Temperature

This article is about the thermodynamic property. For other uses, see Temperature (disambiguation).

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

Annual mean temperature around the world

Body temperature variation

Temperature is a physical quantity expressing hot and cold. Temperature is measured with athermometer, historically calibrated in varioustemperature scales and units of measurement. The most commonly used scales are the Celsius scale, denoted in °C (informally, degrees centigrade), theFahrenheit scale (°F), and the Kelvin scale. The kelvin (K) is the unit of temperature in theInternational System of Units (SI), in which temperature is one of the seven fundamentalbase quantities.

The coldest theoretical temperature isabsolute zero, at which the thermal motion of all fundamental particles in matter reaches a minimum. Although classically described as motionless, particles still possess a finitezero-point energy in the quantum mechanical description. Absolute zero is denoted as 0 K on the Kelvin scale, −273.15 °C on the Celsius scale, and −459.67 °F on the Fahrenheit scale.

Temperature is a proportional measure of the average translational kinetic energy of the random motions of the constituent microscopic particles in a system (such as electrons, atoms, and molecules); based on the historical development of the kinetic theory of gases, but more rigorous definitions include all quantum states of matter.

Temperature is important in all fields ofnatural science, including physics, chemistry,Earth science, medicine, and biology, as well as most aspects of daily life.

Effects of temperatureEdit

Many physical processes are affected by temperature, such as

• physical properties of materials including the phase (solid, liquid, gaseous or plasma),density, solubility, vapor pressure, electrical conductivity
• rate and extent to which chemical reactionsoccur^[1]
• the amount and properties of thermal radiation emitted from the surface of an object
• speed of sound is a function of the square root of the absolute temperature^[2]

Temperature scalesEdit

See also: Scale of temperature

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

The Celsius scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale. It developed by a historical progress, which led to its zero point 0°C being defined by the freezing point of water, with additional degrees defined so that 100°C was the boiling point of water, both at sea-level atmospheric pressure. Because of the 100 degree interval, it is called a centigrade scale.^[3] Since the standardization of the kelvin in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, and so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though they differ by an additive offset of 273.15.

The United States commonly uses theFahrenheit scale, on which water freezes at32°F and boils at 212°F at sea-level atmospheric pressure.

Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the Scottish physicist who first defined it. It is a thermodynamic or absolute temperature scale. Its zero point, 0K, is defined to coincide with the coldest physically-possible temperature (calledabsolute zero). Its degrees are definedthrough thermodynamics. The temperature of absolute zero occurs at 0K = −273.15°C (or−459.67°F), and the freezing point of water at sea-level atmospheric pressure occurs at273.15K = 0°C.

The International System of Units (SI) defines a scale and unit for the kelvin orthermodynamic temperature by using the reliably reproducible temperature of the triple point of water as a second reference point (the first reference point being 0 K at absolute zero). The triple point is a singular state with its own unique and invariant temperature and pressure, along with, for a fixed mass of water in a vessel of fixed volume, an autonomically and stably self-determining partition into three mutually contacting phases, vapour, liquid, and solid, dynamically depending only on the total internal energy of the mass of water. For historical reasons, the triple point temperature of water is fixed at 273.16 units of the measurement increment.

Thermodynamic approach to temperatureEdit

Temperature is one of the principal quantities in the study of thermodynamics.

Kinds of temperature scaleEdit

There is a variety of kinds of temperature scale. It may be convenient to classify them as empirically and theoretically based. Empirical temperature scales are historically older, while theoretically based scales arose in the middle of the nineteenth century.^[4][5]

Empirically based scalesEdit

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

Theoretically based scalesEdit

Theoretically based temperature scales are based directly on theoretical arguments, especially those of thermodynamics, kinetic theory and quantum mechanics. They rely on theoretical properties of idealized devices and materials. They are more or less comparable with practically feasible physical devices and materials. Theoretically based temperature scales are used to provide calibrating standards for practical empirically based thermometers.

The accepted fundamental thermodynamic temperature scale is the Kelvin scale, based on an ideal cyclic process envisaged for aCarnot heat engine.

An ideal material on which a temperature scale can be based is the ideal gas. The pressure exerted by a fixed volume and mass of an ideal gas is directly proportional to its temperature. Some natural gases show so nearly ideal properties over suitable temperature ranges that they can be used for thermometry; this was important during the development of thermodynamics and is still of practical importance today.^[6][7] 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. The physical reason is that the ideal gas law, exactly read, refers to the limit of infinitely high temperature and zero pressure.^[8][9][10]

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

Measurement of the spectrum of noise-power produced by an electrical resistor can also provide an 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 band-width. In a given frequency band, the noise-power has equal contributions from every frequency and is called Johnson noise. If the value of the resistance is known then the temperature can be found.^[11][12]

If molecules, or atoms, or electrons,^[13][14] are emitted from a material and their velocities are measured, the spectrum of their velocities often nearly obeys a theoretical law called theMaxwell–Boltzmann distribution, which gives a well-founded measurement of temperatures for which the law holds.^[15] 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 future.^[16]

Absolute thermodynamic scaleEdit

The Kelvin scale is called 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 Kelvin temperature does in fact have a definite numerical value that has been arbitrarily chosen by tradition and is dependent on the property of a particular materials; it is simply less arbitrary than relative "degrees" scales such as Celsius and Fahrenheit. Being an absolute scale with one fixed point (zero), there is only one degree of freedom left to arbitrary choice, rather than two as in relative scales. For the Kelvin scale in modern times, this choice of convention is made to be that of setting the gas–liquid–solid triple point of water, a point which can be reliably reproduced as a standard experimental phenomenon, at a numerical value of 273.16 kelvins. The Kelvin scale is also called thethermodynamic scale. However, to demonstrate that its numerical value is indeed arbitrary, it is useful to point out that an alternate, less widely used absolute temperature scale exists called the Rankine scale, made to be aligned with the Fahrenheit scale as Kelvin is with Celsius.

Definition of the Kelvin scaleEdit

The thermodynamic definition of temperature is due to Kelvin.

It is framed in terms of an idealized device called a Carnot engine, imagined to define a continuous cycle of states of its working body. 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. There are four limbs in such aCarnot cycle. The engine consists of four bodies. The main one is called the working body. Two of them are 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. 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 two heat reservoirs should be chosen so that they are not in thermal equilibrium with one another. This means that they must be at different fixed temperatures, one, labeled here with the number 1, hotter than the other, labeled here with the number 2. This can be tested by connecting the heat reservoirs successively to an auxiliary empirical thermometric body that starts each time at a convenient fixed intermediate temperature. The thermometric body should be composed of a material that has a strictly monotonic relation between its chosen empirical thermometric variable and the amount of adiabatic isochoric work done 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.

With this set-up, 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 T₂, of the reservoirs are defined so that to be such that

${\displaystyle T_{1}/T_{2}=-Q_{1}/Q_{2}\,\,\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)}$

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. 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.^{[17][18][19][20]}

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).^[21] Also, referring to thermal contact equilibrium, "whenever each of the systems S₁ and S₂ is made to reach equilibrium with a third systemS₃ under identical conditions, the systems S₁and S₂ are in mutual equilibrium."^[22] It may be viewed as a re-statement of the principle stated by Maxwell in the words: "All heat is of the same kind."^[23] This physical idea is also expressed by Bailyn as a possible version of the zeroth law of thermodynamics: "All diathermal walls are equivalent."^[24] Thus the present definition of thermodynamic temperature rests on the zeroth law of thermodynamics. Explicitly, this present definition of thermodynamic temperature also rests on the first law of thermodynamics, for the determination of amounts of energy transferred as heat.

Implicitly for this definition, 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.^[25] 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.

Temperature as an intensive variableEdit

In thermodynamic terms, temperature is anintensive variable because it is equal to adifferential coefficient of one extensive variable with respect to another, for a given body. It thus has the dimensions of a ratio of two extensive variables. In thermodynamics, two bodies are often considered as connected by contact with a common wall, which has some specific permeability properties. Such specific permeability can be referred to a specific intensive variable. An example is a diathermic wall that is permeable only to heat; the intensive variable for this case is temperature. When the two bodies have been in contact 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.^[26][27][28]

In particular, when the body is described by stating its internal energy U, an extensive variable, as a function of its entropy S, also an extensive variable, and other state variablesV, N, with U = U (S, V, N), then the temperature is equal to the partial derivativeof the internal energy with respect to the entropy:

${\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V,N}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)}$^[27][28][29]

Likewise, when the body is described by stating its entropy S as a function of its internal energy U, and other state variablesV, N, with S = S (U, V, N), then the reciprocal of the temperature is equal to the partial derivative of the entropy with respect to the internal energy:

${\displaystyle {\frac {1}{T}}=\left({\frac {\partial S}{\partial U}}\right)_{V,N}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)}$^[27][29][30]

The above definition, equation (1), of the absolute temperature is due to Kelvin. It refers to systems closed to transfer of matter, and has 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.^[31]

Temperature local when local thermodynamic equilibrium prevailsEdit