- Kinetic theory
[

temperature of an idealmonatomic gas is a measure related to the averagekinetic energy of its atoms as they move. In this animation, the size ofhelium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two**trillion**fold).]**Kinetic theory**(or**kinetic theory of gases**) attempts to explainmacroscopic properties ofgas es, such as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as wasIsaac Newton 's conjecture, but due tocollision s between molecules moving at different velocities. Kinetic theory is also known as the**kinetic-molecular theory**or the.collision theory **Postulates**The theory for ideal gases makes the following assumptions:

* The gas consists of very small particles, each of which has amass or weight in SI units, kilograms.

* The number of molecules is large such that statistical treatment can be applied.

* These molecules are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container.

* The collisions of gas particles with the walls of the container holding them are perfectly elastic.

* Theinteraction s among molecules arenegligible . They exert noforce s on one another except during collisions.

* The totalvolume of the individual gas molecules added up isnegligible compared to the volume of the container. This is equivalent to stating that theaverage distance separating the gas particles is relatively large compared to their size.

* The molecules are perfectly spherical in shape, and elastic in nature.

* The averagekinetic energy of the gas particles depends only on the temperature of thesystem .

* Relativistic effects are negligible.

* Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than thethermal de Broglie wavelength and the molecules can be treated as classical objects.

* The time during collision of molecule with the container's wall is negligible as comparable to the time between successive collisions.

* The equations of motion of the molecules are time-reversible.In addition, if the gas is in a container, the collisions with the walls are assumed to be instantaneous and elastic.More modern developments relax these assumptions and are based on the

Boltzmann equation . These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects,molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known asvirial expansions . The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.Fact|date=November 2007 In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in theKnudsen number .The kinetic theory has also been extended to include inelastic collisions in

granular matter by Jenkins and others.Fact|date=November 2007**Pressure**Pressure is explained by kinetic theory as arising from the force exerted by gas molecules impacting on the walls of the container. Consider a gas of "N" molecules, each of mass "m", enclosed in a cuboidal container of volume "V". When a gas molecule collides with the wall of the container perpendicular to the "x" coordinate axis and bounces off in the opposite direction with the same speed (anelastic collision ), then themomentum lost by the particle and gained by the wall is::$Delta\; p\_x\; =\; p\_i\; -\; p\_f\; =\; 2\; m\; v\_x,$

where "v

_{x}" is the "x"-component of the initial velocity of the particle.The particle impacts the wall once every 2"l/v

_{x}" time units (where "l" is the length of the container). Although the particle impacts a side wall once every 1"l/v_{x}" time units, only the momentum change on one wall is considered so that the particle produces a momentum change on a particular wall once every 2"l/v_{x}" time units.:$Delta\; t\; =\; frac\{2l\}\{v\_x\}$The

force due to this particle is::$F\; =\; frac\{Delta\; p\}\{Delta\; t\}\; =\; frac\{2\; m\; v\_x\}\{frac\{2l\}\{v\_x\; =\; frac\{m\; v\_x^2\}\{l\}$

The total force acting on the wall is:

:$F\; =\; frac\{msum\_j\; v\_\{jx\}^2\}\{l\}$

where the summation is over all the gas molecules in the container.

The magnitude of the velocity for each particle will follow:

:$v^2\; =\; v\_x^2\; +\; v\_y^2\; +\; v\_z^2$

Now considering the total force acting on all six walls, adding the contributions from each direction we have:

:$mbox\{Total\; Force\}\; =\; 2\; cdot\; frac\{m\}\{l\}(sum\_j\; v\_\{jx\}^2\; +\; sum\_j\; v\_\{jy\}^2\; +\; sum\_j\; v\_\{jz\}^2)\; =\; 2\; cdot\; frac\{m\}\{l\}\; sum\_j\; (v\_\{jx\}^2\; +\; v\_\{jy\}^2\; +\; v\_\{jz\}^2)\; =\; 2\; cdot\; frac\{m\; sum\_j\; v\_\{j\}^2\}\{l\}$

where the factor of two arises from now considering both walls in a given direction.

Assuming there are a large number of particles moving sufficiently randomly, the force on each of the walls will be approximately the same and now considering the force on only one wall we have:

:$F\; =\; frac\{1\}\{6\}\; left(2\; cdot\; frac\{m\; sum\_j\; v\_\{j\}^2\}\{l\}\; ight)\; =\; frac\{m\; sum\_j\; v\_\{j\}^2\}\{3l\}$

The quantity $sum\_j\; v\_\{j\}^2$ can be written as $\{N\}\; overline\{v^2\}$, where the bar denotes an average, in this case an average over all particles. This quantity is also denoted by $v\_\{rms\}^2$ where $v\_\{rms\}$ is the root-mean-square velocity of the collection of particles.

Thus the force can be written as:

:$F\; =\; frac\{Nmv\_\{rms\}^2\}\{3l\}$

Pressure, which is force per unit area, of the gas can then be written as:

:$P\; =\; frac\{F\}\{A\}\; =\; frac\{Nmv\_\{rms\}^2\}\{3Al\}$

where "A" is the area of the wall of which the force exerted on is considered.

Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure

:$P\; =\; \{Nmv\_\{rms\}^2\; over\; 3V\}$

where "V" is the volume. Also, as "Nm" is the total mass of the gas, and mass divided by volume is density

:$P\; =\; \{1\; over\; 3\}\; ho\; v\_\{rms\}^2$

where ρ is the density of the gas.

This result is interesting and significant, because it relates pressure, a

macroscopic property, to the average (translational)kinetic energy per molecule (1/2"mv_{rms}"^{2}), which is amicroscopic property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy.**Temperature and kinetic energy**From the

ideal gas law ,:Eq.(3)_{1}is one important result of the kinetictheory:*The average molecular kinetic energy is proportional tothe absolute temperature*.From Eq.(1) and Eq.(3)

_{1},we have:Thus, the product of pressure andvolume per mole is proportional to the average(translational) molecular kinetic energy.Eq.(1) and Eq.(4)are called the "classical results", which could also be derived from

statistical mechanics ; for more details, see [*[*] .*http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)*]Since there are $displaystyle\; 3N$

degrees of freedom(dofs)in a monoatomic-gas system with $displaystyle\; N$particles,the kinetic energy per dof is:In the kinetic energy per dof,the constant of proportionality of temperature is 1/2 timesBoltzmann constant . This result is relatedto theequipartition theorem .As noted in the article on

heat capacity , diatomicgases should have 7 degrees of freedom, but the lighter gases actas if they have only 5.Thus the kinetic energy per kelvin (monatomic

ideal gas ) is:

* per mole: 12.47 J

* per molecule: 20.7 yJ = 129 μeVAt standard temperature (273.15 K), we get:

* per mole: 3406 J

* per molecule: 5.65 zJ = 35.2 meV**Number of collisions with wall**One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.

Assuming an ideal gas, a derivation [

*[*] results in an equation for total number of collisions per unit time per area:*http://www.chem.arizona.edu/~salzmanr/480a/480ants/collsurf/collsurf.html Collisions With a Surface*]::$A\; =\; frac\{1\}\{4\}frac\{N\}\{V\}\; v\_\{avg\}\; =\; frac\{\; ho\}\{4\}\; sqrt\{frac\{8\; k\; T\}\{pi\; m\; frac\{1\}\{m\}\; ,$

**RMS speeds of molecules**From the kinetic energy formula it can be shown that

:$v\_\{rms\}^2\; =\; frac\{3RT\}\{mbox\{molar\; mass$

with "v" in m/s, "T" in kelvins, and "R" is the

gas constant . The molar mass is given as kg/mol. The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (distribution of speeds).**History**In 1740

Daniel Bernoulli published "Hydrodynamica", which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience asheat is simply thekinetic energy of their motion. The theory was not immediately accepted, in part becauseconservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.Other pioneers of the kinetic theory (which were neglected by their contemporaries) were

Mikhail Lomonosov (1747), [*Lomovosov 1758*]Georges-Louis Le Sage (ca. 1780, published 1818), [*Le Sage 1780/1818*]John Herapath (1816) [*Herapath 1816, 1821*] andJohn James Waterston (1843), [*Waterston 1843*] which connected their research with the development ofmechanical explanations of gravitation . In 1856August Krönig (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles. [*Krönig 1856*]In 1857

Rudolf Clausius , according to his own words independently of Krönig, developed a similar, but much more sophisticated version of the theory which included translational and contrary to Krönig also rotational and vibrational molecular motions. In this same work he introduced the concept ofmean free path of a particle. [*Clausius 1857*] In 1859, after reading a paper by Clausius,James Clerk Maxwell formulated theMaxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. [*Mahon 2003*] 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 calledpressure of air and other gases.” [*Maxwell 1875*] In 1871,Ludwig Boltzmann generalized Maxwell's achievement and formulated theMaxwell–Boltzmann distribution . Also thelogarithm ic connection betweenentropy andprobability was first stated by him.In the beginning of twentieth century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was

Albert Einstein 's (1905) [*Einstein 1905*] andMarian Smoluchowski 's (1906) [*Smoluchowski 1906*] papers onBrownian motion , which succeeded in making certain accurate quantitative predictions based on the kinetic theory.**See also***

Gas laws

*Heat

*Maxwell-Boltzmann distribution

*Thermodynamics

*Collision theory

*Critical temperature **References***Citation

author=Clausius, R.

title =Ueber die Art der Bewegung, welche wir Wärme nennen

journal =Annalen der Physik

volume =100

pages =353–379

year =1857

url=http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.table

doi=10.1002/andp.18571760302*Citation

author=Einstein, A.

title =Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen

journal =Annalen der Physik

volume =17

pages =549–560

year=1905

url=http://www3.interscience.wiley.com/homepages/5006612/549_560.pdf

doi=10.1002/andp.19053220806*Citation

author= Herapath, J.

title =On the physical properties of gases

journal =Annals of Philosophy

year =1816

pages= 56–60

url =http://books.google.com/books?id=dBkAAAAAMAAJ&pg=PA56*Citation

author=Herapath, J.

year= 1821

title=On the Causes, Laws and Phenomena of Heat, Gases, Gravitation

journal= Annals of Philosophy

volume =9

pages =273–293

url=http://books.google.com/books?id=nCsAAAAAMAAJ&pg=RA1-PA273*Citation

author=Krönig, A.

title =Grundzüge einer Theorie der Gase

journal =Annalen der Physik

volume =99

pages =315–322

year =1856

url=http://gallica.bnf.fr/ark:/12148/bpt6k15184h/f327.table

doi=10.1002/andp.18561751008*Citation

author=Le Sage, G.-L.

year=1818

chapter=Physique Mécanique des Georges-Louis Le Sage

editor=Prévost, Pierre

title=Deux Traites de Physique Mécanique

place=Geneva & Paris

publisher=J.J. Paschoud

pages=1–186

chapter-url=http://resolver.sub.uni-goettingen.de/purl?PPN521099943*Citation

author=Lomonosow, M.

year= 1758/1970

chapter=On the Relation of the Amount of Material and Weight

editor= Henry M. Leicester

journal= Mikhail Vasil'evich Lomonosov on the Corpuscular Theory

place = Cambridge

publisher=Harvard University Press

pages =224–233

chapterurl=http://www.archive.org/details/mikhailvasilevic017733mbp*Citation|author=Mahon, Basil

title=The Man Who Changed Everything – the Life of James Clerk Maxwell

place=Hoboken, NJ

publisher=Wiley

year=2003

isbn= 0-470-86171-1*Citation

author=Maxwell, James Clerk

title =Molecules

journal =Nature

volume =417

year=1873

doi=10.1038/417903a

url=http://www.thecore.nus.edu.sg/landow/victorian/science/science_texts/molecules.html*Citation

author=Smoluchowski, M.

title =Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen

journal =Annalen der Physik

volume =21

pages =756–780

year=1906

url=http://gallica.bnf.fr/ark:/12148/bpt6k15328k/f770.chemindefer

doi=10.1002/andp.19063261405*Citation

author = Waterston, John James

year = 1843

title = Thoughts on the Mental Functions (reprinted in his "Papers",**3**, 167, 183.);Endnotes

The Mathematical Theory of Non-uniform Gases : An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in GasesSydney Chapman, T. G. Cowling

**External links*** [

*http://www.math.umd.edu/~lvrmr/History/EarlyTheories.html Early Theories of Gases*]

* [*http://www.lightandmatter.com/html_books/0sn/ch05/ch05.html Thermodynamics*] - 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*] .

* [*http://www.ucdsb.on.ca/tiss/stretton/chem1/gases9.html Introduction*] to the kinetic molecular theory of gases, from The Upper Canada District School Board

* [*http://comp.uark.edu/~jgeabana/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

* [*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.

* [*http://www.bustertests.co.uk/answer/molecular-kinetic-theory/ Molecular kinetic theory fundamentals*]

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2010.*