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TEMPERATURE[edit]

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[edit]

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[53] 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,[54] 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[edit]

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


from Celsius

to Celsius


Fahrenheit [°F] = [°C] × 9⁄5 + 32 [°C] = ([°F] − 32) × 5⁄9

Kelvin [K] = [°C] + 273.15 [°C] = [K] − 273.15

Rankine [°R] = ([°C] + 273.15) × 9⁄5 [°C] = ([°R] − 491.67) × 5⁄9

Delisle [°De] = (100 − [°C]) × 3⁄2 [°C] = 100 − [°De] × 2⁄3

Newton [°N] = [°C] × 33⁄100 [°C] = [°N] × 100⁄33

Réaumur [°Ré] = [°C] × 4⁄5 [°C] = [°Ré] × 5⁄4

Rømer [°Rø] = [°C] × 21⁄40 + 7.5 [°C] = ([°Rø] − 7.5) × 40⁄21

Plasma physics[edit]

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[edit]

See also: Thermodynamic temperature

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.[55] 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 monatomic 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[edit]



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 times from room temperature).

The Maxwell-Boltzmann kinetic theory of gases contributes to a fundamental understanding of temperature.[56] It also explains the ideal gas law, and under certain circumstances, the heat capacity of gases (especially monatomic or 'noble' gases.) An important result of the kinetic theory of gases is that it relates temperature to the average translational kinetic energy of the molecules in a container of gas in thermodynamic equilibrium.[57][58][59]

The ideal gas law is based on observed empirical relationships between pressure (p) and volume (V), and temperature (T) of a gas. It had been recognized long before the kinetic theory of gases was developed (see Boyle's and Charles's laws). The ideal gas law states:[60] pV = nRT\,\! where n is the number of moles of gas and R = 8.314472(15) Jmol−1K−1 is the gas constant.

The ideal gas law 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 absolutescale such as Kelvins. Moreover, the ideal gas law allows one to measure temperature on this absolute scale using the gas thermometer. 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.



Plots of pressure vs temperature for three different gas samples extrapolate to absolute zero.

Although it is not a particularly convenient device, the gas thermometer provides an essential theoretical basis by which all thermometers can be calibrated. 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 non-negative. 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 to the left.

By showing that the force associated with pressure is caused by collisions of individual atoms with the walls of a container, the kinetic theory is able to relate temperature to energy for an ideal gas. For simplicity, we shall consider only the monatomic ideal gas, for which all energy is translational kinetic energy. From classical mechanics, we have, 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.[58] The temperature of a classical ideal gas is related to its average kinetic energy per degree of freedom Ek via the equation:[61] \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.

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:[62] \displaystyle \frac 1 2 mv_\mathrm{rms}^2 = \frac 3 2 k T. 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.

It should be noted that this direct proportionality between temperature and energy does not hold for all substances. It is, however, generally true that energy is an increasing function of temperature.

Zeroth law of thermodynamics[edit]

Main article: Zeroth law of thermodynamics

Energy can be transferred between to bodies by a wide variety of processes that include conduction, convection, radiation, phase changes, and Joule (ohmic) heating. (See heat transfer.) While all these processes can be called heat, for our purposes it is best to view heat as the transfer of energy via between conduction two bodies in thermal contact. Energy transfer due to compression/decompression is not heat, but is instead called work.

The exchange of energy will, in turn, cause other state variables to change. For example, if one of the two bodies is a thermometer, an important state variable that changes is the volume. Left isolated from other bodies, the two connected bodies eventually reach a state of thermal equilibrium in which no further changes occur.

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 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.[48] 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.

Second law of thermodynamics[edit]

Main article: 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(T2)· g(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[edit]

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[edit]

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,[63] obtained under the hypothesis of ergodicity and orthodicity, allows to express the generalized temperature from the ratio of the average time of occupation \tau1 and \tau2 of the single/double occupancy system:[64] 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[edit]

Main article: 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.[65] 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[edit]

Main article: Orders of magnitude (temperature)


Temperature

Peak emittance wavelength[66]

of black-body radiation


Kelvin

Degrees Celsius

Absolute zero

(precisely by definition) 0 K −273.15 °C cannot be defined 

Coldest temperature

achieved[67] 100 pK −273.149999999900 °C 29,000 km 

Coldest Bose–Einstein

condensate[68] 450 pK −273.14999999955 °C 6,400 km 

One millikelvin

(precisely by definition) 0.001 K −273.149 °C 2.89777 m
(radio, FM band)[69] 

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][70] 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][71] 350 MK 350 million °C 8.3×10−3 nm
(gamma rays) 

Sandia National Labs' Z machine[E][72] 2 GK 2 billion °C 1.4×10−3 nm

(gamma rays)[F] 

Core of a high-mass

star on its last day[E][73] 3 GK 3 billion °C 1×10−3 nm
(gamma rays) 

Merging binary neutron

star system[E][74] 350 GK 350 billion °C 8×10−6 nm
(gamma rays) 

Relativistic Heavy

Ion Collider[E][75] 1 TK 1 trillion °C 3×10−6 nm
(gamma rays) 

CERN's proton vs

nucleus collisions[E][76] 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 1.616×10−27 nm
(Planck Length)[77]