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A map of global long term monthly average surface air temperatures in Mollweide projection.

Temperature is a physical quantity that is a measure of hotness and coldness on a numerical scale.[1] It is a measure of the thermal energy per particle of matter or radiation; it is measured by a thermometer, which may be calibrated in any of various temperature scales, Celsius, Fahrenheit, Kelvin, etc.

Temperature is an intensive property, which means it is independent of the amount of material present; in contrast to energy, an extensive property, which is proportional to the amount of material in the system. For example a spark may well be (very briefly!) as hot as the Sun.

Empirically it is found that an isolated system, one that exchanges no energy or material with its environment, tends to a spatially uniform temperature as time passes. When a path permeable only to heat is open between two bodies, energy always transfers spontaneously as heat from a hotter body to a colder one. The transfer rate depends on the thermal conductivity of the path or boundary between them. Between two bodies with the same temperature no heat flows. These bodies are said to be in thermal equilibrium.

In kinetic theory and in statistical mechanics, temperature is the effect of the thermal energy arising from the motion of microscopic particles[2] such as atoms, molecules and photons. The relation is proportional as given by the Boltzmann constant.

Thermal vibration of a segment of protein alpha helix. The amplitude of the vibrations increases with temperature.

The lowest theoretical temperature is called absolute zero, however it cannot be achieved in any actual physical device. It is denoted by 0 K on the Kelvin scale, −273.15 °C on the Celsius scale. In matter at absolute zero, the motions of microscopic constituents are minimal; moreover their kinetic energies are also minimal.

Temperature is important in all fields of natural science, including physics, geology, chemistry, atmospheric sciences and biology.

## Use in science

Annual mean temperature around the world

Many things depend on temperature, such as

## Temperature scales

Much of the world uses the Celsius scale (°C) for most temperature measurements. It has the same incremental scaling as the Kelvin scale used by scientists, but fixes its null point, at 0°C = 273.15K, approximately the freezing point of water (at one atmosphere of pressure).[note 1] The United States uses the Fahrenheit scale for common purposes, a scale on which water freezes at 32 °F and boils at 212 °F (at one atmosphere of pressure).

For practical purposes of scientific temperature measurement, the International System of Units (SI) defines a scale and unit for the thermodynamic temperature by using the easily reproducible temperature of the triple point of water as a second reference point. The reason for this choice is that, unlike the freezing and boiling point temperatures, the temperature at the triple point is independent of pressure (since the triple point is a fixed point on a two-dimensional plot of pressure vs. temperature). For historical reasons, the triple point temperature of water is fixed at 273.16 units of the measurement increment, which has been named the kelvin in honor of the Scottish physicist who first defined the scale. The unit symbol of the kelvin is K.

Absolute zero is defined as a temperature of precisely 0 kelvins, which is equal to −273.15 °C or −459.67 °F.

## Thermodynamic approach to temperature

Temperature is one of the principal quantities in the study of thermodynamics. Thermodynamics investigates the relation between heat and work, using the absolute temperature scale. In thermodynamic terms, temperature is an intensive variable because it is independent of the amount of elementary entities in the system, be they atoms, molecules, or electrons. Real world systems are often not in thermodynamic equilibrium and not homogeneous. For 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 each 'cell', then it is homogeneous and a temperature exists for it, and local thermodynamic equilibrium is said to prevail throughout the body.

### Definition

The thermodynamic definition of temperature is due to Kelvin. It defines a null point of temperature that is the theoretically lowest temperature. Such temperature scales are called absolute temperature scales and the null point is absolute zero.

It is framed in terms of a Carnot engine, imagined to define a continuous cycle of states of its working body, each point of the cycle being in thermodynamic equilibrium. There are four limbs in such a Carnot cycle, which is called a reversible Carnot cycle. The engine consists of four bodies, two of them called heat reservoirs, so large that their respective non-deformation variables are not changed by transfer of energy as heat through a wall permeable only to heat to the working body of the engine. The fourth body is able to exchange energy with the working body only through adiabatic work; it may be called the work reservoir. The substances and states of the heat reservoirs should be chosen so that they are not in thermal equilibrium with one another. This can be tested by connecting the heat reservoirs successively to an auxiliary thermometric body, which is required to show changes in opposite senses to its non-deformation variable, and which is composed of a material that has a strictly monotonic relation to the amount of work done on it in an isochoric adiabatic process. Typically, such a material expands as the surrounds do isochoric work on it. In order to settle the structure and sense of operation of the Carnot cycle, it is convenient to use such a material also for the working body; because most materials are of this kind, this is hardly a restriction of the generality of this definition. The Carnot cycle is considered to start from an initial condition of the working body that was reached by the completion of a reversible adiabatic compression. From there, the working body is initially connected by a wall permeable only to heat to the heat reservoir number 1, so that during the first limb of the cycle it expands and does work on the work reservoir. The second limb of the cycle sees the working body expand adiabatically and reversibly, with no energy exchanged as heat, but more energy being transferred as work to the work reservoir. The third limb of the cycle sees the working body connected, through a wall permeable only to heat, to the heat reservoir 2, contracting and accepting energy as work from the work reservoir. The cycle is closed by reversible adiabatic compression of the working body, with no energy transferred as heat, but energy being transferred to it as work from the work reservoir.

Then[when?] the four limbs of the reversible Carnot cycle are characterized by amounts of energy transferred, as work from the working body to the work reservoir, and as heat from the heat reservoirs to the working body. The amounts of energy transferred as heat from the heat reservoirs are measured through the changes in the non-deformation variable of the working body, with reference to the previously known properties of that body, the amounts of work done on the work reservoir, and the first law of thermodynamics. The amounts of energy transferred as heat respectively from reservoir 1 and from reservoir 2 may then be denoted respectively Q1 and Q2. Then the absolute or thermodynamic temperatures, T1 and T1, of the reservoirs are defined so that to be such that

$T_1 / T_2 = - Q_1 / Q_2 \,\,\,.$

Kelvin's original work postulating absolute temperature was published in 1848, before the formulation of the first law. His definitive publication, which sets out the definition just stated, was printed in 1853, a paper read in 1851.[3][4][5]

This definition rests on the physical assumption that there are readily available walls permeable only to heat. In his detailed definition of a wall permeable only to heat, Carathéodory includes several ideas. The non-deformation state variable of a closed system is represented as a real number. A state of thermal equilibrium between two closed systems connected by a wall permeable only to heat means that a certain mathematical relation holds between the state variables, including the respective non-deformation variables, of those two systems (that particular mathematical relation is regarded by Buchdahl as a preferred statement of the zeroth law of thermodynamics).[6] Also, referring to thermal contact equilibrium, "whenever each of the systems S1 and S2 is made to reach equilibrium with a third system S3 under identical conditions, the systems S1 and S2 are in mutual equilibrium."[7] It may viewed as a re-statement of the principle stated by Maxwell in the words: "All heat is of the same kind."[8] This physical idea is also expressed by Bailyn as a possible version of the zeroth law of thermodynamics: "All diathermal walls are equivalent."[9] Thus the present definition of thermodynamic temperature rests on the zeroth law of thermodynamics. This present definition of thermodynamic temperature also rests on the first law of thermodynamics, for the determination of amounts of energy transferred as heat. In itself, the present definition of thermodynamic temperature does not rely on the second law of thermodynamics.

The second law of thermodynamics provides information that establishes the virtuous character of the temperature so defined. It provides that any working substance that complies with the requirement stated in this definition will lead to the same ratio of thermodynamic temperatures, which in this sense is universal, or absolute. The second law of thermodynamics also provides that the thermodynamic temperature defined in this way is positive, because this definition requires that the heat reservoirs not be in thermal equilibrium with one another, and the cycle can be imagined to operate only in one sense if net work is to be supplied to the work reservoir.

Numerical details are settled by making one of the heat reservoirs a cell at the triple point of water, which is defined to have an absolute temperature of 273.16 K.[10] 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.

## Statistical mechanics approach to temperature

Statistical mechanics provides a microscopic explanation of temperature, based on macroscopic systems' being composed of many particles, such as molecules and ions of various species, the particles of a species being all alike. It explains macroscopic phenomena in terms of the mechanics of the molecules and ions, and statistical assessments of their joint adventures. In the statistical thermodynamic approach, by the equipartition theorem each classical degree of freedom that the particle has will have an average energy of kT/2 where k is Boltzmann's constant. 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 energy of a particle in an system with temperature T will be 3kT/2.

On the molecular level, temperature is the result of the motion of the particles that constitute the material. Moving particles carry kinetic energy. Temperature increases as this motion and the kinetic energy increase. The motion may be the translational motion of particles, or the energy of the particle due to molecular vibration or the excitation of an electron energy level. Although very specialized laboratory equipment is required to directly detect the translational thermal motions, thermal collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The thermal motions of atoms are very fast and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting low temperature of 700 nK (1 nK = 10−9 K) in 1994, they used laser equipment to create an optical lattice to adiabatically cool caesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7mm per second in order to calculate their temperature.

Molecules, such as oxygen (O2), have more degrees of freedom than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase in temperature due to an increase in the average translational energy of the molecules. Heating will also cause, through equipartitioning, the energy associated with vibrational and rotational modes to increase. Thus a diatomic gas will require a higher energy input to increase its temperature by a certain amount, i.e. it will have a higher heat capacity than a monatomic gas.

The process of cooling involves removing thermal energy from a system. When no more energy can be removed, the system is at absolute zero, which cannot be achieved experimentally. Absolute zero is the null point of the thermodynamic temperature scale, also called absolute temperature. If it were possible to cool a system to absolute zero, all motion of the particles comprising matter would cease and they would be at complete rest in this classical sense. Microscopically in the description of quantum mechanics, however, matter still has zero-point energy even at absolute zero, because of the uncertainty principle.

## Basic theory

Temperature is a measure of a quality of a state of a material [11] 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. Accordingly, temperature is an intensive variable. It is not necessarily the case that a material in a particular place is in a state that is steady and nearly homogeneous enough to allow it to have a well-defined hotness or temperature. Hotness may be represented abstractly as a one-dimensional manifold. Every valid temperature scale has its own one-to-one map into the hotness manifold.[12][13]

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. Heat transfer occurs by conduction or by thermal radiation.[14][15][16][17][18][19][20][21]

Experimental physicists, for example Galileo and Newton,[22] found that there are indefinitely many empirical temperature scales. Nevertheless, the zeroth law of thermodynamics says that they all measure the same quality.

### Temperature for 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.[23] 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.[24][25] A definite sense of greater hotness can be had, independently of calorimetry, of thermodynamics, and of properties of particular materials, from Wien's displacement law 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. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.[12][26][27][28][13]

Except for a system undergoing a first-order phase change such as the melting of ice, as a closed system receives heat, without change in its volume and without change in external force fields acting on it, its temperature rises. For a system undergoing such a phase change so slowly that departure from thermodynamic equilibrium can be neglected, its temperature remains constant as the system is supplied with latent heat. Conversely, a loss of heat from a closed system, without phase change, without change of volume, and without change in external force fields acting on it, decreases its temperature.[29]

### Temperature for 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 the hotter, and if this is so, then at least one of the bodies does not have a well defined absolute thermodynamic temperature. Nevertheless, any one given body and any one suitable empirical thermometer can still support notions of empirical, non-absolute, hotness and temperature, for a suitable range of processes. This is a matter for study in non-equilibrium thermodynamics.

### Temperature for bodies not in a steady state

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

### Thermodynamic equilibrium axiomatics

For axiomatic treatment of thermodynamic equilibrium, since the 1930s, it has become customary to refer to a zeroth law of thermodynamics. The customarily stated minimalist version of such a law postulates only that all bodies, which when thermally connected would be in thermal equilibrium, should be said to have the same temperature by definition, but by itself does not establish temperature as a quantity expressed as a real number on a scale. A more physically informative version of such a law views empirical temperature as a chart on a hotness manifold.[12][28][30] While the zeroth law permits the definitions of many different empirical scales of temperature, the second law of thermodynamics selects the definition of a single preferred, absolute temperature, unique up to an arbitrary scale factor, whence called the thermodynamic temperature.[12][27][31][32][33][34] If internal energy is considered as a function of the volume and entropy of a homogeneous system in thermodynamic equilibrium, thermodynamic absolute temperature appears as the partial derivative of internal energy with respect the entropy at constant volume. Its natural, intrinsic origin or null point is absolute zero at which the entropy of any system is at a minimum. Although this is the lowest absolute temperature described by the model, the third law of thermodynamics postulates that absolute zero cannot be attained by any physical system.

## Heat capacity

When a sample is heated, meaning it receives thermal energy from an external source, some of the introduced heat is converted into kinetic energy, the rest to other forms of internal energy, specific to the material. The amount converted into kinetic energy causes the temperature of the material to rise. The introduced heat ($\Delta Q$) divided by the observed temperature change is the heat capacity (C) of the material.

$C = \frac{\Delta Q}{\Delta T}$

If heat capacity is measured for a well defined amount of substance, the specific heat is the measure of the heat required to increase the temperature of such a unit quantity by one unit of temperature. For example, to raise the temperature of water by one kelvin (equal to one degree Celsius) requires 4186 joules per kilogram (J/kg)..

## Temperature measurement

A typical Celsius thermometer measures a winter day temperature of -17°C.

Temperature measurement using modern scientific thermometers and temperature scales goes back at least as far as the early 18th century, when Gabriel Fahrenheit 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 that may be calibrated to a variety of temperature scales. In most of the world (except for Belize, Myanmar, Liberia and the United States), the Celsius scale is used for most temperature measuring purposes. Most scientists measure temperature using the Celsius scale and thermodynamic temperature using the Kelvin scale, which is the Celsius scale offset so that its null point is 0K = −273.15°C, or absolute zero. Many engineering fields in the U.S., notably high-tech and US federal specifications (civil and military), also use the Kelvin and Celsius scales. Other engineering fields in the U.S. also rely upon the Rankine scale (a shifted Fahrenheit scale) when working in thermodynamic-related disciplines such as combustion.

### Units

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

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

By international agreement[35] the Kelvin and Celsius scales are defined by two fixing points: absolute zero and the triple point of Vienna Standard Mean Ocean Water, which is water specially prepared with a specified blend of hydrogen and oxygen isotopes. Absolute zero is defined as precisely 0K and −273.15°C. It is the temperature at which all classical translational motion of the particles comprising matter ceases and they are at complete rest in the classical model. Quantum-mechanically, however, zero-point motion remains and has an associated energy, the zero-point energy. Matter is in its ground state,[36] and contains no thermal energy. The triple point of water is defined as 273.16K and 0.01°C. This definition serves the following purposes: it fixes 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 establishes that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it establishes the difference between the null points of these scales as being 273.15K (0K = −273.15°C and 273.16K = 0.01°C).

In the United States, the Fahrenheit scale is widely used. On this scale the freezing point of water corresponds to 32 °F and the boiling point to 212 °F. The Rankine scale, still used in fields of chemical engineering in the U.S., is an absolute scale based on the Fahrenheit increment.

#### Conversion

The following table shows the temperature conversion formulas for conversions to and from the Celsius scale.

from Celsius to Celsius
Fahrenheit [°F] = [°C] × 95 + 32 [°C] = ([°F] − 32) × 59
Kelvin [K] = [°C] + 273.15 [°C] = [K] − 273.15
Rankine [°R] = ([°C] + 273.15) × 95 [°C] = ([°R] − 491.67) × 59
Delisle [°De] = (100 − [°C]) × 32 [°C] = 100 − [°De] × 23
Newton [°N] = [°C] × 33100 [°C] = [°N] × 10033
Réaumur [°Ré] = [°C] × 45 [°C] = [°Ré] × 54
Rømer [°Rø] = [°C] × 2140 + 7.5 [°C] = ([°Rø] − 7.5) × 4021

#### Plasma physics

The field of plasma physics deals with phenomena of electromagnetic nature that involve very high temperatures. It is customary to express temperature in electronvolts (eV) or kiloelectronvolts (keV), where 1 eV = 11605K. In the study of QCD matter one routinely encounters temperatures of the order of a few hundred MeV, equivalent to about 1012K.

## Theoretical foundation

Historically, there are several scientific approaches to the explanation of temperature: the classical thermodynamic description based on macroscopic empirical variables that can be measured in a laboratory; the kinetic theory of gases which relates the macroscopic description to the probability distribution of the energy of motion of gas particles; and a microscopic explanation based on statistical physics and quantum mechanics. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics and temperature.[37] Statistical physics provides a deeper understanding by describing the atomic behavior of matter, and derives macroscopic properties from statistical averages of microscopic states, including both classical and quantum states. In the fundamental physical description, using natural units, temperature may be measured directly in units of energy. However, in the practical systems of measurement for science, technology, and commerce, such as the modern metric system of units, the macroscopic and the microscopic descriptions are interrelated by the Boltzmann constant, a proportionality factor that scales temperature to the microscopic mean kinetic energy.

The microscopic description in statistical mechanics is based on a model that analyzes a system into its fundamental particles of matter or into a set of classical or quantum-mechanical oscillators and considers the system as a statistical ensemble of microstates. As a collection of classical material particles, temperature is a measure of the mean energy of motion, called kinetic energy, of the particles, whether in solids, liquids, gases, or plasmas. The kinetic energy, a concept of classical mechanics, is half the mass of a particle times its speed squared. In this mechanical interpretation of thermal motion, the kinetic energies of material particles may reside in the velocity of the particles of their translational or vibrational motion or in the inertia of their rotational modes. In monoatomic perfect gases and, approximately, in most gases, temperature is a measure of the mean particle kinetic energy. It also determines the probability distribution function of the energy. In condensed matter, and particularly in solids, this purely mechanical description is often less useful and the oscillator model provides a better description to account for quantum mechanical phenomena. Temperature determines the statistical occupation of the microstates of the ensemble. The microscopic definition of temperature is only meaningful in the thermodynamic limit, meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model.

In the context of thermodynamics, the kinetic energy is also referred to as thermal energy. The thermal energy may be partitioned into independent components attributed to the degrees of freedom of the particles or to the modes of oscillators in a thermodynamic system. In general, the number of these degrees of freedom that are available for the equipartitioning of energy depend 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, vibrational and rotational motions also contribute degrees of freedom.

### Kinetic theory of gases

The temperature of an ideal monatomic gas is related to the average kinetic energy of its atoms. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These atoms have a certain, average speed (slowed down here two trillion fold from room temperature).

The kinetic theory of gases uses the model of the ideal gas to relate temperature to the average translational kinetic energy of the molecules in a container of gas in thermodynamic equilibrium.[38][39][40]

Classical mechanics defines the translational kinetic energy of a gas molecule as follows:

$E_\text{k} = \begin{matrix} \frac 1 2 \end{matrix} mv^2,$

where m is the particle mass and v its speed, the magnitude of its velocity. The distribution of the speeds (which determine the translational kinetic energies) of the particles in a classical ideal gas is called the Maxwell-Boltzmann distribution.[39] The temperature of a classical ideal gas is related to its average kinetic energy per degree of freedom Ek via the equation:[41]

$\overline{E}_\text{k} = \begin{matrix} \frac 1 2 \end{matrix} kT,$

where the Boltzmann constant $k = R/n$ (n = Avogadro number, R = ideal gas constant). This relation is valid in the ideal gas regime, i.e. when the particle density is much less than $1/\Lambda^{3}$, where $\Lambda$ is the thermal de Broglie wavelength. A monoatomic gas has only the three translational degrees of freedom.

The zeroth law of thermodynamics implies that any two given systems in thermal equilibrium have the same temperature. In statistical thermodynamics, it can be deduced from the second law of thermodynamics that they also have the same average kinetic energy per particle.

In a mixture of particles of various masses, lighter particles move faster than do heavier particles, but have the same average kinetic energy. A neon atom moves slowly relative to a hydrogen molecule of the same kinetic energy. A pollen particle suspended in water moves in a slow Brownian motion among fast-moving water molecules.

### Zeroth law of thermodynamics

It has long been recognized that if two bodies of different temperatures are brought into thermal connection, conductive or radiative, they exchange heat accompanied by changes of other state variables. Left isolated from other bodies, the two connected bodies eventually reach a state of thermal equilibrium in which no further changes occur. This basic knowledge is relevant to thermodynamics. Some approaches to thermodynamics take this basic knowledge as axiomatic, other approaches select only one narrow aspect of this basic knowledge as axiomatic, and use other axioms to justify and express deductively the remaining aspects of it. The one aspect chosen by the latter approaches is often stated in textbooks as the zeroth law of thermodynamics, but other statements of this basic knowledge are made by various writers.

The usual textbook statement of the zeroth law of thermodynamics is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other. This statement is taken to justify a statement that all three systems have the same temperature, but, by itself, it does not justify the idea of temperature as a numerical scale for a concept of hotness which exists on a one-dimensional manifold with a sense of greater hotness. Sometimes the zeroth law is stated to provide the latter justification.[30] For suitable systems, an empirical temperature scale may be defined by the variation of one of the other state variables, such as pressure, when all other coordinates are fixed. The second law of thermodynamics is used to define an absolute thermodynamic temperature scale for systems in thermal equilibrium.

A temperature scale is based on the properties of some reference system to which other thermometers may be calibrated. One such reference system is a fixed quantity of gas. The ideal gas law indicates that the product of the pressure (p) and volume (V) of a gas is directly proportional to the thermodynamic temperature:[42]

$pV = nRT\,\!$

where T is temperature, n is the number of moles of gas and R = 8.314472(15) Jmol-1K-1 is the gas constant. Reformulating the pressure-volume term as the sum of classical mechanical particle energies in terms of particle mass, m, and root-mean-square particle speed v, the ideal gas law directly provides the relationship between kinetic energy and temperature:[43]

$\displaystyle \frac 1 2 mv_{rms}^2 = \frac 3 2 k T.$

Thus, one can define a scale for temperature based on the corresponding pressure and volume of the gas: the temperature in kelvins is the pressure in pascals of one mole of gas in a container of one cubic metre, divided by the gas constant. In practice, such a gas thermometer is not very convenient, but other thermometers can be calibrated to this scale.

The pressure, volume, and the number of moles of a substance are all inherently greater than or equal to zero, suggesting that temperature must also be greater than or equal to zero. As a practical matter it is not possible to use a gas thermometer to measure absolute zero temperature since the gasses tend to 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.

### Second law of thermodynamics

In the previous section certain properties of temperature were expressed by the zeroth law of thermodynamics. It is also possible to define temperature in terms of the second law of thermodynamics which deals with entropy. Entropy is often thought of as a measure of the disorder in a system. 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 that for a perfectly ordered set of coin tosses, there is only one set of toss outcomes possible: the set in which 100% of tosses come up the same. On the other hand, there are multiple combinations that can result in disordered or mixed systems, where some fraction are heads and the rest tails. A disordered system can be 90% heads and 10% tails, or it could be 98% heads and 2% tails, et cetera. As the number of coin tosses increases, the number of possible combinations corresponding to imperfectly ordered systems increases. For a very large number of coin tosses, the combinations to ~50% heads and ~50% tails dominates and obtaining an outcome significantly different from 50/50 becomes extremely unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.

It has been previously stated that temperature governs the transfer of heat between two systems and it was just shown that the universe tends to progress so as to maximize entropy, which is expected of any natural system. Thus, it is expected that there is some relationship between temperature and entropy. To find this relationship, the relationship between heat, work and temperature is first considered. A heat engine is a device for converting thermal energy into mechanical energy, resulting in the performance of work, and analysis of the Carnot heat engine provides the necessary relationships. The work from a heat engine corresponds to the difference between the heat put into the system at the high temperature, qH and the heat ejected at the low temperature, qC. The efficiency is the work divided by the heat put into the system or:

$\textrm{efficiency} = \frac {w_{cy}}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H}$ (2)

where wcy is the work done per cycle. The efficiency depends only on qC/qH. Because qC and qH correspond to heat transfer at the temperatures TC and TH, respectively, qC/qH should be some function of these temperatures:

$\frac{q_C}{q_H} = f(T_H,T_C)$ (3)

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and T2, and the second between T2 and T3. This can only be the case if:

$q_{13} = \frac{q_1 q_2} {q_2 q_3}$

which implies:

$q_{13} = f(T_1,T_3) = f(T_1,T_2)f(T_2,T_3)$

Since the first function is independent of T2, this temperature must cancel on the right side, meaning f(T1,T3) is of the form g(T1)/g(T3) (i.e. f(T1,T3) = f(T1,T2)f(T2,T3) = g(T1)/g(T2g(T2)/g(T3) = g(T1)/g(T3)), where g is a function of a single temperature. A temperature scale can now be chosen with the property that:

$\frac{q_C}{q_H} = \frac{T_C}{T_H}$ (4)

Substituting Equation 4 back into Equation 2 gives a relationship for the efficiency in terms of temperature:

$\textrm{efficiency} = 1 - \frac{q_C}{q_H} = 1 - \frac{T_C}{T_H}$ (5)

Notice that for TC = 0 K the efficiency is 100% and that efficiency becomes greater than 100% below 0 K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0 K is the minimum possible temperature. In fact the lowest temperature ever obtained in a macroscopic system was 20 nK, which was achieved in 1995 at NIST. Subtracting the right hand side of Equation 5 from the middle portion and rearranging gives:

$\frac {q_H}{T_H} - \frac{q_C}{T_C} = 0$

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by:

$dS = \frac {dq_\mathrm{rev}}{T}$ (6)

where the subscript indicates a reversible process. The change of this state function around any cycle is zero, as is necessary for any state function. This function corresponds to the entropy of the system, which was described previously. Rearranging Equation 6 gives a new definition for temperature in terms of entropy and heat:

$T = \frac{dq_\mathrm{rev}}{dS}$ (7)

For a system, where entropy S(E) is a function of its energy E, the temperature T is given by:

${T}^{-1} = \frac{d}{dE} S(E)$ (8),

i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy.

### Definition from statistical mechanics

Statistical mechanics defines temperature based on a system's fundamental degrees of freedom. Eq.(8) is the defining relation of temperature. Eq. (7) can be derived from the principles underlying the fundamental thermodynamic relation.

### Generalized temperature from single particle statistics

It is possible to extend the definition of temperature even to systems of few particles, like in a quantum dot. The generalized temperature is obtained by considering time ensembles instead of configuration space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of fermions (N even less than 10) with a single/double occupancy system. The finite quantum grand canonical ensemble,[44] obtained under the hypothesis of ergodicity and orthodicity, allows to express the generalized temperature from the ratio of the average time of occupation $\tau$1 and $\tau$2 of the single/double occupancy system:[45]

$T = k^{-1} \ln 2\frac{\tau_\mathrm{2}}{\tau_\mathrm{1}} \left(E - E_{F} \left(1+\frac{3}{2N}\right) \right),$

where EF is the Fermi energy which tends to the ordinary temperature when N goes to infinity.

### Negative temperature

On the empirical temperature scales, which are not referenced to absolute zero, a negative temperature is one below the zero-point of the scale used. For example, dry ice has a sublimation temperature of −78.5°C which is equivalent to −109.3°F. On the absolute Kelvin scale, however, this temperature is 194.6 K. On the absolute scale of thermodynamic temperature no material can exhibit a temperature smaller than or equal to 0 K, both of which are forbidden by the third law of thermodynamics.

In the quantum mechanical description of electron and nuclear spin systems that have a limited number of possible states, and therefore a discrete upper limit of energy they can attain, it is possible to obtain a negative temperature, which is numerically indeed less than absolute zero. However, this is not the macroscopic temperature of the material, but instead the temperature of only very specific degrees of freedom, that are isolated from others and do not exchange energy by virtue of the equipartition theorem.

A negative temperature is experimentally achieved with suitable radio frequency techniques that cause a population inversion of spin states from the ground state. As the energy in the system increases upon population of the upper states, the entropy increases as well, as the system becomes less ordered, but attains a maximum value when the spins are evenly distributed among ground and excited states, after which it begins to decrease, once again achieving a state of higher order as the upper states begin to fill exclusively. At the point of maximum entropy, the temperature function shows the behavior of a singularity, because the slope of the entropy function decreases to zero at first and then turns negative. Since temperature is the inverse of the derivative of the entropy, the temperature formally goes to infinity at this point, and switches to negative infinity as the slope turns negative. At energies higher than this point, the spin degree of freedom therefore exhibits formally a negative thermodynamic temperature. As the energy increases further by continued population of the excited state, the negative temperature approaches zero asymptotically.[46] As the energy of the system increases in the population inversion, a system with a negative temperature is not colder than absolute zero, but rather it has a higher energy than at positive temperature, and may be said to be in fact hotter at negative temperatures. When brought into contact with a system at a positive temperature, energy will be transferred from the negative temperature regime to the positive temperature region.

## Examples of temperature

Temperature Peak emittance wavelength[47]
Kelvin Degrees Celsius
Absolute zero
(precisely by definition)
0 K −273.15 °C cannot be defined
Coldest temperature
achieved[48]
100 pK −273.149999999900 °C 29,000 km
Coldest Bose–Einstein
condensate[49]
450 pK −273.14999999955 °C 6,400 km
One millikelvin
(precisely by definition)
0.001 K −273.149 °C 2.89777 m
Water's triple point
(precisely by definition)
273.16 K 0.01 °C 10,608.3 nm
(long wavelength I.R.)
Water's boiling point[A] 373.1339 K 99.9839 °C 7,766.03 nm
(mid wavelength I.R.)
Incandescent lamp[B] 2500 K ≈2,200 °C 1,160 nm
(near infrared)[C]
Sun's visible surface[D][51] 5,778 K 5,505 °C 501.5 nm
(green-blue light)
Lightning bolt's
channel[E]
28 kK 28,000 °C 100 nm
(far ultraviolet light)
Sun's core[E] 16 MK 16 million °C 0.18 nm (X-rays)
Thermonuclear weapon
(peak temperature)[E][52]
350 MK 350 million °C 8.3×10−3 nm
(gamma rays)
Sandia National Labs'
Z machine[E][53]
2 GK 2 billion °C 1.4×10−3 nm
(gamma rays)[F]
Core of a high-mass
star on its last day
[E][54]
3 GK 3 billion °C 1×10−3 nm
(gamma rays)
Merging binary neutron
star
system[E][55]
350 GK 350 billion °C 8×10−6 nm
(gamma rays)
Relativistic Heavy
Ion Collider
[E][56]
1 TK 1 trillion °C 3×10−6 nm
(gamma rays)
CERN's proton vs
nucleus collisions[E][57]
10 TK 10 trillion °C 3×10−7 nm
(gamma rays)
Universe 5.391×10−44 s
after the Big Bang[E]
1.417×1032 K 1.417×1032 °C 1.616×10−26 nm
(Planck Length)[58]
• A For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated strictly per the two-point definition of thermodynamic temperature.
• B The 2500 K value is approximate. The 273.15 K difference between K and °C is rounded to 300 K to avoid false precision in the Celsius value.
• C For a true black-body (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.
• D Effective photosphere temperature. The 273.15 K difference between K and °C is rounded to 273 K to avoid false precision in the Celsius value.
• E The 273.15 K difference between K and °C is without the precision of these values.
• F For a true black-body (which the plasma was not). The Z machine's dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.

## Notes

1. ^ Historically, the Celsius scale was a purely empirical temperature scale defined only by the freezing and boiling points of water. 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.

## References

1. ^ Maxwell, J.C. (1871). Theory of Heat, Longmans, Green, and Co., London. p. 2.
2. ^ Maxwell, J.C. (1871). Theory of Heat, Longmans, Green, and Co., London. p. 2.
3. ^ Thomson, W. (1848). On an absolute thermometric scale founded on Carnot's theory of the motive power of heat, and calculated from Regnault's observations, Proc. Cambridge Phil. Soc. (1843/1863) 1, No. 5: 66–71.
4. ^ Thomson, W. (March 1851). "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule’s equivalent of a Thermal Unit, and M. Regnault’s Observations on Steam". Transactions of the Royal Society of Edinburgh XX (part II): 261–268; 289–298.
5. ^ 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.
6. ^ Buchdahl, H.A (1986). On the redundancy of the zeroth law of thermodynamics, J. Phys. A, Math. Gen., 19: L561–L564.
7. ^ C. Carathéodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 355–386. doi:10.1007/BF01450409. A partly reliable translation is to be found at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.
8. ^ Maxwell, J.C. (1871). Theory of Heat, Longmans, Green, and Co., London, p. 57.
9. ^ Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, ISBN 0-88318-797-3, page 24.
10. ^ Quinn, T.J. (1983). Temperature, Academic Press, London, ISBN0-12-569680-9, pp. 160–162.
11. ^ Bryan, G.H. (1907). Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications, B.G. Teubner, Leipzig, page 3.[1]
12. ^ a b c d Mach, E. (1900). Die Principien der Wärmelehre. Historisch-kritisch entwickelt, Johann Ambrosius Barth, Leipzig, section 22, pages 56-57.
13. ^ a b Serrin, J. (1986). Chapter 1, 'An Outline of Thermodynamical Structure', pages 3-32, especially page 6, in New Perspectives in Thermodynamics, edited by J. Serrin, Springer, Berlin, ISBN 3-540-15931-2.
14. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longmans, Green, London, page 32.
15. ^ Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pages 39-40.
16. ^ Planck, M. (1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green, London, pages 1-2.
17. ^ Planck, M. (1914), The Theory of Heat Radiation, second edition, translated into English by M. Masius, Blakiston's Son & Co., Philadelphia, reprinted by Kessinger.
18. ^ J. S. Dugdale (1996, 1998). Entropy and its Physical Interpretation. Taylor & Francis. p. 13. ISBN 978-0-7484-0569-5.
19. ^ F. Reif (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill. p. 102.
20. ^ M. J. Moran, H. N. Shapiro (2006). Fundamentals of Engineering Thermodynamics (5 ed.). John Wiley & Sons, Ltd. p. 14. ISBN 978-0-470-03037-0. Unknown parameter |section= ignored (help)
21. ^ T.W. Leland, Jr. "Basic Principles of Classical and Statistical Thermodynamics". p. 14. "Consequently we identify temperature as a driving force which causes something called heat to be transferred."
22. ^ Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pages 42, 103-117.
23. ^ Beattie, J.A., Oppenheim, I. (1979). Principles of Thermodynamics, Elsevier Scientific Publishing Company, Amsterdam, 0–444–41806–7, page 29.
24. ^ Landsberg, P.T. (1961). Thermodynamics with Quantum Statistical Illustrations, Interscience Publishers, New York, page 17.
25. ^ Thomsen, J.S. (1962). "A restatement of the zeroth law of thermodynamics". Am. J. Phys. 30: 294–296.
26. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longman's, Green & Co, London, page 45.
27. ^ a b Truesdell, C.A. (1980). The Tragicomical History of Thermodynamics, 1822-1854, Springer, New York, ISBN 0-387-90403-4, Sections 11 B, 11H, pages 306–310, 320-332.
28. ^ a b Pitteri, M. (1984). On the axiomatic foundations of temperature, Appendix G6 on pages 522-544 of Rational Thermodynamics, C. Truesdell, second edition, Springer, New York, ISBN 0-387-90874-9.
29. ^ Truesdell, C., Bharatha, S. (1977). The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, Springer, New York, ISBN 0-387-07971-8, page 20.
30. ^ a b 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 Janiero, August 1977, edited by G.M. de La Penha, L.A.J. Medeiros, North-Holland, Amsterdam, ISBN 0-444-85166-6, pages 411-451.
31. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longmans, Green, London, pages 155-158.
32. ^ Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, Section 95, pages 68-69.
33. ^ H.A. Buchdahl (1966). The Concepts of Classical Thermodynamics. Cambridge University Press. p. 73.
34. ^ Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 978-0-470-01598-8, Section 32., pages 106-108.
35. ^ The kelvin in the SI Brochure
36. ^ "Absolute Zero". Calphad.com. Retrieved 2010-09-16.
37. ^ C. Caratheodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67 (3): 355–386. doi:10.1007/BF01450409.
38. ^ Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York, ISBN 0-471-04600-0, pages 148-154.
39. ^ a b Kittel, Charles; Kroemer, Herbert (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. pp. 391–397. ISBN 0-7167-1088-9.
40. ^ Kondepudi, D.K. (1987). "Microscopic aspects implied by the second law". Foundations of Physics 17: 713–722.
41. ^ Tolman, R.C. (1938). The Principles of Statistical Mechanics, Oxford University Press, London, pp. 93, 655.
42. ^ Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics, Addison-Wesley, Reading MA, volume 1, pages 39–6 to 39–12.
43. ^ Peter Atkins, Julio de Paula (2006). Physical Chemistry (8 ed.). Oxford University Press. p. 9.
44. ^ Prati, E. (2010). "The finite quantum grand canonical ensemble and temperature from single-electron statistics for a mesoscopic device". J. Stat. Mech. 1: P01003. arXiv:1001.2342. Bibcode:2010JSMTE..01..003P. doi:10.1088/1742-5468/2010/01/P01003. arxiv.org
45. ^ Prati, E., et al. (2010). "Measuring the temperature of a mesoscopic electron system by means of single electron statistics". Applied Physics Letters 96 (11): 113109. arXiv:1002.0037. Bibcode:2010ApPhL..96k3109P. doi:10.1063/1.3365204. arxiv.org
46. ^ Kittel, Charles; Kroemer, Herbert (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. pp. Appendix E. ISBN 0-7167-1088-9.
47. ^ The cited emission wavelengths are for black bodies in equilibrium. CODATA 2006 recommended value of 2.8977685(51)×10−3 m K used for Wien displacement law constant b.
48. ^ "World record in low temperatures". Retrieved 2009-05-05.
49. ^ A temperature of 450 ±80 pK in a Bose–Einstein condensate (BEC) of sodium atoms was achieved in 2003 by researchers at MIT. Citation: Cooling Bose–Einstein Condensates Below 500 Picokelvin, A. E. Leanhardt et al., Science 301, 12 Sept. 2003, p. 1515. It's noteworthy that this record's peak emittance black-body wavelength of 6,400 kilometers is roughly the radius of Earth.
50. ^ The peak emittance wavelength of 2.89777 m is a frequency of 103.456 MHz
51. ^ Measurement was made in 2002 and has an uncertainty of ±3 kelvin. A 1989 measurement produced a value of 5,777.0±2.5 K. Citation: Overview of the Sun (Chapter 1 lecture notes on Solar Physics by Division of Theoretical Physics, Dept. of Physical Sciences, University of Helsinki).
52. ^ The 350 MK value is the maximum peak fusion fuel temperature in a thermonuclear weapon of the Teller–Ulam configuration (commonly known as a hydrogen bomb). Peak temperatures in Gadget-style fission bomb cores (commonly known as an atomic bomb) are in the range of 50 to 100 MK. Citation: Nuclear Weapons Frequently Asked Questions, 3.2.5 Matter At High Temperatures. Link to relevant Web page. All referenced data was compiled from publicly available sources.
53. ^ Peak temperature for a bulk quantity of matter was achieved by a pulsed-power machine used in fusion physics experiments. The term bulk quantity draws a distinction from collisions in particle accelerators wherein high temperature applies only to the debris from two subatomic particles or nuclei at any given instant. The >2 GK temperature was achieved over a period of about ten nanoseconds during shot Z1137. In fact, the iron and manganese ions in the plasma averaged 3.58±0.41 GK (309±35 keV) for 3 ns (ns 112 through 115). Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2×109 Kelvin, M. G. Haines et al., Physical Review Letters 96 (2006) 075003. Link to Sandia's news release.
54. ^ Core temperature of a high–mass (>8–11 solar masses) star after it leaves the main sequence on the Hertzsprung–Russell diagram and begins the alpha process (which lasts one day) of fusing silicon–28 into heavier elements in the following steps: sulfur–32 → argon–36 → calcium–40 → titanium–44 → chromium–48 → iron–52 → nickel–56. Within minutes of finishing the sequence, the star explodes as a Type II supernova. Citation: Stellar Evolution: The Life and Death of Our Luminous Neighbors (by Arthur Holland and Mark Williams of the University of Michigan). Link to Web site. More informative links can be found here [2], and here [3], and a concise treatise on stars by NASA is here [4].
55. ^ Based on a computer model that predicted a peak internal temperature of 30 MeV (350 GK) during the merger of a binary neutron star system (which produces a gamma–ray burst). The neutron stars in the model were 1.2 and 1.6 solar masses respectively, were roughly 20 km in diameter, and were orbiting around their barycenter (common center of mass) at about 390 Hz during the last several milliseconds before they completely merged. The 350 GK portion was a small volume located at the pair's developing common core and varied from roughly 1 to 7 km across over a time span of around 5 ms. Imagine two city-sized objects of unimaginable density orbiting each other at the same frequency as the G4 musical note (the 28th white key on a piano). It's also noteworthy that at 350 GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass (m) 5% greater than its rest mass (m0).  Torus Formation in Neutron Star Mergers and Well-Localized Short Gamma-Ray Bursts, R. Oechslin et al. of Max Planck Institute for Astrophysics., arXiv:astro-ph/0507099 v2, 22 Feb. 2006. An html summary.
56. ^ Results of research by Stefan Bathe using the PHENIX detector on the Relativistic Heavy Ion Collider at Brookhaven National Laboratory in Upton, New York, U.S.A.  Bathe has studied gold-gold, deuteron-gold, and proton-proton collisions to test the theory of quantum chromodynamics, the theory of the strong force that holds atomic nuclei together.  Link to news release.
57. ^
58. ^ The Planck frequency equals 1.85487(14)×1043 Hz (which is the reciprocal of one Planck time). Photons at the Planck frequency have a wavelength of one Planck length. The Planck temperature of 1.41679(11)×1032 K equates to a calculated /T = λmax wavelength of 2.04531(16)×10−26 nm. However, the actual peak emittance wavelength quantizes to the Planck length of 1.61624(12)×10−26 nm.

• Chang, Hasok (2004). Inventing Temperature: Measurement and Scientific Progress. Oxford: Oxford University Press. ISBN 978-0-19-517127-3.
• Zemansky, Mark Waldo (1964). Temperatures Very Low and Very High. Princeton, N.J.: Van Nostrand.
• T. J. Quinn (1983), Temperature, Academic Press, London.
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