |pages=3275–3295 |doi=10.1007/s11229-015-0701-9 |s2cid=255075377 |issn=1573-0964}}</ref> However, the classical picture is only true in the cases where temperature is measured in a system that is in [[Thermal equilibrium|equilibrium]], that is, temperature may not be continuous outside these conditions.<ref name=":0" /> For systems outside equilibrium, such as at interfaces between materials (e.g., a metal/non-metal interface or a liquid-vapour interface) temperature measurements may show steep discontinuities in time and space.<ref name=":0" /> For instance, Fang and Ward were some of the first authors to successfully report temperature discontinuities of as much as 7.8 K at the surface of evaporating water droplets.<ref>{{Cite journal |last1=Fang |first1=G. |last2=Ward |first2=C. A. |date=1999-01-01 |title=Temperature measured close to the interface of an evaporating liquid |url=https://link.aps.org/doi/10.1103/PhysRevE.59.417 |journal=Physical Review E |volume=59 |issue=1 |pages=417–428 |doi=10.1103/PhysRevE.59.417|bibcode=1999PhRvE..59..417F }}</ref> This was reported at inter-molecular scales, or at the scale of the [[mean free path]] of molecules which is typically of the order of a few micrometers in gases<ref>{{Cite journal |last=Newell |first=Homer E. |date=1960-02-12 |title=The Space Environment: As man looks forward to flight into space, he finds the outer regions not completely unknown. |url=https://www.science.org/doi/10.1126/science.131.3398.385 |journal=Science |language=en |volume=131 |issue=3398 |pages=385–390 |doi=10.1126/science.131.3398.385 |pmid=14426791 |issn=0036-8075}}</ref> at room temperature. Generally speaking, temperature discontinuities are considered to be norms rather than exceptions in cases of interfacial heat transfer.<ref name=":1">{{Cite journal |last=Chen |first=Gang |date=2022-08-01 |title=On the molecular picture and interfacial temperature discontinuity during evaporation and condensation |url=https://www.sciencedirect.com/science/article/pii/S0017931022003271 |journal=International Journal of Heat and Mass Transfer |language=en |volume=191 |pages=122845 |doi=10.1016/j.ijheatmasstransfer.2022.122845 |issn=0017-9310|arxiv=2201.07318 |s2cid=246036409 }}</ref> This is due to the abrupt change in the vibrational or [[Heat transfer|thermal properties of the materials]] across such interfaces which prevent instantaneous transfer of heat and the establishment of thermal equilibrium (a prerequisite for having a uniform equilibrium temperature across the interface).<ref name=":2">{{Cite journal |last=Cahill |first=D |display-authors=et al |date=27 Dec 2022 |title=Nanoscale thermal transport |url=https://pubs.aip.org/jap/article/93/2/793/485688/Nanoscale-thermal-transport |access-date=2023-08-02 |journal=Journal of Applied Physics |volume=93 |issue=2 |pages=793–818 |doi=10.1063/1.1524305|hdl=2027.42/70161 |s2cid=15327316 |hdl-access=free }}</ref><ref name=":3">{{Cite journal |last1=Chen |first1=Jie |last2=Xu |first2=Xiangfan |last3=Zhou |first3=Jun |last4=Li |first4=Baowen |date=2022-04-22 |title=Interfacial thermal resistance: Past, present, and future |url=https://link.aps.org/doi/10.1103/RevModPhys.94.025002 |journal=Reviews of Modern Physics |volume=94 |issue=2 |pages=025002 |doi=10.1103/RevModPhys.94.025002|bibcode=2022RvMP...94b5002C |s2cid=248350864 }}</ref> Further, temperature measurements at the macro-scale (typical observational scale) may be too coarse-grained as they average out the microscopic thermal information based on the scale of the representative sample volume of the control system, and thus it is likely that temperature discontinuities at the micro-scale may be overlooked in such averages.<ref name=":0" /> Such an averaging may even produce incorrect or misleading results in many cases of temperature measurements, even at macro-scales, and thus it is prudent that one examines the micro-physical information carefully before averaging out or smoothing out any potential temperature discontinuities in a system as such discontinuities cannot always be averaged or smoothed out.<ref name=":0" /><ref name=":4">{{Cite journal |last1=Aursand |first1=Eskil |last2=Ytrehus |first2=Tor |date=2019-07-01 |title=Comparison of kinetic theory evaporation models for liquid thin-films |url=https://www.sciencedirect.com/science/article/pii/S0301932218308188 |journal=International Journal of Multiphase Flow |language=en |volume=116 |pages=67–79 |doi=10.1016/j.ijmultiphaseflow.2019.04.007 |s2cid=146056093 |issn=0301-9322|hdl=11250/2594950 |hdl-access=free }}</ref> Temperature discontiuities, rather than merely being anomalies, have actually substantially improved our understanding and predictive abilities pertaining to heat transfer at small scales.<ref name=":0" /><ref name=":1" /><ref name=":2" /><ref name=":3" /><ref name=":4" />

== Theoretical foundation ==
{{see also|Thermodynamic temperature}}
Historically, there are several scientific approaches to the explanation of temperature: the classical thermodynamic description based on macroscopic empirical variables that can be measured in a laboratory; the [[kinetic theory of gases]] which relates the macroscopic description to the probability distribution of the energy of motion of gas particles; and a microscopic explanation based on [[statistical physics]] and [[quantum mechanics]]. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics and temperature.<ref name=caratheodory>
{{cite journal
 |author=C. Carathéodory
 |title=Untersuchungen über die Grundlagen der Thermodynamik
 |year=1909
 |journal=Mathematische Annalen
 |volume=67
 |pages=355–386
 |doi=10.1007/BF01450409
 |issue=3
 |s2cid=118230148
 |url=https://zenodo.org/record/1428268
}}</ref> Statistical physics provides a deeper understanding by describing the atomic behavior of matter and derives macroscopic properties from statistical averages of microscopic states, including both classical and quantum states. In the fundamental physical description, the temperature may be measured directly in units of energy. However, in the practical systems of measurement for science, technology, and commerce, such as the modern [[metric system]] of units, the macroscopic and the microscopic descriptions are interrelated by the [[Boltzmann constant]], a proportionality factor that scales temperature to the microscopic mean kinetic energy.

The microscopic description in [[statistical mechanics]] is based on a model that analyzes a system into its fundamental particles of matter or into a set of classical or [[quantum mechanics|quantum-mechanical]] oscillators and considers the system as a [[Statistical ensemble (mathematical physics)|statistical ensemble]] of [[State of matter|microstates]]. As a collection of classical material particles, the temperature is a measure of the mean energy of motion, called translational [[kinetic energy]], of the particles, whether in solids, liquids, gases, or plasmas. The kinetic energy, a concept of [[classical mechanics]], is half the [[mass]] of a particle times its [[speed]] squared. In this mechanical interpretation of thermal motion, the kinetic energies of material particles may reside in the velocity of the particles of their translational or vibrational motion or in the inertia of their rotational modes. In monatomic [[perfect gas]]es and, approximately, in most gas and in simple metals, the temperature is a measure of the mean particle translational kinetic energy, 3/2 ''k''<sub>B</sub>''T''. It also determines the probability distribution function of energy. In condensed matter, and particularly in solids, this purely mechanical description is often less useful and the oscillator model provides a better description to account for quantum mechanical phenomena. Temperature determines the statistical occupation of the microstates of the ensemble. The microscopic definition of temperature is only meaningful in the [[thermodynamic limit]], meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model.

Kinetic energy is also considered as a component of [[thermal energy]]. The thermal energy may be partitioned into independent components attributed to the [[Degrees of freedom (physics and chemistry)|degrees of freedom]] of the particles or to the modes of oscillators in a [[thermodynamic system]]. In general, the number of these degrees of freedom that are available for the [[equipartition theorem|equipartitioning]] of energy depends on the temperature, i.e. the energy region of the interactions under consideration. For solids, the thermal energy is associated primarily with the [[Atom vibrations|vibrations]] of its atoms or molecules about their equilibrium position. In an [[ideal gas|ideal monatomic gas]], the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems, [[vibration]]al and [[rotation]]al motions also contribute degrees of freedom.

=== Kinetic theory of gases ===
[[File:Translational motion.gif|thumb|A theoretical understanding of temperature in a hard-sphere model of a gas can be obtained from the [[Kinetic theory of gases|Kinetic theory]].]]
[[James Clerk Maxwell|Maxwell]] and [[Ludwig Boltzmann|Boltzmann]] developed a [[kinetic theory of gases|kinetic theory]] that yields a fundamental understanding of temperature in gases.<ref>{{cite journal|last=Swendsen|first=Robert|title=Statistical mechanics of colloids and Boltzmann's definition of entropy|journal=American Journal of Physics|date=March 2006|volume=74|issue=3|pages=187–190|doi=10.1119/1.2174962|bibcode = 2006AmJPh..74..187S |s2cid=59471273|url=https://pdfs.semanticscholar.org/ff7b/6750c54750d9b13fa4d9adaeaf4b046bc7e6.pdf|archive-url=https://web.archive.org/web/20200228234741/https://pdfs.semanticscholar.org/ff7b/6750c54750d9b13fa4d9adaeaf4b046bc7e6.pdf|url-status=dead|archive-date=2020-02-28}}</ref>
This theory also explains the [[ideal gas]] law and the observed heat capacity of [[Monatomic gas|monatomic]] (or [[Noble gas|'noble']]) gases.<ref>Balescu, R. (1975). ''Equilibrium and Nonequilibrium Statistical Mechanics'', Wiley, New York, {{ISBN|0-471-04600-0}}, pp. 148–154.</ref><ref name="K&K 391 397">
{{cite book
 |title=Thermal Physics
 |last=Kittel |first=Charles
 |author-link=Charles Kittel
 |author2=Kroemer, Herbert |author-link2=Herbert Kroemer
 |year=1980
 |edition=2nd
 |publisher=W.H. Freeman Company
 |isbn=978-0-7167-1088-2
 |pages=391–397
}}</ref><ref>{{cite journal | last1 = Kondepudi | first1 = D.K. | year = 1987 | title = Microscopic aspects implied by the second law | journal = Foundations of Physics | volume = 17 | issue = 7| pages = 713–722 |bibcode = 1987FoPh...17..713K |doi = 10.1007/BF01889544 | s2cid = 120576357 }}</ref>

[[File:Gas thermometer and absolute zero.jpg|thumb|Plots of pressure vs temperature for three different gas samples extrapolated to absolute zero|300x300px]]
The [[ideal gas law]] is based on observed [[empirical relationship]]s between pressure (''p''), volume (''V''), and temperature (''T''), and was recognized long before the kinetic theory of gases was developed (see [[Boyle's law|Boyle's]] and [[Charles's law|Charles's]] laws). The ideal gas law states:<ref>[https://feynmanlectures.caltech.edu/I_39.html#Ch39-S5 The Feynman Lectures on Physics. 39–5 The ideal gas law]</ref>
: <math>pV = nRT,</math>
where ''n'' is the number of [[mole unit|moles]] of gas and {{physconst|R|symbol=yes}} is the [[gas constant]].

This relationship gives us our first hint that there is an [[absolute zero]] on the temperature scale, because it only holds if the temperature is measured on an absolute scale such as Kelvin's. The [[ideal gas law]] allows one to measure temperature on this absolute scale using the [[gas thermometer]]. The temperature in kelvins can be defined as the pressure in pascals of one mole of gas in a container of one cubic meter, divided by the gas constant.

Although it is not a particularly convenient device, the [[gas thermometer]] provides an essential theoretical basis by which all thermometers can be calibrated. As a practical matter, it is not possible to use a gas thermometer to measure absolute zero temperature since the gases condense into a liquid long before the temperature reaches zero. It is possible, however, to extrapolate to absolute zero by using the ideal gas law, as shown in the figure.

The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational [[kinetic energy]]. Using a sophisticated symmetry argument,<ref>{{cite web |url=http://galileo.phys.virginia.edu/classes/252/kinetic_theory.html |title=Kinetic Theory |website=galileo.phys.virginia.edu |access-date=27 January 2018 |url-status=live |archive-url=https://web.archive.org/web/20170716052320/http://galileo.phys.virginia.edu/classes/252/kinetic_theory.html |archive-date=16 July 2017 }}</ref> [[Ludwig Boltzmann|Boltzmann]] deduced what is now called the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann probability distribution]] function for the velocity of particles in an ideal gas. From that [[probability distribution]] function, the average [[kinetic energy]] (per particle) of a [[monatomic]] [[ideal gas]] is<ref name="K&K 391 397"/><ref>Tolman, R.C. (1938). ''The Principles of Statistical Mechanics'', Oxford University Press, London, pp. 93, 655.</ref>
: <math>E_\text{k} = \frac 1 2 mv_\text{rms}^2 = \frac 3 2 k_\text{B} T,</math>
where the [[Boltzmann constant]] {{var|k}}{{sub|B}} is the [[ideal gas constant]] divided by the [[Avogadro number]], and <math display="inline"> v_\text{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\langle \mathbf{v\cdot v} \rangle}</math> is the [[root-mean-square speed]].<ref name=atkins>
{{cite book
 |title=Physical Chemistry
 |author=Peter Atkins, Julio de Paula
 |page=9
 |edition=8
 |publisher=Oxford University Press
 |year=2006
}}</ref> This direct proportionality between temperature and mean molecular kinetic energy is a special case of the [[Equipartition theorem#General formulation of the equipartition theorem|equipartition theorem]], and holds only in the [[classical physics|classical]] limit of a [[perfect gas]]. It does not hold exactly for most substances.

=== Zeroth law of thermodynamics ===
{{main|Zeroth law of thermodynamics}}

When two otherwise isolated bodies are connected together by a rigid physical path impermeable to matter, there is the spontaneous transfer of energy as heat from the hotter to the colder of them. Eventually, they reach a state of mutual [[thermal equilibrium]], in which heat transfer has ceased, and the bodies' respective state variables have settled to become unchanging.<ref>Maxwell, J.C. (1872). ''Theory of Heat'', third edition, Longman's, Green & Co, London, p. 32.</ref><ref>Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, {{ISBN|0-88318-797-3}}, p. 23, "..., if a temperature gradient exists,  ..., then a flow of heat, ..., must occur to achieve a uniform temperature."</ref><ref>[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', [[Elsevier|North-Holland Publishing Company.]], Amsterdam, (1st edition 1949) fifth edition 1965, p. 8, "... will gradually adjust themselves until eventually they do reach mutual equilibrium after which there will of course be no further change."</ref>

One statement of the [[zeroth law of thermodynamics]] is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other.<ref>Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, {{ISBN|0-88318-797-3}}, p. 22.</ref><ref>[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', [[Elsevier|North-Holland Publishing Company.]], Amsterdam, (1st edition 1949) fifth edition 1965, p. 8: "If two systems are both in thermal equilibrium with a third system then they are in thermal equilibrium with each other."</ref><ref>Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge, p. 29: "... if each of two systems is in equilibrium with a third system then they are in equilibrium with each other."</ref>

This statement helps to define temperature but it does not, by itself, complete the definition. An empirical temperature is a numerical scale for the hotness of a thermodynamic system. Such hotness may be defined as existing on a [[Manifold#Motivational examples|one-dimensional manifold]], stretching between hot and cold. Sometimes the zeroth law is stated to include the existence of a unique universal hotness manifold, and of numerical scales on it, so as to provide a complete definition of empirical temperature.<ref name="Serrin 1978"/> To be suitable for empirical thermometry, a material must have a monotonic relation between hotness and some easily measured state variable, such as pressure or volume, when all other relevant coordinates are fixed. An exceptionally suitable system is the [[ideal gas]], which can provide a temperature scale that matches the absolute Kelvin scale. The Kelvin scale is defined on the basis of the second law of thermodynamics.

=== Second law of thermodynamics ===
{{main|Second law of thermodynamics}}
As an alternative to considering or defining the zeroth law of thermodynamics, it was the historical development in thermodynamics to define temperature in terms of the [[second law of thermodynamics]] which deals with [[entropy]]. {{citation needed|date=January 2021}} The second law states that any process will result in either no change or a net increase in the entropy of the universe. This can be understood in terms of probability.

For example, in a series of coin tosses, a perfectly ordered system would be one in which either every toss comes up heads or every toss comes up tails. This means the outcome is always 100% the same result. In contrast, many mixed (''disordered'') outcomes are possible, and their number increases with each toss. Eventually, the combinations of ~50% heads and ~50% tails dominate, and obtaining an outcome significantly different from 50/50 becomes increasingly unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.

As temperature governs the transfer of heat between two systems and the universe tends to progress toward a maximum of entropy, it is expected that there is some relationship between temperature and entropy. A [[heat engine]] is a device for converting thermal energy into mechanical energy, resulting in the performance of work. An analysis of the [[Carnot heat engine]] provides the necessary relationships. According to energy conservation and energy being a [[state function]] that does not change over a full cycle, the work from a heat engine over a full cycle is equal to the net heat, i.e. the sum of the heat put into the system at high temperature, ''q''<sub>H</sub> > 0, and the waste heat given off at the low temperature, ''q''<sub>C</sub> < 0.<ref name="PlanckBook">{{cite book |last=Planck |first=M. |title=Treatise on Thermodynamics |page=§90 & §137 |quote=eqs.(39), (40), & (65) |publisher=Dover Publications |year=1945}}.</ref>

The efficiency is the work divided by the heat input:
{{NumBlk|:|<math>\text{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}},</math>|4}}
where ''w''<sub>cy</sub> is the work done per cycle. The efficiency depends only on |''q''<sub>C</sub>|/''q''<sub>H</sub>. Because ''q''<sub>C</sub> and ''q''<sub>H</sub> correspond to heat transfer at the temperatures ''T''<sub>C</sub> and ''T''<sub>H</sub>, respectively, |''q''<sub>C</sub>|/''q''<sub>H</sub> should be some function of these temperatures:
{{NumBlk|:|<math>\frac{|q_\text{C}|}{q_\text{H}} = f\left(T_\text{H}, T_\text{C}\right).</math>|5}}

[[Carnot's theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient. {{citation needed|date=January 2021}} Thus, a heat engine operating between ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and ''T''<sub>3</sub>. This can only be the case if
: <math>q_{13} = \frac{q_1 q_2}{q_2 q_3},</math>
which implies
: <math>q_{13} = f\left(T_1, T_3\right) = f\left(T_1, T_2\right)f\left(T_2, T_3\right).</math>

Since the first function is independent of ''T''<sub>2</sub>, this temperature must cancel on the right side, meaning ''f''(''T''<sub>1</sub>, ''T''<sub>3</sub>) is of the form ''g''(''T''<sub>1</sub>)/''g''(''T''<sub>3</sub>) (i.e. {{nowrap|''f''(''T''<sub>1</sub>, ''T''<sub>3</sub>)}} = {{nowrap|''f''(''T''<sub>1</sub>, ''T''<sub>2</sub>)''f''(''T''<sub>2</sub>, ''T''<sub>3</sub>)}} = {{nowrap|''g''(''T''<sub>1</sub>)/''g''(''T''<sub>2</sub>) · ''g''(''T''<sub>2</sub>)/''g''(''T''<sub>3</sub>)}} = {{nowrap|''g''(''T''<sub>1</sub>)/''g''(''T''<sub>3</sub>))}}, where ''g'' is a function of a single temperature. A temperature scale can now be chosen with the property that
{{NumBlk|:|<math>\frac{|q_\text{C}|}{q_\text{H}} = \frac{T_\text{C}}{T_\text{H}}.</math>|6}}

Substituting (6) back into (4) gives a relationship for the efficiency in terms of temperature:
{{NumBlk|:|<math>\text{efficiency} = 1 - \frac{|q_\text{C}|}{q_\text{H}} = 1 - \frac{T_\text{C}}{T_\text{H}}.</math>|7}}

For ''T''<sub>C</sub> = 0{{nbsp}}K the efficiency is 100% and that efficiency becomes greater than 100% below 0{{nbsp}}K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0{{nbsp}}K is the minimum possible temperature. In fact, the lowest temperature ever obtained in a macroscopic system was 20{{nbsp}}nK, which was achieved in 1995 at NIST. Subtracting the right hand side of (5) from the middle portion and rearranging gives<ref name="FermiBook"/><ref name="PlanckBook"/>
: <math>\frac{q_\text{H}}{T_\text{H}} + \frac{q_\text{C}}{T_\text{C}} = 0,</math>
where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, ''S'', whose change characteristically vanishes for a complete cycle if it is defined by
{{NumBlk|:|<math>dS = \frac{dq_\text{rev}}{T},</math>|8}}
where the subscript indicates a reversible process. This function corresponds to the entropy of the system, which was described previously. Rearranging (8) gives a formula for temperature in terms of fictive infinitesimal quasi-reversible elements of entropy and heat:
{{NumBlk|:|<math>T = \frac{dq_\text{rev}}{dS}.</math>|9}}

For a constant-volume system where entropy ''S''(''E'') is a function of its energy ''E'', d''E'' = d''q''<sub>rev</sub> and (9) gives
{{NumBlk|:|<math>T^{-1} = \frac{d}{dE} S(E),</math>|10}}
i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy at constant volume.

=== Definition from statistical mechanics ===
[[Statistical mechanics]] defines temperature based on a system's fundamental degrees of freedom.  Eq.(10) is the defining relation of temperature, where the entropy <math>S</math> is defined (up to a constant) by the logarithm of the number of [[Microstate (statistical mechanics)|microstates]] of the system in the given macrostate (as specified in the [[microcanonical ensemble]]): 
: <math> S = k_\mathrm B \ln(W)</math>
where <math>k_\mathrm B</math> is the Boltzmann constant and ''W'' is the number of microstates with the energy ''E'' of the system (degeneracy).

When two systems with different temperatures are put into purely thermal connection, heat will flow from the higher temperature system to the lower temperature one; thermodynamically this is understood by the second law of thermodynamics: The total change in entropy following a transfer of energy <math> \Delta E</math> from system 1 to system 2 is:
: <math>\Delta S = -(dS/dE)_1 \cdot \Delta E + (dS/dE)_2 \cdot \Delta E = \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\Delta E</math>
and is thus positive if <math>T_1 > T_2</math>

From the point of view of statistical mechanics, the total number of microstates in the combined system 1 + system 2 is <math>N_1 \cdot N_2</math>, the logarithm of which (times the Boltzmann constant) is the sum of their entropies; thus a flow of heat from high to low temperature, which brings an increase in total entropy, is more likely than any other scenario (normally it is much more likely), as there are more microstates in the resulting macrostate.

=== Generalized temperature from single-particle statistics ===
It is possible to extend the definition of temperature even to systems of few particles, like in a [[quantum dot]]. The generalized temperature is obtained by considering time ensembles instead of configuration-space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of [[fermion]]s (''N'' even less than 10) with a single/double-occupancy system. The finite quantum [[grand canonical ensemble]],<ref name="finense">{{cite journal |author=Prati, E. |title=The finite quantum grand canonical ensemble and temperature from single-electron statistics for a mesoscopic device |journal=J. Stat. Mech. |volume=1 |issue=1 |page=P01003 |year=2010 |doi=10.1088/1742-5468/2010/01/P01003 |arxiv=1001.2342 |bibcode=2010JSMTE..01..003P |s2cid=118339343 }} [https://arxiv.org/abs/1001.2342v1 arxiv.org] {{webarchive|url=https://web.archive.org/web/20171122152058/https://arxiv.org/abs/1001.2342v1 |date=2017-11-22 }}</ref> obtained under the hypothesis of [[ergodicity]] and orthodicity,<ref>{{cite web |url=http://tnt.phys.uniroma1.it/twiki/pub/TNTgroup/AngeloVulpiani/Dellago.pdf |title=Realizing Boltzmann's dream: computer simulations in modern statistical mechanics |access-date=2014-04-11 |url-status=live |archive-url=https://web.archive.org/web/20140413130129/http://tnt.phys.uniroma1.it/twiki/pub/TNTgroup/AngeloVulpiani/Dellago.pdf |archive-date=2014-04-13 }}</ref> allows expressing the generalized temperature from the ratio of the average time of occupation <math>\tau_1</math> and <math>\tau_2</math> of the single/double-occupancy system:<ref name="singlepart">{{cite journal |author=Prati, E. |title=Measuring the temperature of a mesoscopic electron system by means of single electron statistics |journal=Applied Physics Letters |volume=96 |issue=11 |page=113109 |year=2010 |doi=10.1063/1.3365204 |url=http://link.aip.org/link/?APL/96/113109 |archive-url=http://arquivo.pt/wayback/20160514121637/http://link.aip.org/link/?APL/96/113109 |url-status=dead |archive-date=2016-05-14 |bibcode=2010ApPhL..96k3109P |arxiv=1002.0037 |s2cid=119209143 |display-authors=etal |access-date=2022-03-02 }} [https://arxiv.org/abs/1002.0037v2 arxiv.org] {{webarchive|url=https://web.archive.org/web/20171122182750/https://arxiv.org/abs/1002.0037v2 |date=2017-11-22 }}</ref>
:<math>
T = \frac{E - E_\text{F} \left(1 + \frac{3}{2N}\right)}{k_\text{B} \ln\left(2\frac{\tau_2}{\tau_1}\right)},
</math>
where ''E''<sub>F</sub> is the [[Fermi energy]]. This generalized temperature tends to the ordinary temperature when ''N'' goes to infinity.

=== Negative temperature ===
{{main|Negative temperature}}
On the empirical temperature scales that are not referenced to absolute zero, a negative temperature is one below the zero point of the scale used. For example, [[dry ice]] has a sublimation temperature of {{val|−78.5|u=degC}} which is equivalent to {{val|−109.3|u=degF}}.<ref>{{cite web |last1=Water Science School |title=Frozen carbon dioxide (dry ice) sublimates directly into a vapor. |url=https://www.usgs.gov/media/images/frozen-carbon-dioxide-dry-ice-sublimates-directly-a-vapor |website=USGS}}</ref> On the absolute Kelvin scale this temperature is {{val|194.6|u=K}}. No body of matter can be brought to exactly {{val|0|u=K}} (the temperature of the ideally coldest possible body) by any finite practicable process; this is a consequence of the [[third law of thermodynamics]].<ref>
{{citation
 | last = Guggenheim
 | first = E.A.
 | author-link = Edward A. Guggenheim
 | title = Thermodynamics. An Advanced Treatment for Chemists and Physicists
 | place = Amsterdam
 | publisher = [[Elsevier|North-Holland Publishing Company.]]
 | orig-year = 1949
 | year = 1967
 | edition = fifth
 | page = 157
}}: "It is impossible by any procedure, no matter how idealized, to reduce the temperature of any system to zero temperature in a finite number of finite operations."</ref><ref>[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge, page 51: "''By no finite series of processes is the absolute zero attainable.''"</ref><ref>Tisza, L. (1966). ''Generalized Thermodynamics'', MIT Press,  Cambridge MA, page 96: "It is impossible to reach absolute zero as a result of a finite sequence of operations."</ref>

The internal kinetic theory states that the temperature of a body of matter cannot take negative values. The thermodynamic temperature scale, however, is not so constrained.

A body of matter can sometimes be conceptually defined in terms of microscopic degrees of freedom, namely particle spins, a subsystem with a temperature other than that of the whole body. When the body is in its state of internal thermodynamic equilibrium, the temperatures of the entire body and the subsystem must be the same. The two temperatures can differ when, by work through externally imposed force fields, energy can be transferred to and from the subsystem, separately from the rest of the body; then, the whole body is not in its own state of internal thermodynamic equilibrium. There is an upper limit of energy such a spin subsystem can attain.

Considering the subsystem to be in a temporary state of virtual thermodynamic equilibrium, obtaining a [[negative temperature]] on the thermodynamic scale is possible. Thermodynamic temperature is the inverse of the derivative of the subsystem's entropy for its internal energy. As the subsystem's internal energy increases, the entropy increases for some range but eventually attains a maximum value and then begins to decrease as the highest energy states begin to fill. At the point of maximum entropy, the temperature function shows the behavior of a [[Mathematical singularity|singularity]] because the slope of the entropy as a function of energy decreases to zero and then turns negative. As the subsystem's entropy reaches its maximum, its thermodynamic temperature goes to positive infinity, switching to negative infinity as the slope turns negative. Such negative temperatures are hotter than any positive temperature. Over time, when the subsystem is exposed to the rest of the body, which has a positive temperature, energy is transferred as heat from the negative temperature subsystem to the positive temperature system.<ref>
{{cite book
 |title=Thermal Physics
 |last=Kittel |first=Charles
 |author-link=Charles Kittel
 |author2=Kroemer, Herbert |author-link2=Herbert Kroemer
 |year=1980
 |edition=2nd
 |publisher=W.H. Freeman Company
 |isbn=978-0-7167-1088-2
 |page=Appendix E
}}</ref> The kinetic theory temperature is not defined for such subsystems.

== Examples <span class="anchor" id="Table of common temperatures"></span> ==
[[File:A Guide to Cosmic Temperatures (SVS14374 - Cosmic Temperatures Infographic Final Full).jpg|thumb|An illustration of the range of cosmic temperatures<ref>{{Cite web |last=SVS |date=2023-08-03 |title=NASA Scientific Visualization Studio {{!}} A Guide to Cosmic Temperatures |url=https://svs.gsfc.nasa.gov/14374/ |access-date=2023-08-06 |website=SVS |language=english}}</ref>]]

{{main|Orders of magnitude (temperature)}}
{| class="wikitable" style="text-align:center"
|+Comparisons of temperatures in various scales
|-
! rowspan=2 |
! colspan=2 | Temperature
! rowspan=2 | Peak emittance [[wavelength]]{{efn|The cited emission wavelengths are for black bodies in equilibrium. CODATA 2006 recommended value of {{val|2.8977685|(51)|e=-3|u=m&thinsp;K}} used for Wien displacement law constant ''b''.}}<br /> of [[Wien's displacement law|black-body radiation]]
|-
! [[Kelvin#Usage conventions|Kelvin]]
! Celsius
|-
| style="background:#d9d9d3"|[[Absolute zero]]<br />(precisely by definition)
| {{val|0|u=K}}
| {{val|−273.15|u=degC}}
| {{n/a|Infinity}}
|-
| style="background:#d9d9d3"|Blackbody temperature of the black hole at the centre of our galaxy, [[Sagittarius A*]]{{efn|1=This the [[Hawking Radiation]] for a [[Schwarzschild metric|Schwarzschild black hole]] of mass M = {{val|4.145e6}} [[Solar mass|{{solar mass}}]].  It is too faint to be observed.}}
| {{Val|15|u=fK}}
| {{val|−273.149999999999985|u=degC}}
| {{val|2.5|e=8|u=km}} ({{Val|1.7|ul=AU}})
|-
| style="background:#d9d9d3"|Lowest temperature<br />achieved<ref name="Lowest temperature"/>
| {{Val|38|u=pK}}
| {{val|-273.149999999962|u=degC}}
| {{val|76263|u=km}}
|-
| style="background:#d9d9d3"|Coldest<br /> [[Bose–Einstein condensate]]{{efn|name="recordcold"|A temperature of 450&nbsp;±80&nbsp;pK in a Bose–Einstein condensate (BEC) of sodium atoms was achieved in 2003 by researchers at [[Massachusetts Institute of Technology|MIT]]. Citation: ''Cooling Bose–Einstein Condensates Below 500 Picokelvin'', A.E. Leanhardt ''et al''., Science '''301''', 12 Sept. 2003, p. 1515. This record's peak emittance black-body wavelength of 6,400 kilometers is roughly the radius of Earth.}}
| {{Val|450|u=pK}}
| {{val|−273.14999999955|u=degC}}
| {{val|6400|u=km}}
|-
| style="background:#d9d9d3"|One millikelvin<br />(precisely by definition)
| {{val|0.001|u=K}}
| {{val|−273.149|u=degC}}
| {{val|2.89777|u=m}}<br />(radio, [[FM broadcasting|FM band]]){{efn|The peak emittance wavelength of {{val|2.89777|u=m}} is a frequency of {{val|103.456|u=MHz}}}}
|-
| style="background:#d9d9d3"|[[Cosmic microwave background]]<br />(2013 measurement)
| {{val|2.7260|u=K}}
| {{val|−270.424|u=degC}}
| {{val|0.00106301|u=m}}<br />(millimeter-wavelength microwave)
|-
| style="background:#d9d9d3"|[[Vienna Standard Mean Ocean Water|Water]] [[triple point]]<br />(previously by definition){{efn|Since 2019, Kelvin is now defined on the [[Boltzmann constant]], so that the triple point is {{val|273.16|0.0001|u=K}}}}
| {{val|273.16|u=K}}
| {{val|0.01|u=degC}}
| {{val|10608.3|u=nm}}<br />(long-wavelength [[Infrared|IR]])
|-
| style="background:#d9d9d3"|Water [[boiling point]]{{ref label|water|A|A}}
| {{val|373.1339|u=K}}
| {{val|99.9839|u=degC}}
| {{val|7766.03|u=nm}}<br />(mid-wavelength IR)
|-
| style="background:#d9d9d3"|[[Iron]] melting point
| {{val|1811|u=K}}
| {{val|1538|u=degC}}
| {{val|1600|u=nm}}<br />([[far infrared]])
|-
| style="background:#d9d9d3" |[[incandescent light bulb|Incandescent lamp]]{{ref label|incandescent|B|B}}
| {{val|2500|u=K}}
| ≈{{val|2200|u=degC}}
| {{val|1160|u=nm}}<br />(near [[infrared]]){{ref label|tungsten|C|C}}
|-
| style="background:#d9d9d3"|[[Sun|Sun's]] visible surface{{ref label|sun|D|D}}{{efn|Measurement was made in 2002 and has an uncertainty of ±3 kelvins. A [http://www.kis.uni-freiburg.de/~hw/astroandsolartitles.html 1989 measurement] {{webarchive|url=https://web.archive.org/web/20100211055705/http://www3.kis.uni-freiburg.de/~hw/astroandsolartitles.html |date=2010-02-11 }} produced a value of 5,777.0±2.5&nbsp;K. Citation: [https://web.archive.org/web/20080307112328/http://theory.physics.helsinki.fi/~sol_phys/Sol0601.pdf ''Overview of the Sun''] (Chapter 1 lecture notes on Solar Physics by Division of Theoretical Physics, Dept. of Physical Sciences, University of Helsinki).}}
| {{val|5778|u=K}}
| {{val|5505|u=degC}}
| {{val|501.5|u=nm}}<br />([[color#Spectral colors|green-blue light]])
|-
| style="background:#d9d9d3" |[[lightning|Lightning bolt]]<br />channel{{ref label|celsiuskelvin|E|E}}
| {{Val|28|u=kK}}
| {{val|28000|u=degC}}
| {{val|100|u=nm}}<br />(far [[ultraviolet]] light)
|-
| style="background:#d9d9d3"|[[Sun#Core|Sun's core]]{{ref label|celsiuskelvin|E|E}}
| {{Val|16|u=MK}}
| 16&nbsp;million&nbsp;°C
| {{val|0.18|u=nm}}<br />([[X-ray]]s)
|-
| style="background:#d9d9d3"|[[Nuclear weapon|Thermonuclear weapon]]<br />(peak temperature){{ref label|celsiuskelvin|E|E}}{{efn|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. Citation: ''Nuclear Weapons Frequently Asked Questions, 3.2.5 Matter At High Temperatures.'' [http://nuclearweaponarchive.org/Nwfaq/Nfaq3.html#nfaq3.2 Link to relevant Web page.] {{webarchive|url=https://web.archive.org/web/20070503215509/http://nuclearweaponarchive.org/Nwfaq/Nfaq3.html |date=2007-05-03 }} All referenced data was compiled from publicly available sources.}}
| {{Val|350|u=MK}}
| 350&nbsp;million&nbsp;°C
| {{Val|8.3|u=nm|e=-3}}<br />([[gamma ray]]s)
|-
| style="background:#d9d9d3"|Sandia National Labs'<br />[[Z Pulsed Power Facility|Z machine]]{{ref label|celsiuskelvin|E|E}}{{efn|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 3.58±0.41&nbsp;GK (309±35&nbsp;keV) for 3&nbsp;ns (ns 112 through 115). [http://prl.aps.org/abstract/PRL/v96/i7/e075003 ''Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over {{val|2|e=9}} Kelvin''], M.G. Haines ''et al.'', Physical Review Letters '''96''' (2006) 075003. [http://sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html Link to Sandia's news release.] {{webarchive|url=https://web.archive.org/web/20100530095133/http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html |date=2010-05-30 }}}}
| {{Val|2|u=GK}}
| 2&nbsp;billion&nbsp;°C
| {{Val|1.4|u=nm|e=-3}}<br />(gamma rays){{ref label|zmachine|F|F}}
|-
| style="background:#d9d9d3"|Core of a [[silicon burning process|high-mass<br />star on its last day]]{{ref label|celsiuskelvin|E|E}}{{efn|Core temperature of a high–mass (>8–11 solar masses) star after it leaves the ''main sequence'' on the [[Hertzsprung-Russell diagram|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]]. Citation: {{cite web |url-status=live |title=Stellar Evolution: The Life and Death of Our Luminous Neighbors |first1=Arthur |last1=Holland |first2=Mark |last2=Williams |publisher=University of Michigan |website=GS265 |archive-url=https://web.archive.org/web/20090116111858/http://umich.edu/~gs265/star.htm |archive-date=2009-01-16 |url=http://umich.edu/~gs265/star.htm }} More informative links can be found here {{cite web |url=http://schools.qps.org/hermanga/images/Astronomy/chapter_21___stellar_explosions.htm |title=Chapter 21 Stellar Explosions |access-date=2016-02-08 |url-status=dead |archive-url=https://web.archive.org/web/20130411183756/http://schools.qps.org/hermanga/images/Astronomy/chapter_21___stellar_explosions.htm |archive-date=2013-04-11 }}, and here {{cite web |url=http://cosserv3.fau.edu/~cis/AST2002/Lectures/C13/Trans/Trans.html |title=Trans |access-date=2016-02-08 |url-status=dead |archive-url=https://web.archive.org/web/20110814005149/http://cosserv3.fau.edu/~cis/AST2002/Lectures/C13/Trans/Trans.html |archive-date=2011-08-14 }}, and a concise treatise on stars by NASA is&nbsp;here {{cite web |url=http://nasa.gov/worldbook/star_worldbook.html |title=NASA - Star |access-date=2010-10-12 |url-status=dead |archive-url=https://web.archive.org/web/20101024154533/http://www.nasa.gov/worldbook/star_worldbook.html |archive-date=2010-10-24 }}.}}
| {{Val|3|u=GK}}
| 3&nbsp;billion&nbsp;°C
| {{Val|1|u=nm|e=-3}}<br />(gamma rays)
|-
| style="background:#d9d9d3"|Merging binary [[neutron star|neutron<br />star]] system{{ref label|celsiuskelvin|E|E}}{{efn|Based on a computer model that predicted a peak internal temperature of 30&nbsp;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 {{val|20|u=km}} in diameter, and were orbiting around their barycenter (common center of mass) at about {{val|390|u=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 {{val|1|to|7|u=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). It's also noteworthy that at 350{{nbsp}}GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass (''m'') 5% greater than its rest mass (''m''<sub>0</sub>).&nbsp; [https://arxiv.org/abs/astro-ph/0507099 ''Torus Formation in Neutron Star Mergers and Well-Localized Short Gamma-Ray Bursts''] {{webarchive|url=https://web.archive.org/web/20171122171839/https://arxiv.org/abs/astro-ph/0507099 |date=2017-11-22 }}, R. Oechslin ''et al''. of [http://www.mpa-garching.mpg.de/ Max Planck Institute for Astrophysics.] {{webarchive|url=https://web.archive.org/web/20050403173333/http://www.mpa-garching.mpg.de/ |date=2005-04-03 }}, arXiv:astro-ph/0507099 v2, 22 Feb. 2006. [http://www.mpa-garching.mpg.de/mpa/research/current_research/hl2005-10/hl2005-10-en.html An html summary] {{webarchive|url=https://web.archive.org/web/20101109042335/http://www.mpa-garching.mpg.de/mpa/research/current_research/hl2005-10/hl2005-10-en.html |date=2010-11-09 }}.}}
| {{Val|350|u=GK}}
| 350&nbsp;billion&nbsp;°C
| {{val|8|e=−6|u=nm}}<br />(gamma rays)
|-
| style="background:#d9d9d3"|[[Relativistic Heavy Ion Collider|Relativistic Heavy<br />Ion Collider]]{{ref label|celciuskelvin|E|E}}<ref>Results of research by Stefan Bathe using the [http://www.phenix.bnl.gov/ PHENIX] {{webarchive|url=https://web.archive.org/web/20081120000211/http://www.phenix.bnl.gov/ |date=2008-11-20 }} detector on the [http://www.bnl.gov/rhic/ Relativistic Heavy Ion Collider] {{webarchive|url=https://web.archive.org/web/20160303211827/https://www.bnl.gov/RHIC/ |date=2016-03-03 }} at [http://www.bnl.gov/world/ Brookhaven National Laboratory] {{webarchive|url=https://web.archive.org/web/20120624155651/http://www.bnl.gov/world/ |date=2012-06-24 }} in Upton, New York. Bathe has studied gold-gold, deuteron-gold, and proton-proton collisions to test the theory of quantum chromodynamics, the theory of the strong force that holds atomic nuclei together. [http://bnl.gov/bnlweb/pubaf/pr/PR_display.asp?prID=06-56 Link to news release.] {{webarchive|url=https://web.archive.org/web/20090211202512/http://www.bnl.gov/bnlweb/pubaf/pr/PR_display.asp?prID=06-56 |date=2009-02-11 }}</ref>
| {{Val|1|u=TK}}
| 1&nbsp;trillion&nbsp;°C
| {{val|3|e=−6|u=nm}}<br />(gamma rays)
|-
| style="background:#d9d9d3" |[[CERN|CERN's]] proton vs<br /> nucleus collisions{{ref label|celsiuskelvin|E|E}}<ref>[http://public.web.cern.ch/public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html How do physicists study particles?] {{webarchive|url=https://web.archive.org/web/20071011103924/http://public.web.cern.ch/Public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html |date=2007-10-11 }} by [http://public.web.cern.ch/public/Welcome.html CERN] {{webarchive|url=https://web.archive.org/web/20120707065844/http://public.web.cern.ch/public/Welcome.html |date=2012-07-07 }}.</ref>
| {{Val|10|u=TK}}
| 10&nbsp;trillion&nbsp;°C
| {{val|3|e=−7|u=nm}}<br />(gamma rays)
|-
|}
{{plainlist|* {{note label|water|A|A}} For [[Vienna Standard Mean Ocean Water]] at one standard atmosphere ({{val|101.325|u=kPa}}) when calibrated strictly per the two-point definition of thermodynamic temperature.
* {{note label|incandescent|B|B}} The {{val|2500|u=K}} value is approximate. The {{val|273.15|u=K}} difference between K and °C is rounded to {{val|300|u=K}} to avoid [[false precision]] in the Celsius value.
* {{note label|tungsten|C|C}} For a true black-body (which tungsten filaments are not). Tungsten filament emissivity is greater at shorter wavelengths, which makes them appear whiter.
* {{note label|sun|D|D}} Effective photosphere temperature. The {{val|273.15|u=K}} difference between K and °C is rounded to {{val|273|u=K}} to avoid false precision in the Celsius value.
* {{note label|celsiuskelvin|E|E}} The {{val|273.15|u=K}} difference between K and °C is within the precision of these values.
* {{note label|zmachine|F|F}} For a true black-body (which the plasma was not). The Z machine's dominant emission originated from {{val|40|u=MK}} electrons (soft x-ray emissions) within the plasma.}}

== See also ==
{{cols|colwidth=21em}}
* {{annotated link|Atmospheric temperature}}
* {{annotated link|Thermoregulation|Body temperature}} (thermoregulation)
* {{annotated link|Color temperature}}
* {{annotated link|Dry-bulb temperature}}
* {{annotated link|Thermal conduction}}
* {{annotated link|Convective heat transfer}}
* {{annotated link|Instrumental temperature record}}
* {{annotated link|ISO 1}}
* {{annotated link|International Temperature Scale of 1990|abbreviation=ITS-90}}
* {{annotated link|Laser schlieren deflectometry}}
* [[List of cities by average temperature]]
* {{annotated link|Maxwell's demon}}
* {{annotated link|Orders of magnitude (temperature)}}
* {{annotated link|Outside air temperature}}
* {{annotated link|Planck temperature}}
* {{annotated link|Rankine scale}}
* {{annotated link|Relativistic heat conduction}}
* {{annotated link|Satellite temperature measurements}}
* {{annotated link|Scale of temperature}}
* {{annotated link|Sea surface temperature}}
* {{annotated link|Stagnation temperature}}
* {{annotated link|Thermal radiation}}
* {{annotated link|Thermoception}}
* {{annotated link|Thermodynamic temperature|Thermodynamic (absolute) temperature}}
* {{annotated link|Thermography}}
* {{annotated link|Thermometer}}
* {{annotated link|Virtual temperature}}
* {{annotated link|Wet-bulb globe temperature}}
* {{annotated link|Wet-bulb temperature}}
{{colend}}

==Notes and references==

; Notes :
{{notelist}}

; Citations :
{{reflist}}

=== Bibliography of cited references ===
* Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.
* Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge.
* Jaynes, E.T. (1965). Gibbs vs Boltzmann entropies, ''American Journal of Physics'', '''33'''(5), 391–398.
* Middleton, W.E.K. (1966). ''A History of the Thermometer and its Use in Metrology'', Johns Hopkins Press, Baltimore.
* {{cite journal | last1 = Miller | first1 = J | year = 2013 | title = Cooling molecules the optoelectric way | journal = Physics Today | volume = 66 | issue = 1 | pages = 12–14 | doi = 10.1063/pt.3.1840 | bibcode = 2013PhT....66a..12M | doi-access = free }}
* [[J. R. Partington|Partington, J.R.]] (1949). ''An Advanced Treatise on Physical Chemistry'', volume 1, ''Fundamental Principles. The Properties of Gases'', Longmans, Green & Co., London, pp.&nbsp;175–177.
* [[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.
* Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.
* Schooley, J.F. (1986). ''Thermometry'', CRC Press, Boca Raton, {{ISBN|0-8493-5833-7}}.
* Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.
* [[William Thomson, 1st Baron Kelvin|Thomson, W. (Lord Kelvin)]] (1848). On an absolute thermometric scale founded on Carnot's theory of the motive power of heat, and calculated from Regnault's observations, ''Proc. Camb. Phil. Soc.'' (1843/1863) '''1''', No. 5: 66–71.
* {{cite journal|last=Thomson|first=W. (Lord Kelvin)|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|date=March 1851|volume=XX|issue=part II|pages=261–268, 289–298}}
* Truesdell, C.A. (1980). ''The Tragicomical History of Thermodynamics, 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.
* Tschoegl, N.W. (2000). ''Fundamentals of Equilibrium and Steady-State Thermodynamics'', Elsevier, Amsterdam, {{ISBN|0-444-50426-5}}.
* {{cite journal | last1 = Zeppenfeld | first1 = M. | last2 = Englert | first2 = B.G.U. | last3 = Glöckner | first3 = R. | last4 = Prehn | first4 = A. | last5 = Mielenz | first5 = M. | last6 = Sommer | first6 = C. | last7 = van Buuren | first7 = L.D. | last8 = Motsch | first8 = M. | last9 = Rempe | first9 = G. | year = 2012 | title = Sysiphus cooling of electrically trapped polyatomic molecules | journal = Nature | volume = 491 | issue = 7425| pages = 570–573 | doi=10.1038/nature11595|arxiv = 1208.0046 |bibcode = 2012Natur.491..570Z | pmid=23151480| s2cid = 4367940 }}

== Further reading ==
* Chang, Hasok (2004). ''Inventing Temperature: Measurement and Scientific Progress''. Oxford: Oxford University Press. {{ISBN|978-0-19-517127-3}}.
* Zemansky, Mark Waldo (1964). ''Temperatures Very Low and Very High''. Princeton, NJ: Van Nostrand.

== External links ==
{{sister project links|d=Q11466|c=Category:Temperature|wikt=temperature|n=no|voy=no|m=no|mw=no|q=no|s=no|b=no|v=Temperature}}
* [https://earth.nullschool.net/#current/wind/surface/level/overlay=temp/winkel3 Current map of global surface temperatures]

{{Scales of temperature}}
{{Meteorological variables}}
{{SI base quantities}}
{{authority control}}
{{Portal bar|Energy}}

[[Category:Temperature| ]]