# Temperature

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

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot. Heat spontaneously flows from bodies of a higher temperature to bodies of lower temperature, and no net heat will be exchanged between bodies of the same temperature. Such bodies are said to be in "thermal equilibrium".

The temperature of a substance varies with the microscopic speed of the fundamental particles that it contains, raised to the second power; that is, it is proportional to the mean kinetic energy of its particles. However any increase in temperature upon receiving external kinetic energy as heat is also inversely proportional to heat capacity. Temperature can be thought of as the "concentration" of kinetic energy relative to its heat capacity. A higher heat capacity implies a higher entropy in which to "spread out" the kinetic energy. A hot object must at least either have a smaller heat capacity or a larger kinetic energy than a cold object.

Quantitatively, temperature is measured with thermometers, which may be calibrated to a variety of temperature scales.

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

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

## Contents

### Use in science

Annual mean temperature around the world

Many physical properties of materials including the phase (solid, liquid, gaseous or plasma), density, solubility, vapor pressure, and electrical conductivity depend on the temperature. Temperature also plays an important role in determining the rate and extent to which chemical reactions occur. This is one reason why the human body has several elaborate mechanisms for maintaining the temperature at 310 K, since temperatures only a few degrees higher can result in harmful reactions with serious consequences. Temperature also controls the thermal radiation emitted from a surface. One application of this effect is the incandescent light bulb, in which a tungsten filament is electrically heated to a temperature at which significant quantities of visible light are emitted.

### Temperature scales

Some 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, the freezing point of water.[note 1] A few countries, most notably the United States, use the Fahrenheit scale for common purposes, a historical scale on which water freezes at 32 °F and boils at 212 °F.

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. For historical reasons, the triple point 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 −459.68 °F.

### Thermodynamic approach to temperature

Temperature is one of the principal quantities studied in the field of thermodynamics. Thermodynamics investigates the relation between heat and work, using a special scale of temperature called the absolute temperature, and thus relates temperature to work, as considered below. In thermodynamic terms, temperature is a macroscopic scale intensive variable because it is independent of the bulk amount of elementary entities contained inside, be they atoms, molecules, or electrons. Real world systems are not homogeneous and for study are usually spatially and temporally divided conceptually into imagined 'cells' of small size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation (local thermodynamic equilibrium).

### 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, degrees of freedom are used instead of particles.

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

As distinct from a quantity of heat, temperature may be viewed as a measure of a quality of a body [1] or of heat.[2][3][4][5] The quality is called hotness by some writers.[6][7]

When two systems are at the same temperature, no net heat transfer occurs spontanteously, by conduction or radiation, between them. When a temperature difference does exist, and there is a thermally conductive or radiative connection between them, heat transfers spontaneously from the warmer system to the colder system, until they are at mutual thermal equilibrium. This transfer occurs by heat conduction or by thermal radiation.[8][9][10][11][12][13][14][15]

Experimental physicists, for example Galileo and Newton[16], found that there are indefinitely many empirical temperature scales.

### Temperature for bodies in thermodynamic equilibrium

For experimental physics, the fact of hotness means that, when comparing any two given bodies in their respective separate thermodynamic equilibria, any two suitably given empirical thermometers with numerical scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature. This does not require the two thermometers to have a linear relation between their numerical scale readings, but it does require that the relation between their numerical readings shall be strictly monotonic.[17] 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.[18][6][19][20][7]

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.[21]

### 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 safely 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 1930's, 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.[6][22][20] 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.[23][24][6][25][19][26] 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 (ΔQ) 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 scientist measures temperature using the Celsius scale and the 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[27] 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,[28] 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 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 theory. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics and temperature.[29] 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 decomposes 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. Kinetic energy, a concept of classical mechanics, is one half the product of mass and the square of a particle's velocity. 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 kinetic energy of the atoms in a container of gas. Classical mechanics defines the kinetic energy as follows:

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

where m is the particle mass and v its velocity. The distribution of energies (and thus speeds) of the particles in any gas are given by the Maxwell-Boltzmann distribution. The temperature of a classical ideal gas is related to its average kinetic energy via the equation[30]:

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

for each degree of freedom, where k = R / n (n= Avogadro number, R= ideal gas constant). This relation is valid in the classical regime, i.e. when the particle density is much less than 1 / Λ3, where Λ is the thermal de Broglie wavelength. A monoatomic gas has only the three translational degrees of freedom.

The second law of thermodynamics states that any two given systems when interacting with each other will later reach the same average energy per particle and hence the same temperature.

In a mixture of particles of various masses, the heaviest particles will move slower than lighter particles, but have the same average kinetic energy. A neon atom moves slower 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.[22] 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[30]:

$pV = nRT\,\!$

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:[31]

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

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 controls the flow 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:

q13 = f(T1,T3) = f(T1,T2)f(T2,T3)

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

The previous section elaborated the historical derivation relating entropy and heat. A modern definition of temperature is given by statistical mechanics. It is defined in terms of the fundamental degrees of freedom of a system. Eq.(8) of the previous section is taken to be the defining relation of the temperature. Eq. (7) can be derived from first principles.

### 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 partition ensemble[32], obtained under the hypothesis of ergodicity and orthodicity, allows to express the generalized temperature from the ratio of the average time of occupation τ1 and τ2 of the single/double occupancy system [33]:

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

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 0K, 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.[34] 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[35]
Kelvin Degrees Celsius
Absolute zero
(precisely by definition)
0 K −273.15 °C Infinite
Coldest measured
temperature[36]
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][38] 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][39]
350 MK 350 million °C 8.3×10−3 nm
(gamma rays)
Sandia National Labs'
Z machine[E][40]
2 GK 2 billion °C 1.4×10−3 nm
(gamma rays)[F]
Core of a high-mass
star on its last day
[E][41]
3 GK 3 billion °C 1×10−3 nm
(gamma rays)
Merging binary neutron
star
system[E][42]
350 GK 350 billion °C 8×10−6 nm
(gamma rays)
Relativistic Heavy
Ion Collider
[E][43]
1 TK 1 trillion °C 3×10−6 nm
(gamma rays)
CERN's proton vs
nucleus collisions[E][44]
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)[45]
• 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. ^ Bryan, G.H. (1907). Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications, B.G. Teubner, Leipzig, page 3.[1]
2. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longman's, Green & Co, London, page 44.
3. ^ Planck, M. (1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green, London, page 31.
4. ^ Gibbs, J.W. (1875). Graphical Methods in the Thermodynamics of Fluids, Collected Works, Vol. 1, page 10, cited by Serrin, J. (1986). Chapter 1, 'An Outline of Thermodynamical Structure', page 7, in New Perspectives in Thermodynamics, edited by J. Serrin, Springer, Berlin, ISBN 3-540-15931-2.
5. ^ Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, New York, ISBN 0-88318-797-3, page 14.
6. ^ a b c d Mach, E. (1900). Die Principien der Wärmelehre. Historisch-kritisch entwickelt, Johann Ambrosius Barth, Leipzig, section 22, pages 56-57.
7. ^ 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.
8. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longmans, Green, London, page 32.
9. ^ Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pages 39-40.
10. ^ Planck, M. (1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green, London, pages 1-2.
11. ^ Planck, M. (1914), The Theory of Heat Radiation, second edition, translated into English by M. Masius, Blakiston's Son & Co., Philadelphia, reprinted by Kessinger.
12. ^ J. S. Dugdale (1996, 1998). Entropy and its Physical Interpretation. Tayler & Francis. p. 13. ISBN 9-7484-0569-0.
13. ^ F. Reif (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill. p. 102.
14. ^ 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.
15. ^ 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."
16. ^ Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pages 42, 103-117.
17. ^ Thomsen, J.S. (1962). A restatement of the zeroth law of thermodynamics, Am. J. Phys. 30: 294-296.
18. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longman's, Green & Co, London, page 45.
19. ^ a b Truesdell, C.A. (1980). The Tragicomical History of Thermodynamics, 1822-1854, Springer, New York, ISBN 0-387-90403-4, Section 11H, pages 320-332.
20. ^ 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.
21. ^ 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.
22. ^ 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.
23. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longmans, Green, London, pages 155-158.
24. ^ Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, Section 95, pages 68-69.
25. ^ H.A. Buchdahl (1966). The Concepts of Classical Thermodynamics. Cambridge University Press. p. 73.
26. ^ Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 978-0-470-01598-8, Section 32., pages 106-108.
27. ^ The kelvin in the SI Brochure
28. ^ "Absolute Zero". Calphad.com. Retrieved 2010-09-16.
29. ^ C. Caratheodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 355–386.
30. ^ a b Vu-Quoc, L., Configuration integral (statistical mechanics), 2008
31. ^ Peter Atkins, Julio de Paula (2006). Physical Chemistry (8 ed.). Oxford University Press. p. 9.
32. ^ 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.org
33. ^ Prati, E., et al. (2010). "Measuring the temperature of a mesoscopic electron system by means of single electron statistics". Applied Physics Letters 96: 113109. Bibcode 2010ApPhL..96k3109P. doi:10.1063/1.3365204.  arxiv.org
34. ^ Kittel, Charles; Kroemer, Herbert (1980). Thermal Physics (2nd ed. ed.). W. H. Freeman Company. pp. Appendix E. ISBN 0-7167-1088-9.
35. ^ 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.
36. ^ A record cold 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.
37. ^ The peak emittance wavelength of 2.89777 m is a frequency of 103.456 MHz
38. ^ 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).
39. ^ 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.
40. ^ 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.
41. ^ 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, and here, and a concise treatise on stars by NASA is here.
42. ^ 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.
43. ^ 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.
44. ^ How do physicists study particles? by CERN.
45. ^ 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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