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Temperature is a
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
that expresses quantitatively the perceptions of hotness and coldness. Temperature is
measured Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on various reference points and thermometric substances for definition. The most common scales are the
Celsius The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
scale with the unit symbol °C (formerly called ''centigrade''), the
Fahrenheit The Fahrenheit scale () is a 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 originally defined his ...
scale (°F), and the Kelvin scale (K), the latter being used predominantly for scientific purposes. The kelvin is one of the seven base units in the International System of Units (SI).
Absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
, i.e., zero kelvin or −273.15 °C, is the lowest point in the
thermodynamic temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
scale. Experimentally, it can be approached very closely but not actually reached, as recognized in the
third law of thermodynamics The third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
. It would be impossible to extract energy as heat from a body at that temperature. Temperature is important in all fields of natural science, including physics, chemistry, Earth science,
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
, medicine,
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, ecology, material science, metallurgy,
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, an ...
and geography 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 ( solid, liquid,
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
eous or plasma), density, solubility,
vapor pressure Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
,
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. A low resistivity indicates a material that readily allows ...
, hardness,
wear resistance Wear is the damaging, gradual removal or deformation of material at solid surfaces. Causes of wear can be mechanical (e.g., erosion) or chemical (e.g., corrosion). The study of wear and related processes is referred to as tribology. Wear in ...
, thermal conductivity,
corrosion resistance Corrosion is a natural process that converts a refined metal into a more chemically stable oxide. It is the gradual deterioration of materials (usually a metal) by chemical or electrochemical reaction with their environment. Corrosion engin ...
, 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 substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breakin ...
s occur * the amount and properties of thermal radiation emitted from the surface of an object *
air temperature Atmospheric temperature is a measure of temperature at different levels of the Earth's atmosphere. It is governed by many factors, including incoming solar radiation, humidity and altitude. When discussing surface air temperature, the annual ...
affects all living organisms * the speed of sound, which in a gas is proportional to the square root of the absolute temperature


Scales

Temperature scales need two values for definition: the point chosen as zero degrees and the magnitudes of the incremental unit of temperature. The
Celsius The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that developed historically, which led to its zero point being defined as the freezing point of water, and as the boiling point of water, both at
atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, ...
at sea level. It was called a centigrade scale because of the 100-degree interval. Since the standardization of the kelvin in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though numerically the scales differ by an exact offset of 273.15. The
Fahrenheit The Fahrenheit scale () is a 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 originally defined his ...
scale is in common use in the United States. 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 scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
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, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
. At this temperature, matter contains no macroscopic thermal energy, but still has quantum-mechanical
zero-point energy Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...
as predicted by the uncertainty principle, although this does not enter into the definition of absolute temperature. Experimentally, absolute zero can be approached only very closely; it can never be reached (the lowest temperature attained by experiment is 100 pK). Theoretically, in a body at a temperature of absolute zero, all classical motion of its particles has ceased and they are at complete rest in this classical sense. 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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, to the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
, and to the Boltzmann statistical mechanical definition of entropy, 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, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and p ...
, widely used in science and technology. The kelvin (the unit name is spelled with a lower-case 'k') is the unit of temperature in the International System of Units (SI). The temperature of a body in a state of thermodynamic equilibrium is always positive relative to the absolute zero. Besides the internationally agreed Kelvin scale, there is also a thermodynamic temperature scale, invented by Lord Kelvin, also with its numerical zero at the absolute zero of temperature, but directly relating to purely macroscopic
thermodynamic Thermodynamics is a branch of physics that deals with heat, 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 the four laws of ther ...
concepts, including the macroscopic entropy, 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 that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the hea ...
, 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 thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at which the ...
of water, the numerical value of which is defined by measurements using the aforementioned internationally agreed Kelvin scale.


Kelvin scale

Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the physicist who first defined it. It is an
absolute Absolute 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 manag ...
scale. Its numerical zero point, , is at the absolute zero of temperature. Since May, 2019, the kelvin has been defined through particle kinetic theory, and statistical mechanics. In the International System of Units (SI), the magnitude of the kelvin is defined in terms of the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, 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, but 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 or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
, by using the reliably reproducible temperature of the
triple point In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at which the ...
of water as a second reference point, the first reference point being at absolute zero. Historically, the temperature of the triple point of water was defined as exactly 273.16 K. Today it is an empirically measured quantity. The freezing point of water at sea-level atmospheric pressure occurs at very close to ().


Classification of scales

There are various kinds of temperature scale. It may be convenient to classify them as empirically and theoretically based. Empirical temperature scales are historically older, while theoretically based scales arose in the middle of the nineteenth century.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 In chemistry and thermodynamics, 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 example, to chemical re ...
, 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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
. 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 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 of state, and is a ...
, 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, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
, 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 distribution 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 A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bio ...
character of the gas, from its temperature and pressure, and from the value of the Boltzmann constant. For a gas of known molecular character and pressure, this provides a relation between temperature and the Boltzmann constant. Those quantities can be known or measured more precisely than can the thermodynamic variables that define the state of a sample of water at its triple point. Consequently, taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.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 body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
can provide an accurate temperature measurement because the frequency of maximum spectral radiance of black-body radiation is directly proportional to the temperature of the black body; this is known as Wien's displacement law and has a theoretical explanation in
Planck's law In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. At ...
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 Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
. 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 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 of state, and is a ...
. 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 entropy of an ideal gas at its absolute zero of temperature is not a positive semi-definite quantity, which puts the gas in violation of the third law of thermodynamics. In contrast to real materials, the ideal gas does not liquefy or solidify, no matter how cold it is. Alternatively thinking, the ideal gas law, refers to the limit of infinitely high temperature and zero pressure; these conditions guarantee non-interactive motions of the constituent molecules.


Kinetic theory approach

The magnitude of the kelvin is now defined in terms of kinetic theory, derived from 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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
. Kinetic theory provides a microscopic account of temperature for some bodies of material, especially gases, based on macroscopic systems' being composed of many microscopic particles, such as molecules and ions of various species, the particles of a species being all alike. It explains macroscopic phenomena through the
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
of the microscopic particles. The
equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
of kinetic theory asserts that each classical degree of freedom of a freely moving particle has an average kinetic energy of where denotes the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
. 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 . Molecules, such as oxygen (O2), have more
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
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 molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. O ...
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 of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
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 the Boltzmann constant. Taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas. It is possible to measure the average kinetic energy of constituent
microscopic The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale be ...
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, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
. 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 the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
and
Fahrenheit The Fahrenheit scale () is a 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 originally defined hi ...
. 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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, 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 The Rankine scale () is an absolute scale of thermodynamic temperature named after the University of Glasgow engineer and physicist Macquorn Rankine, who proposed it in 1859. History Similar to the Kelvin scale, which was first proposed in 184 ...
, made to be aligned with the Fahrenheit scale as
Kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
is with Celsius. The thermodynamic definition of temperature is due to Kelvin. It is framed in terms of an idealized device called a Carnot engine, imagined to run in a fictive continuous cycle of successive processes 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 Entropy production (or generation) is the amount of entropy which is produced in any irreversible processes such as heat and mass transfer processes including motion of bodies, heat exchange, fluid flow, substances expanding or mixing, anelastic ...
. 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 Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
because it is equal to a differential coefficient of one
extensive variable Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
with respect to another, for a given body. It thus has the dimensions of a
ratio In mathematics, a ratio shows 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 the ...
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 total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
, an extensive variable, as a function of its entropy , also an extensive variable, and other state variables , with ), then the temperature is equal to the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
of the internal energy with respect to the entropy: Callen, H.B. (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 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 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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 systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
. 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) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
and Newton, found that there are indefinitely many empirical temperature scales. Nevertheless, the
zeroth law of thermodynamics The zeroth law of thermodynamics is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fow ...
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 In chemistry and thermodynamics, 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 example, to chemical re ...
, of thermodynamics, and of properties of particular materials, from Wien's displacement law of thermal radiation: the temperature of a bath of thermal radiation is proportional, by a universal constant, to the frequency of the maximum of its frequency spectrum; 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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 In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
phase change such as the melting of ice, as a closed system receives heat, without a change in its volume and without a change in external force fields acting on it, its temperature rises. For a system undergoing such a phase change so slowly that departure from thermodynamic equilibrium can be neglected, its temperature remains constant as the system is supplied with
latent heat Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a first-order phase transition. Latent heat can be underst ...
. 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 ext ...
.


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 ext ...
.


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 is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fow ...
. 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 The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
selects the definition of a single preferred,
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
, 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 or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
.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 total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
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 total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
with respect the entropy at constant volume. Its natural, intrinsic origin or null point is
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
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, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
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 of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
at constant volume: : C_V = \frac. If heat capacity is measured for a well-defined
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, io ...
, the
specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
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 The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. ...
(J/kg).


Measurement

Temperature measurement using modern scientific thermometers and temperature scales goes back at least as far as the early 18th century, when
Daniel Gabriel Fahrenheit Daniel Gabriel Fahrenheit FRS (; ; 24 May 1686 – 16 September 1736) was a physicist, inventor, and scientific instrument maker. Born in Poland to a family of German extraction, he later moved to the Dutch Republic at age 15, where he spen ...
adapted a thermometer (switching to mercury) and a scale both developed by Ole Christensen Rømer. Fahrenheit's scale is still in use in the United States for non-scientific applications. Temperature is measured with
thermometers A thermometer is a device that measures temperature or a temperature gradient (the degree of hotness or coldness of an object). A thermometer has two important elements: (1) a temperature sensor (e.g. the bulb of a mercury-in-glass thermomete ...
that may be calibrated to a variety of temperature scales. In most of the world (except for
Belize Belize (; bzj, Bileez) is a Caribbean and Central American country on the northeastern coast of Central America. It is bordered by Mexico to the north, the Caribbean Sea to the east, and Guatemala to the west and south. It also shares a wa ...
,
Myanmar Myanmar, ; UK pronunciations: US pronunciations incl. . Note: Wikipedia's IPA conventions require indicating /r/ even in British English although only some British English speakers pronounce r at the end of syllables. As John Wells explai ...
,
Liberia Liberia (), officially the Republic of Liberia, is a country on the West African coast. It is bordered by Sierra Leone to Liberia–Sierra Leone border, its northwest, Guinea to Guinea–Liberia border, its north, Ivory Coast to Ivory Coast� ...
and the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
), 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 scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
. 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 The Rankine scale () is an absolute scale of thermodynamic temperature named after the University of Glasgow engineer and physicist Macquorn Rankine, who proposed it in 1859. History Similar to the Kelvin scale, which was first proposed in 184 ...
(a shifted Fahrenheit scale) when working in thermodynamic-related disciplines such as
combustion Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...
.


Units

The basic unit of temperature in the International System of Units (SI) is the kelvin. It has the symbol K. For everyday applications, it is often convenient to use the Celsius scale, in which corresponds very closely to the
freezing point The melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depend ...
of water and is its
boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding env ...
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 scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
and the
triple point In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at which the ...
of
Vienna Standard Mean Ocean Water Vienna Standard Mean Ocean Water (VSMOW) is an isotopic standard for water. Despite the name, VSMOW is pure water with no salt or other chemicals found in the oceans. The VSMOW standard was promulgated by the International Atomic Energy Agency ( ...
, 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 energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...
. Matter is in its
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
, and contains no
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
. 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 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.


Historical scales

The following temperature scales are in use or have historically been used for measuring temperature: *
Kelvin scale The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and p ...
*
Celsius scale The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
*
Fahrenheit scale The Fahrenheit scale () is a 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 originally defined h ...
*
Rankine scale The Rankine scale () is an absolute scale of thermodynamic temperature named after the University of Glasgow engineer and physicist Macquorn Rankine, who proposed it in 1859. History Similar to the Kelvin scale, which was first proposed in 184 ...
* Delisle scale * Newton scale * Réaumur scale * Rømer scale


Plasma physics

The field of
plasma physics Plasma ()πλάσμα
, Henry George Liddell, R ...
deals with phenomena of electromagnetic nature that involve very high temperatures. It is customary to express temperature as energy in a unit related to the
electronvolt In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacu ...
or kiloelectronvolt ( eV/''k''B or keV/''k''B). The corresponding energy, which is dimensionally distinct 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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
and temperature, E = k_\text T. Then, 1eV/''k''B is . In the study of QCD matter one routinely encounters temperatures of the order of a few hundred MeV/''k''B, 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 Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ent ...
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 Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxim ...
and
quantum mechanics Quantum mechanics is a fundamental 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 quantum physics including quantum chemistry, ...
. 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, 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 The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Intern ...
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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, a proportionality factor that scales temperature to the microscopic mean kinetic energy. The microscopic description in
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
is based on a model that analyzes a system into its fundamental particles of matter or into a set of classical or
quantum-mechanical Quantum mechanics is a fundamental 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 quantum physics including quantum chemistry, qu ...
oscillators and considers the system as a statistical ensemble of microstates. As a collection of classical material particles, the temperature is a measure of the mean energy of motion, called translational
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
, of the particles, whether in solids, liquids, gases, or plasmas. The kinetic energy, a concept of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, is half the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
of a particle times its
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quant ...
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 In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
, meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model. Kinetic energy is also considered as a component of
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
. The thermal energy may be partitioned into independent components attributed to the
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of the particles or to the modes of oscillators in a
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 surroundings may include other thermodynamic systems, or physical systems that are ...
. In general, the number of these degrees of freedom that are available for the 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 vibrations of its atoms or molecules about their equilibrium position. In an ideal monatomic gas, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems,
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
al and
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
al motions also contribute degrees of freedom.


Kinetic theory of gases

Maxwell Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of ...
and Boltzmann developed a 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 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 of state, and is a ...
law and the observed heat capacity of monatomic (or 'noble') gases. The
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
is based on observed
empirical relationship In science, an empirical relationship or phenomenological relationship is a relationship or correlation that is supported by experiment and observation but not necessarily supported by theory. Analytical solutions without a theory An empirical rel ...
s between pressure (''p''), volume (''V''), and temperature (''T''), and was recognized long before the kinetic theory of gases was developed (see Boyle's and Charles's laws). The ideal gas law states: :pV = nRT, where ''n'' is the number of
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain *The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People * Abraham Moles, French engin ...
of gas and is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
. This relationship gives us our first hint that there is an
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
on the temperature scale, because it only holds if the temperature is measured on an absolute scale such as Kelvin's. The
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
allows one to measure temperature on this absolute scale using the gas thermometer. The temperature in kelvins can be defined as the pressure in pascals of one mole of gas in a container of one cubic meter, divided by the gas constant. Although it is not a particularly convenient device, the gas thermometer provides an essential theoretical basis by which all thermometers can be calibrated. As a practical matter, it is not possible to use a gas thermometer to measure absolute zero temperature since the gases condense into a liquid long before the temperature reaches zero. It is possible, however, to extrapolate to absolute zero by using the ideal gas law, as shown in the figure. The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
. Using a sophisticated symmetry argument, Boltzmann deduced what is now called the Maxwell–Boltzmann probability distribution function for the velocity of particles in an ideal gas. From that
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
function, the average
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
(per particle) of a monatomic
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 of state, and is a ...
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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
is the
ideal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
divided by the Avogadro number, and v_\text = \sqrt = \sqrt is the
root-mean-square speed In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
. This direct proportionality between temperature and mean molecular kinetic energy is a special case of the
equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
, and holds only in the 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 systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, 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 is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fow ...
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 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 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 of state, and is a ...
, 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 The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
which deals with entropy. 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 In thermodynamics and engineering, a heat engine is a system that converts heat to mechanical energy, which can then be used to do mechanical work. It does this by bringing a working substance from a higher state temperature to a lower stat ...
is a device for converting thermal energy into mechanical energy, resulting in the performance of work. An analysis of the
Carnot heat engine A Carnot heat engine is a heat engine that operates on the Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in 1824. The Carnot engine model was graphically expanded by Benoît Paul Émile Clapeyron in 1 ...
provides the necessary relationships. According to energy conservation and energy being a
state function In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
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 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(T_1, T_3\right) = f\left(T_1, T_2\right)f\left(T_2, T_3\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 In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
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 microstates of the system in the given macrostate (as specified in the
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
): : S = k_\mathrm B \ln(W) where k_\mathrm B is the Boltzmann constant and ''W'' is the number of microstates with the energy ''E'' of the system (degeneracy). When two systems with different temperatures are put into purely thermal connection, heat will flow from the higher temperature system to the lower temperature one; thermodynamically this is understood by the second law of thermodynamics: The total change in entropy following a transfer of energy \Delta E from system 1 to system 2 is: :\Delta S = -(dS/dE)_1 \cdot \Delta E + (dS/dE)_2 \cdot \Delta E = \left(\frac - \frac\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 the Boltzmann constant) is the sum of their entropies; thus a flow of heat from high to low temperature, which brings an increase in total entropy, is more likely than any other scenario (normally it is much more likely), as there are more microstates in the resulting macrostate.


Generalized temperature from single-particle statistics

It is possible to extend the definition of temperature even to systems of few particles, like in a
quantum dot Quantum dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the ...
. The generalized temperature is obtained by considering time ensembles instead of configuration-space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s (''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 In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
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 The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi ga ...
. 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 Dry ice is the solid form of carbon dioxide. It is commonly used for temporary refrigeration as CO2 does not have a liquid state at normal atmospheric pressure and sublimates directly from the solid state to the gas state. It is used primarily ...
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, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
. 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 Certain systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers ...
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 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


See also

* * (thermoregulation) * * * * * * * * *
List of cities by average temperature {{Average temperature table/row/C , Algeria , Tamanrasset , 12.8 15.0 18.1 22.2 26.1 28.9 28.7 28.2 26.5 22.4 17.3 13.9 21.7 , ref= {{Average temperature table/row/C , Algeria , Reggane , 16.0 18.2 23.1 27.9 32.2 36.4 39.8 38.4 35.5 29.2 22.0 17.8 ...
* * * * * * * * * * * * * * * * * *


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. * * 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. * 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. * 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, . *


Further reading

* Chang, Hasok (2004). ''Inventing Temperature: Measurement and Scientific Progress''. Oxford: Oxford University Press. . * Zemansky, Mark Waldo (1964). ''Temperatures Very Low and Very High''. Princeton, NJ: Van Nostrand.


External links


Current map of global surface temperatures
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