Thermodynamic temperature

Thermodynamic temperature is the measure of absolute temperature and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero denotes the point at which the fundamental physical property that imbues matter with a temperature, transferable kinetic energy due to atomic motion, begins. In science, thermodynamic temperature is measured on the "Kelvin scale" and the unit of measure is the "kelvin" (unit symbol: K). For comparison, a temperature of 295 K is a comfortable one, equal to 21.85 °C and 71.33 °F.
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 kelvin (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 zero-point energy. 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.”
The Kelvin scale is also used in everyday life—often without people realizing it—due to color film photography and the need for films that were balanced for two kinds of tungsten-filament studio lights as well as noonday sun. Since the temperature of objects that emit black-body radiation, like the sun and tungsten filaments, had long been measured in kelvins, the color temperature of photographic lighting—and even regular LED bulbs for room lighting nowadays—are measured in kelvins, as per the following example usage: “Photographers who don’t own a color temperature meter may illuminate their scenes with ordinary 3200 kelvin (warm-white) LED bulbs and set their digital cameras’ white point to ‘"tungsten’".”
Overview.
The International System of Units (SI) specifies the "Kelvin scale" for measuring thermodynamic 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 particle motion. The redefinition fixed the Boltzmann constant at precisely (J/K). The compound unit of measure for the Boltzmann constant is often also given as J·K−1, which may seem abstract due the multiplication dot (·) and a kelvin symbol that is followed by a superscripted "negative 1" exponent, however this is merely another mathematical syntax denoting the same measure: "joules" (the SI unit for energy, including kinetic energy) "per kelvin."
The property that imbues any 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 perfectly elastic and spherical billiard balls that move only in a specific subset of the possible vibrational 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 in the X, Y, and Z axis 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 monoatomic 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 atoms’ and molecules’ three translational degrees of freedom.
Fixing the Boltzmann constant at specific value, along with other rule making, had the effect of precisely establishing the magnitude of the unit interval of thermodynamic 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.
The 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 kelvins and 0.01 °C. One effect of the aforementioned resolutions was that the melting point of water, while "very" close to 273.15 kelvin 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 affect 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 273.1600 ±0.0001 K (0.0100 ±0.0001 °C). 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.
The relationship of temperature, motions, conduction, and thermal energy.
The 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 "pV" = "nRT" 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 "Ek" = "mv"2. 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: "kB"). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an "individual" particles’ translational motion as follows:
where:
While the Boltzmann constant is useful for finding the mean kinetic energy in a sample of particles, it's 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 "" below. In both graphs, zero on the "x"–axis represents infinite temperature. Additionally, the "x" and "y"–axis on both graphs are scaled proportionally.
The high speeds of translational motion.
Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles 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 cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool cesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature.  Formulas for calculating the velocity and speed of translational motion are given in the following footnote.
1777: In his book "Pyrometrie" (Berlin: Haude & Spener, 1779) completed four months before his death, Johann Heinrich Lambert (1728–1777), sometimes incorrectly referred to as Joseph Lambert, proposed an absolute temperature scale based on the pressure/temperature relationship of a fixed volume of gas. This is distinct from the volume/temperature relationship of gas under constant pressure that Guillaume Amontons discovered 75 years earlier. Lambert stated that absolute zero was the point where a simple straight-line extrapolation reached zero gas pressure and was equal to −270 °C.
Circa 1787: Notwithstanding the work of Guillaume Amontons 85 years earlier, Jacques Alexandre César Charles (1746–1823) is often credited with discovering, but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was "V"1/"T"1 = "V"2/"T"2.
1802: Joseph Louis Gay-Lussac (1778–1850) published work (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called Charles's Law and is one of the gas laws. His are the first known formulas to use the number "273" for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C).
1848: William Thomson, (1824–1907) also known as Lord Kelvin, wrote in his paper, "On an Absolute Thermometric Scale," of the need for a scale whereby "infinite cold" (absolute zero) was the scale's zero point, and which used the degree Celsius for its unit increment. Like Gay-Lussac, Thomson calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the kelvin thermodynamic temperature scale. It's noteworthy that Thomson's value of "−273" was actually derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of −0.00366 expressed to five significant digits is −273.22 °C which is remarkably close to the true value of −273.15 °C.
1859: Macquorn Rankine (1820–1872) proposed a thermodynamic temperature scale similar to William Thomson's but which used the degree Fahrenheit for its unit increment. This absolute scale is known today as the Rankine thermodynamic temperature scale.
1877–1884: Ludwig Boltzmann (1844–1906) made major contributions to thermodynamics through an understanding of the role that particle kinetics and black body radiation played. His name is now attached to several of the formulas used today in thermodynamics.
Circa 1930s: Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed that absolute zero was equivalent to −273.15 °C.
1948: Resolution 3 of the 9th CGPM (Conférence Générale des Poids et Mesures, also known as the General Conference on Weights and Measures) fixed the triple point of water at precisely 0.01 °C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared "will be fixed at a later date". The implication is that "if" the value of absolute zero measured in the 1930s was truly −273.15 °C, then the triple point of water (0.01 °C) was equivalent to 273.16 K. Additionally, both the CIPM (Comité international des poids et mesures, also known as the International Committee for Weights and Measures) and the CGPM formally adopted the name "Celsius" for the "degree Celsius" and the "Celsius temperature scale".
1954: Resolution 3 of the 10th CGPM gave the kelvin scale its modern definition by choosing the triple point of water as its upper defining point (with no change to absolute zero being the null point) and assigning it a temperature of precisely 273.16 kelvins (what was actually written 273.16 "degrees Kelvin" at the time). This, in combination with Resolution 3 of the 9th CGPM, had the effect of defining absolute zero as being precisely zero kelvins and −273.15 °C.
1967/1968: Resolution 3 of the 13th CGPM renamed the unit increment of thermodynamic temperature "kelvin", symbol K, replacing "degree absolute", symbol °K. Further, feeling it useful to more explicitly define the magnitude of the unit increment, the 13th CGPM also decided in Resolution 4 that "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water".
2005: The CIPM (Comité International des Poids et Mesures, also known as the International Committee for Weights and Measures) affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin thermodynamic temperature scale would refer to water having an isotopic composition defined as being precisely equal to the nominal specification of Vienna Standard Mean Ocean Water.
2019: In November 2018, the 26th General Conference on Weights and Measures (CGPM) changed the definition of the Kelvin by fixing the Boltzmann constant to when expressed in the unit J/K. This change (and other changes in the definition of SI units) was made effective on the 144th anniversary of the Metre Convention, 20 May 2019.