TheInfoList

Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of
thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concepts, such as the internal energy of a system; heat or sensible heat, which are defined as types of energy transfer (as is ...
, present in all matter, which is the source of the occurrence of
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

, a flow of energy, when a body is in contact with another that is colder or hotter. Temperature is with a
thermometer (mercury-in-glass thermometer) for measurement of room temperature. A thermometer is a device that temperature measurement, measures temperature or a temperature gradient A temperature gradient is a physical quantity that describes in which dir ...

. Thermometers are calibrated in various temperature scales that historically have used various reference points and thermometric substances for definition. The most common scales are the
Celsius scale The degree Celsius is a unit of temperature on the Celsius scale, a Scale of temperature, temperature scale originally known as the centigrade scale. The degree Celsius (symbol: °C) can refer to a specific temperature on the Celsius scale or a ...

(formerly called ''centigrade'', denoted as °C), the
Fahrenheit scale The Fahrenheit scale ( or ) is a Scale of temperature, temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686–1736). It uses the degree Fahrenheit (symbol: °F) as the unit. Several accounts of how he ...

(denoted as °F), and the
Kelvin scale The kelvin is the base unit of temperature in the International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonasticall ...

(denoted as K), the last of which is predominantly used for scientific purposes by conventions of the
International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
(SI). The lowest theoretical temperature is
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
, at which no more thermal energy can be extracted from a body. Experimentally, it can only be approached very closely (100 pK), but not reached, which is recognized in the
third law of thermodynamics The third law of thermodynamics states as follows, regarding the properties of closed systems in thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals wit ...
. Temperature is important in all fields of
natural science Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer r ...

, including
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

,
chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

,
Earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Ta ...
,
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, physi ...
,
medicine Medicine is the science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations ...

,
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

,
ecology Ecology (from el, οἶκος, "house" and el, -λογία, label=none, "study of") is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers at the individual, , , , an ...
,
material science The Interdisciplinarity, interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials science ste ...
,
metallurgy Metallurgy is a domain of Materials science, materials science and engineering that studies the physical and chemical behavior of metallic Chemical element, elements, their Inter-metallic alloy, inter-metallic compounds, and their mixtures, which ...
,
mechanical engineering Mechanical engineering is an engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discip ...

and
geography Geography (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10. ...

as well as most aspects of daily life.

# Effects

Many physical processes are related to temperature, some of them are given below: * the physical properties of materials including the
phase Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter) In the physical sciences, a phase is a region of space (a thermodynamic system A thermodynamic system is a ...
(
solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied ...

,
liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) refers to a fluid flow, flow in which the material density is constant within a fluid parc ...

,
gas Gas is one of the four fundamental states of matter (the others being solid Solid is one of the four fundamental states of matter (the others being liquid, gas and plasma). The molecules in a solid are closely packed together and c ...

eous or
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood plasma ...
),
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

,
solubility In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds composed of atom ...

,
vapor pressure 280px, The ''pistol test tube'' experiment. The tube contains alcohol and is closed with a piece of cork. By heating the alcohol, the vapors fill in the space, increasing the pressure in the tube to the point of the cork popping out. Vapor pre ...

,
electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current An electric current is a stream of charged particles, suc ...
,
hardness Hardness (antonym: softness) is a measure of the resistance to localized induced by either mechanical or . In general, different materials differ in their hardness; for example hard metals such as and are harder than soft metals such as and ...
,
wear resistance Wear is the damaging, gradual removal or deformation of material at solid surfaces. Causes of wear can be mechanical (e.g., erosion In earth science, erosion is the action of surface processes (such as Surface runoff, water flow or wind) tha ...

,
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal c ...

,
corrosion resistance Corrosion is a Erosion, natural process that converts a refined metal into a more chemically stable form such as oxide, hydroxide, or sulfide. It is the gradual destruction of materials (usually a metal) by chemical and/or electrochemical reactio ...

, strength * the rate and extent to which
chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substance A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and t ...

s occur * the amount and properties of
thermal radiation Thermal radiation in visible light can be seen on this hot metalwork. Its emission in the infrared is invisible to the human eye. Infrared cameras are capable of capturing this infrared emission (see Thermography). Thermal radiation is electrom ...
emitted from the surface of an object *
air temperature Temperature is a physical quantity that expresses hot and cold. 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 i ...
affects all living organisms * the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends s ...
which is a function of the square root of the absolute temperature.

# Scales

Temperature scales differ in two ways: the point chosen as zero degrees and the magnitudes of incremental units or degrees on the scale.

## Commonly used scales

The
Celsius The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...

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 being defined by the freezing point of water, and additional degrees defined so that was the boiling point of water, both at sea-level atmospheric pressure. Because of the 100-degree interval, it was called a centigrade scale. 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 The Fahrenheit scale ( or ) is a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to convenient and stable parameters, such as ...

scale, on which water freezes at and boils at at sea-level atmospheric pressure.

## Absolute zero

At the
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
of temperature, no energy can be removed from matter as heat, a fact expressed in the
third law of thermodynamics The third law of thermodynamics states as follows, regarding the properties of closed systems in thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals wit ...
. At this temperature, matter contains no macroscopic thermal energy, but still has quantum-mechanical
zero-point energy Zero point may refer to: *Origin (mathematics), a fixed point of reference for a coordinate system *Zero Point (film), ''Zero Point'' (film), an Estonian film *Zero point (photometry), a calibration mechanism for magnitude in astronomy *Zero Point ...
as predicted by the
uncertainty principle In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quant ...

, 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 (least temperature attained by experiment is 100 pK). Theoretically, in a body at absolute zero temperature, all classical motion of its particles has ceased and they are at complete rest in this classical sense. The absolute zero, defined as , is exactly equal to , or .

## Absolute scales

Referring to the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, to the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used for describing particle speeds i ...
, and to the Boltzmann statistical mechanical definition of
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics ...

, as distinct from the Gibbs definition,Jaynes, E.T. (1965), pp. 391–398. 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 the absolute zero, it does not have a reference temperature. It is known as the
Kelvin scale The kelvin is the base unit of temperature in the International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonasticall ...
, widely used in science and technology. The kelvin (the word is spelled with a
lower-case Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet), a written element of an alphabet * Letterform, a typographic term for alphabetical letter shapes * Rehea ...
k) is the unit of temperature in the
International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
(SI). The temperature of a body in its own state of thermodynamic equilibrium is always positive, relative to the
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
. Besides the internationally agreed Kelvin scale, there is also a , invented by
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...

, also with its numerical zero at the absolute zero of temperature, but directly relating to purely macroscopic
thermodynamic Thermodynamics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ent ...

concepts, including the macroscopic
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics ...

, though microscopically referable to the Gibbs statistical mechanical definition of entropy for the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble (mathematical physics), statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. Th ...
, 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. This scale has a reference temperature at the
triple point In , the triple point of a substance is the and at which the three (, , and ) of that substance coexist in .. It is that temperature and pressure at which the curve, curve and the curve meet. For example, the triple point of occurs at a tem ...
of water, the numerical value of which is defined by measurements using the aforementioned internationally agreed Kelvin scale.

## International Kelvin scale

Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the physicist who first defined it. It is an
absoluteAbsolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk managem ...
scale. Its numerical zero point, , is at the
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
of temperature. Since May, 2019, its degrees have been defined through particle kinetic theory, and statistical mechanics. In the
International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
(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 Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, the value of which is defined as fixed by international convention.Cryogenic Society
(2019).

## 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, since 1954, the International System of Units defined a scale and unit for the kelvin as a
thermodynamic temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. A thermodynamic temperature reading of zero is of particular importance for the third law of therm ...

, by using the reliably reproducible temperature of the
triple point In , the triple point of a substance is the and at which the three (, , and ) of that substance coexist in .. It is that temperature and pressure at which the curve, curve and the curve meet. For example, the triple point of occurs at a tem ...
of water as a second reference point, the first reference point being at absolute zero. Historically, the triple point temperature of water was defined as exactly 273.16 units of the measurement increment. Today it is an empirically measured quantity. The freezing point of water at sea-level atmospheric pressure occurs at approximately = .

# Classification of scales

There is a variety of 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.Truesdell, C.A. (1980), Sections 11 B, 11H, pp. 306–310, 320–332.

## 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, 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 upSnellen direct calorimetry chamber, University of Ottawa. Calorimetry is the science or act of measuring changes in ''state variables'' of a body for the purpose of deriving the heat transfer associated with changes of its state due, for exam ...
, 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, 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 The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
. 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 An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter (the others being solid, liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mecha ...
, 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, 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 In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used for describing particle speeds i ...
, which gives a well-founded measurement of temperatures for which the law holds. There have not yet been successful experiments of this same kind that directly use the
Fermi–Dirac distributionFermi–Dirac may refer to: * Fermi–Dirac statistics or Fermi–Dirac distribution * Fermi–Dirac integral (disambiguation) ** Complete Fermi–Dirac integral ** Incomplete Fermi–Dirac integral See also

* Fermi (disambiguation) * Dirac (di ...
for thermometry, but perhaps that will be achieved in the future. The speed of sound in a gas can be calculated theoretically from the molecular character of the gas, from its temperature and pressure, and from the value of Boltzmann's constant. For a gas of known molecular character and pressure, this provides a relation between temperature and Boltzmann's 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 Boltzmann's 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.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 Measurement of the spectrum of electromagnetic radiation from an ideal three-dimensional
black body A black body or blackbody is an idealized physical object, physical body that absorption (electromagnetic radiation), absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence (optics), angle of incidence. The ...

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 upright=1.45, Black-body radiation as a function of wavelength for various temperatures. Each temperature curve peaks at a different wavelength and Wien's law describes the shift of that peak. Wien's displacement law states that the black-body r ...
and has a theoretical explanation in
Planck's law Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of th ...
and the 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 Johnson is a surname of English name, English and Scottish name, Scottish origin.. The name itself is a patronym of the given name ''John (first name), John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', wh ...
. If the value of the resistance is known then the temperature can be found.

### 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. 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 . 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 An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter (the others being solid, liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mecha ...
. 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. The ideal gas thermometer is, however, not theoretically perfect for thermodynamics. This is because the 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.

# Kinetic theory approach

The magnitude of the kelvin is now defined in terms of kinetic theory, derived from the value of
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
.
Kinetic theory of the ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation o ...

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
molecule A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically Electricity is the set of physical phenomena associated with the presence and motion I ...

s and
ion An ion () is an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ...
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 peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the ''average'' kinetic energy In physics, the kinetic energy of an object is the e ...
of kinetic theory asserts that each classical
degree of freedom Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
of a freely moving particle has an average kinetic energy of where denotes
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
. 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 will be .
Molecule A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically Electricity is the set of physical phenomena associated with the presence and motion I ...

s, such as oxygen (O2), have more
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation Translation is the communicatio ...
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 equipartitioning, the energy associated with vibrational and rotational modes to increase. Thus a
diatomic Diatomic molecules are molecule A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically neutral group of two or more atoms held together by chemical b ...
gas will require more energy input to increase its temperature by a certain amount, i.e. it will have a greater
heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on t ...
than a monatomic gas. As noted above, the speed of sound in a gas can be calculated from the molecular character of the gas, from its temperature and pressure, and from the value of Boltzmann's constant. Taking the value of Boltzmann's 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. It is possible to measure the average kinetic energy of constituent 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

Temperature is one of the principal quantities in the study of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by ...

. 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 The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...
and
Fahrenheit The Fahrenheit scale ( or ) is a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to convenient and stable parameters, such as ...
. 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 The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, 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 __NOTOC__ The Rankine scale () is an absolute scale An absolute scale is a system of measurement that begins at a minimum, or zero point, and progresses in only one direction. An absolute scale differs from an arbitrary, or "relative", scale, whic ...
, made to be aligned with the
Fahrenheit scale The Fahrenheit scale ( or ) is a Scale of temperature, temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686–1736). It uses the degree Fahrenheit (symbol: °F) as the unit. Several accounts of how he ...
as
Kelvin The kelvin is the base unit of temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat, a flow of energy, ...
is with
Celsius The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...
. The thermodynamic definition of temperature is due to Kelvin. It is framed in terms of an idealized device called a , imagined to run in a fictive continuous that traverse a cycle of states of its working body. The engine takes in a quantity of heat from a hot reservoir and passes out a lesser quantity of waste heat 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, and , of the reservoirs are defined such that. 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. 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. 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 variable because it is equal to a differential coefficient of one extensive variable with respect to another, for a given body. It thus has the
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
of a
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

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.Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, , pp. 49, 69.Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, , pp. 14–15, 214. In particular, when the body is described by stating its
internal energy The internal energy of a thermodynamic system is the energy contained within it. It is the energy necessary to create or prepare the system in any given internal state. It does not include the kinetic energy of motion of the system as a whole, ...
, an extensive variable, as a function of its
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics ...

, also an extensive variable, and other state variables , with ), then the temperature is equal to the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of the internal energy with respect to the entropy: (1960/1985), ''Thermodynamics and an Introduction to Thermostatistics'', (first edition 1960), second edition 1985, John Wiley & Sons, New York, , pp. 146–148. Likewise, when the body is described by stating its entropy as a function of its internal energy , and other state variables , with , then the reciprocal of the temperature is equal to the partial derivative of the entropy with respect to the internal energy: 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.

## 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
local thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of ma ...
is said to prevail throughout the body. It makes good sense, for example, to say of the extensive variable , or of the extensive variable , 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

Temperature is a measure of a
quality Quality may refer to: Concepts *Quality (business), the ''non-inferiority'' or ''superiority'' of something *Quality (philosophy), an attribute or a property *Quality (physics), in response theory *Energy quality, used in various science disciplin ...
of a state of a material. 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. 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

. Every valid temperature scale has its own one-to-one map into the hotness manifold.Mach, E. (1900). ''Die Principien der Wärmelehre. Historisch-kritisch entwickelt'', Johann Ambrosius Barth, Leipzig, section 22, pp. 56–57.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, . 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 Two physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment (systems), environment. The enviro ...

. Such heat transfer occurs by conduction or by thermal radiation.Maxwell, J.C. (1872). ''Theory of Heat'', third edition, Longmans, Green, London, p. 32.Tait, P.G. (1884). ''Heat'', Macmillan, London, Chapter VII, pp. 39–40.Planck, M. (1897/1903). ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans, Green, London, pp. 1–2. Experimental physicists, for example
Galileo Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific qu ...

and
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
, found that there are indefinitely many empirical temperature scales. Nevertheless, the
zeroth law of thermodynamics The zeroth law of thermodynamics states that if two thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundi ...

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 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. 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 strictly monotonic. A definite sense of greater hotness can be had, independently of
calorimetry upSnellen direct calorimetry chamber, University of Ottawa. Calorimetry is the science or act of measuring changes in ''state variables'' of a body for the purpose of deriving the heat transfer associated with changes of its state due, for exam ...
, of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by ...

, and of properties of particular materials, from
Wien's displacement law upright=1.45, Black-body radiation as a function of wavelength for various temperatures. Each temperature curve peaks at a different wavelength and Wien's law describes the shift of that peak. Wien's displacement law states that the black-body r ...
of
thermal radiation Thermal radiation in visible light can be seen on this hot metalwork. Its emission in the infrared is invisible to the human eye. Infrared cameras are capable of capturing this infrared emission (see Thermography). Thermal radiation is electrom ...
: the temperature of a bath of
thermal radiation Thermal radiation in visible light can be seen on this hot metalwork. Its emission in the infrared is invisible to the human eye. Infrared cameras are capable of capturing this infrared emission (see Thermography). Thermal radiation is electrom ...
is
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
, by a universal constant, to the frequency of the maximum of its
frequency spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System ...

; this frequency is always positive, but can have values that 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.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, . Except for a system undergoing a first-order
phase changePhase change may refer to: * Phase transition, the transformation from one thermodynamic state to another. * Phase-change memory, a type of random-access memory. * Phase change (waves), concerning the physics of waves. {{disambiguation ...
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 Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any su ...
. 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.

## 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, anyone has 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 Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an extr ...
.

## 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 Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an extr ...
.

## 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 zeroth law of thermodynamics states that if two thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundi ...

. 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.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, , pp. 411–451. 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 Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. A thermodynamic temperature reading of zero is of particular importance for the third law of therm ...

.Maxwell, J.C. (1872). ''Theory of Heat'', third edition, Longmans, Green, London, pp. 155–158.Tait, P.G. (1884). ''Heat'', Macmillan, London, Chapter VII, Section 95, pp. 68–69. If
internal energy The internal energy of a thermodynamic system is the energy contained within it. It is the energy necessary to create or prepare the system in any given internal state. It does not include the kinetic energy of motion of the system as a whole, ...
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 The internal energy of a thermodynamic system is the energy contained within it. It is the energy necessary to create or prepare the system in any given internal state. It does not include the kinetic energy of motion of the system as a whole, ...
with respect the
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics ...

at constant volume. Its natural, intrinsic origin or null point is
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
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 The third law of thermodynamics states as follows, regarding the properties of closed systems in thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals wit ...
postulates that absolute zero cannot be attained by any physical system.

# Heat capacity

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. 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, , divided by the observed temperature change, , is the body's
heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on t ...
at constant volume: : $C_V = \frac.$ 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

Temperature measurement using modern scientific
thermometer (mercury-in-glass thermometer) for measurement of room temperature. A thermometer is a device that temperature measurement, measures temperature or a temperature gradient A temperature gradient is a physical quantity that describes in which dir ...

s and temperature scales goes back at least as far as the early 18th century, when 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 = , or
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
. 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 __NOTOC__ The Rankine scale () is an absolute scale An absolute scale is a system of measurement that begins at a minimum, or zero point, and progresses in only one direction. An absolute scale differs from an arbitrary, or "relative", scale, whic ...
(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 International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
(SI) is the Kelvin. It has the symbol K. For everyday applications, it is often convenient to use the Celsius scale, in which corresponds very closely to the freezing point of water and is its boiling point at sea level. Because liquid droplets commonly exist in clouds at sub-zero temperatures, is better defined as the melting point of ice. In this scale, a temperature difference of 1 degree Celsius is the same as a increment, but the scale is offset by the temperature at which ice melts (). By international agreement, until May 2019, the Kelvin and Celsius scales were defined by two fixing points:
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
and the
triple point In , the triple point of a substance is the and at which the three (, , and ) of that substance coexist in .. It is that temperature and pressure at which the curve, curve and the curve meet. For example, the triple point of occurs at a tem ...
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 and . 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 Zero point may refer to: *Origin (mathematics), a fixed point of reference for a coordinate system *Zero Point (film), ''Zero Point'' (film), an Estonian film *Zero point (photometry), a calibration mechanism for magnitude in astronomy *Zero Point ...
. Matter is in its ground state, and contains no
thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concepts, such as the internal energy of a system; heat or sensible heat, which are defined as types of energy transfer (as is ...
. The temperatures and 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 ( = and = ). Since 2019, there has been a new definition based on the Boltzmann constant, but the scales are scarcely changed. In the United States, the
Fahrenheit The Fahrenheit scale ( or ) is a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to convenient and stable parameters, such as ...

scale is the most widely used. On this scale the freezing point of water corresponds to and the boiling point to . The Rankine scale, still used in fields of chemical engineering in the US, is an absolute scale based on the Fahrenheit increment.

### Conversion

The following table shows the temperature conversion formulas for conversions to and from the Celsius 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 units of electronvolts (eV) or kiloelectronvolts (keV). The energy, which has a different dimensional analysis, dimension from temperature, is then calculated as the product of the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
and temperature, $E = k_\text T$. Then, 1eV corresponds to . In the study of QCD matter one routinely encounters temperatures of the order of a few hundred MeV, equivalent to about .

# Theoretical foundation

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. 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, using natural units, 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 The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, 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 gases and, approximately, in most gas and in simple metals, the temperature is a measure of the mean particle translational kinetic energy, 3/2 ''k''B''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 Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concepts, such as the internal energy of a system; heat or sensible heat, which are defined as types of energy transfer (as is ...
. The thermal energy may be partitioned into independent components attributed to the
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation Translation is the communicatio ...
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, vibrational and rotational motions also contribute degrees of freedom.

## Kinetic theory of gases

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. This theory also explains the
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter (the others being solid, liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mecha ...
law and the observed heat capacity of Monatomic gas, monatomic (or Noble gas, 'noble') gases. The ideal gas law is based on observed empirical relationships 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: :$pV = nRT,$ where ''n'' is the number of mole unit, moles of gas and is the gas constant. This relationship gives us our first hint that there is an
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
on the temperature scale, because it only holds if the temperature is measured on an absolute temperature, absolute scale such as Kelvin's. The ideal gas law allows one to measure temperature on this absolute temperature, 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, 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 An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter (the others being solid, liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mecha ...
is :$E_\text = \frac 1 2 mv_\text^2 = \frac 3 2 k_\text T,$ where the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
is the ideal gas constant divided by the Avogadro number, and $v_\text = \sqrt = \sqrt$ is the root-mean-square speed. 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

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 Two physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment (systems), environment. The enviro ...

, in which heat transfer has ceased, and the bodies' respective state variables have settled to become unchanging. One statement of the
zeroth law of thermodynamics The zeroth law of thermodynamics states that if two thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundi ...

is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other. 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. 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 An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter (the others being solid, liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mecha ...
, 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

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 Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics ...

. 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''H > 0, and the waste heat given off at the low temperature, ''q''C < 0.. The efficiency is the work divided by the heat input: where ''w''cy is the work done per cycle. The efficiency depends only on , ''q''C, /''q''H. Because ''q''C and ''q''H correspond to heat transfer at the temperatures ''T''C and ''T''H, respectively, , ''q''C, /''q''H should be some function of these temperatures: Carnot's theorem (thermodynamics), Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between ''T''1 and ''T''3 must have the same efficiency as one consisting of two cycles, one between ''T''1 and ''T''2, and the second between ''T''2 and ''T''3. This can only be the case if :$q_ = \frac,$ which implies :$q_ = f\left\left(T_1, T_3\right\right) = f\left\left(T_1, T_2\right\right)f\left\left(T_2, T_3\right\right).$ Since the first function is independent of ''T''2, this temperature must cancel on the right side, meaning ''f''(''T''1, ''T''3) is of the form ''g''(''T''1)/''g''(''T''3) (i.e. = = = , where ''g'' is a function of a single temperature. A temperature scale can now be chosen with the property that Substituting (6) back into (4) gives a relationship for the efficiency in terms of temperature: For ''T''C = 0K the efficiency is 100% and that efficiency becomes greater than 100% below 0K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0K is the minimum possible temperature. In fact, the lowest temperature ever obtained in a macroscopic system was 20nK, which was achieved in 1995 at NIST. Subtracting the right hand side of (5) from the middle portion and rearranging gives :$\frac + \frac = 0,$ 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 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: For a constant-volume system where entropy ''S''(''E'') is a function of its energy ''E'', d''E'' = d''q''rev and (9) gives 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 $S$ 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): :$S = k_\mathrm B \ln\left(W\right)$ where $k_\mathrm B$ is Boltzmann's 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 $\Delta E$ from system 1 to system 2 is: :$\Delta S = -\left(dS/dE\right)_1 \cdot \Delta E + \left(dS/dE\right)_2 \cdot \Delta E = \left\left(\frac - \frac\right\right)\Delta E$ and is thus positive if $T_1 > T_2$ From the point of view of statistical mechanics, the total number of microstates in the combined system 1 + system 2 is $N_1 \cdot N_2$, the logarithm of which (times Boltzmann's 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 fermions (''N'' even less than 10) with a single/double-occupancy system. The finite quantum grand canonical ensemble,arxiv.org
obtained under the hypothesis of ergodicity and orthodicity, allows expressing the generalized temperature from the ratio of the average time of occupation $\tau_1$ and $\tau_2$ of the single/double-occupancy system:
arxiv.org
:$T = \frac,$ where ''E''F is the Fermi energy. This generalized temperature tends to the ordinary temperature when ''N'' goes to infinity.

## 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 which is equivalent to . On the absolute Kelvin scale this temperature is . No body can be brought to exactly (the temperature of the ideally coldest possible body) by any finite practicable process; this is a consequence of the
third law of thermodynamics The third law of thermodynamics states as follows, regarding the properties of closed systems in thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals wit ...
. The international kinetic theory temperature of a body cannot take negative values. The thermodynamic temperature scale, however, is not so constrained. For a body of matter, there 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 own state of internal thermodynamic equilibrium, the temperatures of the whole body and of 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, it is possible to obtain a negative temperature on the thermodynamic scale. Thermodynamic temperature is the inverse of the derivative of the subsystem's entropy with respect to 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. The kinetic theory temperature is not defined for such subsystems.

# Examples

* * (thermoregulation) * * * * * * * * * List of cities by average temperature * * * * * * * * * * * * * * * * * *

# Notes and references

## Bibliography of cited references

* Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, . * 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. * * 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. * Quinn, T.J. (1983). ''Temperature'', Academic Press, London, . * Schooley, J.F. (1986). ''Thermometry'', CRC Press, Boca Raton, . * 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. * * Truesdell, C.A. (1980). ''The Tragicomical History of Thermodynamics, 1822–1854'', Springer, New York, . * Tschoegl, N.W. (2000). ''Fundamentals of Equilibrium and Steady-State Thermodynamics'', Elsevier, Amsterdam, . *