Absolute temperature

Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.

Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic work and heat transfer as defined in thermodynamics, but the kelvin was redefined by international agreement in 2019 in terms of phenomena that are now understood as manifestations of the kinetic energy of free motion of microscopic particles such as atoms, molecules, and electrons. From the thermodynamic viewpoint, for historical reasons, because of how it is defined and measured, this microscopic kinetic definition is regarded as an "empirical" temperature. It was adopted because in practice it can generally be measured more precisely than can Kelvin's thermodynamic temperature.

A thermodynamic temperature reading of zero is of particular importance for the third law of thermodynamics. By convention, it is reported on the '' Kelvin scale'' of temperature in which the unit of measurement is the ''kelvin'' (unit symbol: K). For comparison, a temperature of 295 K is equal to 21.85 °C and 71.33 °F.

Overview

Thermodynamic temperature, as distinct from SI temperature, is defined in terms of a macroscopic Carnot cycle. Thermodynamic temperature is of importance in thermodynamics because it is defined in purely thermodynamic terms. SI temperature is conceptually far different from thermodynamic temperature. Thermodynamic temperature was rigorously defined historically long before there was a fair knowledge of microscopic particles such as atoms, molecules, and electrons.

The International System of Units (SI) specifies the international absolute scale for measuring temperature, and the unit of measure ''kelvin'' (unit symbol: K) for specific values along the scale. The kelvin is also used for denoting temperature ''intervals'' (a span or difference between two temperatures) as per the following example usage: "A 60/40 tin/lead solder is non-eutectic and is plastic through a range of 5 kelvins as it solidifies." A temperature interval of one degree Celsius is the same magnitude as one kelvin.

The magnitude of the kelvin was redefined in 2019 in relation to the ''very physical property'' underlying thermodynamic temperature: the kinetic energy of atomic free particle motion. The redefinition fixed the Boltzmann constant at precisely (J/K).
CODATA Value: Boltzmann constant
'. ''The NIST Reference on Constants, Units, and Uncertainty''. National Institute of Standards and Technology. The compound unit of measure for the Boltzmann constant is often also written as J·K⁻¹, with a multiplication dot (·) and a kelvin symbol that is followed by a superscripted ''negative 1'' exponent. This is another mathematical syntax denoting the same measure: '' joules'' (the SI unit for energy, including kinetic energy) ''per kelvin''.

The microscopic property that imbues material substances with a temperature can be readily understood by examining the ideal gas law, which relates, per the Boltzmann constant, how heat energy causes precisely defined changes in the pressure and temperature of certain gases. This is because monatomic gases like helium and argon behave kinetically like freely moving perfectly elastic and spherical billiard balls that move only in a specific subset of the possible motions that can occur in matter: that comprising the ''three translational'' degrees of freedom. The translational degrees of freedom are the familiar billiard ball-like movements along the X, Y, and Z axes of 3D space (see ''Fig. 1'', below). This is why the noble gases all have the same specific heat capacity per atom and why that value is lowest of all the gases.

Molecules (two or more chemically bound atoms), however, have ''internal structure'' and therefore have additional ''internal'' degrees of freedom (see ''Fig. 3'', below), which makes molecules absorb more heat energy for any given amount of temperature rise than do the monatomic gases. Heat energy is born in all available degrees of freedom; this is in accordance with the equipartition theorem, so all available internal degrees of freedom have the same temperature as their three external degrees of freedom. However, the property that gives all gases their pressure, which is the net force per unit area on a container arising from gas particles recoiling off it, is a function of the kinetic energy borne in the freely moving atoms’ and molecules’ three translational degrees of freedom.

Fixing the Boltzmann constant at a specific value, along with other rule making, had the effect of precisely establishing the magnitude of the unit interval of SI temperature, the kelvin, in terms of the average kinetic behavior of the noble gases. Moreover, the ''starting point'' of the thermodynamic temperature scale, absolute zero, was reaffirmed as the point at which ''zero average kinetic energy'' remains in a sample; the only remaining particle motion being that comprising random vibrations due to zero-point energy.

Absolute zero of temperature

Temperature scales are numerical. The numerical zero of a temperature scale is not bound to the absolute zero of temperature. Nevertheless, some temperature scales have their numerical zero coincident with the absolute zero of temperature. Examples are the International SI temperature scale, the Rankine temperature scale, and the thermodynamic temperature scale. Other temperature scales have their numerical zero far from the absolute zero of temperature. Examples are the Fahrenheit scale and the Celsius scale.

At the zero point of thermodynamic temperature, absolute zero, the particle constituents of matter have minimal motion and can become no colder. Absolute zero, which is a temperature of zero kelvins (0 K), is precisely equal to −273.15 °C and −459.67 °F. Matter at absolute zero has no remaining transferable average kinetic energy and the only remaining particle motion is due to an ever-pervasive quantum mechanical phenomenon called ZPE ( Zero-Point Energy). While scientists are achieving temperatures ever closer to absolute zero, they can not fully achieve a state of ''zero'' temperature. However, even if scientists could remove ''all'' kinetic thermal energy from matter, quantum mechanical '' zero-point energy'' (ZPE) causes particle motion that can never be eliminated. Encyclopædia Britannica Onlin
defines zero-point
energy as the "vibrational energy that molecules retain even at the absolute zero of temperature". ZPE is the result of all-pervasive energy fields in the vacuum between the fundamental particles of nature; it is responsible for the Casimir effect and other phenomena. See
Zero Point Energy and Zero Point Field
'. See also

'' by the University of Alberta's Department of Physics to learn more about ZPE's effect on Bose–Einstein condensates of helium.

Although absolute zero () is not a state of zero molecular motion, it ''is'' the point of zero temperature and, in accordance with the Boltzmann constant, is also the point of zero particle kinetic energy and zero kinetic velocity. To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two helium atoms in zero gravity are carefully positioned and observed to have an average separation of 620  pm between them (a gap of ten atomic diameters). It is an "average" separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83 yoctokelvins (1 yK = ). This is done in a way that directs this atom's velocity vector at the other atom. With 83 yK of kinetic energy between them, the 620 pm gap through their common barycenter would close at a rate of 719 pm/s and they would collide after 0.862 second. This is the same speed as shown in the '' Fig. 1 ''animation above. Before being given the kinetic kick, both atoms had zero kinetic energy and zero kinetic velocity because they could persist indefinitely in that state and relative orientation even though both were being jostled by ZPE. At , no kinetic energy is available for transfer to other systems.

Note too that absolute zero serves as the baseline atop which thermodynamics and its equations are founded because they deal with the exchange of thermal energy between "''systems''" (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPE-induced particle motion ''within'' a system that is at absolute zero but there can never be a net outflow of thermal energy from such a system. Also, the peak emittance wavelength of black-body radiation shifts to infinity at absolute zero; indeed, a peak no longer exists and black-body photons can no longer escape. Because of ZPE, however, ''virtual'' photons are still emitted at . Such photons are called "virtual" because they can't be intercepted and observed. Furthermore, this ''zero-point radiation'' has a unique ''zero-point spectrum''. However, even though a system emits zero-point radiation, no net heat flow ''Q'' out of such a system can occur because if the surrounding environment is at a temperature greater than , heat will flow inward, and if the surrounding environment is at ', there will be an equal flux of ZP radiation both inward and outward. A similar ''Q ''equilibrium exists at with the ZPE-induced spontaneous emission of photons (which is more properly called a ''stimulated'' emission in this context). The graph at upper right illustrates the relationship of absolute zero to zero-point energy. The graph also helps in the understanding of how zero-point energy got its name: it is the vibrational energy matter retains at the ''zero-kelvin point''
''Derivation of the classical electromagnetic zero-point radiation spectrum via a classical thermodynamic operation involving van der Waals forces''
Daniel C. Cole, Physical Review A, 42 (1990) 1847. Though the atoms in, for instance, a container of liquid helium that was ''precisely'' at absolute zero would still jostle slightly due to zero-point energy, a theoretically perfect heat engine with such helium as one of its working fluids could never transfer any net kinetic energy ( heat energy) to the other working fluid and no thermodynamic work could occur.

Temperature is generally expressed in absolute terms when scientifically examining temperature's interrelationships with certain other physical properties of matter such as its volume or pressure (see Gay-Lussac's law), or the wavelength of its emitted black-body radiation. Absolute temperature is also useful when calculating chemical reaction rates (see Arrhenius equation). Furthermore, absolute temperature is typically used in cryogenics and related phenomena like superconductivity, as per the following example usage:
"Conveniently, tantalum's transition temperature (''T'') of 4.4924 kelvin is slightly above the 4.2221 K boiling point of helium."

Boltzmann constant

The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE ( zero-point energy) is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of '' chaos'', i.e., unpredictability, to rebound kinetics; it is as likely that there will be ''less'' ZPE-induced particle motion after a given collision as ''more''. This random nature of ZPE is why it has no net effect upon either the pressure or volume of any ''bulk quantity'' (a statistically significant quantity of particles) of gases. However, in temperature condensed matter; e.g., solids and liquids, ZPE causes inter-atomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the real-world effects that ZPE has on substances can vary as one alters a thermodynamic system (for example, due to ZPE, helium won't freeze unless under a pressure of at least 2.5  MPa (25 bar)), ZPE is very much a form of thermal energy and may properly be included when tallying a substance's internal energy.

Rankine scale

Though there have been many other temperature scales throughout history, there have been only two scales for measuring thermodynamic temperature where absolute zero is their null point (0): The Kelvin scale and the Rankine scale.

Throughout the scientific world where modern measurements are nearly always made using the International System of Units, thermodynamic temperature is measured using the Kelvin scale. The Rankine scale is part of English engineering units in the United States and finds use in certain engineering fields, particularly in legacy reference works. The Rankine scale uses the ''degree Rankine'' (symbol: °R) as its unit, which is the same magnitude as the degree Fahrenheit (symbol: °F).

A unit increment of one degree Rankine is precisely 1.8 times smaller in magnitude than one kelvin; thus, to convert a specific temperature on the Kelvin scale to the Rankine scale, , and to convert from a temperature on the Rankine scale to the Kelvin scale, . Consequently, absolute zero is "0" for both scales, but the melting point of water ice (0 °C and 273.15 K) is 491.67 °R.

To convert temperature ''intervals'' (a span or difference between two temperatures), one uses the same formulas from the preceding paragraph; for instance, a range of 5 kelvins is precisely equal to a range of 9 degrees Rankine.

Modern redefinition of the kelvin

For 65 years, between 1954 and the 2019 redefinition of the SI base units, a temperature interval of one kelvin was defined as the difference between the triple point of water and absolute zero. The 1954 resolution by the International Bureau of Weights and Measures (known by the French-language acronym BIPM), plus later resolutions and publications, defined the triple point of water as precisely 273.16 K and acknowledged that it was "common practice" to accept that due to previous conventions (namely, that 0 °C had long been defined as the melting point of water and that the triple point of water had long been experimentally determined to be indistinguishably close to 0.01 °C), the difference between the Celsius scale and Kelvin scale is accepted as 273.15 kelvins; which is to say, 0 °C equals 273.15 kelvins. The net effect of this as well as later resolutions was twofold: 1) they defined absolute zero as precisely 0 K, and 2) they defined that the triple point of special isotopically controlled water called Vienna Standard Mean Ocean Water was precisely 273.16 K and 0.01 °C. One effect of the aforementioned resolutions was that the melting point of water, while ''very'' close to 273.15 K and 0 °C, was not a defining value and was subject to refinement with more precise measurements.

The 1954 BIPM standard did a good job of establishing—within the uncertainties due to isotopic variations between water samples—temperatures around the freezing and triple points of water, but required that ''intermediate values'' between the triple point and absolute zero, as well as extrapolated values from room temperature and beyond, to be experimentally determined via apparatus and procedures in individual labs. This shortcoming was addressed by the International Temperature Scale of 1990, or ITS90, which defined 13 additional points, from 13.8033 K, to 1,357.77 K. While definitional, ITS90 had—and still has—some challenges, partly because eight of its extrapolated values depend upon the melting or freezing points of metal samples, which must remain exceedingly pure lest their melting or freezing points be affected—usually depressed.

The 2019 redefinition of the SI base units was primarily for the purpose of decoupling much of the SI system's definitional underpinnings from the kilogram, which was the last physical artifact defining an SI base unit (a platinum/iridium cylinder stored under three nested bell jars in a safe located in France) and which had highly questionable stability. The solution required that four physical constants, including the Boltzmann constant, be definitionally fixed.

Assigning the Boltzmann constant a precisely defined value had no practical effect on modern thermometry except for the most exquisitely precise measurements. Before the redefinition, the triple point of water was exactly 273.16 K and 0.01 °C and the Boltzmann constant was experimentally determined to be , where the "(51)" denotes the uncertainty in the two least significant digits (the 03) and equals a relative standard uncertainty of 0.37 ppm. Afterwards, by defining the Boltzmann constant as exactly , the 0.37 ppm uncertainty was transferred to the triple point of water, which became an experimentally determined value of (). That the triple point of water ended up being exceedingly close to 273.16 K after the SI redefinition was no accident; the final value of the Boltzmann constant was determined, in part, through clever experiments with argon and helium that used the triple point of water for their key reference temperature. 

Notwithstanding the 2019 redefinition, water triple-point cells continue to serve in modern thermometry as exceedingly precise calibration references at 273.16 K and 0.01 °C. Moreover, the triple point of water remains one of the 14 calibration points comprising ITS90, which spans from the triple point of hydrogen (13.8033 K) to the freezing point of copper (1,357.77 K), which is a nearly hundredfold range of thermodynamic temperature.

Relationship of temperature, motions, conduction, and thermal energy

Nature of kinetic energy, translational motion, and temperature

The thermodynamic temperature of any ''bulk quantity'' of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as ''translational motion''. These simple movements in the three X, Y, and Z–axis dimensions of space means the particles move in the three spatial ''degrees of freedom''. This particular form of kinetic energy is sometimes referred to as ''kinetic temperature''. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the ideal gas law's formula and is embodied in the gas laws.

Though the kinetic energy borne exclusively in the three translational degrees of freedom comprise the thermodynamic temperature of a substance, molecules, as can be seen in ''Fig. 3'', can have other degrees of freedom, all of which fall under three categories: bond length, bond angle, and rotational. All three additional categories are not necessarily available to all molecules, and even for molecules that ''can'' experience all three, some can be "frozen out" below a certain temperature. Nonetheless, all those degrees of freedom that are available to the molecules under a particular set of conditions contribute to the specific heat capacity of a substance; which is to say, they increase the amount of heat (kinetic energy) required to raise a given amount of the substance by one kelvin or one degree Celsius.

The relationship of kinetic energy, mass, and velocity is given by the formula ''E_k'' = ''mv''. Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy, and precisely the same temperature, as those with four times the mass but half the velocity.

The extent to which the kinetic energy of translational motion in a statistically significant collection of atoms or molecules in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the Boltzmann constant (symbol: ''k''_B). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an ''individual'' particles’ translational motion as follows:
$\tilde = \frac k_\text T$
where:
* $\tilde$ is the mean kinetic energy for an individual particle
* ''k''_B =
* ''T'' is the thermodynamic temperature of the bulk quantity of the substance

While the Boltzmann constant is useful for finding the mean kinetic energy in a sample of particles, it is important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in '' Fig. 1 ''above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in ''Fig. 2''  shows the speed distribution of 5500 K helium atoms. They have a ''most probable'' speed of 4.780 km/s (0.2092 s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the ''x''–axis to the right). This graph uses ''inverse speed'' for its ''x''–axis so the shape of the curve can easily be compared to the curves in '' Fig. 5'' below. In both graphs, zero on the ''x''–axis represents infinite temperature. Additionally, the ''x'' and ''y''–axis on both graphs are scaled proportionally.

High speeds of translational motion

Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collis