Editing Thermodynamic temperature (section)

== Relationship of temperature, motions, conduction, and thermal energy ==

[[Image:Translational motion.gif|thumb|upright=1.4|'''Figure 1''' The ''translational motion'' of fundamental particles of nature such as atoms and molecules is directly related to temperature. Here, the size of [[helium]] atoms relative to their spacing is shown to scale under 1950 [[atmosphere (unit)|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 ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->

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 (physics and chemistry)|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 {{math|1=''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 [[#Internal motions of molecules and internal energy|''Fig.&nbsp;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 {{math|1=''E''<sub>k</sub>&nbsp;=&nbsp;{{sfrac|2}}''mv''{{i sup|2}}}}.<ref>At non-[[Special relativity|relativistic]] temperatures of less than about 30&nbsp;GK, [[classical mechanics]] are sufficient to calculate the velocity of particles. At 30&nbsp;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&nbsp;GK.</ref> 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:&nbsp;{{math|''k''<sub>B</sub>}}). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an ''individual'' particles' translational motion as follows:
<math display="block">\tilde{E} = \frac{3}{2} k_\text{B} T</math> 
where:
* <math display="inline"> \tilde{E}</math> is the mean kinetic energy for an individual particle
* {{math|1=''k''<sub>B</sub> = {{val|1.380649|e=-23|u=J/K}}}}
* {{mvar|T}} is the thermodynamic temperature of the bulk quantity of the substance

[[Image:Maxwell Dist-Inverse Speed.png|thumb|upright=1.4|'''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 ''[[#Diffusion of thermal energy: black-body radiation|Fig.&nbsp;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 ''[[#Nature of kinetic energy, translational motion, and temperature|Fig.&nbsp;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.&nbsp;2'' shows the speed distribution of 5500&nbsp;K helium atoms. They have a ''most probable'' speed of 4.780&nbsp;km/s (0.2092&nbsp;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 ''[[#Diffusion of thermal energy: black-body radiation|Fig.&nbsp;5]]'' below. In both graphs, zero on the ''x''-axis represents infinite temperature. Additionally, the ''x''- and ''y''-axes 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<ref>Even room–temperature air has an average molecular translational ''speed'' (not vector-isolated velocity) of 1822&nbsp;km/hour. This is relatively fast for something the size of a molecule considering there are roughly {{val|2.42|e=16}} of them crowded into a single cubic millimeter. Assumptions: Average molecular weight of wet air = 28.838 g/mol and {{mvar|T}} = 296.15&nbsp;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&nbsp;°C, a dew point of 9&nbsp;°C (40.85% relative humidity), and {{cvt|lk=in|760|mmHg|kPa}} sea level–corrected barometric pressure.</ref> and temperatures close to [[absolute zero]] are required to directly observe them. For instance, when scientists at the [[National Institute of Standards and Technology|NIST]] achieved a record-setting cold temperature of 700&nbsp;nK (billionths of a kelvin) in 1994, they used [[optical lattice]] laser equipment to [[Adiabatic process|adiabatically]] cool [[caesium|cesium]] atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7&nbsp;mm per second to in order to calculate their temperature.<ref>{{cite journal |title=Adiabatic Cooling of Cesium to 700&nbsp;nK in an Optical Lattice |first=A. |last=Kastberg |display-authors=etal|journal=Physical Review Letters |volume=74 |issue=9 |date=27 February 1995 |pages=1542–1545 |doi=10.1103/PhysRevLett.74.1542 |pmid=10059055 |bibcode=1995PhRvL..74.1542K }} A record cold temperature of 450&nbsp;[[Kelvin#SI prefixes|pK]] in a Bose–Einstein condensate of sodium atoms (achieved by A. E. Leanhardt ''et al.''. of [[Massachusetts Institute of Technology|MIT]]){{cn|{{subst:DATE}} equates to an average vector-isolated atom velocity of 0.4&nbsp;mm/s and an average atom speed of 0.7&nbsp;mm/s.</ref> Formulas for calculating the velocity and speed of translational motion are given in the following footnote.<ref name="Boltzmann">The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature as follows:
<math display="block">\tilde{v} = \sqrt{\frac {\frac{k_\text{B}}{2} \cdot T}{\frac{m}{2}}}</math>
where
* <math display="inline">\tilde{v}</math> is the vector-isolated mean velocity of translational particle motion in m/s
* {{math|''k''<sub>B</sub>}} ([[Boltzmann constant]]) = {{val|1.380649|e=-23|u=J/K}}
* {{mvar|T}} is the thermodynamic temperature in kelvins
* {{mvar|m}} is the molecular mass of substance in kg/particle

In the above formula, molecular mass, {{mvar|m}}, in kg/particle is the quotient of a substance's [[molar mass]] (also known as ''atomic weight'', ''[[atomic mass]]'', ''relative atomic mass'', and ''[[Atomic mass unit|unified atomic mass units]]'') in [[Gram|g]]/[[Mole (unit)|mol]] or [[Atomic mass unit|daltons]] divided by {{val|6.02214076|e=26}} (which is the [[Avogadro constant]] times one thousand). For [[diatomic]] molecules such as [[hydrogen|H<sub>2</sub>]], [[nitrogen|N<sub>2</sub>]], and [[oxygen|O<sub>2</sub>]], 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:
<math display="block">\tilde{s} = \tilde{v} \cdot \sqrt{3}</math>
where <math display="inline">\tilde{s}</math> is the mean speed of translational particle motion in m/s.

The mean energy of the translational motions of a substance's constituent particles correlates to their mean ''speed'', not velocity. Thus, substituting <math display="inline">\tilde{s}</math> for {{mvar|v}} in the classic formula for kinetic energy, {{math|1=''E''<sub>k</sub> = {{sfrac|2}}''mv''{{i sup|2}}}} produces precisely the same value as does {{math|1=''E''<sub>mean</sub> = 3/2''k''<sub>B</sub>''T''}} (as shown in {{section link|#Nature of kinetic energy, translational motion, and temperature}}). 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 ''[[#Notes|Note 1]]'' above).</ref>

[[File:Argon atom at 1E-12 K.gif|thumb|left|upright=1.4|'''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&nbsp;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&nbsp;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.&nbsp;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&nbsp;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,<ref>One-trillionth of a kelvin is to one kelvin as the thickness of two sheets of kitchen aluminum foil (0.04&nbsp;mm) is to the distance around Earth at the equator.</ref> 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&nbsp;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 ====
[[Image:Thermally Agitated Molecule.gif|thumb|upright=1.1|'''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, [[molecule]]s are complex objects; they are a population of atoms and thermal agitation can strain their internal [[chemical bond]]s 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 gas]]es [[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 (chemistry)|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.<ref>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).</ref> 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 (unit)|mole]] (a specific number of molecules) as do the monatomic gases.<ref>When measured at constant-volume since different amounts of work must be performed if measured at constant-pressure. Nitrogen's {{math|''C<sub>v</sub>H''}} (100&nbsp;kPa, 20&nbsp;°C) equals {{val|20.8|u=J⋅mol<sup>–1</sup>⋅K<sup>–1</sup>}} vs. the monatomic gases, which equal 12.4717&nbsp;J&nbsp;mol<sup>–1</sup>&nbsp;K<sup>–1</sup>. {{cite book |first=W. H. |last=Freeman |title=Physical Chemistry |url=http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf |chapter=Part 3: Change|archive-url=https://wayback.archive-it.org/all/20070927061428/http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf |archive-date=2007-09-27 |at=Exercise 21.20b, p.&nbsp;787}} See also {{cite web |first=R. |last=Nave |publisher=Georgia State University |url=http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html |title=Molar Specific Heats of Gases |website=HyperPhysics}}</ref> Another example is [[gasoline]] (see [[Specific heat capacity#Table of specific heat capacities|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 ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->

[[Image:1D normal modes (280 kB).gif|thumb|upright=1.2|'''Figure 4''' The temperature-induced translational motion of particles in solids takes the form of ''[[phonon]]s''. Shown here are phonons with identical [[amplitude]]s but with [[wavelength]]s 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 ''[[#Relationship of temperature, motions, conduction, and thermal energy|Fig.&nbsp;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 (heat transfer)|convection]] speeds this process even more.<ref>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 [[insulated glazing|multi-pane windows]] have insulating qualities.</ref>

Translational motion in ''solids'', however, takes the form of ''[[phonon]]s'' (see ''Fig.&nbsp;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<ref>[[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]].</ref> 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.

[[Metal]]s 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 [[electron]]s''. This is why there is a near-perfect correlation between metals' [[thermal conductivity]] and their [[electrical conductivity]].<ref>Correlation is 752&nbsp;(W⋅m<sup>−1</sup>⋅K<sup>−1</sup>)/(MS⋅cm), {{mvar|σ}}&nbsp;=&nbsp;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. Data from ''CRC Handbook of Chemistry and Physics'', 1st Student Edition.</ref> Conduction electrons imbue metals with their extraordinary conductivity because they are ''[[Delocalized electron|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 ''[[#Notes|Note 1]]'' below). Furthermore, electrons are relatively light with a rest mass only {{frac|1836}} that of a [[proton]]. This is about the same ratio as a [[.22 Short]] bullet (29 [[grain (measure)|grains]] or 1.88&nbsp;[[gram|g]]) compared to the rifle that shoots it. As [[Isaac Newton]] wrote with his [[Newton's laws of motion#Newton's third law|third law of motion]],

{{Quote|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 are delocalized and ''very'' fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.

=== Diffusion of thermal energy: black-body radiation ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->

[[Image:Wiens law.svg|thumb|upright=1.4|'''Figure 5''' The spectrum of black-body radiation has the form of a Planck curve. A 5500&nbsp;K black-body has a peak emittance wavelength of 527&nbsp;nm. Compare the shape of this curve to that of a Maxwell distribution in ''[[#Nature of kinetic energy, translational motion, and temperature|Fig.&nbsp;2]]''&nbsp;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 [[photon]]s (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 [[wavelength]]s in a spectrum that has a bell curve-like shape called a [[Planck's law of black body radiation|Planck curve]] (see graph in ''Fig.&nbsp;5'' at right). The top of a Planck curve ([[Wien's displacement law|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 [[cryogenics|cryogenic]] temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short [[gamma ray]]s (see {{section link|#Table of thermodynamic 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&nbsp;K (just short of glowing dull red) emits 60 times the radiant [[Power (physics)|power]] as it does at 296&nbsp;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 light bulb|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.

{| class="wikitable" style="text-align:center"
|-
!
!Kelvin
!Peak emittance<br />[[wavelength]]<ref>The cited emission wavelengths are for true black bodies in equilibrium. In this table, only the sun so qualifies. [https://physics.nist.gov/cgi-bin/cuu/Value?bwien CODATA recommended value] of {{val|2.897771955|end=...|e=-3|u=m⋅K}} used for Wien displacement law constant ''b''.</ref> of<br />[[Wien's displacement law|black-body photons]]
|-
|style="text-align:right"|[[Absolute zero]]<br />(precisely by definition)
|0 K
|{{resize|140%|[[Infinity|∞]]}}<ref name="T0"/>
|-
|style="text-align:right"|Coldest measured<br />temperature<ref name="recordcold">A record cold temperature of 450&nbsp;±80&nbsp;pK in a Bose–Einstein condensate (BEC) of sodium (<sup>23</sup>Na) atoms was achieved in 2003 by researchers at [[Massachusetts Institute of Technology|MIT]]. {{cite journal |title=Cooling Bose–Einstein Condensates Below 500 Picokelvin |first=A. E. |last=Leanhardt |display-authors=etal |journal=Science |volume=301 |issue=5639 |date=12 September 2003 |page=1515|doi=10.1126/science.1088827 |pmid=12970559 |bibcode=2003Sci...301.1513L }} The thermal velocity of the atoms averaged about 0.4&nbsp;mm/s. This record's peak emittance black-body radiation wavelength of 6,400 kilometers is roughly the radius of Earth.</ref>
|450 [[Orders of magnitude (temperature)#SI multiples|pK]]
|6,400 [[Kilometre|km]]
|-
|style="text-align:right"|One [[Orders of magnitude (temperature)#SI multiples|millikelvin]]<br />(precisely by definition)
|0.001 K
|2.897&nbsp;77 [[Metre|m]]<br /> (radio, [[FM broadcasting|FM band]])<ref>The peak emittance wavelength of 2.897&nbsp;77&nbsp;m is a frequency of 103.456&nbsp;MHz.</ref>
|-
|style="text-align:right"|[[Cosmic microwave background|cosmic microwave<br />background radiation]]
|2.725&nbsp;K
|1.063&nbsp;[[Metre|mm]] (peak wavelength)
|-
|style="text-align:right"|[[Vienna Standard Mean Ocean Water|Water]]'s [[triple point]]
|273.16 K
|10.6083 [[Metre#SI prefixed forms of metre|μm]]<br />(long wavelength [[Infrared|I.R.]])
|-
|style="text-align:right"|[[ISO 1]] standard temperature<br />for precision [[metrology]]<br />(precisely 20&nbsp;°C by definition)
|293.15 K
|{{val|9.88495}}&nbsp;μm<br />(long wavelength [[Infrared|I.R.]])
|-
|-
|style="text-align:right"|[[Incandescent light bulb|Incandescent lamp]]{{efn-ua|For a true black body (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.}}
|2500 K{{efn-ua|The 2500&nbsp;K value is approximate.}}
|1.16&nbsp;μm<br />(near [[infrared]]){{efn-ua|name=Photosphere|Effective photosphere temperature.}}<!--Should this be here?-->
|-
|[[Sun]]'s visible surface<ref>{{Cite web |year=2015 |title=Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties |url=https://iau.org/static/resolutions/IAU2015_English.pdf}}</ref><ref>{{Cite book |last1=Hertel |first1=Ingolf V. |url=https://books.google.com/books?id=vr0UBQAAQBAJ&dq=5772+K+sun&pg=PA35 |title=Atoms, Molecules and Optical Physics 1: Atoms and Spectroscopy |last2=Schulz |first2=Claus-Peter |date=2014-10-24 |publisher=Springer |isbn=978-3-642-54322-7 |pages=35 |language=en}}</ref><ref>{{Cite book |last1=Vignola |first1=Frank |url=https://books.google.com/books?id=q9WlDwAAQBAJ&dq=5772+K+sun&pg=PP26 |title=Solar and Infrared Radiation Measurements |edition=2nd |last2=Michalsky |first2=Joseph |last3=Stoffel |first3=Thomas |date=2019-07-30 |publisher=CRC Press |isbn=978-1-351-60020-0 |pages=chapter 2.1, 2.2 |language=en}}</ref><ref>{{Cite web |title=Sun Fact Sheet |url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html |access-date=2023-08-27 |website=NASA Space Science Center Coordinated Archive}}</ref>
|5772 K
|502&nbsp;[[Metre#SI prefixed forms of metre|nm]]<br />([[Color#Spectral colors|green light]])
|-
|style="text-align:right"|[[Lightning|Lightning bolt's]]<br />channel
|28,000 K
|100&nbsp;nm<br />(far [[ultraviolet]] light)
|-
|style="text-align:right"|[[Sun#Core|Sun's core]]
|16 [[Orders of magnitude (temperature)#SI multiples|MK]]
|0.18&nbsp;nm ([[X-ray]]s)
|-
|style="text-align:right"|[[Thermonuclear explosion]]<br />(peak temperature)<ref>The 350&nbsp;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&nbsp;MK. {{cite web |title=Nuclear Weapons Frequently Asked Questions |at=3.2.5 Matter At High Temperatures |url=http://nuclearweaponarchive.org/Nwfaq/Nfaq3.html#nfaq3.2}}{{fcn|{{subst:DATE}}|date=September 2024}} All referenced data was compiled from publicly available sources.</ref>
|align="center"|350 MK
|align="center"|8.3 × 10<sup>−3</sup> nm<br />([[gamma ray]]s)
|-
|style="text-align:right"|Sandia National Labs'<br />[[Z Pulsed Power Facility|Z machine]]{{efn-ua|For a true black body (which the plasma was not). The Z machine's dominant emission originated from 40&nbsp;MK electrons (soft x–ray emissions) within the plasma.}}<ref>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&nbsp;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&nbsp;GK (309 ±35&nbsp;keV) for 3&nbsp;ns (ns 112 through 115). {{cite journal |title=Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2&nbsp;×&nbsp;10<sup>9</sup> Kelvin |first=M. G. |last=Haines |display-authors=etal |journal=Physical Review Letters |volume=96 |issue=7 |id=No. 075003 |year=2006 |page=075003 |doi=10.1103/PhysRevLett.96.075003|pmid=16606100 |bibcode=2006PhRvL..96g5003H }} For a press summary of this article, see {{cite web |date=March 8, 2006 |title=Sandia's Z machine exceeds two billion degrees Kelvin |publisher=Sandia |url=http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html |archive-url=https://web.archive.org/web/20060702185740/http://www.sandia.gov/news-center/news-releases/2006/physics-astron/hottest-z-output.html |archive-date=2006-07-02 }}</ref>
|2 [[Orders of magnitude (temperature)#SI multiples|GK]]
|1.4 × 10<sup>−3</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|Core of a [[Silicon-burning process|high-mass star on its last day]]<ref>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 reactions|alpha process]] (which lasts one day) of [[Silicon burning process|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&nbsp;II [[supernova]].</ref>
|align="center"|3 GK
|align="center"|1 × 10<sup>−3</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|Merging binary [[neutron star]] system<ref>Based on a computer model that predicted a peak internal temperature of 30 MeV (350&nbsp;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&nbsp;km in diameter, and were orbiting around their barycenter (common center of mass) at about 390&nbsp;Hz during the last several milliseconds before they completely merged. The 350&nbsp;GK portion was a small volume located at the pair's developing common core and varied from roughly 1 to 7&nbsp;km across over a time span of around 5&nbsp;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). At 350&nbsp;GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass 5% greater than its rest mass. {{cite journal|arxiv=astro-ph/0507099v2|doi=10.1111/j.1365-2966.2006.10238.x|title=Torus formation in neutron star mergers and well-localized short gamma-ray bursts|journal=Monthly Notices of the Royal Astronomical Society|volume=368|issue=4|pages=1489–1499|year=2006|last1=Oechslin|first1=R.|last2=Janka|first2=H.-T.|doi-access=free |bibcode=2006MNRAS.368.1489O|s2cid=15036056}} For a summary, see {{cite web|url=http://www.mpa-garching.mpg.de/mpa/research/current_research/hl2005-10/hl2005-10-en.html |title=Short Gamma-Ray Bursts: Death Throes of Merging Neutron Stars |publisher=Max-Planck-Institut für Astrophysik |access-date=24 September 2024}}</ref>
|350 GK
|8 × 10<sup>−6</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|[[Gamma-ray burst progenitors]]<ref>{{cite magazine |magazine=New Scientist |url=https://www.newscientist.com/article/mg20928026.300-eight-extremes-the-hottest-thing-in-the-universe.html |title=Eight extremes: The hottest thing in the universe |first=Stephen |last=Battersby |date=2 March 2011 |quote=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.}}</ref>
|1 [[Orders of magnitude (temperature)#SI multiples|TK]]
|3 × 10<sup>−6</sup> nm<br />(gamma rays)
|-
|style="text-align:right"|[[CERN]]'s proton vs. nucleus collisions<ref>{{cite web |url=http://public.web.cern.ch/public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html |url-status=dead |title=How do physicists study particles? |archive-url=https://web.archive.org/web/20071011103924/http://public.web.cern.ch/Public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html |archive-date=2007-10-11 |publisher=CERN}}</ref>
|10 TK
|3 × 10<sup>−7</sup> nm<br />(gamma rays)
|-
|}
{{notelist-ua}}

=== Heat of phase changes ===
[[Image:IceBlockNearJoekullsarlon.jpg|thumb|left|upright=1.4|'''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 transition]]s'', which are the [[potential energy]] of molecular bonds that can form in a substance as it cools (such as during [[condensation|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&nbsp;°C can cause severe burns much faster than the 100&nbsp;°C air from a [[blowdryer|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 element]]s, [[chemical compound|compounds]], and [[eutectic point|eutectic]] [[alloy]]s exhibit no temperature change whatsoever while they undergo them (see ''Fig.&nbsp;7'', below right). Consider one particular type of phase transition: melting. When a solid is melting, [[Crystal structure|crystal lattice]] [[chemical bond]]s are being broken apart; the substance is transitioning from what is known as a ''more ordered state'' to a ''less ordered state''. In ''Fig.&nbsp;7'', the melting of ice is shown within the lower left box heading from blue to green.

[[File:Energy through phase changes.png|thumb|upright=1.8|'''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&nbsp;kJ.]]

At one specific thermodynamic point, the [[melting point]] (which is 0&nbsp;°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,<ref>Water's enthalpy of fusion (0&nbsp;°C, 101.325&nbsp;kPa) equates to {{val|0.062284|u=eV}}&nbsp;per molecule so adding one joule of thermal energy to 0&nbsp;°C water ice causes {{val|1.0021|e=20}} water molecules to break away from the crystal lattice and become liquid.</ref> 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 cannot 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 is called ''[[Standard enthalpy change of fusion|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&nbsp;kJ per mole for water and most of the metallic elements.<ref>Water's enthalpy of fusion is {{val|6.0095|u=kJ⋅mol<sup>−1</sup>}} K<sup>−1</sup> (0&nbsp;°C, 101.325&nbsp;kPa). {{cite web |website=Water Structure and Science |title=Water Properties (including isotopologues) |first=Martin |last=Chaplin |publisher=London South Bank University |url=http://www.lsbu.ac.uk/water/data.html |url-status=dead |archive-url=https://web.archive.org/web/20201121051504/http://www.lsbu.ac.uk/water/water_properties.html |archive-date=2020-11-21}} The only metals with enthalpies of fusion ''not'' in the range of 6–30&nbsp;J&nbsp;mol<sup>−1</sup>&nbsp;K<sup>−1</sup> 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.</ref> 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&nbsp;kJ per mole.<ref>For xenon,  available values range from 2.3 to 3.1 kJ/mol. {{cite web |url=http://www.webelements.com/webelements/elements/text/Xe/heat.html |website=WebElements |title=Xenon – 54Xe: the essentials |access-date=24 September 2024}} Helium's heat of fusion of only 0.021&nbsp;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.</ref> Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0&nbsp;°C into water at 0&nbsp;°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.<ref>''CRC Handbook of Chemistry and Physics'', 1st Student Edition{{fcn|{{subst:DATE}}|date=September 2024}}</ref> 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 ''[[standard enthalpy change of vaporization|enthalpy of vaporization]]'') is roughly 540 times that required for a one-degree increase.<ref>H<sub>2</sub>O specific heat capacity, {{math|''C<sub>p</sub>''}}&nbsp;= {{val|0.075327|u=kJ⋅mol<sup>−1</sup>⋅K<sup>−1</sup>}} (25&nbsp;°C); enthalpy of fusion&nbsp;= 6.0095&nbsp;kJ/mol (0&nbsp;°C, 101.325&nbsp;kPa); enthalpy of vaporization (liquid)&nbsp;= 40.657&nbsp;kJ/mol (100&nbsp;°C). {{cite web |website=Water Structure and Science |title=Water Properties (including isotopologues) |first=Martin |last=Chaplin |publisher=London South Bank University |url=http://www.lsbu.ac.uk/water/data.html |url-status=dead |archive-url=https://web.archive.org/web/20201121051504/http://www.lsbu.ac.uk/water/water_properties.html |archive-date=2020-11-21}}</ref>

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.&nbsp;7''&nbsp;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 pool]]s 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&nbsp;mm of water from a 1.29-meter-deep pool chills its water {{convert|8.4|C-change}}.

=== 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]]<ref name="T0"/> of a substance comprise the ''[[internal energy]]'' of it.

[[Image:Close-packed spheres, with umbrella light & camerea.jpg|thumb|left|266px|'''Figure 8''' When many of the chemical elements, such as the [[noble gas]]es and [[platinum group|platinum-group metals]], freeze to a solid — the most ordered state of matter — their [[crystal structures]] have a ''[[close-packing|close-packed arrangement]]''. This yields the greatest possible packing density and the lowest energy state.]]

=== Internal energy at absolute zero ===
<!-- NOTE TO EDITORS: This section is internally linked from elsewhere within the article. -->
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;<ref>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 {{val|1.6|e=6|u=m/s}}. Kinetic thermal energy adds to this speed and also causes delocalized electrons to travel farther away from the nuclei.</ref> 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 ({{mvar|T}}&nbsp;=&nbsp;0).

[[Image:Liquid helium superfluid phase.jpg|thumb|right|upright=1.1|'''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 will not freeze unless under 25 bar of pressure (c.&nbsp;25 atmospheres).]]
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, {{mvar|T}}&nbsp;=&nbsp;0 [[helium]] remains liquid at room pressure (''Fig.&nbsp;9'' at right) and must be under a pressure of at least {{convert|25|bar|MPa|abbr=on|lk=on}} to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21&nbsp;joules&nbsp;per&nbsp;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 {{mvar|T}}&nbsp;=&nbsp;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 ''[[close-packing|closest-packed arrangement]]'' (see ''Fig.&nbsp;8'', above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy.<ref name="T0"/><ref>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 crystal system|hexagonal close packed]]'' (HCP) and ''[[cubic crystal system|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]]'' (MgAl<sub>2</sub>O<sub>4</sub>). It has a face-centered cubic crystal lattice and no change in pressure can produce a lattice with a lower energy state.</ref> 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).<ref>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: {{math|''H''{{=}}''U''&nbsp;+&nbsp;''pV''}}), internal energy may exclude different sources of thermal energy (particularly ZPE) depending on the nature of the analysis. Accordingly, all {{mvar|T}}&nbsp;=&nbsp;0 closest-packed matter under a perfect vacuum has either minimal or zero enthalpy, depending on the nature of the analysis. {{cite journal |url=http://iupac.org/publications/pac/2001/pdf/7308x1349.pdf |title=Use of Legendre Transforms In Chemical Thermodynamics |first=Robert A. |last=Alberty |journal=Pure and Applied Chemistry |volume=73 |issue=8 |year=2001 |page=1349|doi=10.1351/pac200173081349 }}</ref> Lastly, all {{mvar|T}}&nbsp;=&nbsp;0 substances contain zero kinetic thermal energy.<ref name="T0"/><ref name="Boltzmann"/>
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