Thermodynamic 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.


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 1.380649×10−23 joules per kelvin (J/K).[1] 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.[2]

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.[3][4] 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).[5] 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 (Tc) 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 T = 0 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, K × 1.8 = °R, and to convert from a temperature on the Rankine scale to the Kelvin scale, °R / 1.8 = K. 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 1/273.16 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.[6] 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 1.38064903(51)×10−23 J/K, where the "(51)" denotes the uncertainty in the two least significant digits (the 03) and equals a relative standard uncertainty of 0.37 ppm.[7] Afterwards, by defining the Boltzmann constant as exactly 1.380649×10−23 J/K, 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.[8][9]

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

Figure 1 The translational motion of fundamental particles of nature such as atoms and molecules are directly related to temperature. Here, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain average speed (slowed down here two trillion-fold). At any given instant however, a particular helium atom may be moving much faster than average while another may be nearly motionless. Five atoms are colored red to facilitate following their motions. This animation illustrates statistical mechanics, which is the science of how the group behavior of a large collection of microscopic objects emerges from the kinetic properties of each individual object.

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 = 1/2mv2.[10] 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:


  • is the mean kinetic energy for an individual particle
  • kB = 1.380649×10−23 J/K
  • T is the thermodynamic temperature of the bulk quantity of the substance
Figure 2 The translational motions of helium atoms occur across a range of speeds. Compare the shape of this curve to that of a Planck curve in Fig. 5 below.

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 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[11] 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.[12]  Formulas for calculating the velocity and speed of translational motion are given in the following footnote.[13]

Figure 2.5 This simulation illustrates an argon atom as it would appear through a 400-power optical microscope featuring a reticle graduated with 50-micron (0.05 mm) tick marks. This atom is moving with a velocity of 14.43 microns per second, which gives the atom a kinetic temperature of one-trillionth of a kelvin. The atom requires 13.9 seconds to travel 200 microns (0.2 mm). Though the atom is being invisibly jostled due to zero-point energy, its translational motion seen here comprises all its kinetic energy.

It is neither difficult to imagine atomic motions due to kinetic temperature, nor distinguish between such motions and those due to zero-point energy. Consider the following hypothetical thought experiment, as illustrated in Fig. 2.5 at left, with an atom that is exceedingly close to absolute zero. Imagine peering through a common optical microscope set to 400 power, which is about the maximum practical magnification for optical microscopes. Such microscopes generally provide fields of view a bit over 0.4 mm in diameter. At the center of the field of view is a single levitated argon atom (argon comprises about 0.93% of air) that is illuminated and glowing against a dark backdrop. If this argon atom was at a beyond-record-setting one-trillionth of a kelvin above absolute zero,[14] and was moving perpendicular to the field of view towards the right, it would require 13.9 seconds to move from the center of the image to the 200-micron tick mark; this travel distance is about the same as the width of the period at the end of this sentence on modern computer monitors. As the argon atom slowly moved, the positional jitter due to zero-point energy would be much less than the 200-nanometer (0.0002 mm) resolution of an optical microscope. Importantly, the atom's translational velocity of 14.43 microns per second constitutes all its retained kinetic energy due to not being precisely at absolute zero. Were the atom precisely at absolute zero, imperceptible jostling due to zero-point energy would cause it to very slightly wander, but the atom would perpetually be located, on average, at the same spot within the field of view. This is analogous to a boat that has had its motor turned off and is now bobbing slightly in relatively calm and windless ocean waters; even though the boat randomly drifts to and fro, it stays in the same spot in the long term and makes no headway through the water. Accordingly, an atom that was precisely at absolute zero would not be "motionless", and yet, a statistically significant collection of such atoms would have zero net kinetic energy available to transfer to any other collection of atoms. This is because regardless of the kinetic temperature of the second collection of atoms, they too experience the effects of zero-point energy. Such are the consequences of statistical mechanics and the nature of thermodynamics.

Internal motions of molecules and internal energy

Figure 3 Molecules have internal structures because they are composed of atoms that have different ways of moving within molecules. Being able to store kinetic energy in these internal degrees of freedom contributes to a substance's specific heat capacity, or internal energy, allowing it to contain more internal energy at the same temperature.

As mentioned above, there are other ways molecules can jiggle besides the three translational degrees of freedom that imbue substances with their kinetic temperature. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain their internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements; these are all types of internal degrees of freedom. This makes molecules distinct from monatomic substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom (the X, Y, and Z axis). Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an internal temperature. Even though these motions are called "internal", the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as internal energy is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-local thermodynamic equilibrium (LTE) phenomena such as combustion, the sublimation of solids, and the diffusion of hot gases in a partial vacuum.

The kinetic energy stored internally in molecules causes substances to contain more heat energy at any given temperature and to absorb additional internal energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions, is not contributing to the molecules' translational motions at that same instant .[15] This extra kinetic energy simply increases the amount of internal energy that substance absorbs for a given temperature rise. This property is known as a substance's specific heat capacity.

Different molecules absorb different amounts of internal energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances’ molecules possess more internal degrees of freedom than others do. For instance, room-temperature nitrogen, which is a diatomic molecule, has five active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has five-thirds the specific heat capacity per mole (a specific number of molecules) as do the monatomic gases.[16] Another example is gasoline (see table showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.

Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons

Figure 4 The temperature-induced translational motion of particles in solids takes the form of phonons. Shown here are phonons with identical amplitudes but with wavelengths ranging from 2 to 12 average inter-molecule separations (a).

Heat conduction is the diffusion of thermal energy from hot parts of a system to cold parts. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term bulk in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever thermal energy diffuses within an isolated system, temperature differences within the system decrease (and entropy increases).

One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in Fig. 1. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can move forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; convection speeds this process even more.[17]

Translational motion in solids, however, takes the form of phonons (see Fig. 4 at right). Phonons are constrained, quantized wave packets that travel at the speed of sound of a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is usually inefficient[18] and such solids are considered thermal insulators (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.

Metals however, are not restricted to only phonon-based heat conduction. Thermal energy conducts through metals extraordinarily quickly because instead of direct molecule-to-molecule collisions, the vast majority of thermal energy is mediated via very light, mobile conduction electrons. This is why there is a near-perfect correlation between metals' thermal conductivity and their electrical conductivity.[19] Conduction electrons imbue metals with their extraordinary conductivity because they are delocalized (i.e., not tied to a specific atom) and behave rather like a sort of quantum gas due to the effects of zero-point energy (for more on ZPE, see Note 1 below). Furthermore, electrons are relatively light with a rest mass only 11836 that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Isaac Newton wrote with his third law of motion,

Law #3: All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.

However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, thermal energy is readily borne by mobile conduction electrons. Additionally, because they're delocalized and very fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.

Diffusion of thermal energy: black-body radiation

Figure 5 The spectrum of black-body radiation has the form of a Planck curve. A 5500 K black-body has a peak emittance wavelength of 527 nm. Compare the shape of this curve to that of a Maxwell distribution in Fig. 2 above.

Thermal radiation is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal photons (known as black-body radiation). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even individual molecules with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. In any bulk quantity of a substance at equilibrium, black-body photons are emitted across a range of wavelengths in a spectrum that has a bell curve-like shape called a Planck curve (see graph in Fig. 5 at right). The top of a Planck curve (the peak emittance wavelength) is located in a particular part of the electromagnetic spectrum depending on the temperature of the black-body. Substances at extreme cryogenic temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short gamma rays (see Table of common temperatures).

Black-body radiation diffuses thermal energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.

As established by the Stefan–Boltzmann law, the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black-body at 824 K (just short of glowing dull red) emits 60 times the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, black-body radiation can be the principal mechanism by which thermal energy escapes a system.

Table of thermodynamic temperatures

The table below shows various points on the thermodynamic scale, in order of increasing temperature.

kelvin Peak emittance
wavelength[20] of
black-body photons
Absolute zero
(precisely by definition)
0 K    [5]
Coldest measured
temperature [21]
450 pK 6,400 km
One millikelvin
(precisely by definition)
0.001 K 2.897 77 m
(radio, FM band)[22]
cosmic microwave
background radiation
2.725 K 1.063 mm (peak wavelength)
Water's triple point 273.16 K 10.6083 µm
(long wavelength I.R.)
ISO 1 standard temperature
for precision metrology
(precisely 20 °C by definition)
293.15 K 9.88495 µm
(long wavelength I.R.)
Incandescent lamp[upper-alpha 1] 2500 K[upper-alpha 2] 1.16 µm
(near infrared)[upper-alpha 3]
Sun's visible surface[upper-alpha 3][23] 5778 K 501.5 nm
(green light)
Lightning bolt's
28,000 K 100 nm
(far ultraviolet light)
Sun's core 16 MK 0.18 nm (X-rays)
Thermonuclear explosion
(peak temperature)[24]
350 MK 8.3 × 10−3 nm
(gamma rays)
Sandia National Labs'
Z machine[upper-alpha 4][25]
2 GK 1.4 × 10−3 nm
(gamma rays)
Core of a high-mass
star on its last day
3 GK 1 × 10−3 nm
(gamma rays)
Merging binary neutron
system [27]
350 GK 8 × 10−6 nm
(gamma rays)
Gamma-ray burst progenitors[28] 1 TK 3 × 10−6 nm
(gamma rays)
CERN's proton vs.
nucleus collisions[29]
10 TK 3 × 10−7 nm
(gamma rays)
  1. For a true blackbody (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.
  2. The 2500 K value is approximate.
  3. Effective photosphere temperature.
  4. For a true blackbody (which the plasma was not). The Z machine's dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.

Heat of phase changes

Figure 6  Ice and water: two phases of the same substance

The kinetic energy of particle motion is just one contributor to the total thermal energy in a substance; another is phase transitions, which are the potential energy of molecular bonds that can form in a substance as it cools (such as during condensing and freezing). The thermal energy required for a phase transition is called latent heat. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to break chemical bonds (such as during evaporation and melting). Almost everyone is familiar with the effects of phase transitions; for instance, steam at 100 °C can cause severe burns much faster than the 100 °C air from a hair dryer. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.

Even though thermal energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic alloys exhibit no temperature change whatsoever while they undergo them (see Fig. 7, below right). Consider one particular type of phase transition: melting. When a solid is melting, crystal lattice chemical bonds are being broken apart; the substance is transitioning from what is known as a more ordered state to a less ordered state. In Fig. 7, the melting of ice is shown within the lower left box heading from blue to green.

Figure 7 Water's temperature does not change during phase transitions as heat flows into or out of it. The total heat capacity of a mole of water in its liquid phase (the green line) is 7.5507 kJ.

At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are, on average, at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are all-or-nothing forces: they either hold fast, or break; there is no in-between state. Consequently, when a substance is at its melting point, every joule of added thermal energy only breaks the bonds of a specific quantity of its atoms or molecules,[30] converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional thermal energy can't make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), thermal energy must be removed from a substance.

As stated above, the thermal energy required for a phase transition is called latent heat. In the specific cases of melting and freezing, it's called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements.[31] If the substance is one of the monatomic gases, (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole.[32] Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the thermal energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals' ratios are even greater, typically in the range of 400 to 1200 times.[33] And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a one-degree increase.[34]

Water's sizable enthalpy of vaporization is why one's skin can be burned so quickly as steam condenses on it (heading from red to green in Fig. 7 above); water vapors (gas phase) are liquefied on the skin with releasing a large amount of energy (enthalpy) to the environment including the skin, resulting in skin damage. In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient wet-bulb temperature that is dependent on relative humidity); the water evaporation on the skin takes a large amount of energy from the environment including the skin, reducing the skin temperature. Water's highly energetic enthalpy of vaporization is also an important factor underlying why solar pool covers (floating, insulated blankets that cover swimming pools when the pools are not in use) are so effective at reducing heating costs: they prevent evaporation. (In other words, taking energy from water when it is evaporated is limited.) For instance, the evaporation of just 20 mm of water from a 1.29-meter-deep pool chills its water 8.4 degrees Celsius (15.1 °F).

Internal energy

The total energy of all translational and internal particle motions, including that of conduction electrons, plus the potential energy of phase changes, plus zero-point energy[5] of a substance comprise the internal energy of it.

Figure 8 When many of the chemical elements, such as the noble gases and platinum-group metals, freeze to a solid — the most ordered state of matter — their crystal structures have a close-packed arrangement. This yields the greatest possible packing density and the lowest energy state.

Internal energy at absolute zero

As a substance cools, different forms of internal energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions is liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic energy or temperature decreases); the internal motions of molecules diminish (their internal energy or temperature decreases); conduction electrons (if the substance is an electrical conductor) travel somewhat slower;[35] and black-body radiation's peak emittance wavelength increases (the photons' energy decreases). When particles of a substance are as close as possible to complete rest and retain only ZPE (Zero Point Energy)-induced quantum mechanical motion, the substance is at the temperature of absolute zero (T = 0).

Figure 9 Due to the effects of zero-point energy, helium at ambient pressure remains a superfluid even when exceedingly close to absolute zero; it won't freeze unless under 25 bar of pressure (~25 atmospheres).

Note that whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero internal energy; one must be very precise with what one means by internal energy. Often, all the phase changes that can occur in a substance, will have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, T = 0 helium remains liquid at room pressure (Fig. 9 at right) and must be under a pressure of at least 25 bar (2.5 MPa) to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21 joules per mole) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures.

A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars, or hundreds of gigapascals). These are known as solid–solid phase transitions wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one.

The above complexities make for rather cumbersome blanket statements regarding the internal energy in T = 0 substances. Regardless of pressure though, what can be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a closest-packed arrangement (see Fig. 8, above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy.[5] [36] One can also say that for a given substance at constant pressure, absolute zero is the point of lowest enthalpy (a measure of work potential that takes internal energy, pressure, and volume into consideration).[37] Lastly, it is always true to say that all T = 0 substances contain zero kinetic thermal energy.[5] [13]

Practical applications for thermodynamic temperature

Thermodynamic temperature is useful not only for scientists, it can also be useful for lay-people in many disciplines involving gases. By expressing variables in absolute terms and applying Gay-Lussac's law of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a cold gage[38] pressure of 200 kPa, then its absolute pressure is 300 kPa.[39][40] Room temperature ("cold" in tire terms) is 296 K. If the tire temperature is 20 °C hotter (20 kelvins), the solution is calculated as 316 K/296 K = 6.8% greater thermodynamic temperature and absolute pressure; that is, an absolute pressure of 320 kPa, which is a gage pressure of 220 kPa.

Relationship to ideal gas law

The thermodynamic temperature is closely linked to the ideal gas law and its consequences. It can be linked also to the second law of thermodynamics. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratio T2/T1 of two temperatures T1 and T2 is the same in all absolute scales.

Strictly speaking, the temperature of a system is well-defined only if it is at thermal equilibrium. From a microscopic viewpoint, a material is at thermal equilibrium if the quantity of heat between its individual particles cancel out. There are many possible scales of temperature, derived from a variety of observations of physical phenomena.

Loosely stated, temperature differences dictate the direction of heat between two systems such that their combined energy is maximally distributed among their lowest possible states. We call this distribution "entropy". To better understand the relationship between temperature and entropy, consider the relationship between heat, work and temperature illustrated in the Carnot heat engine. The engine converts heat into work by directing a temperature gradient between a higher temperature heat source, TH, and a lower temperature heat sink, TC, through a gas filled piston. The work done per cycle is equal in magnitude to net heat taken up, which is sum of the heat qH taken up by the engine from the high-temperature source, plus the waste heat given off by the engine, qC < 0.[41] The efficiency of the engine is the work divided by the heat put into the system or

where is the work done per cycle. Thus the efficiency depends only on |qC| / |qH|.

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, that is to say, the efficiency is the function of only temperatures

In addition, a reversible heat engine operating between a pair of thermal reservoirs at temperatures T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2 andT3. If this were not the case, then energy (in the form of q) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles as an engine design choice, and any reversible engine between the same reservoir at T1 and T3 must be equally efficient regardless of the engine design.

If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as the below.


Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at T2. We also have used the fact that the heat passes through the intermediate thermal reservoir at without losing its energy. (I.e., is not lost during its passage through the reservoir at .) This fact can be proved by the following.

In order to have the consistency in the last equation, the heat flown from the engine 2 to the intermediate reservoir must be equal to the heat flown out from the reservoir to the engine 3.

With this understanding of q1, q2 and q3, mathematically,

But since the first function is NOT a function of T2, the product of the final two functions MUST result in the removal of T2 as a variable. The only way is therefore to define the function f as follows:


so that

I.e. the ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our ;[42] it is a matter of convenience and convention that we choose . Choosing then one fixed reference temperature (i.e. triple point of water), we establish the thermodynamic temperature scale.

Such a definition coincides with that of the ideal gas derivation; also it is this definition of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of TH and TC, and hence derive that the (complete) Carnot cycle is isentropic:

Substituting this back into our first formula for efficiency yields a relationship in terms of temperature:

Note that for TC = 0 the efficiency is 100% and that efficiency becomes greater than 100% for TC < 0, which is unrealistic. Subtracting 1 from the right hand side of the Equation (4) and the middle portion gives and thus [43][41]

The generalization of this equation is the Clausius theorem, which proposes the existence of a state function (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by






where the subscript rev indicates heat transfer in a reversible process. The function is the entropy of the system, mentioned previously, and the change of around any cycle is zero (as is necessary for any state function). The Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid a logic loop, we should first define entropy through statistical mechanics):

For a constant-volume system (so no mechanical work ) in which the entropy is a function of its internal energy , and the thermodynamic temperature is therefore given by

so that the reciprocal of the thermodynamic temperature is the rate of change of entropy with respect to the internal energy at the constant volume.


1702–1703: Guillaume Amontons (1663–1705) published two papers that may be used to credit him as being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. He made the discovery while endeavoring to improve upon the air thermometers in use at the time. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation led him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240 °C—only 33.15 degrees short of the true value of −273.15 °C. Amonton's discovery of a one-to-one relationship between absolute temperature and absolute pressure was rediscovered a century later and popularized within the scientific community by Joseph Louis Gay-Lussac. Today, this principle of thermodynamics is commonly known as Gay-Lussac's law but is also known as Amonton's law.

1742: Anders Celsius (1701–1744) created a "backwards" version of the modern Celsius temperature scale. In Celsius's original scale, zero represented the boiling point of water and 100 represented the melting point of ice. In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that ice's melting point was effectively unaffected by pressure. He also determined with remarkable precision how water's boiling point varied as a function of atmospheric pressure. He proposed that zero on his temperature scale (water's boiling point) would be calibrated at the mean barometric pressure at mean sea level.

1744: Coincident with the death of Anders Celsius, the famous botanist Carl Linnaeus (1707–1778) effectively reversed[44] Celsius's scale upon receipt of his first thermometer featuring a scale where zero represented the melting point of ice and 100 represented water's boiling point. The custom-made linnaeus-thermometer, for use in his greenhouses, was made by Daniel Ekström, Sweden's leading maker of scientific instruments at the time. For the next 204 years, the scientific and thermometry communities worldwide referred to this scale as the centigrade scale. Temperatures on the centigrade scale were often reported simply as degrees or, when greater specificity was desired, degrees centigrade. The symbol for temperature values on this scale was °C (in several formats over the years). Because the term centigrade was also the French-language name for a unit of angular measurement (one-hundredth of a right angle) and had a similar connotation in other languages, the term "centesimal degree" was used when very precise, unambiguous language was required by international standards bodies such as the International Bureau of Weights and Measures (Bureau international des poids et mesures) (BIPM). The 9th CGPM (General Conference on Weights and Measures (Conférence générale des poids et mesures) and the CIPM (International Committee for Weights and Measures (Comité international des poids et mesures) formally adopted[45] degree Celsius (symbol: °C) in 1948.[46]

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 V1/T1 = V2/T2.

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. [46]

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 1.380649×10−23 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.

See also


In the following notes, wherever numeric equalities are shown in concise form, such as 1.85487(14)×1043, the two digits between the parentheses denotes the uncertainty at 1-σ (1 standard deviation, 68% confidence level) in the two least significant digits of the significand.
  1. CODATA Value: Boltzmann constant. The NIST Reference on Constants, Units, and Uncertainty. National Institute of Standards and Technology.
  2. Georgia State University, HyperPhysics Project, “Equipartition of Energy"
  3. Rankine, W. J. M., "A manual of the steam engine and other prime movers", Richard Griffin and Co., London (1859), p. 306–307.
  4. William Thomson, 1st Baron Kelvin, "Heat", Adam and Charles Black, Edinburgh (1880), p. 39.
  5. Absolute zero's relationship to 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 Online 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 Solid Helium Archived 2008-02-12 at the Wayback Machine 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 (T = 0) 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 T = 0 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 = 1×10−24 K). 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 T = 0 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 T = 0, 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 T = 0. 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 T = 0 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 T = 0, heat will flow inward, and if the surrounding environment is at 'T = 0, there will be an equal flux of ZP radiation both inward and outward. A similar Q equilibrium exists at T = 0 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.
  6. "SI brochure, section". International Bureau of Weights and Measures. Archived from the original on 26 September 2007. Retrieved 9 May 2008.
  7. Newell, D B; Cabiati, F; Fischer, J; Fujii, K; Karshenboim, S G; Margolis, H S; de Mirandés, E; Mohr, P J; Nez, F; Pachucki, K; Quinn, T J; Taylor, B N; Wang, M; Wood, B M; Zhang, Z; et al. (Committee on Data for Science and Technology (CODATA) Task Group on Fundamental Constants) (29 January 2018). "The CODATA 2017 values of h, e, k, and NA for the revision of the SI". Metrologia. 55 (1): L13–L16. Bibcode:2018Metro..55L..13N. doi:10.1088/1681-7575/aa950a.
  8. "SI Redefinition – Kelvin: Boltzmann Constant". National Institute of Standards and Technology. Archived from the original on 1 July 2020. Retrieved 13 Dec 2020.
  9. "Acoustic Thermometry". National Institute of Standards and Technology. Archived from the original on 23 September 2020. Retrieved 13 Dec 2020.
  10. At non-relativistic temperatures of less than about 30 GK, classical mechanics are sufficient to calculate the velocity of particles. At 30 GK, individual neutrons (the constituent of neutron stars and one of the few materials in the universe with temperatures in this range) have a 1.0042 γ (gamma or Lorentz factor). Thus, the classic Newtonian formula for kinetic energy is in error less than half a percent for temperatures less than 30 GK.
  11. Even room–temperature air has an average molecular translational speed (not vector-isolated velocity) of 1822 km/hour. This is relatively fast for something the size of a molecule considering there are roughly 2.42×1016 of them crowded into a single cubic millimeter. Assumptions: Average molecular weight of wet air = 28.838 g/mol and T = 296.15 K. Assumption's primary variables: An altitude of 194 meters above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C (40.85% relative humidity), and 760 mmHg (101.325 kPa) sea level–corrected barometric pressure.
  12. Citation: Adiabatic Cooling of Cesium to 700 nK in an Optical Lattice, A. Kastberg et al., Physical Review Letters 74, No. 9, 27 Feb. 1995, Pg. 1542. It's noteworthy that a record cold temperature of 450 pK in a Bose–Einstein condensate of sodium atoms (achieved by A. E. Leanhardt et al.. of MIT) equates to an average vector-isolated atom velocity of 0.4 mm/s and an average atom speed of 0.7 mm/s.
  13. The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature as follows:
    • is the vector-isolated mean velocity of translational particle motion in m/s
    • kB (Boltzmann constant) = 1.380649×10−23 J/K
    • T is the thermodynamic temperature in kelvins
    • m is the molecular mass of substance in kg/particle
    In the above formula, molecular mass, m, in kg/particle is the quotient of a substance's molar mass (also known as atomic weight, atomic mass, relative atomic mass, and unified atomic mass units) in g/mol or daltons divided by 6.02214076×1026 (which is the Avogadro constant times one thousand). For diatomic molecules such as H2, N2, and O2, multiply atomic weight by two before plugging it into the above formula. The mean speed (not vector-isolated velocity) of an atom or molecule along any arbitrary path is calculated as follows:
    where is the mean speed of translational particle motion in m/s. Note that the mean energy of the translational motions of a substance's constituent particles correlates to their mean speed, not velocity. Thus, substituting for v in the classic formula for kinetic energy, Ek = 12mv2 produces precisely the same value as does Emean = 3/2kBT (as shown in the section titled The nature of kinetic energy, translational motion, and temperature).  Note too that the Boltzmann constant and its related formulas establish that absolute zero is the point of both zero kinetic energy of particle motion and zero kinetic velocity (see also Note 1 above).
  14. One-trillionth of a kelvin is to one kelvin as two sheets of kitchen aluminum foil (0.04 mm) are to the distance around Earth at the equator.
  15. The internal degrees of freedom of molecules cause their external surfaces to vibrate and can also produce overall spinning motions (what can be likened to the jiggling and spinning of an otherwise stationary water balloon). If one examines a single molecule as it impacts a containers’ wall, some of the kinetic energy borne in the molecule's internal degrees of freedom can constructively add to its translational motion during the instant of the collision and extra kinetic energy will be transferred into the container's wall. This would induce an extra, localized, impulse-like contribution to the average pressure on the container. However, since the internal motions of molecules are random, they have an equal probability of destructively interfering with translational motion during a collision with a container's walls or another molecule. Averaged across any bulk quantity of a gas, the internal thermal motions of molecules have zero net effect upon the temperature, pressure, or volume of a gas. Molecules’ internal degrees of freedom simply provide additional locations where kinetic energy is stored. This is precisely why molecular-based gases have greater specific internal capacity than monatomic gases (where additional internal energy must be added to achieve a given temperature rise).
  16. When measured at constant-volume since different amounts of work must be performed if measured at constant-pressure. Nitrogen's CvH (100 kPa, 20 °C) equals 20.8 J⋅mol–1⋅K–1 vs. the monatomic gases, which equal 12.4717 J mol–1 K–1. Citations: W.H. Freeman's Physical Chemistry, Part 3: Change (422 kB PDF, here), Exercise 21.20b, Pg. 787. Also Georgia State University's Molar Specific Heats of Gases.
  17. The speed at which thermal energy equalizes throughout the volume of a gas is very rapid. However, since gases have extremely low density relative to solids, the heat flux (the thermal power passing per area) through gases is comparatively low. This is why the dead-air spaces in multi-pane windows have insulating qualities.
  18. Diamond is a notable exception. Highly quantized modes of phonon vibration occur in its rigid crystal lattice. Therefore, not only does diamond have exceptionally poor specific heat capacity, it also has exceptionally high thermal conductivity.
  19. Correlation is 752 (W⋅m−1⋅K−1)/(MS⋅cm), σ = 81, through a 7:1 range in conductivity. Value and standard deviation based on data for Ag, Cu, Au, Al, Ca, Be, Mg, Rh, Ir, Zn, Co, Ni, Os, Fe, Pa, Pt, and Sn. Citation: Data from CRC Handbook of Chemistry and Physics, 1st Student Edition and this link to Web Elements' home page.
  20. The cited emission wavelengths are for true black bodies in equilibrium. In this table, only the sun so qualifies. CODATA recommended value of 2.897771955...×10−3 m⋅K used for Wien displacement law constant b.
  21. A record cold temperature of 450 ±80 pK in a Bose–Einstein condensate (BEC) of sodium (23Na) 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, Pg. 1515. The thermal velocity of the atoms averaged about 0.4 mm/s. It is noteworthy that this record's peak emittance black-body wavelength of 6,400 kilometers is roughly the radius of Earth.
  22. The peak emittance wavelength of 2.897 77 m is a frequency of 103.456 MHz
  23. Measurement was made in 2002 and has an uncertainty of ±3 kelvins. A 1989 measurement produced a value of 5777 ±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). Download paper (252 kB PDF Archived 2014-08-23 at the Wayback Machine)
  24. 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.
  25. 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). Citation: Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2 × 109 Kelvin, M. G. Haines et al., Physical Review Letters 96, Issue 7, id. 075003. Link to Sandia's news release. Archived 2006-07-02 at the Wayback Machine
  26. 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 Archived 2011-08-14 at the Wayback Machine, and a concise treatise on stars by NASA is here. Archived July 20, 2015, at the Wayback Machine
  27. 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 is 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).  Citation: Oechslin, R.; Janka, H.- T. (2006). "Torus formation in neutron star mergers and well-localized short gamma-ray bursts". Monthly Notices of the Royal Astronomical Society. 368 (4): 1489–1499. arXiv:astro-ph/0507099v2. Bibcode:2006MNRAS.368.1489O. doi:10.1111/j.1365-2966.2006.10238.x. S2CID 15036056. To view a browser-based summary of the research, click here.
  28. NewScientist: Eight extremes: The hottest thing in the universe, 07 March 2011, which stated "While the details of this process are currently unknown, it must involve a fireball of relativistic particles heated to something in the region of a trillion kelvin."
  29. Citation: How do physicists study particles? Archived 2007-10-11 at the Wayback Machine by CERN.
  30. Water's enthalpy of fusion (0 °C, 101.325 kPa) equates to 0.062284 eV per molecule so adding one joule of thermal energy to 0 °C water ice causes 1.0021×1020 water molecules to break away from the crystal lattice and become liquid.
  31. Water's enthalpy of fusion is 6.0095 kJ⋅mol−1 K−1 (0 °C, 101.325 kPa). Citation: Water Structure and Science, Water Properties, Enthalpy of fusion, (0 °C, 101.325 kPa) (by London South Bank University). Link to Web site. The only metals with enthalpies of fusion not in the range of 6–30 J mol−1 K−1 are (on the high side): Ta, W, and Re; and (on the low side) most of the group 1 (alkaline) metals plus Ga, In, Hg, Tl, Pb, and Np. Citation: This link to Web Elements' home page.
  32. Xenon value citation: This link to WebElements' xenon data (available values range from 2.3 to 3.1 kJ/mol). It is also noteworthy that helium's heat of fusion of only 0.021 kJ/mol is so weak of a bonding force that zero-point energy prevents helium from freezing unless it is under a pressure of at least 25 atmospheres.
  33. CRC Handbook of Chemistry and Physics, 1st Student Edition and Web Elements.
  34. H2Ospecific heat capacity, Cp = 0.075327 kJ⋅mol−1⋅K−1 (25 °C); Enthalpy of fusion = 6.0095 kJ/mol (0 °C, 101.325 kPa); Enthalpy of vaporization (liquid) = 40.657 kJ/mol (100 °C). Citation: Water Structure and Science, Water Properties (by London South Bank University). Link to Web site.
  35. Mobile conduction electrons are delocalized, i.e. not tied to a specific atom, and behave rather like a sort of quantum gas due to the effects of zero-point energy. Consequently, even at absolute zero, conduction electrons still move between atoms at the Fermi velocity of about 1.6×106 m/s. Kinetic thermal energy adds to this speed and also causes delocalized electrons to travel farther away from the nuclei.
  36. No other crystal structure can exceed the 74.048% packing density of a closest-packed arrangement.  The two regular crystal lattices found in nature that have this density are hexagonal close packed (HCP) and face-centered cubic (FCC). These regular lattices are at the lowest possible energy state. Diamond is a closest-packed structure with an FCC crystal lattice. Note too that suitable crystalline chemical compounds, although usually composed of atoms of different sizes, can be considered as closest-packed structures when considered at the molecular level. One such compound is the common mineral known as magnesium aluminum spinel (MgAl2O4). It has a face-centered cubic crystal lattice and no change in pressure can produce a lattice with a lower energy state.
  37. Nearly half of the 92 naturally occurring chemical elements that can freeze under a vacuum also have a closest-packed crystal lattice. This set includes beryllium, osmium, neon, and iridium (but excludes helium), and therefore have zero latent heat of phase transitions to contribute to internal energy (symbol: U). In the calculation of enthalpy (formula: H = U + pV), internal energy may exclude different sources of thermal energy (particularly ZPE) depending on the nature of the analysis. Accordingly, all T = 0 closest-packed matter under a perfect vacuum has either minimal or zero enthalpy, depending on the nature of the analysis. Use Of Legendre Transforms In Chemical Thermodynamics, Robert A. Alberty, Pure Appl.Chem., 73 (2001) 1349.
  38. Regarding the spelling "gage" vs. "gauge" in the context of pressures measured relative to atmospheric pressure, the preferred spelling varies by country and even by industry. Further, both spellings are often used within a particular industry or country. Industries in British English-speaking countries typically use the spelling "gauge pressure" to distinguish it from the pressure-measuring instrument, which in the U.K., is spelled pressure gage. For the same reason, many of the largest American manufacturers of pressure transducers and instrumentation use the spelling gage pressure (the convention used here) in their formal documentation to distinguish it from the instrument, which is spelled pressure gauge. (see Honeywell-Sensotec's FAQ page and Fluke Corporation's product search page).
  39. Pressure also must be in absolute terms. The air still in a tire at a gage pressure of 0 kPa expands too as it gets hotter. It's not uncommon for engineers to overlook that one must work in terms of absolute pressure when compensating for temperature. For instance, a dominant manufacturer of aircraft tires published a document on temperature-compensating tire pressure, which used gage pressure in the formula. However, the high gage pressures involved (180 psi; 12.4 bar; 1.24 MPa) means the error would be quite small. With low-pressure automobile tires, where gage pressures are typically around 2 bar (200 kPa), failing to adjust to absolute pressure results in a significant error. Referenced document: Aircraft Tire Ratings (155 kB PDF, here).
  40. A difference of 100 kPa is used here instead of the 101.325 kPa value of one standard atmosphere. In 1982, the International Union of Pure and Applied Chemistry (IUPAC) recommended that for the purposes of specifying the physical properties of substances, the standard pressure (atmospheric pressure) should be defined as precisely 100 kPa (≈ 750.062 Torr). Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100 kPa equates to the mean pressure at an altitude of about 112 meters, which is closer to the 194–meter, worldwide median altitude of human habitation. For especially low-pressure or high-accuracy work, true atmospheric pressure must be measured. Citation:, Gold Book, Standard Pressure
  41. Planck, M. (1945). Treatise on Thermodynamics. Dover Publications. p. §90 & §137. eqs.(39), (40), & (65).
  42. Here, need to add a reason of requiring the function g(T) to be a monotonic function. The Carnot efficiency (efficiency of all reversible engines) may be a reason.
  43. Fermi, E. (1956). Thermodynamics. Dover Publications (still in print). p. 48. eq.(64).
  44. A Brief History of Temperature Measurement and; Uppsala University (Sweden), Linnaeus' thermometer
  46. According to The Oxford English Dictionary (OED), the term "Celsius's thermometer" had been used at least as early as 1797. Further, the term "The Celsius or Centigrade thermometer" was again used in reference to a particular type of thermometer at least as early as 1850. The OED also cites this 1928 reporting of a temperature: "My altitude was about 5,800 metres, the temperature was 28° Celsius". However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards the terminology used throughout the history of science. According to several writings of Dr. Terry Quinn CBE FRS, Director of the BIPM (1988–2004), including Temperature Scales from the early days of thermometry to the 21st century (150 kB PDF, here) as well as Temperature (2nd Edition / 1990 / Academic Press / 0125696817), the term Celsius in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM wasn't even aware that degree Celsius was in sporadic, non-scientific use before that time. It's also noteworthy that the twelve-volume, 1933 edition of OED did not even have a listing for the word Celsius (but did have listings for both centigrade and centesimal in the context of temperature measurement). The 1948 adoption of Celsius accomplished three objectives:
    1. All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.
    2. Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius's name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name centigrade could continue to be used and would simultaneously inherit an intuitive association with the new name.
    3. The new name eliminated the ambiguity of the term centigrade, freeing it to refer exclusively to the French-language name for the unit of angular measurement.
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