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Temperature is a physical quantity expressing hot and cold. It is measurement with a thermometer Calibration in one or more temperature scales. The most commonly used scales are the Celsius (formerly called centigrade) (denoted °C), Fahrenheit (denoted °F), and Kelvin (denoted K). The kelvin (the word is spelled with a Letter case k) is the unit of temperature in the International System of Units (SI), in which temperature is one of the seven fundamental base quantities. The Kelvin scale is widely used in science and technology.
Theoretically, the coldest a system can be is when its temperature is absolute zero, at which point the thermal motion in matter would be zero. However, an actual physical system or object can never attain a temperature of absolute zero. Absolute zero is denoted as on the Kelvin scale, on the Celsius scale, and on the Fahrenheit scale.
For an ideal gas, temperature is proportional to the average kinetic energy of the random microscopic motions of the constituent microscopic particles.
Temperature is important in all fields of natural science, including physics, chemistry, Earth science, medicine, and biology, as well as most aspects of daily life.
The Celsius scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that was developed by a historical progress, which led to its zero point being defined by the freezing point of water, and additional degrees defined so that was the boiling point of water, both at sealevel atmospheric pressure. Because of the 100degree interval, it was called a centigrade scale.Middleton, W.E.K. (1966), pp. 89–105. 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 the Fahrenheit scale, on which water freezes at and boils at at sealevel atmospheric pressure.
Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the ScotsIrish physicist who first defined it. It is a thermodynamic or absolute temperature scale. Its zero point, , is defined to coincide with the coldest physicallypossible temperature (called absolute zero). Its degrees are defined through thermodynamics. The temperature of absolute zero occurs at = (or ), and the freezing point of water at sealevel atmospheric pressure occurs at = .
The International System of Units (SI) defines a scale and unit for the kelvin or thermodynamic temperature by using the reliably reproducible temperature of the triple point of water as a second reference point (the first reference point being 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 selfdetermining 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.
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 recalibrated, by use of theoretical physical reasoning, and this can extend their range of adequacy.
The accepted fundamental thermodynamic temperature scale is the Kelvin scale, based on an ideal cyclic process envisaged for a Carnot 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.Quinn, T.J. (1983), pp. 61–83.Schooley, J.F. (1986), pp. 115–138. 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 semidefinite 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.Adkins, C.J. (1968/1983), pp. 119–120.Buchdahl, H.A. (1966), pp. 137–138.Tschoegl, N.W. (2000), p. 88.
Measurement of the spectrum of electromagnetic radiation from an ideal threedimensional black body can provide an accurate temperature measurement because the frequency of maximum spectral radiance of blackbody 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 the Bose–Einstein law.
Measurement of the spectrum of noisepower produced by an electrical resistor can also provide an accurate temperature measurement. The resistor has two terminals and is in effect a onedimensional body. The BoseEinstein law for this case indicates that the noisepower 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 noisepower 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.Quinn, T.J. (1983), pp. 98–107.Schooley, J.F. (1986), pp. 138–143.
If molecules, or atoms, or electrons,Germer, L.H. (1925). 'The distribution of initial velocities among thermionic electrons', Phys. Rev., 25: 795–807. hereTurvey, K. (1990). 'Test of validity of Maxwellian statistics for electrons thermionically emitted from an oxide cathode', European Journal of Physics, 11(1): 51–59. here are emitted from a material and their velocities are measured, the spectrum of their velocities often nearly obeys a theoretical law called the Maxwell–Boltzmann distribution, which gives a wellfounded measurement of temperatures for which the law holds.Zeppenfeld, M., Englert, B.G.U., Glöckner, R., Prehn, A., Mielenz, M., Sommer, C., van Buuren, L.D., Motsch, M., Rempe, G. (2012). 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.Miller, J. (2013).
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 a Carnot 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 nondeformation 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 setup, 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 nondeformation 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 and . Then the absolute or thermodynamic temperatures, and , of the reservoirs are defined so that to be such that
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.Thomson, W. (Lord Kelvin) (1848).Thomson, W. (Lord Kelvin) (1851).Partington, J.R. (1949), pp. 175–177.Roberts, J.K., Miller, A.R. (1928/1960), pp. 321–322.
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 nondeformation 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 nondeformation variables, of those two systems (that particular mathematical relation is regarded by Buchdahl as a preferred statement of the zeroth law of thermodynamics).Buchdahl, H. A. (1986). On the redundancy of the zeroth law of thermodynamics, J. Phys. A: Math. Gen., 19: L561–L564. Also, referring to thermal contact equilibrium, "whenever each of the systems and is made to reach equilibrium with a third system under identical conditions, the systems and are in mutual equilibrium." A partly reliable translation is to be found at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA. It may be viewed as a restatement of the principle stated by Maxwell in the words: "All heat is of the same kind."Maxwell, J.C. (1871). Theory of Heat, Longmans, Green, and Co., London, p. 57. This physical idea is also expressed by Bailyn as a possible version of the zeroth law of thermodynamics: "All diathermal walls are equivalent."Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, , p. 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.Quinn, T.J. (1983). Temperature, Academic Press, London, , pp. 160–162. 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.
In particular, when the body is described by stating its internal energy , 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 of the internal energy with respect to the entropy:Herbert Callen (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:Kondepudi, D., Ilya Prigogine (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures, John Wiley, Chichester, , pp. 115–116.
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.Tisza, L. (1966). Generalized Thermodynamics, M.I.T. Press, Cambridge MA, p. 58.
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 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, 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, temperature can be regarded as a spatially varying local property in that body, and this is because temperature is an intensive variable.
Kinetic theory provides a microscopic explanation of temperature, based on macroscopic systems' being composed of many microscopic particles, such as and of various species, the particles of a species being all alike. It explains macroscopic phenomena through the classical mechanics of the microscopic particles. The equipartition theorem of kinetic theory asserts that each classical degree of freedom of a freely moving particle has an average kinetic energy of where denotes 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 kinetic energy of a freely moving particle in a system with temperature will be .
It is possible to measure the average kinetic energy of constituent microscopic particles if they are allowed to escape from the bulk of the system. 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.
, such as oxygen (O_{2}), 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 kinetic energy of the molecules. Heating will also cause, through , the energy associated with vibrational and rotational modes to increase. Thus a diatomic gas will require more energy input to increase its temperature by a certain amount, i.e. it will have a greater heat capacity than a monatomic gas.
The process of cooling involves removing internal energy from a system. When no more energy can be removed, the system is at absolute zero, though this 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 classical motion of its particles would cease and they would be at complete rest in this classical sense. Microscopically in the description of quantum mechanics, however, matter still has zeropoint energy even at absolute zero, because of the uncertainty principle.
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.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.Planck, M. (1914), The Theory of Heat Radiation , second edition, translated into English by M. Masius, Blakiston's Son & Co., Philadelphia, reprinted by Kessinger.
Experimental physicists, for example Galileo and Newton,Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pp. 42, 103–117. found that there are indefinitely many empirical temperature scales. Nevertheless, the zeroth law of thermodynamics says that they all measure the same quality.
Except for a system undergoing a order parameter phase transition 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.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, , p. 20.
Temperature is measured with thermometers that may be calibration 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 = , or absolute zero. Many engineering fields in the US, notably hightech 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 (a shifted Fahrenheit scale) when working in thermodynamicrelated disciplines such as combustion.
For everyday applications, it is often convenient to use the Celsius scale, in which corresponds very closely to the freezing point of water and is its boiling point at sea level. Because liquid droplets commonly exist in clouds at subzero 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 The kelvin in the SI Brochure 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 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. Quantummechanically, however, zeropoint motion remains and has an associated energy, the zeropoint energy. Matter is in its ground state, and contains no thermal energy. The triple point of water is defined as and . 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 ( = and = ).
In the United States, the Fahrenheit scale is 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.
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 quantummechanical 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 monatomic 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 depends on the temperature, i.e. the energy region of the interactions under consideration. For solids, the thermal energy is associated primarily with the Atom vibrations of its atoms or molecules about their equilibrium position. In an ideal gas, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems, and motions also contribute degrees of freedom.
The ideal gas law is based on observed empirical relationships between pressure ( p), volume ( V), and temperature ( T), and was recognized long before the kinetic theory of gases was developed (see Boyle's and Charles's laws). The ideal gas law states:Richard Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics, Addison–Wesley, Reading MA, volume 1, pp. 396 to 3912.
This relationship gives us our first hint that there is an absolute zero 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 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 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, as shown in the figure.
The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational kinetic energy. Using a sophisticated symmetry argument, Ludwig Boltzmann deduced what is now called the Maxwell–Boltzmann probability distribution function for the velocity of particles in an ideal gas. From that probability distribution function, the average kinetic energy (per particle) of a monatomic ideal gas isTolman, R.C. (1938). The Principles of Statistical Mechanics, Oxford University Press, London, pp. 93, 655.
where the Boltzmann constant is the ideal gas constant divided by the Avogadro number, and $v\_\backslash text\{rms\}\; =\; \backslash sqrt\{\backslash langle\; v^2\; \backslash rangle\}$ is the rootmeansquare speed. Thus the ideal gas law states that internal energy is directly proportional to temperature. This direct proportionality between temperature and internal energy is a special case of the equipartition theorem, and holds only in the classical limit of an ideal gas. It does not hold for most substances, although it is true that temperature is a monotonic (nondecreasing) function of internal energy.
One statement of the zeroth law of thermodynamics is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other.
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 onedimensional 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, 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.
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, etcetera. 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 dominate 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, q_{H} and the heat ejected at the low temperature, q_{C}. The efficiency is the work divided by the heat put into the system:
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
which implies
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. f( T_{1}, T_{3}) = f( T_{1}, T_{2}) f( T_{2}, T_{3}) = g( T_{1})/ g( T_{2})· g( T_{2})/ g( T_{3}) = g( T_{1})/ g( T_{3})), 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} = 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 (5) from the middle portion and rearranging gives
where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by
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 (8) gives a formula for temperature in terms of fictive infinitesimal quasireversible elements of entropy and heat:
For a system, where entropy S( E) is a function of its energy E, the temperature T is given by
i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy.
[https://arxiv.org/abs/1002.0037v2 arxiv.org]
However, when there is an upper limit of energy a system attain, it is possible to obtain a negative temperature on the absolute scale. Such negative temperatures are in fact hotter than any positive temperature and can be achieved by heating the system past a point of infinite temperature. As the energy in such systems increases, the entropy increases for some range, but eventually attains a maximum value at a critical temperature 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 function decreases to zero and then turns negative. Since temperature is the inverse of the derivative of the entropy, the temperature goes to positive infinity at this point, switching to negative infinity as the slope turns negative.
When brought into contact with a system at a positive temperature, energy will be transferred as heat from the negative temperature system to the positive temperature system.

