Editing Thermodynamic temperature (section)

=== Nature of kinetic energy, translational motion, and temperature ===
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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.