View source for Kinetic theory of gases

{{Short description|Understanding of gas properties in terms of molecular motion}}
[[Image:Translational motion.gif|thumb|upright=1.4|The [[temperature]] of the [[ideal gas]] is proportional to the average [[kinetic energy]] of its particles. The [[Bohr radius|size]] of [[helium]] atoms relative to their spacing is shown to scale under 1,950 [[Atmosphere (unit)|atmospheres]] of pressure. The atoms have an average speed relative to their size slowed down here two [[1000000000000 (number)|trillion]] fold from that at room temperature.]]

The '''kinetic theory of gases''' is a simple [[classical mechanics|classical]] model of the [[thermodynamics|thermodynamic]] behavior of [[gas]]es. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small to be seen with a microscope, in constant, random motion.<!-- not Brownian motion, so don't link to it --> These particles are now known to be the [[atom]]s or [[molecule]]s of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explain [[Ideal gas law#Combined gas law|the relationship]] between the [[macroscopic scale|macroscopic]] properties of gases, such as [[volume]], [[pressure]], and [[temperature]], as well as [[Transport phenomena|transport properties]] such as [[viscosity]], [[thermal conductivity]] and [[mass diffusivity]].

The basic version of the model describes an [[ideal gas]]. It treats the collisions as [[elastic collision|perfectly elastic]] and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.

Due to the [[time reversibility]] of microscopic dynamics ([[microscopic reversibility]]), the kinetic theory is also connected to the principle of [[detailed balance]], in terms of the [[fluctuation-dissipation theorem]] (for [[Brownian motion]]) and the [[Onsager reciprocal relations]].

The theory was historically significant as the first explicit exercise of the ideas of [[statistical mechanics]].

== History ==
{{See also|Heat#History|Atomism|History of thermodynamics}}

=== Kinetic theory of matter ===

==== Antiquity ====
In about 50 [[before common era|BCE]], the Roman philosopher [[Lucretius]] proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other.<ref>{{Cite journal|last1=Maxwell|first1=J. C.|year=1867|title=On the Dynamical Theory of Gases|journal=Philosophical Transactions of the Royal Society of London|volume=157|pages=49–88|doi=10.1098/rstl.1867.0004|s2cid=96568430}}</ref> This [[Epicureanism|Epicurean]] atomistic point of view was rarely considered in the subsequent centuries, when [[Aristotle]]an ideas were dominant.{{citation needed|date=December 2024}}

==== Modern era ====

===== "Heat is motion" =====
[[File:Portrait of Francis Bacon (cropped).jpg|thumb|196x196px|Francis Bacon]]
One of the first and boldest statements on the relationship between motion of particles and [[heat]] was by the English philosopher [[Francis Bacon]] in 1620. "It must not be thought that heat generates motion, or motion heat (though in some respects this be true), but that the very essence of heat ... is motion and nothing else."<ref>{{Cite book |last=Bacon |first=F. |author-link=Francis Bacon |url=https://www.gutenberg.org/cache/epub/45988/pg45988-images.html |title=Novum Organum: Or True Suggestions for the Interpretation of Nature |publisher=P. F. Collier & son |year=1902 |editor-last=Dewey |editor-first=J. |pages=153 |orig-date=1620}}</ref> "not a ... motion of the whole, but of the small particles of the body."<ref>{{Cite book |last=Bacon |first=F. |author-link=Francis Bacon |url=https://www.gutenberg.org/cache/epub/45988/pg45988-images.html |title=Novum Organum: Or True Suggestions for the Interpretation of Nature |publisher=P. F. Collier & son |year=1902 |editor-last=Dewey |editor-first=J. |pages=156 |orig-date=1620}}</ref> In 1623, in ''[[The Assayer]]'', [[Galileo Galilei]], in turn, argued that heat, pressure, smell and other phenomena perceived by our senses are apparent properties only, caused by the movement of particles, which is a real phenomenon.{{Sfn|Galilei|1957|p=273-4}}<ref>{{Citation |last=Adriaans |first=Pieter |title=Information |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |page=3.4 Physics |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/sum2024/entries/information/#Phys |edition=Summer 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>
[[File:John_Locke._Portrait_by_Herman_Verelst_(cropped).jpg|thumb|184x184px|John Locke]]
In 1665, in ''[[Micrographia]]'', the English polymath [[Robert Hooke]] repeated Bacon's assertion,<ref>{{Cite book |last=Hooke |first=Robert |url=https://ttp.royalsociety.org/ttp/ttp.html?id=a9c4863d-db77-42d1-b294-fe66c85958b3&type=book |title=Micrographia: Or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses with Observations and Inquiries Thereupon |publisher=Printed by Jo. Martyn, and Ja. Allestry, Printers to the Royal Society |year=1665 |pages=12 |postscript=. (Facsimile, with pagination)}}</ref><ref>{{Cite book |last=Hooke |first=Robert |url=https://www.gutenberg.org/files/15491/15491-h/15491-h.htm |title=Micrographia: Or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses with Observations and Inquiries Thereupon |publisher=Printed by Jo. Martyn, and Ja. Allestry, Printers to the Royal Society |year=1665 |pages=12 |postscript=. (Machine-readable, no pagination)}}</ref> and in 1675, his colleague, Anglo-Irish scientist [[Robert Boyle]] noted that a hammer's "impulse" is transformed into the motion of a nail's constituent particles, and that this type of motion is what heat consists of.<ref>{{Cite book |last=Boyle |first=Robert |url=https://archive.org/details/experimentsnotes00boyl/page/n100/mode/1up |title=Experiments, notes, &c., about the mechanical origine or production of divers particular qualities: Among which is inserted a discourse of the imperfection of the chymist's doctrine of qualities; together with some reflections upon the hypothesis of alcali and acidum |publisher=Printed by E. Flesher, for R. Davis |year=1675 |pages=61–62}}</ref> Boyle also believed that all macroscopic properties, including color, taste and elasticity, are caused by and ultimately consist of nothing but the arrangement and motion of indivisible particles of matter.<ref>{{Citation |last=Chalmers |first=Alan |title=Atomism from the 17th to the 20th Century |date=2019 |encyclopedia=The Stanford Encyclopedia of Philosophy |page=2.1 Atomism and the Mechanical Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/atomism-modern/#AtoMec |edition=Spring 2019 |publisher=Metaphysics Research Lab, Stanford University}}</ref> In a lecture of 1681, Hooke asserted a direct relationship between the temperature of an object and the speed of its internal particles. "Heat ... is nothing but the internal Motion of the Particles of [a] Body; and the hotter a Body is, the more violently are the Particles moved."<ref>{{Cite book |last=Hooke |first=Robert |url=https://archive.org/details/b30454621_0001/page/116/mode/1up |title=The posthumous works of Robert Hooke ... containing his Cutlerian lectures, and other discourses, read at the meetings of the illustrious Royal Society ... Illustrated with sculptures. To these discourses is prefixt the author's life, giving an account of his studies and employments, with an enumeration of the many experiments, instruments, contrivances and inventions, by him made and produc'd as Curator of Experiments to the Royal Society |publisher=Publish'd by Richard Waller. Printed by Sam. Smith and Benj. Walford, (Printers to the Royal Society) |year=1705 |pages=116 |language=en |orig-date=1681}}</ref> In a manuscript published 1720, the English philosopher [[John Locke]] made a very similar statement: "What in our sensation is ''heat'', in the object is nothing but ''motion''."<ref>{{Cite book |last=Locke |first=John |url=https://archive.org/details/acollectionseve01desmgoog/page/n291/mode/1up |title=A collection of several pieces of Mr. John Locke, never before printed, or not extant in his works. |publisher=Printed by J. Bettenham for R. Francklin |year=1720 |pages=224 |via=Internet Archive}}</ref><ref>{{Cite book |last=Locke |first=John |url=https://play.google.com/books/reader?id=QqxsP-VKrpkC&pg=GBS.PA224 |title=A collection of several pieces of Mr. John Locke, never before printed, or not extant in his works. |publisher=Printed by J. Bettenham for R. Francklin |year=1720 |pages=224 |via=Google Play Books}}</ref> Locke too talked about the motion of the internal particles of the object, which he referred to as its "insensible parts".[[File:Catherine II visiting Mikhail Lomonosov by Ivan Feodorov 1884.jpg|thumb|[[Catherine the Great]] visiting Mikhail Lomonosov]]
In his 1744 paper ''Meditations on the Cause of Heat and Cold'', Russian polymath [[Mikhail Lomonosov]] made a relatable appeal to everyday experience to gain acceptance of the microscopic and kinetic nature of matter and heat:<ref>{{Cite book |last=Lomonosov |first=Mikhail Vasil'evich |url=https://archive.org/details/mikhailvasilevic017733mbp |title=Mikhail Vasil'evich Lomonosov on the Corpuscular Theory |publisher=Harvard University Press |year=1970 |editor-last=Leicester |editor-first=Henry M. |pages=100 |translator-last= |translator-first= |chapter=Meditations on the Cause of Heat and Cold |orig-date=1750 |chapter-url=https://archive.org/details/mikhailvasilevic017733mbp/page/n115/mode/1up}}</ref>{{Blockquote|text=Movement should not be denied based on the fact it is not seen. Who would deny that the leaves of trees move when rustled by a wind, despite it being unobservable from large distances? Just as in this case motion remains hidden due to perspective, it remains hidden in warm bodies due to the extremely small sizes of the moving particles. In both cases, the viewing angle is so small that neither the object nor their movement can be seen.}}Lomonosov also insisted that movement of particles is necessary for the processes of [[Solvation|dissolution]], [[Extraction (chemistry)|extraction]] and [[diffusion]], providing as examples the dissolution and diffusion of salts by the action of water particles on the of the “molecules of salt”, the dissolution of metals in mercury, and the extraction of plant pigments by alcohol.<ref>{{Cite book |last=Lomonosov |first=Mikhail Vasil'evich |url=https://archive.org/details/mikhailvasilevic017733mbp |title=Mikhail Vasil'evich Lomonosov on the Corpuscular Theory |publisher=Harvard University Press |year=1970 |editor-last=Leicester |editor-first=Henry M. |pages=102–3 |translator-last= |translator-first= |chapter=Meditations on the Cause of Heat and Cold |orig-date=1750 |chapter-url=https://archive.org/details/mikhailvasilevic017733mbp/page/n115/mode/1up}}</ref>

Also the [[Heat transfer|transfer of heat]] was explained by the motion of particles. Around 1760, Scottish physicist and chemist [[Joseph Black]] wrote: "Many have supposed that heat is a tremulous ... motion of the particles of matter, which ... motion they imagined to be communicated from one body to another."<ref>{{Cite book |last=Black |first=Joseph |author-link=Joseph Black |url=https://books.google.com/books?id=lqI9AQAAMAAJ&pg=PA80 |title=Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh |date=1807 |publisher=Mathew Carey |editor-last=Robinson |editor-first=John |edition= |pages=80 |language=en}}</ref>

=== Kinetic theory of gases ===
[[File:Porträt des Daniel Bernoulli (cropped).jpg|thumb|167x167px|Daniel Bernoulli]]
[[Image:HYDRODYNAMICA, Danielis Bernoulli.png|thumb|upright|''Hydrodynamica'' front cover]]

In 1738 [[Daniel Bernoulli]] published ''[[Hydrodynamica]]'', which laid the basis for the [[Kinetic energy|kinetic]] theory of [[gas]]es. In this work, Bernoulli posited the argument, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the pressure of the gas, and that their average [[kinetic energy]] determines the temperature of the gas. The theory was not immediately accepted, in part because [[conservation of energy]] had not yet been established, and it was not obvious to [[physicist]]s how the collisions between molecules could be perfectly elastic.<ref name="PonomarevKurchatov1993">{{cite book|title=The Quantum Dice|author1=L.I Ponomarev|author2=I.V Kurchatov|date=1 January 1993|publisher=CRC Press|isbn=978-0-7503-0251-7}}</ref>{{rp|36–37}}

Pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747),<ref>Lomonosov 1758</ref> [[Georges-Louis Le Sage]] (ca. 1780, published 1818),<ref>Le Sage 1780/1818</ref> [[John Herapath]] (1816)<ref>Herapath 1816, 1821</ref> and [[John James Waterston]] (1843),<ref>Waterston 1843</ref> which connected their research with the development of [[mechanical explanations of gravitation]].

In 1856 [[August Krönig]] created a simple gas-kinetic model, which only considered the [[Translation (geometry)|translational motion]] of the particles.<ref>Krönig 1856</ref> In 1857 [[Rudolf Clausius]] developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also [[rotation]]al and vibrational molecular motions. In this same work he introduced the concept of [[mean free path]] of a particle.<ref>Clausius 1857</ref> In 1859, after reading a paper about the [[diffusion]] of molecules by Clausius, Scottish physicist [[James Clerk Maxwell]] formulated the [[Maxwell distribution]] of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.<ref>See:
* Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&pg=PA19 "Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"] ''Philosophical Magazine'', 4th series, '''19''' : 19–32.
* Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&pg=PA21 "Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another,"] ''Philosophical Magazine'', 4th series, '''20''' : 21–37.</ref> This was the first-ever statistical law in physics.<ref>{{cite book|title=The Man Who Changed Everything – the Life of James Clerk Maxwell|author=Mahon, Basil|publisher=Wiley|year=2003|isbn=0-470-86171-1|location=Hoboken, NJ|oclc=52358254}}</ref> Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.<ref>{{Cite journal|last1=Gyenis|first1=Balazs|year=2017|title=Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium|journal=Studies in History and Philosophy of Modern Physics|volume=57|pages=53–65|arxiv=1702.01411|bibcode=2017SHPMP..57...53G|doi=10.1016/j.shpsb.2017.01.001|s2cid=38272381}}</ref> In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called [[pressure]] of air and other gases."<ref>Maxwell 1873</ref>
In 1871, [[Ludwig Boltzmann]] generalized Maxwell's achievement and formulated the [[Maxwell–Boltzmann distribution]]. The [[logarithm]]ic connection between [[entropy]] and [[probability]] was also first stated by Boltzmann.

At the beginning of the 20th century, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was [[Albert Einstein]]'s (1905)<ref>Einstein 1905</ref> and [[Marian Smoluchowski]]'s (1906)<ref>Smoluchowski 1906</ref> papers on [[Brownian motion]], which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

Following the development of the [[Boltzmann equation]], a framework for its use in developing transport equations was developed independently by [[David Enskog]] and [[Sydney Chapman (mathematician)|Sydney Chapman]] in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as [[Chapman–Enskog theory]]. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.

== Assumptions ==
The application of kinetic theory to ideal gases makes the following assumptions:
* The gas consists of very small particles. This smallness of their size is such that the sum of the [[volume]] of the individual gas molecules is negligible compared to the volume of the container of the gas. This is equivalent to stating that the average distance separating the gas particles is large compared to their [[atomic radius|size]], and that the elapsed time during a collision between particles and the container's wall is negligible when compared to the time between successive collisions.
* The number of particles is so large that a statistical treatment of the problem is well justified. This assumption is sometimes referred to as the [[thermodynamic limit]].
* The rapidly moving particles constantly collide among themselves and with the walls of the container, and all these collisions are perfectly elastic.
* Interactions (i.e. collisions) between particles are strictly binary and [[Uncorrelatedness (probability theory)|uncorrelated]], meaning that there are no three-body (or higher) interactions, and the particles have no memory.
* Except during collisions, the interactions among molecules are negligible. They exert no other [[force]]s on one another.

Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.

As a simplifying assumption, the particles are usually assumed to have the same [[mass]] as one another; however, the theory can be generalized to a mass distribution, with each mass type contributing to the gas properties independently of one another in agreement with [[Dalton's law|Dalton's law of partial pressures]]. Many of the model's predictions are the same whether or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below).<ref>{{cite book |last1=Chang |first1=Raymond | last2=Thoman | first2=John W. Jr. |title=Physical Chemistry for the Chemical Sciences |date=2014 |publisher=University Science Books |location=New York, NY |page=37}}</ref>

More modern developments, such as the [[revised Enskog theory]] and the extended [[Bhatnagar–Gross–Krook operator|Bhatnagar–Gross–Krook]] model,<ref>{{Cite journal |last1=van Enk |first1=Steven J. |author-link=Steven J. van Enk |last2=Nienhuis |first2=Gerard |date=1991-12-01 |title=Inelastic collisions and gas-kinetic effects of light |url=https://link.aps.org/doi/10.1103/PhysRevA.44.7615 |journal=Physical Review A |volume=44 |issue=11 |pages=7615–7625 |doi=10.1103/PhysRevA.44.7615|pmid=9905900 |bibcode=1991PhRvA..44.7615V |url-access=subscription }}</ref> relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with [[Molecular vibration|internal degrees of freedom]], because they include the volume of the particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation.<ref>{{cite book |last1=McQuarrie |first1=Donald A. |title=Statistical Mechanics |date=1976 |publisher=University Science Press |location=New York, NY}}</ref> While theories relaxing the assumptions that the gas particles occupy negligible volume and that collisions are strictly elastic have been successful, it has been shown that relaxing the requirement of interactions being binary and uncorrelated will eventually lead to divergent results.<ref>{{Cite journal |last=Cohen |first=E. G. D. |date=1993-03-15 |title=Fifty years of kinetic theory |url=https://dx.doi.org/10.1016/0378-4371%2893%2990357-A |journal=Physica A: Statistical Mechanics and Its Applications |volume=194 |issue=1 |pages=229–257 |doi=10.1016/0378-4371(93)90357-A |bibcode=1993PhyA..194..229C |issn=0378-4371|url-access=subscription }}</ref>

== Equilibrium properties ==
=== Pressure and kinetic energy ===
<!-- This section is linked from [[Pressure]] -->
In the kinetic theory of gases, the [[pressure]] is assumed to be equal to the force (per unit area) exerted by the individual gas atoms or molecules hitting and rebounding from the gas container's surface.

Consider a gas particle traveling at velocity, <math display="inline">v_i</math>, along the <math>\hat{i}</math>-direction in an enclosed volume with [[characteristic length]], <math>L_i</math>, cross-sectional area, <math>A_i</math>, and volume, <math>V = A_i L_i</math>. The gas particle encounters a boundary after characteristic time
<math display="block"> t = L_i / v_i.</math>

The [[momentum]] of the gas particle can then be described as
<math display="block"> p_i = m v_i = m L_i / t .</math>

We combine the above with [[Newton's second law]], which states that the force experienced by a particle is related to the time rate of change of its momentum, such that
<math display="block">F_i = \frac{\mathrm{d}p_i}{\mathrm{d}t} = \frac{m L_i}{t^2}=\frac{m v_i^2}{L_i}.</math>

Now consider a large number, <math>N</math>, of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed, <math display='inline'> v </math>, in every direction is identical
<math display="block">v_x^2 = v_y^2 = v_z^2.</math>

Further, assume that the volume is symmetrical about its three dimensions, <math>\hat{i}, \hat{j}, \hat{k}</math>, such that
<math display="block">\begin{align}
V ={}& V_i = V_j = V_k, \\
F ={}& F_i = F_j = F_k, \\
     & A_i=A_j=A_k.
\end{align}</math>
The total surface area on which the gas particles act is therefore
<math display="block">A = 3 A_i.</math>

The pressure exerted by the collisions of the <math>N</math> gas particles with the surface can then be found by adding the force contribution of every particle and dividing by the interior surface area of the volume,
<math display="block">P = \frac{N \overline{F}}{A}=\frac{NLF}{V} </math>
<math display="block"> \Rightarrow PV = NLF = \frac{N}{3} m v^2.</math>

The total translational [[kinetic energy]] <math>K_\text{t} </math> of the gas is defined as
<math display="block">K_\text{t} = \frac{N}{2} m v^2 ,</math>
providing the result
<math display="block">PV = \frac{2}{3} K_\text{t} .</math>

This is an important, non-trivial result of the kinetic theory because it relates pressure, a [[macroscopic]] property, to the translational kinetic energy of the molecules, which is a [[microscopic]] property.

The mass density of a gas <math>\rho </math> is expressed through the total mass of gas particles and through volume of this gas: <math> \rho = \frac {N m}{V}</math>. Taking this into account, the pressure is equal to
<math display="block">P = \frac{\rho v^2}{3} .</math>

Relativistic expression for this formula is <ref>{{Cite journal |last=Fedosin |first=Sergey G. | date=2021 |title= The potentials of the acceleration field and pressure field in rotating relativistic uniform system |journal= Continuum Mechanics and Thermodynamics |volume= 33|issue= 3|pages= 817–834|language=en |doi= 10.1007/s00161-020-00960-7|s2cid= 230076346 |arxiv=2410.17289 |bibcode= 2021CMT....33..817F}}</ref>

<math display="block"> P = \frac {2 \rho c^2 }{3} \left({\left(1 - \overline{v^2} / c^2\right)}^{-1/2} - 1 \right) , </math>
where <math> c </math> is [[speed of light]]. In the limit of small speeds, the expression becomes <math>P \approx \rho \overline{v^2}/3</math>.

=== Temperature and kinetic energy ===
Rewriting the above result for the pressure as <math display="inline">PV = \frac{1}{3}Nmv^2 </math>, we may combine it with the [[ideal gas law]]
{{NumBlk||<math display="block"> PV = N k_\mathrm{B} T ,</math>|{{EquationRef|1}}}}
where <math> k_\mathrm{B}</math> is the [[Boltzmann constant]] and <math> T</math> is the [[Thermodynamic temperature|absolute]] [[temperature]] defined by the ideal gas law, to obtain
<math display="block">k_\mathrm{B} T = \frac{1}{3} m v^2, </math>
which leads to a simplified expression of the average translational kinetic energy per molecule,<ref>The average kinetic energy of a fluid is proportional to the [[Root-mean-square speed|root mean-square velocity]], which always exceeds the mean velocity - {{usurped|1=[https://web.archive.org/web/20071011213546/http://www.mikeblaber.org/oldwine/chm1045/notes/Gases/Kinetic/Gases08.htm Kinetic Molecular Theory]}}</ref>
<math display="block"> \frac{1}{2} m v^2 = \frac{3}{2} k_\mathrm{B} T.</math>
The translational kinetic energy of the system is <math>N</math> times that of a molecule, namely <math display="inline"> K_\text{t} = \frac{1}{2} N m v^2 </math>. The temperature, <math> T</math> is related to the translational kinetic energy by the description above, resulting in
{{NumBlk||<math display="block"> T = \frac{1}{3} \frac{m v^2}{k_\mathrm{B} } </math>|{{EquationRef|2}}}}
which becomes
{{NumBlk||<math display="block"> T = \frac{2}{3} \frac{K_\text{t}}{N k_\mathrm{B} }. </math>|{{EquationRef|3}}}}
Equation ({{EquationNote|3}}) is one important result of the kinetic theory:
''The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature''.
From equations ({{EquationNote|1}}) and ({{EquationNote|3}}), we have
{{NumBlk||<math display="block"> PV = \frac{2}{3} K_\text{t}.</math>|{{EquationRef|4}}}}
Thus, the product of pressure and volume per [[Mole (unit)|mole]] is proportional to the average
translational molecular kinetic energy.

Equations ({{EquationNote|1}}) and ({{EquationNote|4}}) are called the "classical results", which could also be derived from [[statistical mechanics]];
for more details, see:<ref>
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)] {{webarchive|url=https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 |date=2012-04-28 }}
</ref>

The [[equipartition theorem]] requires that kinetic energy is partitioned equally between all kinetic [[degrees of freedom]], ''D''. A monatomic gas is axially symmetric about each spatial axis, so that ''D'' = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that ''D'' = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like [[water]], is not radially symmetric about any axis, resulting in ''D'' = 6, comprising 3 translational and 3 rotational degrees of freedom.

Because the [[equipartition theorem]] requires that kinetic energy is partitioned equally, the total kinetic energy is
<math display="block"> K =D K_\text{t} = \frac{D}{2} N m v^2. </math>

Thus, the energy added to the system per gas particle kinetic degree of freedom is
<math display="block"> \frac{K}{ND} = \frac{1}{2} k_\text{B} T . </math>

Therefore, the kinetic energy per kelvin of one mole of monatomic [[ideal gas]] (''D'' = 3) is
<math display="block"> K = \frac{D}{2} k_\text{B} N_\text{A} = \frac{3}{2} R, </math>
where <math>N_\text{A}</math> is the [[Avogadro constant]], and ''R'' is the [[ideal gas constant]].

Thus, the ratio of the kinetic energy to the absolute temperature of an ideal monatomic gas can be calculated easily:
* per mole: 12.47&nbsp;J/K
* per molecule: 20.7&nbsp;[[yoctojoule|yJ]]/K = 129&nbsp;μeV/K

At [[Standard temperature and pressure|standard temperature]] (273.15&nbsp;K), the kinetic energy can also be obtained:
* per mole: 3406&nbsp;J
* per molecule: 5.65&nbsp;[[zeptojoule|zJ]] = 35.2&nbsp;meV.

At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on ''D'' and the total molecular energy. Quantum [[statistical mechanics]] is needed to accurately compute these contributions.<ref>{{cite book |last1=Chang |first1=Raymond | last2=Thoman | first2=John W. Jr. |title=Physical Chemistry for the Chemical Sciences |date=2014 |publisher=University Science Books |location=New York |pages=56–61}}</ref>

=== Collisions with container wall ===
For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated<ref name="OCW">{{cite web |title=5.62 Physical Chemistry II |url=https://ocw.mit.edu/courses/chemistry/5-62-physical-chemistry-ii-spring-2008/lecture-notes/29_562ln08.pdf |website=MIT OpenCourseWare}}</ref> based on naive kinetic theory, and the results can be used for analyzing [[Effusion#Physics in Effusion|effusive flow rate]]s, which is useful in applications such as the [[Gaseous diffusion#Technology|gaseous diffusion]] method for [[Isotope separation#Diffusion|isotope separation]].

Assume that in the container, the number density (number per unit volume) is <math>n = N/V</math> and that the particles obey [[Maxwell-Boltzmann distribution|Maxwell's velocity distribution]]:
<math display="block">f_\text{Maxwell}(v_x,v_y,v_z) \, dv_x \, dv_y \, dv_z = \left(\frac{m}{2 \pi k_\text{B} T}\right)^{3/2} e^{- \frac{mv^2}{2k_\text{B}T}} \, dv_x \, dv_y \, dv_z</math>

Then for a small area <math>dA</math> on the container wall, a particle with speed <math>v</math> at angle <math>\theta</math> from the normal of the area <math>dA</math>, will collide with the area within time interval <math>dt</math>, if it is within the distance <math>v\,dt</math> from the area <math>dA</math>. Therefore, all the particles with speed <math>v</math> at angle <math>\theta</math> from the normal that can reach area <math>dA</math> within time interval <math>dt</math> are contained in the tilted pipe with a height of <math>v\cos (\theta) dt</math> and a volume of <math>v\cos (\theta) \,dA\,dt</math>.

The total number of particles that reach area <math>dA</math> within time interval <math>dt</math> also depends on the velocity distribution; All in all, it calculates to be:<math display="block">n v \cos(\theta) \, dA\, dt \times\left(\frac{m}{2 \pi k_\text{B}T}\right)^{3/2} e^{- \frac{mv^2}{2k_\text{B}T}} \left( v^2 \sin(\theta) \, dv \, d\theta \, d\phi \right).</math>

Integrating this over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
<math display="block">J_\text{collision} = \frac{\displaystyle\int_0^{\pi/2} \cos(\theta) \sin(\theta) \, d\theta}{\displaystyle\int_0^\pi \sin(\theta) \, d\theta}\times n \bar v
= \frac{1}{4} n \bar v
= \frac{n}{4} \sqrt{\frac{8 k_\mathrm{B} T}{\pi m}}. </math>

This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed <math>\bar{v}</math> of the Maxwell's velocity distribution, one has to integrate over <math>v > 0 </math>, <math>0 < \theta < \pi </math>, <math>0 < \phi < 2\pi</math>.

The momentum transfer to the container wall from particles hitting the area <math>dA</math> with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is:
<math display="block">[2mv \cos(\theta)]\times n v \cos(\theta) \, dA\, dt \times\left(\frac{m}{2 \pi k_\text{B}T}\right)^{3/2} e^{- \frac{mv^2}{2k_\text{B}T}} \left( v^2 \sin(\theta) \, dv \, d\theta \, d\phi \right).</math>
Integrating this over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the [[pressure]] (consistent with [[Ideal gas law]]):
<math display="block">P = \frac{\displaystyle 2\int_0^{\pi/2} \cos^2(\theta) \sin(\theta) \, d\theta}{\displaystyle \int_0^\pi \sin(\theta) \, d\theta}\times n mv_\text{rms}^2
= \frac{1}{3} n mv_\text{rms}^2
= \frac{2}{3} n\langle E_\text{kin}\rangle
= n k_\mathrm{B} T </math>
If this small area <math>A</math> is punched to become a small hole, the [[Effusion#Physics in Effusion|effusive flow rate]] will be:
<math display="block">\Phi_\text{effusion} = J_\text{collision} A= n A \sqrt{\frac{k_\mathrm{B} T}{2 \pi m}}. </math>

Combined with the [[ideal gas law]], this yields
<math display="block">\Phi_\text{effusion} = \frac{P A}{\sqrt{2 \pi m k_\mathrm{B} T}}. </math>

The above expression is consistent with [[Graham's law]].

To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with <math>(v,\theta,\phi)</math> that hit the area <math>dA</math> within the time interval <math>dt</math> are contained in the tilted pipe with a height of <math>v\cos (\theta) \, dt</math> and a volume of <math>v\cos (\theta) \, dA \, dt</math>; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of <math>v\cos \theta</math>:
<math display="block">\begin{align}
f(v,\theta,\phi) \, dv \, d\theta \, d\phi
&= \lambda v\cos{\theta} \left(\frac{m}{2 \pi k T}\right)^{3/2} e^{- \frac{mv^2}{2k_\mathrm{B} T}}(v^2\sin{\theta} \, dv \, d\theta \, d\phi)
\end{align}</math>
with the constraint <math display="inline">v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math>. The constant <math>\lambda</math> can be determined by the normalization condition <math display="inline">\int f(v,\theta,\phi) \, dv \, d\theta \, d\phi=1</math> to be <math display="inline">4/\bar{v} </math>, and overall:
<math display="block">\begin{align}
f(v,\theta,\phi) \, dv \, d\theta \, d\phi
&= \frac{1}{2\pi} \left(\frac{m}{k_\mathrm{B} T}\right)^2e^{- \frac{mv^2}{2k_\mathrm{B} T}}
(v^3\sin{\theta}\cos{\theta} \, dv \, d\theta \, d\phi) \\
\end{align};\quad v>0,\, 0<\theta<\frac \pi 2,\, 0<\phi<2\pi</math>

=== Speed of molecules ===
From the kinetic energy formula it can be shown that
<math display="block">v_\text{p} = \sqrt{2 \cdot \frac{k_\mathrm{B} T}{m}},</math>
<math display="block">\bar{v} = \frac {2}{\sqrt{\pi}} v_p = \sqrt{\frac {8}{\pi} \cdot \frac{k_\mathrm{B} T}{m}},</math>
<math display="block">v_\text{rms} = \sqrt{\frac{3}{2}} v_p = \sqrt{{3} \cdot \frac {k_\mathrm{B} T}{m}},</math>
where ''v'' is in m/s, ''T'' is in kelvin, and ''m'' is the mass of one molecule of gas in kg. The most probable (or mode) speed <math>v_\text{p}</math> is 81.6% of the root-mean-square speed <math>v_\text{rms}</math>, and the mean (arithmetic mean, or average) speed <math>\bar{v}</math> is 92.1% of the rms speed ([[isotropy|isotropic]] [[Maxwell–Boltzmann distribution#Distribution for the speed|distribution of speeds]]).

See:
* [[Average]],
* [[Root-mean-square speed]]
* [[Arithmetic mean]]
* [[Mean]]
* [[Mode (statistics)]]

=== Mean free path ===
{{Main|Mean free path}}
In kinetic theory of gases, the [[Mean free path#Kinetic theory|mean free path]] is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision. Let <math> \sigma </math> be the collision [[Cross section (physics)#Collision among gas particles|cross section]] of one molecule colliding with another. As in the previous section, the number density <math> n </math> is defined as the number of molecules per (extensive) volume, or <math> n = N/V </math>. The collision cross section per volume or collision cross section density is <math> n \sigma </math>, and it is related to the mean free path <math>\ell</math> by<math display="block">\ell = \frac {1} {n \sigma \sqrt{2}} </math>

Notice that the unit of the collision cross section per volume <math> n \sigma </math> is reciprocal of length.

== Transport properties ==
{{See also|Transport phenomena}}

The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means using Kinetic Theory to consider what are known as "transport properties", such as [[viscosity]], [[thermal conductivity]], [[mass diffusivity]] and [[Thermophoresis|thermal diffusion]].

In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of Kinetic gas theory to dense gas mixtures, [[Chapman–Enskog theory#Revised Enskog Theory|Revised Enskog Theory]], was developed in 1983-1987 by [[E. G. D. Cohen]], [[J. M. Kincaid]] and [[M. Lòpez de Haro]],<ref>{{cite journal |last1=Lòpez de Haro |first1=M. |last2=Cohen |first2=E. G. D. |last3=Kincaid |first3=J. M. |title=The Enskog theory for multicomponent mixtures. I. Linear transport theory |journal=The Journal of Chemical Physics |date=1983 |volume=78 |issue=5 |pages=2746–2759 |doi=10.1063/1.444985 |bibcode=1983JChPh..78.2746L |url=https://doi.org/10.1063/1.444985|url-access=subscription }}</ref><ref>{{cite journal |last1=Kincaid |first1=J. M. |last2=Lòpez de Haro |first2=M. |last3=Cohen |first3=E. G. D. |title=The Enskog theory for multicomponent mixtures. II. Mutual diffusion |journal=The Journal of Chemical Physics |date=1983 |volume=79 |issue=9 |pages=4509–4521 |doi=10.1063/1.446388 |url=https://doi.org/10.1063/1.446388|url-access=subscription }}</ref><ref>{{cite journal |last1=Lòpez de Haro |first1=M. |last2=Cohen |first2=E. G. D. |title=The Enskog theory for multicomponent mixtures. III. Transport properties of dense binary mixtures with one tracer component |journal=The Journal of Chemical Physics |date=1984 |volume=80 |issue=1 |pages=408–415 |doi=10.1063/1.446463 |bibcode=1984JChPh..80..408L |url=https://doi.org/10.1063/1.446463|url-access=subscription }}</ref><ref>{{cite journal |last1=Kincaid |first1=J. M. |last2=Cohen |first2=E. G. D. |last3=Lòpez de Haro |first3=M. |title=The Enskog theory for multicomponent mixtures. IV. Thermal diffusion |journal=The Journal of Chemical Physics |date=1987 |volume=86 |issue=2 |pages=963–975 |doi=10.1063/1.452243 |bibcode=1987JChPh..86..963K |url=https://doi.org/10.1063/1.452243|url-access=subscription }}</ref> building on work by [[H. van Beijeren]] and [[M. H. Ernst]].<ref>{{cite journal |last1=van Beijeren |first1=H. |last2=Ernst |first2=M. H. |title=The non-linear Enskog-Boltzmann equation |journal=Physics Letters A |date=1973 |volume=43 |issue=4 |pages=367–368 |doi=10.1016/0375-9601(73)90346-0 |bibcode=1973PhLA...43..367V |hdl=1874/36979 |url=https://doi.org/10.1016/0375-9601(73)90346-0|hdl-access=free }}</ref>

=== Viscosity and kinetic momentum ===
{{See also|Viscosity#Momentum transport}}

In books on elementary kinetic theory<ref name="Sears1975">{{cite book|last1=Sears|first1=F.W.|last2=Salinger|first2=G.L.|year=1975|title=Thermodynamics, Kinetic Theory, and Statistical Thermodynamics|publisher=Addison-Wesley Publishing Company, Inc.|location= Reading, Massachusetts, USA |edition=3|chapter=10|pages=286–291|isbn=978-0201068948}}</ref> one can find results for dilute gas modeling that are used in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a [[Couette flow]] where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force ''F''. The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component <math>u</math> which increase uniformly with distance <math>y</math> above the lower plate. The non-equilibrium flow is superimposed on a [[Maxwell-Boltzmann distribution|Maxwell-Boltzmann equilibrium distribution]] of molecular motions.

Inside a dilute gas in a [[Couette flow]] setup, let <math> u_0 </math> be the forward velocity of the gas at a horizontal flat layer (labeled as <math>y=0</math>); <math> u_0 </math> is along the horizontal direction. The number of molecules arriving at the area <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block">nv\cos({\theta})\, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B} T}\right)^{3/2} \, e^{- \frac{mv^2}{2 k_\mathrm{B} T}} (v^2\sin{\theta} \, dv \, d\theta \, d\phi)</math>

These molecules made their last collision at <math>y = \pm \ell\cos \theta</math>, where <math>\ell</math> is the [[Mean free path#Kinetic theory|mean free path]]. Each molecule will contribute a forward momentum of
<math display="block">p_x^{\pm} = m \left( u_0 \pm \ell \cos \theta \frac{du}{dy} \right), </math>
where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient <math>du/dy</math> can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the forward momentum transfer per unit time per unit area (also known as [[shear stress]]):
<math display="block">\tau^{\pm} = \frac{1}{4} \bar{v} n \cdot m \left( u_0 \pm \frac{2}{3} \ell \frac{du}{dy} \right) </math>

The net rate of momentum per unit area that is transported across the imaginary surface is thus
<math display="block">\tau = \tau^{+} - \tau^{-} = \frac {1}{3} \bar v n m \cdot \ell \frac{du}{dy} </math>

Combining the above kinetic equation with [[Newton's law of viscosity]]
<math display="block">\tau = \eta \frac{du}{dy} </math>
gives the equation for shear viscosity, which is usually denoted <math> \eta_0 </math> when it is a dilute gas:
<math display="block">\eta_0 = \frac{1}{3} \bar{v} n m \ell </math>

Combining this equation with the equation for mean free path gives
<math display="block">\eta_0 = \frac {1} {3 \sqrt{2}} \frac{m \bar{v}}{\sigma}</math>

Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
<math display="block">\bar v = \frac{2}{\sqrt{\pi}} v_p = 2 \sqrt{\frac{2}{\pi} \frac {k_\mathrm{B}T}{m}} </math>
where <math>v_p</math> is the most probable speed. We note that
<math display="block">k_\text{B} N_\text{A} = R \quad \text{and} \quad M = m N_\text{A} </math>

and insert the velocity in the viscosity equation above. This gives the well known equation <ref name="Hildebrand1976">{{cite journal| journal = Proc Natl Acad Sci U S A |last1=Hildebrand |first1=J.H.|year=1976 | volume=76 |title=Viscosity of dilute gases and vapors|issue=12 | pages= 4302–4303 |doi=10.1073/pnas.73.12.4302 |pmid=16592372 |pmc=431439 |bibcode=1976PNAS...73.4302H |doi-access=free }}</ref> (with <math>\sigma</math> subsequently estimated below) for [[Viscosity models for mixtures#Dilute gas limit and scaled variables|shear viscosity for dilute gases]]:
<math display="block">\eta_0
= \frac {2} {3 \sqrt{\pi} } \cdot \frac {\sqrt{m k_\mathrm{B} T}} { \sigma }
= \frac {2} {3 \sqrt{\pi} } \cdot \frac {\sqrt{M R T}} { \sigma N_\text{A} } </math>
and <math> M </math> is the [[molar mass]]. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in [[Revised Enskog theory]], which relaxes the requirement of a gas being dilute. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by
<math display="block">\sigma = \pi \left( 2 r \right)^2 = \pi d^2 </math>

The radius <math>r</math> is called collision cross section radius or kinetic radius, and the diameter <math>d</math> is called collision cross section diameter or [[kinetic diameter]] of a molecule in a monomolecular gas. There are no simple general relation between the collision [[Cross section (physics)#Collision among particles|cross section]] and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e. a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the [[Lennard-Jones potential]] or [[Morse potential]] which have a negative part that attracts the other molecule from distances longer than the hard core radius. [[Mean free path#Mean free path in kinetic theory|The radius for zero Lennard-Jones potential]] may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required [[Cross section (physics)|collision integrals]].

The expression for viscosity obtained from [[Revised Enskog theory|Revised Enskog Theory]] reduces to the above expression in the limit of infinite dilution, and can be written as
<math display="block">\eta = (1 + \alpha_\eta)\eta_0 + \eta_c </math>
where <math>\alpha_\eta</math> is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and <math>\eta_c</math> is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.

=== Thermal conductivity and heat flux ===
{{See also|Thermal conductivity}}
Following a similar logic as above, one can derive the kinetic model for [[thermal conductivity]]<ref name="Sears1975" /> of a dilute gas:

Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as [[thermal reservoir]]s. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy <math>\varepsilon</math> which increases uniformly with distance <math>y</math> above the lower plate. The non-equilibrium energy flow is superimposed on a [[Maxwell-Boltzmann distribution|Maxwell-Boltzmann equilibrium distribution]] of molecular motions.

Let <math> \varepsilon_0 </math> be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block"> nv \cos(\theta)\, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B}T}\right)^{3 / 2} e^{- \frac{mv^2}{2k_\text{B}T}} (v^2 \sin(\theta) \, dv \, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>\ell\cos \theta</math> above and below the gas layer, and each will contribute a molecular kinetic energy of
<math display="block"> \varepsilon^{\pm} = \left( \varepsilon_0 \pm m c_v \ell \cos \theta \, \frac{dT}{dy} \right), </math>
where <math>c_v</math> is the [[specific heat capacity]]. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient <math>dT/dy</math> can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint <math>v > 0 </math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the energy transfer per unit time per unit area (also known as [[heat flux]]):
<math display="block"> q_y^{\pm} = -\frac{1}{4} \bar v n \cdot \left( \varepsilon_0 \pm \frac {2}{3} m c_v \ell \frac{dT}{dy} \right) </math>

Note that the energy transfer from above is in the <math>-y</math> direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
<math display="block"> q = q_y^{+} - q_y^{-} = -\frac{1}{3} \bar{v} n m c_v \ell \,\frac{dT}{dy} </math>

Combining the above kinetic equation with [[Fourier's law]]
<math display="block"> q = -\kappa \, \frac{dT}{dy} </math>
gives the equation for thermal conductivity, which is usually denoted <math> \kappa_0 </math> when it is a dilute gas:
<math display="block"> \kappa_0 = \frac{1}{3} \bar{v} n m c_v \ell </math>

Similarly to viscosity, [[Revised Enskog theory]] yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as
<math display="block"> \kappa = \alpha_\kappa \kappa_0 + \kappa_c </math>
where <math> \alpha_\kappa </math> is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and <math> \kappa_c </math> is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.

=== Diffusion coefficient and diffusion flux ===
{{See also|Fick's laws of diffusion}}Following a similar logic as above, one can derive the kinetic model for [[mass diffusivity]]<ref name="Sears1975" /> of a dilute gas:

Consider a [[Steady state|steady]] diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform [[Number density|number densities]], but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density <math>n</math> in the layer increases uniformly with distance <math>y</math> above the lower plate. The non-equilibrium molecular flow is superimposed on a [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann equilibrium distribution]] of molecular motions.

Let <math> n_0 </math> be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block"> nv\cos(\theta) \, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B}T}\right)^{3 / 2} e^{- \frac{mv^2}{2k_\text{B}T}} (v^2\sin(\theta) \, dv\, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>\ell\cos \theta</math> above and below the gas layer, where the local number density is
<math display="block"> n^{\pm} = \left( n_0 \pm \ell \cos \theta \, \frac{dn}{dy} \right) </math>

Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient <math>dn/dy</math> can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2} </math>, <math>0 < \phi < 2\pi</math> yields the molecular transfer per unit time per unit area (also known as [[diffusion flux]]):
<math display="block"> J_y^{\pm} = - \frac{1}{4} \bar v \cdot \left( n_0 \pm \frac{2}{3} \ell \, \frac{dn}{dy} \right) </math>

Note that the molecular transfer from above is in the <math>-y</math> direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
<math display="block"> J = J_y^{+} - J_y^{-} = -\frac {1}{3} \bar{v} \ell \frac{dn}{dy} </math>

Combining the above kinetic equation with [[Fick's laws of diffusion#Fick's first law|Fick's first law of diffusion]]
<math display="block"> J = - D \frac{dn}{dy} </math>
gives the equation for mass diffusivity, which is usually denoted <math> D_0 </math> when it is a dilute gas:
<math display="block"> D_0 = \frac{1}{3} \bar{v} \ell </math>

The corresponding expression obtained from [[Revised Enskog theory|Revised Enskog Theory]] may be written as
<math display="block"> D = \alpha_D D_0 </math>
where <math> \alpha_D </math> is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation [[chemical potential]]s with density.

== Detailed balance ==

=== Fluctuation and dissipation ===
{{Main|Fluctuation-dissipation theorem}}
The kinetic theory of gases entails that due to the [[microscopic reversibility]] of the gas particles' detailed dynamics, the system must obey the principle of [[detailed balance]]. Specifically, the [[fluctuation-dissipation theorem]] applies to the [[Brownian motion]] (or [[diffusion]]) and the [[Drag (physics)|drag force]], which leads to the [[Einstein relation (kinetic theory)|Einstein–Smoluchowski equation]]:<ref>{{ cite book | last1 = Dill | first1 = Ken A. | url = https://books.google.com/books?id=hdeODhjp1bUC&pg=PA327 | title = Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology | last2 = Bromberg | first2 = Sarina | date = 2003 | publisher = Garland Science | isbn = 9780815320517 | pages = 327 | language = en}}</ref>
<math display="block"> D = \mu \, k_\text{B} T, </math> where
* {{mvar|D}} is the [[mass diffusivity]];
* {{mvar|μ}} is the "mobility", or the ratio of the particle's [[Terminal velocity|terminal]] [[drift velocity]] to an applied [[force]], {{math|1=''μ'' = ''v''<sub>d</sub>/''F''}};
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]];
* {{mvar|T}} is the [[absolute temperature]].
Note that the mobility {{math|1=''μ'' = ''v''<sub>d</sub>/''F''}} can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.

=== Onsager reciprocal relations ===
{{Main|Onsager reciprocal relations}}
The mathematical similarities between the expressions for shear viscocity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of the [[Onsager reciprocal relations]] (i.e. the detailed balance of the [[Microscopic reversibility|reversible dynamics]] of the particles), when applied to the [[convection]] (matter flow due to temperature gradient, and heat flow due to pressure gradient) and [[Advection#Distinction between advection and convection|advection]] (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.

== See also ==
{{Statistical mechanics}}
* [[BBGKY hierarchy|Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations]]
* [[Boltzmann equation]]
* [[Chapman–Enskog theory]]
* [[Collision theory]]
* [[Critical temperature]]
* [[Gas laws]]
* [[Heat]]
* [[Interatomic potential]]
* [[Magnetohydrodynamics]]
* [[Maxwell–Boltzmann distribution]]
* [[Mixmaster universe]]
* [[Thermodynamics]]
* [[Vicsek model]]
* [[Vlasov equation]]

== References ==
=== Citations ===
{{reflist|2}}

=== Sources cited ===
{{refbegin}}
* {{citation
 | author=Clausius, R.
 | title =Ueber die Art der Bewegung, welche wir Wärme nennen
 | journal =Annalen der Physik
 | volume =176
 | pages =353–379
 | year =1857
 | url=http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.table
 | doi=10.1002/andp.18571760302|bibcode = 1857AnP...176..353C
 | issue=3
 | ref =none
}}
* de Groot, S. R., W. A. van Leeuwen and Ch. G. van Weert (1980), Relativistic Kinetic Theory, North-Holland, Amsterdam.
* {{Cite book
|last=Galilei
|first=Galileo
|url=https://www.mercaba.es/renacimiento/escritos_menores_de_galileo.pdf
|title=Discoveries and Opinions of Galileo
|publisher=Doubleday
|year=1957
|editor-last=Drake
|editor-first=Stillman
|chapter=The Assayer
|orig-date=1623
}}
* {{citation
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 | volume =17
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 | url=http://www.cdvandt.org/Band%2017%20Kap%205.pdf
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* {{citation
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* {{citation
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* {{citation
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* {{citation
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* {{citation
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* Liboff, R. L. (1990), Kinetic Theory, Prentice-Hall, Englewood Cliffs, N. J.
* {{citation
 | author=Lomonosov, M. |author-link=Mikhail Lomonosov
 | orig-year=1758
 | year= 1970
 | chapter=On the Relation of the Amount of Material and Weight
 | editor= Henry M. Leicester |editor-link=Henry M. Leicester
 | title= Mikhail Vasil'evich Lomonosov on the Corpuscular Theory
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 | publisher=Harvard University Press
 | pages =224–233
 | chapter-url=https://archive.org/details/mikhailvasilevic017733mbp
 | ref =none
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* {{citation
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 | title=The Man Who Changed Everything – the Life of James Clerk Maxwell
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* {{citation
 | author=Maxwell, James Clerk
 | title=Molecules
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 | year=1873
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* {{citation
 | author=Smoluchowski, M.
 | title =Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen
 | journal =Annalen der Physik
 | volume =21
 | pages =756–780
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* {{citation
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* {{cite book
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{{refend}}

== Further reading ==
* [[Sydney Chapman (mathematician)|Sydney Chapman]] and [[Thomas Cowling|Thomas George Cowling]] (1939/1970), ''The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', (first edition 1939, second edition 1952), third edition 1970 prepared in co-operation with D. Burnett, Cambridge University Press, London
* [[Joseph O. Hirschfelder|Joseph Oakland Hirschfelder]], [https://history.aip.org/phn/11901002.html Charles Francis Curtiss], and [[Robert Byron Bird]] (1964), ''Molecular Theory of Gases and Liquids'', revised edition (Wiley-Interscience), ISBN 978-0471400653
* [[Richard Liboff|Richard Lawrence Liboff]] (2003), ''Kinetic Theory: Classical, Quantum, and Relativistic Descriptions'', third edition (Springer), ISBN 978-0-387-21775-8
* [[Behnam Rahimi]] and [https://www.uvic.ca/engineering/mechanical/faculty-and-staff/faculty/strucht.php Henning Struchtrup] {{Webarchive|url=https://web.archive.org/web/20210725051027/https://www.uvic.ca/engineering/mechanical/faculty-and-staff/faculty/strucht.php |date=2021-07-25 }} (2016), "[https://dx.doi.org/10.1017/jfm.2016.604 Macroscopic and kinetic modelling of rarefied polyatomic gases]", ''Journal of Fluid Mechanics'', '''806''', 437–505, DOI 10.1017/jfm.2016.604

== External links ==
{{Wikiquote}}
* {{usurped|1=[https://web.archive.org/web/20211122221844/https://www.peruzinasi.com/chemistry-form-5-physical-chemistry-gases/ PHYSICAL CHEMISTRY – Gases]}}
* [https://www.math.umd.edu/~lvrmr/History/EarlyTheories.html Early Theories of Gases]
* [http://www.lightandmatter.com/html_books/0sn/ch05/ch05.html Thermodynamics] {{Webarchive|url=https://web.archive.org/web/20170228144433/http://www.lightandmatter.com/html_books/0sn/ch05/ch05.html |date=2017-02-28 }} - a chapter from an online textbook
* [http://physnet.org/modules/pdfmodules/m156.pdf ''Temperature and Pressure of an Ideal Gas: The Equation of State''] on [http://www.physnet.org Project PHYSNET].
* [https://web.archive.org/web/20141103020057/http://www2.ucdsb.on.ca/tiss/stretton/CHEM1/gases9.html Introduction] to the kinetic molecular theory of gases, from The Upper Canada District School Board
* [https://web.archive.org/web/20160331040112/http://comp.uark.edu/%7Ejgeabana/mol_dyn/ Java animation] illustrating the kinetic theory from University of Arkansas
* [http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/ktcon.html Flowchart] linking together kinetic theory concepts, from HyperPhysics
* [https://web.archive.org/web/20050409211009/http://www.ewellcastle.co.uk/science/pages/kinetics.html Interactive Java Applets] allowing high school students to experiment and discover how various factors affect rates of chemical reactions.
* https://www.youtube.com/watch?v=47bF13o8pb8&list=UUXrJjdDeqLgGjJbP1sMnH8A A demonstration apparatus for the thermal agitation in gases.
{{Authority control}}

{{DEFAULTSORT:Kinetic Theory of Gasses}}
[[Category:Gases]]
[[Category:Thermodynamics]]
[[Category:Classical mechanics]]