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{{short description|Physical quantity that expresses hot and cold}} | |||
{{About|the physical quantity||Temperature (disambiguation)}} | {{About|the physical quantity||Temperature (disambiguation)}} | ||
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{{Infobox physical quantity | {{Infobox physical quantity | ||
| name = Temperature | | name = Temperature | ||
| image = | | image = Thermally Agitated Molecule.gif | ||
| caption = | | caption = Thermal vibration of a segment of protein [[alpha helix]]. Its [[amplitude]] increases with temperature | ||
| unit = [[kelvin|K]] | | 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]] | | 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]] | ||
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| derivations = <math>\frac{pV}{nR}</math>, <math>\frac{dq_\text{rev}}{dS}</math> | | derivations = <math>\frac{pV}{nR}</math>, <math>\frac{dq_\text{rev}}{dS}</math> | ||
}} | }} | ||
{{Thermodynamics|cTopic=[[List of thermodynamic properties|System properties]]}}'''Temperature''' is a physical quantity that expresses hot and cold or a measure of the average [[kinetic energy]] of the atoms or molecules in the system. It is the manifestation of [[thermal energy]], present in all [[matter]], which is the source of the occurrence of heat, a flow of energy, when a body is in contact with another that is colder or hotter. Temperature should not be confused with [[heat]]. | |||
{{Thermodynamics|cTopic=[[List of thermodynamic properties|System properties]]}} | |||
'''Temperature''' | |||
Temperature is [[measurement|measured]] with a [[thermometer]]. Thermometers are calibrated in various [[Conversion of units of temperature|temperature scales]] that historically have used various reference points and thermometric substances for definition. The most common scales are the [[Celsius|Celsius scale]] (formerly called ''centigrade'', denoted as °C), the [[Fahrenheit|Fahrenheit scale]] (denoted as °F), and the [[Kelvin|Kelvin scale]] (denoted as K), the last of which is predominantly used for scientific purposes by conventions of the [[International System of Units]] (SI). | Temperature is [[measurement|measured]] with a [[thermometer]]. Thermometers are calibrated in various [[Conversion of units of temperature|temperature scales]] that historically have used various reference points and thermometric substances for definition. The most common scales are the [[Celsius|Celsius scale]] (formerly called ''centigrade'', denoted as °C), the [[Fahrenheit|Fahrenheit scale]] (denoted as °F), and the [[Kelvin|Kelvin scale]] (denoted as K), the last of which is predominantly used for scientific purposes by conventions of the [[International System of Units]] (SI). | ||
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==Effects== | ==Effects== | ||
Many physical processes are related to temperature, some of them are given below: | [[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 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 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 | * the amount and properties of [[thermal radiation]] emitted from the surface of an object | ||
* [[air temperature]] affects all living organisms | * [[air temperature]] affects all living organisms | ||
* the [[speed of sound]] which is a function of the square root of the absolute temperature | * the [[speed of sound]] which is a function of 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== | ==Scales== | ||
{{main|Scale of temperature}} | {{main|Scale of temperature}} | ||
[[File:Thermometer CF.svg|thumb|Two thermometers showing temperature in Celsius and Fahrenheit]] | |||
{{more citations needed section|date=January 2021}} | {{more citations needed section|date=January 2021}} | ||
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===Commonly used scales=== | ===Commonly used scales=== | ||
The [[Celsius]] scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that was developed by historical progress, which led to its zero point {{val|0|u=degC}} being defined by the freezing point of water, and additional degrees defined so that {{val|100|u=degC}} was the boiling point of water, both at sea-level [[ | The [[Celsius]] scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that was developed by historical progress, which led to its zero point {{val|0|u=degC}} being defined by the freezing point of water, and additional degrees defined so that {{val|100|u=degC}} was the boiling point of water, both at sea-level [[atmospheric pressure]]. Because of the 100-degree interval, it was called a centigrade scale.<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, and so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though they differ by an additive offset of exactly 273.15. | ||
The United States commonly uses the [[Fahrenheit]] scale, on which water freezes at {{val|32|u=degF}} and boils at {{val|212|u=degF}} at sea-level atmospheric pressure. | The United States commonly uses the [[Fahrenheit]] scale, on which water freezes at {{val|32|u=degF}} and boils at {{val|212|u=degF}} at sea-level atmospheric pressure. | ||
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===International Kelvin scale=== | ===International 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, its degrees have 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 through various empirical measurements of the average kinetic energies of microscopic particles. It is numerically evaluated 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] (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}}</ref> | 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, its degrees have 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 through various empirical measurements of the average kinetic energies of microscopic particles. It is numerically evaluated 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] (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=== | ===Statistical mechanical ''versus'' thermodynamic temperature scales=== | ||
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===Theoretical scales=== | ===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 | 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==== | ====Microscopic statistical mechanical scale==== | ||
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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 scale|Celsius]]. | 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 scale|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> | 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}} | {{NumBlk|:|<math>\frac{T_1}{T_2} = -\frac{Q_1}{Q_2}.</math>|1}} | ||
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==Heat capacity== | ==Heat capacity== | ||
{{see also|Heat capacity|Calorimetry}} | {{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. {{ | 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 |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: | 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: | ||
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|archive-url = https://web.archive.org/web/20110708112153/http://www.calphad.com/absolute_zero.html | |archive-url = https://web.archive.org/web/20110708112153/http://www.calphad.com/absolute_zero.html | ||
|archive-date = 2011-07-08 | |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)] in November 2018, implemented 20 May 2019</ref> but the scales are scarcely changed. | }}</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. | 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. | ||
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{{main|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> | 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> | 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> | ||
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===Generalized temperature from single-particle statistics=== | ===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=Archived copy |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"> | 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=Archived copy |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> | ||
{{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= | |||
:<math> | :<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)}, | T = \frac{E - E_\text{F} \left(1 + \frac{3}{2N}\right)}{k_\text{B} \ln\left(2\frac{\tau_2}{\tau_1}\right)}, | ||
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===Negative temperature=== | ===Negative temperature=== | ||
{{main|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}}. {{ | 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 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 | | last = Guggenheim | ||
| first = E.A. | | first = E.A. | ||
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| {{n/a|Cannot be defined}} | | {{n/a|Cannot be defined}} | ||
|- | |- | ||
| style="background:#d9d9d3"|Blackbody temperature of the black hole at<br />the centre of our galaxy, [[Sagittarius A*]]<ref>This the [[Hawking Radiation]] for a [[Schwarzschild metric|Schwarzschild black hole]] of mass M = {{val| | | style="background:#d9d9d3"|Blackbody temperature of the black hole at<br />the centre of our galaxy, [[Sagittarius A*]]<ref>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.</ref> | ||
| | | 15 fK | ||
| {{val|−273. | | {{val|−273.149999999999985|u=degC}} | ||
| {{val| | | {{val|2.5|e=8|u=km}} (1.7 [[Astronomical unit|AU]]) | ||
|- | |- | ||
| style="background:#d9d9d3"|Lowest temperature<br />achieved<ref name="ltl">{{cite web |url = http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures |title = World record in low temperatures |access-date = 2009-05-05 |url-status=live |archive-url = https://web.archive.org/web/20090618075820/http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures |archive-date = 2009-06-18 }}</ref> | | style="background:#d9d9d3"|Lowest temperature<br />achieved<ref name="ltl">{{cite web |url = http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures |title = World record in low temperatures |access-date = 2009-05-05 |url-status=live |archive-url = https://web.archive.org/web/20090618075820/http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures |archive-date = 2009-06-18 }}</ref> | ||
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* Jaynes, E.T. (1965). Gibbs vs Boltzmann entropies, ''American Journal of Physics'', '''33'''(5), 391–398. | * 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. | * 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 | url = http://www.physicstoday.org/resource/1/phtoad/v66/i1/p12_s1 | archive-url = | * {{cite journal | last1 = Miller | first1 = J | year = 2013 | title = Cooling molecules the optoelectric way | url = http://www.physicstoday.org/resource/1/phtoad/v66/i1/p12_s1 | archive-url = http://arquivo.pt/wayback/20160515074555/http://www.physicstoday.org/resource/1/phtoad/v66/i1/p12_s1 | url-status = dead | archive-date = 2016-05-15 | journal = Physics Today | volume = 66 | issue = 1 | pages = 12–14 | doi = 10.1063/pt.3.1840 | bibcode = 2013PhT....66a..12M | access-date = 2013-07-25 }} | ||
* [[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. 175–177. | * [[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. 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. | * [[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. | ||