Temperature

 ( Scalar Physical Quantities ) 

For example, if the change is an increase in temperature at constant volume, with no phase change and no chemical change, then the temperature of the body rises and its pressure increases. The quantity of heat transferred, , divided by the observed temperature change, , is the body's heat capacity at constant volume:

$C\_V\; =\; \backslash frac\{\backslash Delta\; Q\}\{\backslash Delta\; T\}.$

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 high-tech 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 thermodynamic-related 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 sub-zero 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 until May 2019, the Kelvin and Celsius scales were 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 was 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. Quantum-mechanically, however, zero-point motion remains and has an associated energy, the zero-point energy. Matter is in its ground state, and contains no thermal energy. The temperatures and were defined as those of the triple point of water. This definition served the following purposes: it fixed 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 established that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it established the difference between the null points of these scales as being ( = and = ). Since 2019, there has been a new definition based on the Boltzmann constant, Definition agreed by the 26th General Conference on Weights and Measures (CGPM) in November 2018, implemented 20 May 2019 but the scales are scarcely changed.

In the United States, the Fahrenheit scale is the most 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.

• Kelvin scale
• Celsius scale
• Fahrenheit scale
• Rankine scale
• Delisle scale
• Newton scale
• Réaumur scale
• Rømer scale

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, the temperature is a measure of the mean energy of motion, called translational 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 gas and in simple metals, the temperature is a measure of the mean particle translational kinetic energy, 3/2 k_B T. It also determines the probability distribution function of 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.

Kinetic energy is also considered as a component of 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: The Feynman Lectures on Physics. 39–5 The ideal gas law

$pV\; =\; nRT,$

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 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.

$E\_\backslash text\{k\}\; =\; \backslash frac\; 1\; 2\; mv\_\backslash text\{rms\}^2\; =\; \backslash frac\; 3\; 2\; k\_\backslash text\{B\}\; T,$

where the Boltzmann constant is the ideal gas constant divided by the Avogadro number, and $v\_\backslash text\{rms\}\; =\; \backslash sqrt\{\backslash langle\; v^2\; \backslash rangle\}\; =\; \backslash sqrt\{\backslash langle\; \backslash mathbf\{v\backslash cdot\; v\}\; \backslash rangle\}$ is the root-mean-square speed. This direct proportionality between temperature and mean molecular kinetic energy is a special case of the equipartition theorem, and holds only in the classical limit of a perfect gas. It does not hold exactly for most substances.

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.Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, , p. 22.Guggenheim, E.A. (1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, Elsevier, Amsterdam, (1st edition 1949) fifth edition 1965, p. 8: "If two systems are both in thermal equilibrium with a third system then they are in thermal equilibrium with each other."Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, Cambridge, p. 29: "... if each of two systems is in equilibrium with a third system then they are in 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 one-dimensional 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 the outcome is always 100% the same result. In contrast, many mixed ( disordered) outcomes are possible, and their number increases with each toss. Eventually, the combinations of ~50% heads and ~50% tails dominate, and obtaining an outcome significantly different from 50/50 becomes increasingly unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.

As temperature governs the transfer of heat between two systems and the universe tends to progress toward a maximum of entropy, it is expected that there is some relationship between temperature and entropy. A heat engine is a device for converting thermal energy into mechanical energy, resulting in the performance of work. An analysis of the Carnot heat engine provides the necessary relationships. According to energy conservation and energy being a state function that does not change over a full cycle, the work from a heat engine over a full cycle is equal to the net heat, i.e. the sum of the heat put into the system at high temperature, q_H > 0, and the waste heat given off at the low temperature, q_C < 0..

The efficiency is the work divided by the heat input:

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:

= f\left(T_\text{H}, T_\text{C}\right).|5}}
     

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

$q\_\{13\}\; =\; \backslash frac\{q\_1\; q\_2\}\{q\_2\; q\_3\},$

which implies

$q\_\{13\}\; =\; f\backslash left(T\_1,\; T\_3\backslash right)\; =\; f\backslash left(T\_1,\; T\_2\backslash right)f\backslash left(T\_2,\; T\_3\backslash right).$

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. = = = , where g is a function of a single temperature. A temperature scale can now be chosen with the property that

= \frac{T_\text{C}}{T_\text{H}}.|6}}
     

Substituting (6) back into (4) gives a relationship for the efficiency in terms of temperature:

= 1 - \frac{T_\text{C}}{T_\text{H}}.|7}}
     

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

$\backslash frac\{q\_\backslash text\{H\}\}\{T\_\backslash text\{H\}\}\; +\; \backslash frac\{q\_\backslash text\{C\}\}\{T\_\backslash text\{C\}\}\; =\; 0,$

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, whose change characteristically vanishes for a complete cycle if it is defined by

where the subscript indicates a reversible process. 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 quasi-reversible elements of entropy and heat:

For a constant-volume system where entropy S( E) is a function of its energy E, d E = d q_rev and (9) gives

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

$S\; =\; k\_\backslash mathrm\; B\; \backslash ln(W)$

When two systems with different temperatures are put into purely thermal connection, heat will flow from the higher temperature system to the lower temperature one; thermodynamically this is understood by the second law of thermodynamics: The total change in entropy following a transfer of energy $\backslash Delta\; E$ from system 1 to system 2 is:

$\backslash Delta\; S\; =\; -(dS/dE)\_1\; \backslash cdot\; \backslash Delta\; E\; +\; (dS/dE)\_2\; \backslash cdot\; \backslash Delta\; E\; =\; \backslash left(\backslash frac\{1\}\{T\_2\}\; -\; \backslash frac\{1\}\{T\_1\}\backslash right)\backslash Delta\; E$

and is thus positive if $T\_1\; >\; T\_2$

From the point of view of statistical mechanics, the total number of microstates in the combined system 1 + system 2 is $N\_1\; \backslash cdot\; N\_2$, the logarithm of which (times the Boltzmann constant) is the sum of their entropies; thus a flow of heat from high to low temperature, which brings an increase in total entropy, is more likely than any other scenario (normally it is much more likely), as there are more microstates in the resulting macrostate.

The international kinetic theory temperature of a body cannot take negative values. The thermodynamic temperature scale, however, is not so constrained.

For a body of matter, there can sometimes be conceptually defined, in terms of microscopic degrees of freedom, namely particle spins, a subsystem, with a temperature other than that of the whole body. When the body is in its own state of internal thermodynamic equilibrium, the temperatures of the whole body and of the subsystem must be the same. The two temperatures can differ when, by work through externally imposed force fields, energy can be transferred to and from the subsystem, separately from the rest of the body; then the whole body is not in its own state of internal thermodynamic equilibrium. There is an upper limit of energy such a spin subsystem can attain.

Considering the subsystem to be in a temporary state of virtual thermodynamic equilibrium, it is possible to obtain a negative temperature on the thermodynamic scale. Thermodynamic temperature is the inverse of the derivative of the subsystem's entropy with respect to its internal energy. As the subsystem's internal energy increases, the entropy increases for some range, but eventually attains a maximum value 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 as a function of energy decreases to zero and then turns negative. As the subsystem's entropy reaches its maximum, its thermodynamic temperature goes to positive infinity, switching to negative infinity as the slope turns negative. Such negative temperatures are hotter than any positive temperature. Over time, when the subsystem is exposed to the rest of the body, which has a positive temperature, energy is transferred as heat from the negative temperature subsystem to the positive temperature system.