View source for Physics:Kinetic theory of gases

{{Short description|Historic physical model of gases}}
[[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 size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have an average speed relative to their size slowed down here two trillion fold from that at room temperature.]]

The '''kinetic theory of gases''' is a simple, historically significant [[Physics:Classical mechanics|classical]] model of the [[Physics:Thermodynamics|thermodynamic]] behavior of [[Physics:Gas|gas]]es, with which many principal concepts of thermodynamics were established.  The model describes a gas as a large number of identical submicroscopic [[Physics:Particle|particle]]s ([[Physics:Atom|atom]]s or [[Physics:Molecule|molecule]]s), all of which are in constant, rapid, [[Brownian motion|random motion]]. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random [[Physics:Elastic collision|elastic collision]]s between themselves and with the enclosing walls of the container. The basic version of the model describes the [[Physics:Ideal gas|ideal gas]], and considers no other interactions between the particles.

The kinetic theory of gases explains the [[Physics:Macroscopic scale|macroscopic]] properties of gases, such as [[Physics:Volume|volume]], [[Physics:Pressure|pressure]], and [[Physics:Temperature|temperature]], as well as [[Physics:Transport phenomena|transport properties]] such as [[Physics:Viscosity|viscosity]], [[Physics:Thermal conductivity|thermal conductivity]] and [[Physics:Mass diffusivity|mass diffusivity]]. Due to the [[Time reversibility|time reversibility]] of microscopic dynamics ([[Physics:Microscopic reversibility|microscopic reversibility]]), the kinetic theory is also connected to the principle of [[Detailed balance|detailed balance]], in terms of the [[Physics:Fluctuation-dissipation theorem|fluctuation-dissipation theorem]] (for [[Brownian motion]]) and the [[Physics:Onsager reciprocal relations|Onsager reciprocal relations]].

Historically, the kinetic theory of gases was the first explicit exercise of the ideas of [[Physics:Statistical mechanics|statistical mechanics]].

==History==
In about 50 BCE, the Roman philosopher [[Biography:Lucretius|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 [[Philosophy:Epicureanism|Epicurean]] atomistic point of view was rarely considered in the subsequent centuries, when [[Biography:Aristotle|Aristotle]]an ideas were dominant.

[[Image:HYDRODYNAMICA, Danielis Bernoulli.png|thumb|upright|Hydrodynamica front cover]]

In 1738 [[Biography:Daniel Bernoulli|Daniel Bernoulli]] published ''[[Physics:Hydrodynamica|Hydrodynamica]]'', which laid the basis for the [[Physics:Kinetic energy|kinetic]] theory of [[Physics:Gas|gas]]es.  In this work, [[Bernoulli's principle|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 [[Physics:Kinetic energy|kinetic energy]] determines the temperature of the gas. The theory was not immediately accepted, in part because [[Physics:Conservation of energy|conservation of energy]] had not yet been established, and it was not obvious to [[Physicist|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}}

Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were [[Biography:Mikhail Lomonosov|Mikhail Lomonosov]] (1747),<ref>Lomonosov 1758</ref> Georges-Louis Le Sage (ca. 1780, published 1818),<ref>Le Sage 1780/1818</ref> [[Biography:John Herapath|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 [[Physics:Mechanical explanations of gravitation|mechanical explanations of gravitation]]. In 1856 [[Biography:August Krönig|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 [[Biography:Rudolf Clausius|Rudolf Clausius]] developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also [[Physics:Rotation|rotation]]al and vibrational molecular motions. In this same work he introduced the concept of [[Physics:Mean free path|mean free path]] of a particle.<ref>Clausius 1857</ref> In 1859, after reading a paper about the [[Physics:Diffusion|diffusion]] of molecules by  Clausius, Scottish physicist [[Biography:James Clerk Maxwell|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 [[Physics:Pressure|pressure]] of air and other gases."<ref>Maxwell 1873</ref>
In 1871, [[Biography:Ludwig Boltzmann|Ludwig Boltzmann]] generalized Maxwell's achievement and formulated the [[Maxwell–Boltzmann distribution]]. The [[Logarithm|logarithm]]ic connection between [[Physics:Entropy|entropy]] and [[Probability|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 [[Biography:Albert Einstein|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 [[Biography:David Enskog|David Enskog]] and [[Biography: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 [[Physics:Chapman–Enskog theory|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 [[Physics:Volume|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 [[Physics: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 [[Physics:Thermodynamic limit|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 [[Physics:Force|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 [[Physics:Mass|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 [[Physics: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 Revised Enskog Theory and the Extended BGK model,<ref>{{Cite journal |last=van Enk |first=Steven J. |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}}</ref> relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with [[Physics: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://www.sciencedirect.com/science/article/pii/037843719390357A |journal=Physica A: Statistical Mechanics and its Applications |volume=194 |issue=1 |pages=229–257 |doi=10.1016/0378-4371(93)90357-A |issn=0378-4371}}</ref>

==Equilibrium properties==
=== {{anchor|Pressure and Kinetic Energy}}Pressure and kinetic energy===
<!-- This section is linked from [[Physics:Pressure|Pressure]] -->
In the kinetic theory of gases, the [[Physics:Pressure|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 [[Physics:Characteristic length|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 [[Physics:Momentum|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, ''N'', 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^2  = {\vec{v}_x^2} = {\vec{v}_y^2} = {\vec{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">v = v_i=v_j=v_k,</math>
<math display="block">F = F_i = F_j = F_k,</math>
<math display="block">A_i=A_j=A_k.</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 ''N'' 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 [[Physics:Kinetic energy|kinetic energy]] <math>K_t </math> of the gas is defined as
<math display="block">K_t = \frac{N}{2} m v^2 ,</math>
providing the result
<math display="block">PV = \frac{2}{3} K_t .</math>

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

===Temperature and kinetic energy===
Rewriting the above result for the pressure as <math display="inline">PV = \frac{Nmv^2}{3} </math>, we may combine it with the [[Physics:Ideal gas law|ideal gas law]]

{{NumBlk|:|<math> PV = N k_\mathrm{B} T ,</math>|{{EquationRef|1}}}}

where <math> k_\mathrm{B}</math> is the [[Physics:Boltzmann constant|Boltzmann constant]] and <math> T</math> the [[Physics:Thermodynamic temperature|absolute]] [[Physics:Temperature|temperature]] defined by the ideal gas law, to obtain

<math display="block">k_\mathrm{B} T = {m v^2\over 3}, </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 [[Physics:Root-mean-square speed|root mean-square velocity]], which always exceeds the mean velocity - [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_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> T = \frac{1}{3} {m v^2 \over k_\mathrm{B} } </math>|{{EquationRef|2}}}}

which becomes

{{NumBlk|:|<math> T = \frac{2}{3} \frac{K_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> PV = \frac{2}{3} K_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 [[Physics:Statistical mechanics|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 [[Physics:Equipartition theorem|equipartition theorem]] requires that kinetic energy is partitioned equally between all kinetic [[Degrees of freedom|degrees of freedom]], ''D''. A monotatomic 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 [[Chemistry:Water|water]], is not radially symmetric about any axis, resulting in ''D'' = 6, comprising 3 translational and 3 rotational degrees of freedom.

Because the [[Physics:Equipartition theorem|equipartition theorem]] requires that kinetic energy is partitioned equally, the total kinetic energy is

<math display="block"> K =D K_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_B T . </math>

Therefore, the kinetic energy per kelvin of one mole of monatomic [[Physics:Ideal gas|ideal gas]] (''D'' = 3) is

<math display="block"> K = \frac{D}{2} k_B N_A T = \frac{3}{2} R, </math>

where <math>N_A</math> is the [[Physics:Avogadro constant|Avogadro constant]], and ''R'' is the [[Physics:Ideal gas constant|ideal gas constant]].

Thus, the kinetic energy per unit kelvin of an ideal monoatomic gas can be calculated easily:
* per mole: 12.47 J / K
* per molecule: 20.7 [[Orders of magnitude (energy)|yJ]] / K = 129 μeV / K

At [[Standard conditions for temperature and pressure|standard temperature]] (273.15 K), the kinetic energy can also be obtained:
* per mole: 3406 J
* per molecule: 5.65 [[Orders of magnitude (energy)|zJ]] = 35.2 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 [[Physics:Statistical mechanics|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 [[Physics:Effusion#Physics in Effusion|effusive flow rate]]s, which is useful in applications such as the [[Physics:Gaseous diffusion#Technology|gaseous diffusion]] method for [[Physics: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'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_B T}\right)^{3/2} e^{- \frac{mv^2}{2k_BT}} \, 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>vdt</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) dAdt</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_BT}\right)^{3/2}  e^{- \frac{mv^2}{2k_BT}} \left( v^2 \sin(\theta) \, dv \, d\theta \, d\phi \right).</math>

Integrating this over all appropriate velocities within the constraint <math>v>0,0<\theta<\pi/2,0<\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{\int_0^{\pi/2}\cos \theta \sin \theta d\theta}{\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>0,0<\theta<\pi,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_BT}\right)^{3/2}  e^{- \frac{mv^2}{2k_BT}} \left( v^2 \sin(\theta) \, dv \, d\theta \, d\phi \right).</math>Integrating this over all appropriate velocities within the constraint <math>v>0,0<\theta<\pi/2,0<\phi<2\pi</math> yields the [[Physics:Pressure|pressure]] (consistent with [[Physics:Ideal gas law|Ideal gas law]]):<math display="block">P=\frac{2\int_0^{\pi/2}\cos^2 \theta \sin \theta d\theta}{\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_{kin}\rangle=nk_\mathrm{B} T </math>If this small area <math>A</math> is punched to become a small hole, the [[Physics: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 [[Physics:Ideal gas law|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 [[Physics:Graham's law|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) dAdt</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}{\times} \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,\, 0<\theta<\frac \pi 2,\, 0<\phi<2\pi</math>. The constant <math>\lambda</math> can be determined by the normalization condition <math>\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]], 
* [[Physics:Root-mean-square speed|Root-mean-square speed]]
* [[Arithmetic mean]]
* [[Mean]]
* [[Mode (statistics)]]

=== Mean free path ===
{{Main|Physics:Mean free path}}
In kinetic theory of gases, the [[Physics: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 [[Physics:Cross section#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>l</math> by<math display="block">l = \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==

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 [[Physics:Viscosity|viscosity]], [[Physics:Thermal conductivity|thermal conductivity]], [[Physics:Mass diffusivity|mass diffusivity]] and [[Physics: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, [[Physics:Chapman–Enskog theory#Revised Enskog Theory|Revised Enskog Theory]], was developed in 1983-1987 by [[Biography:E. G. D. Cohen|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}}</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}}</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}}</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}}</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===

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 [[Physics:Couette flow|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 equilibrium distribution of molecular motions.

Inside a dilute gas in a [[Physics:Couette flow|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 l\cos \theta</math>, where <math>l</math> is the [[Physics: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 l \cos \theta \,{d u \over 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 display="block">\begin{cases}
v>0\\
0<\theta<\pi/2\\
0<\phi<2\pi
\end{cases}</math> 
yields the forward momentum transfer per unit time per unit area (also known as [[Physics:Shear stress|shear stress]]):
<math display="block">\tau^{\pm} = \frac {1}{4} \bar v n \cdot m \left( u_{0} \pm \frac {2}{3} l \,{d u \over 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 l \,{d u \over dy} </math>

Combining the above kinetic equation with Newton's law of viscosity
<math display="block">\tau = \eta \,{d u \over 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 l </math>

Combining this equation with the equation for mean free path gives
<math display="block">\eta_{0} = \frac {1} {3 \sqrt{2} } \frac {m \cdot \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} \cdot \frac {k_\mathrm{B}T}{m}} </math>
where <math>v_{p}</math> is the most probable speed. We note that
<math display="block">k_{B} \cdot N_{A} = R \quad \text{and} \quad M = m \cdot N_{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 [[Physics: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{MRT}} { \sigma \cdot N_{A} } </math>

and <math> M </math> is the [[Physics:Molar mass|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 [[Physics:Kinetic diameter|kinetic diameter]] of a molecule in a monomolecular gas. There are no simple general relation between the collision [[Physics:Cross section#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 [[Physics:Lennard-Jones potential|Lennard-Jones potential]] or [[Physics:Morse potential|Morse potential]] which have a negative part that attracts the other molecule from distances longer than the hard core radius. [[Physics: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 [[Physics:Cross section|collision integrals]].

The expression for viscosity obtained from 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 [[Physics:Thermal conductivity|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 [[Physics:Thermal reservoir|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 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_BT}} (v^2 \sin(\theta) \, dv \, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>l\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 l \cos \theta \, {d T \over dy} \right), </math>
where <math>c_v</math> is the [[Physics:Specific heat capacity|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 display="block">\begin{cases}
v>0\\
0<\theta<\pi/2\\
0<\phi<2\pi
\end{cases}</math>

yields the energy transfer per unit time per unit area (also known as [[Physics:Heat flux|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 l \,{d T \over 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 l \,{d T \over dy} </math>

Combining the above kinetic equation with Fourier's law
<math display="block"> q = -\kappa \,{d T \over 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 l </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 [[Physics:Mass diffusivity|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 [[Physics: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 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_BT}} (v^2\sin(\theta) \, dv\, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>l\cos \theta</math> above and below the gas layer, where the local number density is
<math display="block"> n^{\pm} = \left( n_{0} \pm l \cos \theta \, {d n \over 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 display="block">\begin{cases}
v>0\\
0<\theta<\pi/2\\
0<\phi<2\pi
\end{cases}</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} l \, {d n \over 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 l {d n \over dy} </math>

Combining the above kinetic equation with [[Physics:Fick's laws of diffusion#Fick's first law|Fick's first law of diffusion]]
<math display="block"> J = -D {d n \over 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 l </math>

The corresponding expression obtained from 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 [[Chemistry:Chemical potential|chemical potential]]s with density.

== Detailed balance ==

=== Fluctuation and dissipation ===
{{Main|Physics:Fluctuation-dissipation theorem}}
The kinetic theory of gases entails that due to the [[Physics:Microscopic reversibility|microscopic reversibility]] of the gas particles' detailed dynamics, the system must obey the principle of [[Detailed balance|detailed balance]]. Specifically, the [[Physics:Fluctuation-dissipation theorem|fluctuation-dissipation theorem]] applies to the [[Brownian motion]] (or [[Physics:Diffusion|diffusion]]) and the [[Physics:Drag|drag force]], which leads to the [[Physics: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 diffusion coefficient;
* ''{{mvar|μ}}'' is the "mobility", or the ratio of the particle's [[Physics:Terminal velocity|terminal]] [[Physics:Drift velocity|drift velocity]] to an applied [[Physics:Force|force]], {{math|1=''μ'' = ''v''<sub>d</sub>/''F''}};
* {{math|''k''<sub>B</sub>}} is the [[Physics:Boltzmann constant|Boltzmann constant]];
* ''{{mvar|T}}'' is the [[Physics:Absolute temperature|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|Physics: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 [[Physics:Onsager reciprocal relations|Onsager reciprocal relations]] (i.e. the detailed balance of the [[Physics:Microscopic reversibility|reversible dynamics]] of the particles), when applied to the [[Astronomy:Convection|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}}
* [[Physics:BBGKY hierarchy|Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations]]
* [[Boltzmann equation]]
* [[Physics:Chapman–Enskog theory|Chapman–Enskog theory]]
* [[Chemistry:Collision theory|Collision theory]]
* [[Physics:Critical temperature|Critical temperature]]
* [[Physics:Gas laws|Gas laws]]
* [[Physics:Heat|Heat]]
* [[Physics:Interatomic potential|Interatomic potential]]
* [[Physics:Magnetohydrodynamics|Magnetohydrodynamics]]
* [[Maxwell–Boltzmann distribution]]
* [[Mixmaster universe]]
* [[Physics:Thermodynamics|Thermodynamics]]
* [[Vicsek model]]
* [[Vlasov equation]]

==Notes==
{{Reflist|2}}

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==Further reading==
* [[Biography:Sydney Chapman (mathematician)|Sydney Chapman]] and 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
* [[Biography:Joseph O. Hirschfelder|Joseph Oakland Hirschfelder]], [https://history.aip.org/phn/11901002.html Charles Francis Curtiss], and [[Biography:Robert Byron Bird|Robert Byron Bird]] (1964), ''Molecular Theory of Gases and Liquids'', revised edition (Wiley-Interscience), ISBN 978-0471400653
* [[Biography: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 ==
* [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.

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

{{Sourceattribution|Kinetic theory of gases}}