188:
421:, 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. 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.
5723:
10841:
5428:
7973:, 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
5219:
4579:
3833:
7969:. 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
4984:
6837:
5718:{\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,d\theta \,d\phi &={\frac {1}{2\pi }}\left({\frac {m}{k_{\mathrm {B} }T}}\right)^{2}e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{3}\sin {\theta }\cos {\theta }\,dv\,d\theta \,d\phi )\\\end{aligned}};\quad v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }
4349:
7941:
3627:
4263:
2952:
3547:
9880:
8631:
287:
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. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency
8103:
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
288:
towards equilibrium. 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
6634:
9064:
7809:
4023:
7674:
5897:
71:. It treats a gas as composed of numerous particles, too small to see with a microscope, which are constantly in random motion. Their collisions with each other and with the walls of their container are used to explain physical properties of the gas—for example,
8090:
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
2703:
868:
3340:
9678:
8429:
7247:
10254:
5990:
4692:
6348:
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
5214:{\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,d\theta \,d\phi &=\lambda v\cos {\theta }{\times }\left({\frac {m}{2\pi kT}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )\\\end{aligned}}}
8764:
4574:{\displaystyle P={\frac {2\int _{0}^{\pi /2}\cos ^{2}\theta \sin \theta d\theta }{\int _{0}^{\pi }\sin \theta d\theta }}\times nmv_{\text{rms}}^{2}={\frac {1}{3}}nmv_{\text{rms}}^{2}={\frac {2}{3}}n\langle E_{kin}\rangle =nk_{\mathrm {B} }T}
3828:{\displaystyle 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 {8k_{\mathrm {B} }T}{\pi m}}}.}
2646:
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 based on naive kinetic theory, and the results can be used for analyzing
10128:
8922:
7125:
7353:
7571:
9213:
4769:
1013:
6998:
10386:
5797:
5293:
9997:
8933:
7776:
1906:
2159:
362:
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
7578:
5801:
4336:
3622:
9428:
1370:
5384:
3929:
2071:
9366:
8183:
2550:
740:
8038:
10805:
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
9515:, 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
7134:
1436:
10139:
6406:
In books on elementary kinetic theory 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
7493:
2463:
5901:
5433:
4989:
4608:
1835:
6307:
1074:
10522:
8667:
2398:
406:. 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).
6832:{\displaystyle nv\cos({\theta })\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )}
1524:
20:
7405:
9268:
10791:
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.
1982:
217:
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
2251:
1573:
10692:
7936:{\displaystyle \eta _{0}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {mk_{\mathrm {B} }T}}{\sigma }}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {MRT}}{\sigma \cdot N_{A}}}}
6414:. 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
1697:
727:
402:
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
4258:{\displaystyle \times nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}\left(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi \right).}
10043:
8837:
7040:
1642:
1199:
1138:
6877:
10571:
4950:
3286:
1254:
10440:
9457:
7254:
4816:
8338:
4903:
3239:
2947:{\displaystyle 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_{B}T}}}\,dv_{x}\,dv_{y}\,dv_{z}}
1742:
8212:
315:
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
7500:
6056:
3542:{\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}\left(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi \right).}
649:
11612:
9911:
9875:{\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi )}
9094:
8662:
8626:{\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi )}
4979:
4703:
8284:
6029:
901:
596:
10598:
9484:
9295:
5423:
6085:
3864:
1296:
495:
8239:
7432:
2630:
At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on
8262:
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
6906:
6332:
6242:
5313:
9650:
8401:
7798:
6606:
6161:
3995:
3152:
3017:
10467:
10033:
9587:
8829:
8791:
7701:
7032:
6543:
6490:
6219:
2694:
2579:
1468:
553:
526:
10284:
5735:
6516:
3089:
10279:
9916:
9673:
9610:
9089:
8424:
8361:
6629:
6566:
4862:
4839:
4018:
3955:
3335:
3312:
3198:
3175:
3112:
3063:
3040:
2977:
466:
346:. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.
9630:
9553:
9533:
8381:
8304:
8080:
8060:
7963:
6897:
6586:
6452:
6432:
6262:
6185:
4599:
3975:
3132:
2997:
2002:
1926:
1766:
367:, 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.
12404:
7706:
438:
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.
1304:
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,
894:
6381:
1840:
9511:
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
9377:
1309:
6385:
377:
The rapidly moving particles constantly collide among themselves and with the walls of the container, and all these collisions are perfectly elastic.
12433:
11357:"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,"
9374:
yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as
7801:
370:
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
9300:
8120:
6091:
5224:
2480:
12397:
11127:
6377:
8340:
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
12341:
7978:
11411:
Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium".
1374:
6389:
12366:, (first edition 1939, second edition 1952), third edition 1970 prepared in co-operation with D. Burnett, Cambridge University Press, London
11643:
176:
proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This
7437:
2407:
6368:
In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of
Kinetic gas theory to dense gas mixtures,
12446:
10921:
2097:
1780:
876:, of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed,
6267:
94:
and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.
4268:
3554:
12486:
12364:
The
Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases
12468:
10472:
6410:
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
5318:
3869:
2335:
2012:
4780:
To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with
12019:"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen"
9589:
be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area
9059:{\displaystyle q_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}n\cdot \left(\varepsilon _{0}\pm {\frac {2}{3}}mc_{v}l\,{dT \over dy}\right)}
11609:
6143:
is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision. Let
1473:
8266:. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy
7363:
342:
in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as
9223:
6164:
417:
model, relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with
249:
10391:
267:
developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also
11736:"The Enskog theory for multicomponent mixtures. III. Transport properties of dense binary mixtures with one tracer component"
11269:
10281:
direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
1529:
381:
10654:
662:
8087:
6369:
10821:(matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.
7669:{\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}=2{\sqrt {{\frac {2}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}}}
5892:{\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}
1142:
1081:
11120:
735:, which states that the force experienced by a particle is related to the time rate of change of its momentum, such that
10600:
is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation
10533:
9091:
direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
1203:
12474:
11875:
11186:
10955:
10648:
10396:
297:
12013:
de Groot, S. R., W. A. van
Leeuwen and Ch. G. van Weert (1980), Relativistic Kinetic Theory, North-Holland, Amsterdam.
11343:"Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"
2311:= 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that
1020:
12325:
12213:
11961:
11387:
2612:
863:{\displaystyle F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.}
603:
12401:
10963:
10883:
2315:= 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like
395:
Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.
10818:
3837:
This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed
10632:
10618:
9486:
is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.
8241:
is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.
7242:{\displaystyle \tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}l\,{du \over dy}\right)}
142:
12492:
12489:
allowing high school students to experiment and discover how various factors affect rates of chemical reactions.
10249:{\displaystyle J_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}\cdot \left(n_{0}\pm {\frac {2}{3}}l\,{dn \over dy}\right)}
8100:
1259:
12355:
11113:
9556:
8307:
7000:
where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient
6455:
2697:
339:
97:
The theory's introduction allowed many principal concepts of thermodynamics to be established. It explains the
10001:
Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient
8797:. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient
8099:
which have a negative part that attracts the other molecule from distances longer than the hard core radius.
5985:{\displaystyle v_{\text{rms}}={\sqrt {\frac {3}{2}}}v_{p}={\sqrt {{3}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}
2207:
1931:
12165:
11598:
4687:{\displaystyle \Phi _{\text{effusion}}=J_{\text{collision}}A=nA{\sqrt {\frac {k_{\mathrm {B} }T}{2\pi m}}}.}
3624:
yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
1655:
12518:
10888:
10806:
10800:
7358:
2623:
2604:
150:
4952:; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of
232:
Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were
10878:
10873:
9495:
6844:
1596:
8759:{\displaystyle \varepsilon ^{\pm }=\left(\varepsilon _{0}\pm mc_{v}l\cos \theta \,{dT \over dy}\right),}
4908:
3244:
10893:
9435:
4783:
8316:
8214:
is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and
4867:
3203:
1718:
11870:(3 ed.). Reading, Massachusetts, USA: Addison-Wesley Publishing Company, Inc. pp. 286–291.
11556:
11151:
10810:
10624:
9459:
is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and
8190:
1769:
343:
134:
10052:
8846:
7049:
6034:
10947:
9887:
8638:
7780:
and insert the velocity in the viscosity equation above. This gives the well known equation (with
4955:
358:
The gas consists of very small particles. This smallness of their size is such that the sum of the
327:, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.
214:
12102:
8269:
6007:
558:
384:, meaning that there are no three-body (or higher) interactions, and the particles have no memory.
12513:
11009:
10931:
10576:
9462:
9273:
8105:
8092:
8086:
of a molecule in a monomolecular gas. There are no simple general relation between the collision
2656:
1837:
which leads to a simplified expression of the average translational kinetic energy per molecule,
12480:
12257:
7251:
The net rate of momentum per unit area that is transported across the imaginary surface is thus
6061:
5389:
3840:
2652:
2185:
The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature
471:
221:
determines the temperature of the gas. The theory was not immediately accepted, in part because
12369:
11594:
8794:
8217:
7410:
6107:
732:
387:
Except during collisions, the interactions among molecules are negligible. They exert no other
260:
222:
48:
12443:
11951:
6900:
6314:
6224:
6140:
5298:
1577:
This is an important, non-trivial result of the kinetic theory because it relates pressure, a
72:
12438:
12157:
12085:
11356:
11342:
11176:
11026:
10832:
10527:
10123:{\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}
9635:
9371:
8917:{\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}
8386:
8112:
7970:
7783:
7120:{\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}
6591:
6146:
3980:
3137:
3002:
2635:
2596:
Thus, the kinetic energy per unit kelvin of an ideal monatomic gas can be calculated easily:
2327:
2296:
2289:
498:
414:
410:
237:
157:
12290:
12269:
12233:
12136:
12090:
12033:
11993:
11904:
11825:
11786:
11747:
11677:
11430:
11181:
11161:
11069:
10767:
10445:
10004:
9565:
8800:
8769:
8256:
8250:
7679:
7003:
6521:
6468:
6354:
6190:
2665:
2557:
1446:
531:
504:
334:, a framework for its use in developing transport equations was developed independently by
284:
245:
122:
12161:
12124:
11981:
7348:{\displaystyle \tau =\tau ^{+}-\tau ^{-}={\frac {1}{3}}{\bar {v}}nm\cdot l\,{du \over dy}}
8:
11064:
11054:
10866:
8115:
reduces to the above expression in the limit of infinite dilution, and can be written as
6495:
6343:
3068:
418:
371:
320:
280:
114:
60:
47:
of pressure. The atoms have an average speed relative to their size slowed down here two
12273:
12237:
12186:
12140:
12037:
11997:
11908:
11829:
11790:
11751:
11681:
11434:
10261:
9655:
9592:
9071:
8406:
8343:
6611:
6548:
6401:
4844:
4821:
4000:
3937:
3317:
3294:
3180:
3157:
3094:
3045:
3022:
2959:
12335:
12314:
12190:
11446:
11420:
11242:
11191:
11146:
10939:
10755:
10601:
9615:
9538:
9518:
8366:
8289:
8065:
8045:
7948:
7566:{\displaystyle \eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m\cdot {\bar {v}}}{\sigma }}}
6882:
6571:
6437:
6417:
6247:
6170:
6167:
of one molecule colliding with another. As in the previous section, the number density
4602:
4584:
3960:
3117:
2982:
2648:
2590:
2300:
1987:
1911:
1751:
1745:
444:
331:
156:
The theory was historically significant as the first explicit exercise of the ideas of
130:
11927:
11892:
10817:(matter flow due to temperature gradient, and heat flow due to pressure gradient) and
9208:{\displaystyle q=q_{y}^{+}-q_{y}^{-}=-{\frac {1}{3}}{\bar {v}}nmc_{v}l\,{dT \over dy}}
8664:
above and below the gas layer, and each will contribute a molecular kinetic energy of
4764:{\displaystyle \Phi _{\text{effusion}}={\frac {PA}{\sqrt {2\pi mk_{\mathrm {B} }T}}}.}
2004:
is related to the translational kinetic energy by the description above, resulting in
12471:
to the kinetic molecular theory of gases, from The Upper Canada
District School Board
12377:
12321:
12209:
12182:
11957:
11932:
11871:
11837:
11576:
11572:
11506:
11393:
11383:
11265:
11089:
11031:
10716:
8263:
6122:
2582:
1008:{\displaystyle v^{2}={{\vec {v}}_{x}^{2}}={{\vec {v}}_{y}^{2}}={{\vec {v}}_{z}^{2}}.}
233:
169:
98:
91:
75:
between its temperature, pressure, and volume. The particles are now known to be the
44:
11510:
11450:
11246:
187:
12277:
12241:
12144:
12068:
12041:
12001:
11922:
11912:
11841:
11833:
11794:
11755:
11716:
11685:
11568:
11522:
11438:
11234:
11156:
11084:
11059:
10908:
10704:
10628:
9501:
9218:
8083:
6358:
2402:
Thus, the energy added to the system per gas particle kinetic degree of freedom is
879:
293:
196:
138:
126:
12120:
7575:
Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
354:
The application of kinetic theory to ideal gases makes the following assumptions:
271:
and vibrational molecular motions. In this same work he introduced the concept of
256:
12456:
12450:
12408:
12018:
11616:
11206:
11141:
11074:
11036:
11016:
10636:
8096:
6993:{\displaystyle p_{x}^{\pm }=m\left(u_{0}\pm l\cos \theta \,{du \over dy}\right),}
6112:
4774:
2330:
requires that kinetic energy is partitioned equally, the total kinetic energy is
403:
324:
316:
264:
146:
11442:
7127:
yields the forward momentum transfer per unit time per unit area (also known as
6545:
is along the horizontal direction. The number of molecules arriving at the area
398:
As a simplifying assumption, the particles are usually assumed to have the same
180:
atomistic point of view was rarely considered in the subsequent centuries, when
12508:
12387:
12359:
12258:"Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"
11813:
11196:
11079:
10986:
10898:
10850:
10719:
10644:
10627:
of the gas particles' detailed dynamics, the system must obey the principle of
10133:
9555:
above the lower plate. The non-equilibrium molecular flow is superimposed on a
9512:
6373:
6362:
6221:. The collision cross section per volume or collision cross section density is
6134:
1441:
272:
218:
206:
64:
32:
12502:
12493:
https://www.youtube.com/watch?v=47bF13o8pb8&list=UUXrJjdDeqLgGjJbP1sMnH8A
12281:
12148:
12081:
12046:
12005:
11953:
Molecular
Driving Forces: Statistical Thermodynamics in Chemistry and Biology
11580:
11526:
10381:{\displaystyle J=J_{y}^{+}-J_{y}^{-}=-{\frac {1}{3}}{\bar {v}}l{dn \over dy}}
8062:
is called collision cross section radius or kinetic radius, and the diameter
5792:{\displaystyle v_{\text{p}}={\sqrt {2\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}
5288:{\textstyle v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }
4697:
4343:
2662:
Assume that in the container, the number density (number per unit volume) is
1645:
364:
241:
201:
12373:
12178:
Liboff, R. L. (1990), Kinetic Theory, Prentice-Hall, Englewood Cliffs, N. J.
11917:
11397:
8306:
above the lower plate. The non-equilibrium energy flow is superimposed on a
6004:
is the mass of one molecule of gas in kg. The most probable (or mode) speed
3934:
The momentum transfer to the container wall from particles hitting the area
1017:
Further, assume that the volume is symmetrical about its three dimensions,
12072:
11936:
11666:"The Enskog theory for multicomponent mixtures. I. Linear transport theory"
11238:
11201:
11099:
9992:{\displaystyle n^{\pm }=\left(n_{0}\pm l\cos \theta \,{dn \over dy}\right)}
9508:
7128:
6462:
6407:
3337:
also depends on the velocity distribution; All in all, it calculates to be:
1581:
property, to the translational kinetic energy of the molecules, which is a
335:
177:
12412:
6492:
be the forward velocity of the gas at a horizontal flat layer (labeled as
12056:
11094:
10996:
10991:
10981:
10132:
yields the molecular transfer per unit time per unit area (also known as
9500:
Following a similar logic as above, one can derive the kinetic model for
8255:
Following a similar logic as above, one can derive the kinetic model for
2270:
1772:
1582:
1578:
380:
Interactions (i.e. collisions) between particles are strictly binary and
309:
110:
36:
24:
7771:{\displaystyle k_{B}\cdot N_{A}=R\quad {\text{and}}\quad M=m\cdot N_{A}}
2307:. A monatomic gas is axially symmetric about each spatial axis, so that
2299:
requires that kinetic energy is partitioned equally between all kinetic
10814:
7966:
4265:
Integrating this over all appropriate velocities within the constraint
3551:
Integrating this over all appropriate velocities within the constraint
2288:) are called the "classical results", which could also be derived from
2273:
is proportional to the average translational molecular kinetic energy.
19:
11846:
11775:"The Enskog theory for multicomponent mixtures. IV. Thermal diffusion"
10840:
9270:
gives the equation for thermal conductivity, which is usually denoted
8926:
yields the energy transfer per unit time per unit area (also known as
12246:
11798:
11759:
11720:
11705:"The Enskog theory for multicomponent mixtures. II. Mutual diffusion"
11689:
10640:
8927:
6454:
above the lower plate. The non-equilibrium flow is superimposed on a
6350:
2468:
2323:= 6, comprising 3 translational and 3 rotational degrees of freedom.
1901:{\displaystyle {\frac {1}{2}}mv^{2}={\frac {3}{2}}k_{\mathrm {B} }T.}
301:
276:
226:
181:
173:
118:
87:
28:
10035:
can be considered to be constant over a distance of mean free path.
8831:
can be considered to be constant over a distance of mean free path.
7034:
can be considered to be constant over a distance of mean free path.
1256:
The total surface area on which the gas particles act is therefore
11774:
11735:
11704:
11665:
11425:
11166:
11021:
8834:
Integrating over all appropriate velocities within the constraint
7497:
Combining this equation with the equation for mean free path gives
7037:
Integrating over all appropriate velocities within the constraint
6088:
4339:
2154:{\displaystyle T={\frac {2}{3}}{\frac {K_{t}}{Nk_{\mathrm {B} }}}.}
654:
598:. The gas particle encounters a boundary after characteristic time
435:
289:
268:
106:
80:
10442:
gives the equation for mass diffusivity, which is usually denoted
10038:
Integrating over all appropriate velocities within the constraint
6187:
is defined as the number of molecules per (extensive) volume, or
2467:
Therefore, the kinetic energy per kelvin of one mole of monatomic
12392:
Kinetic Theory: Classical, Quantum, and
Relativistic Descriptions
9913:
above and below the gas layer, where the local number density is
7407:
gives the equation for shear viscosity, which is usually denoted
6101:
4331:{\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }
3866:
of the
Maxwell's velocity distribution, one has to integrate over
3617:{\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }
305:
229:
how the collisions between molecules could be perfectly elastic.
12206:
The Man Who
Changed Everything – the Life of James Clerk Maxwell
11380:
The Man Who Changed Everything – the Life of James Clerk Maxwell
9423:{\displaystyle \kappa =\alpha _{\kappa }\kappa _{0}+\kappa _{c}}
1365:{\displaystyle P={\frac {N{\overline {F}}}{A}}={\frac {NLF}{V}}}
12458:
Temperature and Pressure of an Ideal Gas: The Equation of State
12103:"On the Causes, Laws and Phenomena of Heat, Gases, Gravitation"
6311:
Notice that the unit of the collision cross section per volume
5379:{\displaystyle \int f(v,\theta ,\phi )\,dv\,d\theta \,d\phi =1}
3924:{\displaystyle v>0,0<\theta <\pi ,0<\phi <2\pi }
2066:{\displaystyle T={\frac {1}{3}}{mv^{2} \over k_{\mathrm {B} }}}
359:
248:(1843), which connected their research with the development of
102:
40:
12413:
Macroscopic and kinetic modelling of rarefied polyatomic gases
11868:
Thermodynamics, Kinetic Theory, and Statistical Thermodynamics
259:
created a simple gas-kinetic model, which only considered the
43:
atoms relative to their spacing is shown to scale under 1,950
12495:
A demonstration apparatus for the thermal agitation in gases.
11593:
The average kinetic energy of a fluid is proportional to the
10903:
10723:
2316:
388:
9361:{\displaystyle \kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}l}
8178:{\displaystyle \eta =(1+\alpha _{\eta })\eta _{0}+\eta _{c}}
2545:{\displaystyle K={\frac {D}{2}}k_{B}N_{A}T={\frac {3}{2}}R,}
137:), the kinetic theory is also connected to the principle of
16:
Understanding of gas properties in terms of molecular motion
12483:
linking together kinetic theory concepts, from HyperPhysics
12477:
illustrating the kinetic theory from University of Arkansas
12384:, revised edition (Wiley-Interscience), ISBN 978-0471400653
11773:
Kincaid, J. M.; Cohen, E. G. D.; Lòpez de Haro, M. (1987).
11703:
Kincaid, J. M.; Lòpez de Haro, M.; Cohen, E. G. D. (1983).
11664:
Lòpez de Haro, M.; Cohen, E. G. D.; Kincaid, J. M. (1983).
11225:
Maxwell, J. C. (1867). "On the Dynamical Theory of Gases".
11171:
10116:
9489:
8910:
7113:
6117:
399:
76:
11259:
11142:
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations
12462:
11772:
11702:
11663:
11227:
Philosophical Transactions of the Royal Society of London
8033:{\displaystyle \sigma =\pi \left(2r\right)^{2}=\pi d^{2}}
2319:, is not radially symmetric about any axis, resulting in
210:
68:
9884:
These molecules made their last collision at a distance
8635:
These molecules made their last collision at a distance
1431:{\displaystyle \Rightarrow PV=NLF={\frac {N}{3}}mv^{2}.}
275:
of a particle. In 1859, after reading a paper about the
225:
had not yet been established, and it was not obvious to
12197:, Cambridge: Harvard University Press, pp. 224–233
12195:
Mikhail Vasil'evich Lomonosov on the Corpuscular Theory
11511:"Inelastic collisions and gas-kinetic effects of light"
12187:"On the Relation of the Amount of Material and Weight"
10258:
Note that the molecular transfer from above is in the
6903:. Each molecule will contribute a forward momentum of
5392:
5227:
1934:
1599:
882:
447:
11982:"Ueber die Art der Bewegung, welche wir Wärme nennen"
11632:. New York: University Science Books. pp. 56–61.
11561:
Physica A: Statistical Mechanics and its Applications
11495:. New York, NY: University Science Books. p. 37.
11253:
10657:
10579:
10536:
10475:
10448:
10399:
10287:
10264:
10142:
10046:
10007:
9919:
9890:
9681:
9658:
9638:
9618:
9595:
9568:
9541:
9521:
9465:
9438:
9380:
9303:
9276:
9226:
9097:
9074:
8936:
8840:
8803:
8772:
8670:
8641:
8432:
8409:
8389:
8369:
8346:
8319:
8292:
8272:
8244:
8220:
8193:
8123:
8068:
8048:
7981:
7951:
7812:
7786:
7709:
7682:
7581:
7503:
7488:{\displaystyle \eta _{0}={\frac {1}{3}}{\bar {v}}nml}
7440:
7413:
7366:
7257:
7137:
7043:
7006:
6909:
6885:
6847:
6637:
6614:
6594:
6574:
6551:
6524:
6498:
6471:
6440:
6420:
6317:
6270:
6250:
6227:
6193:
6173:
6149:
6064:
6037:
6010:
5904:
5804:
5738:
5732:
From the kinetic energy formula it can be shown that
5431:
5321:
5301:
4987:
4958:
4911:
4870:
4847:
4824:
4786:
4706:
4611:
4587:
4352:
4271:
4026:
4003:
3983:
3963:
3940:
3872:
3843:
3630:
3557:
3343:
3320:
3297:
3247:
3206:
3183:
3160:
3140:
3120:
3097:
3071:
3048:
3025:
3005:
2985:
2962:
2706:
2668:
2638:
is needed to accurately compute these contributions.
2615:(273.15 K), the kinetic energy can also be obtained:
2560:
2483:
2458:{\displaystyle {\frac {K}{ND}}={\frac {1}{2}}k_{B}T.}
2410:
2338:
2210:
2100:
2015:
1990:
1914:
1843:
1783:
1754:
1721:
1658:
1532:
1476:
1449:
1377:
1312:
1262:
1206:
1145:
1084:
1023:
904:
743:
665:
606:
561:
534:
507:
474:
296:
generalized Maxwell's achievement and formulated the
12059:(1949), "On the Kinetic Theory of Rarefied Gases.",
10623:
The kinetic theory of gases entails that due to the
12172:, Geneva & Paris: J.J. Paschoud, pp. 1–186
11413:
Studies in History and Philosophy of Modern Physics
9068:Note that the energy transfer from above is in the
6058:, and the mean (arithmetic mean, or average) speed
1830:{\displaystyle k_{\mathrm {B} }T={mv^{2} \over 3},}
12394:, third edition (Springer), ISBN 978-0-387-21775-8
12313:
12311:
11733:
10715:is the "mobility", or the ratio of the particle's
10686:
10592:
10565:
10516:
10461:
10434:
10380:
10273:
10248:
10122:
10027:
9991:
9905:
9874:
9667:
9644:
9624:
9604:
9581:
9547:
9527:
9478:
9451:
9422:
9360:
9289:
9262:
9207:
9083:
9058:
8916:
8823:
8785:
8758:
8656:
8625:
8418:
8395:
8375:
8355:
8332:
8298:
8278:
8233:
8206:
8177:
8074:
8054:
8032:
7957:
7935:
7792:
7770:
7695:
7668:
7565:
7487:
7426:
7399:
7347:
7241:
7119:
7026:
6992:
6891:
6871:
6831:
6623:
6600:
6580:
6560:
6537:
6510:
6484:
6446:
6426:
6326:
6302:{\displaystyle l={\frac {1}{n\sigma {\sqrt {2}}}}}
6301:
6256:
6236:
6213:
6179:
6155:
6079:
6050:
6023:
5984:
5891:
5791:
5717:
5417:
5378:
5307:
5287:
5213:
4973:
4944:
4897:
4864:are contained in the tilted pipe with a height of
4856:
4833:
4810:
4763:
4686:
4593:
4573:
4330:
4257:
4012:
3989:
3969:
3949:
3923:
3858:
3827:
3616:
3541:
3329:
3306:
3280:
3233:
3200:are contained in the tilted pipe with a height of
3192:
3169:
3146:
3126:
3106:
3083:
3057:
3042:, will collide with the area within time interval
3034:
3011:
2991:
2971:
2946:
2688:
2573:
2544:
2457:
2392:
2245:
2153:
2065:
1996:
1976:
1920:
1908:The translational kinetic energy of the system is
1900:
1829:
1760:
1736:
1691:
1636:
1567:
1518:
1462:
1430:
1364:
1290:
1248:
1193:
1132:
1068:
1007:
888:
862:
721:
643:
590:
547:
520:
489:
460:
12316:Mathematical Methods in Particle Transport Theory
11811:
6395:
5315:can be determined by the normalization condition
2641:
2183:) is one important result of the kinetic theory:
1588:
12500:
1069:{\displaystyle {\hat {i}},{\hat {j}},{\hat {k}}}
11505:
11260:L.I Ponomarev; I.V Kurchatov (1 January 1993).
10794:
10517:{\displaystyle D_{0}={\frac {1}{3}}{\bar {v}}l}
9535:in the layer increases uniformly with distance
2651:s, which is useful in applications such as the
2393:{\displaystyle K=DK_{t}={\frac {D}{2}}Nmv^{2}.}
1593:Rewriting the above result for the pressure as
441:Consider a gas particle traveling at velocity,
12162:"Physique Mécanique des Georges-Louis Le Sage"
12061:Communications on Pure and Applied Mathematics
11884:
11610:Configuration integral (statistical mechanics)
10612:
8082:is called collision cross section diameter or
3291:The total number of particles that reach area
1300:The pressure exerted by the collisions of the
429:
279:of molecules by Clausius, Scottish physicist
11865:
11121:
6841:These molecules made their last collision at
2979:on the container wall, a particle with speed
2269:Thus, the product of pressure and volume per
657:of the gas particle can then be described as
11949:
11861:
11859:
11857:
11630:Physical Chemistry for the Chemical Sciences
11628:Chang, Raymond; Thoman, John W. Jr. (2014).
11493:Physical Chemistry for the Chemical Sciences
11491:Chang, Raymond; Thoman, John W. Jr. (2014).
11377:
8101:The radius for zero Lennard-Jones potential
4547:
4528:
86:The basic version of the model describes an
11734:Lòpez de Haro, M.; Cohen, E. G. D. (1984).
11627:
11597:, which always exceeds the mean velocity -
11490:
10526:The corresponding expression obtained from
8111:The expression for viscosity obtained from
1519:{\displaystyle K_{t}={\frac {N}{2}}mv^{2},}
12340:: CS1 maint: location missing publisher (
12224:Maxwell, James Clerk (1873), "Molecules",
11890:
11866:Sears, F.W.; Salinger, G.L. (1975). "10".
11814:"The non-linear Enskog-Boltzmann equation"
11128:
11114:
10839:
10390:Combining the above kinetic equation with
9557:Maxwell-Boltzmann equilibrium distribution
9217:Combining the above kinetic equation with
8308:Maxwell-Boltzmann equilibrium distribution
7400:{\displaystyle \tau =\eta \,{du \over dy}}
7357:Combining the above kinetic equation with
6456:Maxwell-Boltzmann equilibrium distribution
6244:, and it is related to the mean free path
3114:. Therefore, all the particles with speed
12245:
12045:
11926:
11916:
11854:
11845:
11544:. New York, NY: University Science Press.
11539:
11424:
10667:
10220:
9963:
9862:
9855:
9848:
9710:
9703:
9612:on one side of the gas layer, with speed
9263:{\displaystyle q=-\kappa \,{dT \over dy}}
9239:
9184:
9030:
8727:
8613:
8606:
8599:
8461:
8454:
8363:on one side of the gas layer, with speed
7703:is the most probable speed. We note that
7376:
7324:
7213:
6961:
6819:
6812:
6805:
6731:
6668:
6661:
6568:on one side of the gas layer, with speed
5696:
5670:
5640:
5633:
5626:
5474:
5467:
5460:
5363:
5356:
5349:
5266:
5240:
5197:
5190:
5183:
5030:
5023:
5016:
4240:
4233:
4226:
4088:
4081:
3524:
3517:
3510:
3372:
3365:
2930:
2916:
2902:
2787:
2773:
2759:
424:
9490:Diffusion Coefficient and diffusion flux
8286:which increases uniformly with distance
4773:The above expression is consistent with
2634:and the total molecular energy. Quantum
1977:{\textstyle K_{t}={\frac {1}{2}}Nmv^{2}}
1775:defined by the ideal gas law, to obtain
186:
18:
11950:Dill, Ken A.; Bromberg, Sarina (2003).
11812:van Beijeren, H.; Ernst, M. H. (1973).
11224:
10813:of the particles), when applied to the
6434:which increase uniformly with distance
6337:
6031:is 81.6% of the root-mean-square speed
4601:is punched to become a small hole, the
2246:{\displaystyle PV={\frac {2}{3}}K_{t}.}
1568:{\displaystyle PV={\frac {2}{3}}K_{t}.}
12501:
11893:"Viscosity of dilute gases and vapors"
11410:
10687:{\displaystyle D=\mu \,k_{\text{B}}T,}
1692:{\displaystyle PV=Nk_{\mathrm {B} }T,}
722:{\displaystyle p_{i}=mv_{i}=mL_{i}/t.}
497:-direction in an enclosed volume with
250:mechanical explanations of gravitation
12382:Molecular Theory of Gases and Liquids
12086:"On the physical properties of gases"
11554:
5727:
12113:, Baldwin, Cradock, and Joy: 273–293
11037:Grand potential / Landau free energy
3154:from the normal that can reach area
2201:
2091:
2006:
1649:
1637:{\textstyle PV={\frac {Nmv^{2}}{3}}}
1194:{\displaystyle F=F_{i}=F_{j}=F_{k},}
1133:{\displaystyle v=v_{i}=v_{j}=v_{k},}
434:In the kinetic theory of gases, the
312:was also first stated by Boltzmann.
12453:- a chapter from an online textbook
12423:, 437–505, DOI 10.1017/jfm.2016.604
11636:
10607:
6872:{\displaystyle y=\pm l\cos \theta }
6402:Viscosity § Momentum transport
51:fold from that at room temperature.
13:
12349:
12170:Deux Traites de Physique Mécanique
12125:"Grundzüge einer Theorie der Gase"
10809:(i.e. the detailed balance of the
10566:{\displaystyle D=\alpha _{D}D_{0}}
9743:
9652:from the normal, in time interval
8494:
8403:from the normal, in time interval
8245:Thermal conductivity and heat flux
7860:
7800:subsequently estimated below) for
7649:
6767:
6701:
6608:from the normal, in time interval
5962:
5869:
5769:
5577:
5520:
5145:
4945:{\displaystyle v\cos(\theta )dAdt}
4746:
4708:
4657:
4613:
4562:
3997:from the normal, in time interval
3801:
3281:{\displaystyle v\cos(\theta )dAdt}
2139:
2055:
1886:
1790:
1728:
1677:
1249:{\displaystyle A_{i}=A_{j}=A_{k}.}
896:, in every direction is identical
778:
761:
409:More modern developments, such as
292:of air and other gases." In 1871,
14:
12530:
12427:
11509:; Nienhuis, Gerard (1991-12-01).
10435:{\displaystyle J=-D{dn \over dy}}
9452:{\displaystyle \alpha _{\kappa }}
6128:
4811:{\displaystyle (v,\theta ,\phi )}
1928:times that of a molecule, namely
404:Dalton's Law of partial pressures
330:Following the development of the
12295:Thoughts on the Mental Functions
11956:. Garland Science. p. 327.
8333:{\displaystyle \varepsilon _{0}}
7802:shear viscosity for dilute gases
6372:, was developed in 1983-1987 by
6139:In kinetic theory of gases, the
4898:{\displaystyle v\cos(\theta )dt}
3234:{\displaystyle v\cos(\theta )dt}
1737:{\displaystyle k_{\mathrm {B} }}
11943:
11805:
11779:The Journal of Chemical Physics
11766:
11740:The Journal of Chemical Physics
11727:
11709:The Journal of Chemical Physics
11696:
11670:The Journal of Chemical Physics
11657:
11621:
11603:
11587:
11557:"Fifty years of kinetic theory"
11548:
11533:
11499:
11484:
11475:
11466:
11457:
11404:
11371:
10633:fluctuation-dissipation theorem
10619:Fluctuation-dissipation theorem
8207:{\displaystyle \alpha _{\eta }}
7745:
7739:
5657:
3065:, if it is within the distance
2698:Maxwell's velocity distribution
205:, which laid the basis for the
143:fluctuation-dissipation theorem
31:is proportional to the average
12208:, Hoboken, New Jersey: Wiley,
11555:Cohen, E. G. D. (1993-03-15).
11332:
11323:
11314:
11305:
11296:
11287:
11278:
11218:
11187:Maxwell–Boltzmann distribution
10649:Einstein–Smoluchowski equation
10505:
10349:
10180:
9869:
9845:
9839:
9820:
9700:
9694:
9333:
9159:
8974:
8620:
8596:
8590:
8571:
8451:
8445:
8149:
8130:
7588:
7551:
7470:
7306:
7167:
6826:
6781:
6658:
6650:
6396:Viscosity and kinetic momentum
6071:
6051:{\displaystyle v_{\text{rms}}}
5811:
5647:
5591:
5457:
5439:
5409:
5346:
5328:
5204:
5159:
5013:
4995:
4927:
4921:
4886:
4880:
4805:
4787:
4223:
4217:
4078:
4072:
4054:
4051:
4045:
4027:
3850:
3769:
3741:
3507:
3501:
3362:
3356:
3263:
3257:
3222:
3216:
2756:
2717:
2642:Collisions with container wall
1589:Temperature and kinetic energy
1378:
1060:
1045:
1030:
984:
955:
926:
644:{\displaystyle t=L_{i}/v_{i}.}
481:
349:
298:Maxwell–Boltzmann distribution
90:. It treats the collisions as
1:
11973:
11540:McQuarrie, Donald A. (1976).
10392:Fick's first law of diffusion
9906:{\displaystyle l\cos \theta }
8657:{\displaystyle l\cos \theta }
4974:{\displaystyle v\cos \theta }
1644:, we may combine it with the
872:Now consider a large number,
101:properties of gases, such as
11838:10.1016/0375-9601(73)90346-0
11644:"5.62 Physical Chemistry II"
11573:10.1016/0378-4371(93)90357-A
10807:Onsager reciprocal relations
10801:Onsager reciprocal relations
10795:Onsager reciprocal relations
8279:{\displaystyle \varepsilon }
6024:{\displaystyle v_{\text{p}}}
3019:from the normal of the area
2696:and that the particles obey
1330:
591:{\displaystyle V=A_{i}L_{i}}
240:(ca. 1780, published 1818),
151:Onsager reciprocal relations
7:
12370:Joseph Oakland Hirschfelder
12312:Williams, M. M. R. (1971).
11443:10.1016/j.shpsb.2017.01.001
10879:Indistinguishable particles
10824:
10613:Fluctuation and dissipation
10593:{\displaystyle \alpha _{D}}
9479:{\displaystyle \kappa _{c}}
9290:{\displaystyle \kappa _{0}}
6087:is 92.1% of the rms speed (
5418:{\textstyle {4}/{\bar {v}}}
2284:
2278:
2259:
2195:
2189:
2179:
2167:
2079:
1705:
430:Pressure and kinetic energy
419:internal degrees of freedom
10:
12535:
12434:PHYSICAL CHEMISTRY – Gases
12417:Journal of Fluid Mechanics
10798:
10616:
9493:
8248:
6399:
6341:
6132:
6080:{\displaystyle {\bar {v}}}
3859:{\displaystyle {\bar {v}}}
1291:{\displaystyle A=3*A_{i}.}
731:We combine the above with
490:{\displaystyle {\hat {i}}}
263:of the particles. In 1857
163:
12256:Smoluchowski, M. (1906),
11891:Hildebrand, J.H. (1976).
11595:root mean-square velocity
10625:microscopic reversibility
10469:when it is a dilute gas:
9297:when it is a dilute gas:
8234:{\displaystyle \eta _{c}}
7434:when it is a dilute gas:
7427:{\displaystyle \eta _{0}}
7359:Newton's law of viscosity
6461:Inside a dilute gas in a
6334:is reciprocal of length.
4841:within the time interval
2292:; for more details, see:
1470:of the gas is defined as
191:Hydrodynamica front cover
135:microscopic reversibility
133:of microscopic dynamics (
12487:Interactive Java Applets
12320:. Butterworths, London.
12282:10.1002/andp.19063261405
12149:10.1002/andp.18561751008
12047:10.1002/andp.19053220806
12006:10.1002/andp.18571760302
11897:Proc Natl Acad Sci U S A
11599:Kinetic Molecular Theory
11527:10.1103/PhysRevA.44.7615
11212:
9496:Fick's laws of diffusion
9370:Similarly to viscosity,
6327:{\displaystyle n\sigma }
6237:{\displaystyle n\sigma }
5308:{\displaystyle \lambda }
1440:The total translational
528:, cross-sectional area,
172:, the Roman philosopher
12439:Early Theories of Gases
12388:Richard Lawrence Liboff
12374:Charles Francis Curtiss
12094:, Robert Baldwin: 56–60
11918:10.1073/pnas.73.12.4302
10922:Thermodynamic ensembles
10874:Spin–statistics theorem
10774:Note that the mobility
9645:{\displaystyle \theta }
8396:{\displaystyle \theta }
8093:Lennard-Jones potential
7793:{\displaystyle \sigma }
6601:{\displaystyle \theta }
6156:{\displaystyle \sigma }
3990:{\displaystyle \theta }
3147:{\displaystyle \theta }
3012:{\displaystyle \theta }
57:kinetic theory of gases
12073:10.1002/cpa.3160020403
11382:. Hoboken, NJ: Wiley.
11360:Philosophical Magazine
11346:Philosophical Magazine
11239:10.1098/rstl.1867.0004
10688:
10594:
10567:
10518:
10463:
10436:
10382:
10275:
10250:
10124:
10029:
9993:
9907:
9876:
9669:
9646:
9626:
9606:
9583:
9559:of molecular motions.
9549:
9529:
9480:
9453:
9424:
9362:
9291:
9264:
9209:
9085:
9060:
8918:
8825:
8795:specific heat capacity
8787:
8760:
8658:
8627:
8420:
8397:
8377:
8357:
8334:
8310:of molecular motions.
8300:
8280:
8235:
8208:
8179:
8076:
8056:
8034:
7959:
7937:
7794:
7772:
7697:
7670:
7567:
7489:
7428:
7401:
7349:
7243:
7121:
7028:
6994:
6893:
6873:
6833:
6625:
6602:
6582:
6562:
6539:
6512:
6486:
6458:of molecular motions.
6448:
6428:
6384:, building on work by
6328:
6303:
6258:
6238:
6215:
6181:
6157:
6108:Root-mean-square speed
6092:distribution of speeds
6081:
6052:
6025:
5986:
5893:
5793:
5719:
5419:
5380:
5309:
5289:
5215:
4975:
4946:
4899:
4858:
4835:
4812:
4765:
4688:
4595:
4575:
4332:
4259:
4014:
3991:
3971:
3951:
3925:
3860:
3829:
3618:
3543:
3331:
3308:
3282:
3235:
3194:
3171:
3148:
3128:
3108:
3085:
3059:
3036:
3013:
2993:
2973:
2956:Then for a small area
2948:
2690:
2575:
2546:
2459:
2394:
2247:
2155:
2067:
1998:
1978:
1922:
1902:
1831:
1762:
1738:
1693:
1638:
1569:
1520:
1464:
1432:
1366:
1292:
1250:
1195:
1134:
1070:
1009:
890:
864:
723:
645:
592:
549:
522:
491:
462:
425:Equilibrium properties
223:conservation of energy
192:
52:
35:of its particles. The
12360:Thomas George Cowling
12291:Waterston, John James
12204:Mahon, Basil (2003),
12101:Herapath, J. (1821),
12017:Einstein, A. (1905),
11980:Clausius, R. (1857),
11542:Statistical Mechanics
11378:Mahon, Basil (2003).
11355:Maxwell, J.C. (1860)
11341:Maxwell, J.C. (1860)
11177:Interatomic potential
11152:Chapman–Enskog theory
11027:Helmholtz free energy
10956:Isoenthalpic–isobaric
10833:Statistical mechanics
10689:
10647:, which leads to the
10595:
10568:
10528:Revised Enskog Theory
10519:
10464:
10462:{\displaystyle D_{0}}
10437:
10383:
10276:
10251:
10125:
10030:
10028:{\displaystyle dn/dy}
9994:
9908:
9877:
9670:
9647:
9627:
9607:
9584:
9582:{\displaystyle n_{0}}
9550:
9530:
9481:
9454:
9425:
9372:Revised Enskog Theory
9363:
9292:
9265:
9210:
9086:
9061:
8919:
8826:
8824:{\displaystyle dT/dy}
8788:
8786:{\displaystyle c_{v}}
8761:
8659:
8628:
8421:
8398:
8378:
8358:
8335:
8301:
8281:
8236:
8209:
8180:
8113:Revised Enskog Theory
8077:
8057:
8035:
7971:Revised Enskog theory
7960:
7938:
7795:
7773:
7698:
7696:{\displaystyle v_{p}}
7671:
7568:
7490:
7429:
7402:
7350:
7244:
7122:
7029:
7027:{\displaystyle du/dy}
6995:
6894:
6874:
6834:
6626:
6603:
6583:
6563:
6540:
6538:{\displaystyle u_{0}}
6513:
6487:
6485:{\displaystyle u_{0}}
6449:
6429:
6370:Revised Enskog Theory
6329:
6304:
6259:
6239:
6216:
6214:{\displaystyle n=N/V}
6182:
6158:
6082:
6053:
6026:
5987:
5894:
5794:
5720:
5420:
5381:
5310:
5290:
5216:
4976:
4947:
4900:
4859:
4836:
4813:
4766:
4689:
4596:
4576:
4333:
4260:
4015:
3992:
3972:
3952:
3926:
3861:
3830:
3619:
3544:
3332:
3314:within time interval
3309:
3283:
3236:
3195:
3177:within time interval
3172:
3149:
3129:
3109:
3086:
3060:
3037:
3014:
2994:
2974:
2949:
2691:
2689:{\displaystyle n=N/V}
2636:statistical mechanics
2600:per mole: 12.47 J / K
2576:
2574:{\displaystyle N_{A}}
2547:
2460:
2395:
2328:equipartition theorem
2297:equipartition theorem
2290:statistical mechanics
2248:
2156:
2068:
1999:
1979:
1923:
1903:
1832:
1763:
1739:
1694:
1639:
1570:
1526:providing the result
1521:
1465:
1463:{\displaystyle K_{t}}
1433:
1367:
1293:
1251:
1196:
1135:
1071:
1010:
891:
865:
724:
646:
593:
550:
548:{\displaystyle A_{i}}
523:
521:{\displaystyle L_{i}}
499:characteristic length
492:
463:
411:Revised Enskog Theory
344:Chapman–Enskog theory
238:Georges-Louis Le Sage
190:
184:ideas were dominant.
158:statistical mechanics
22:
12107:Annals of Philosophy
12091:Annals of Philosophy
11182:Magnetohydrodynamics
11162:Critical temperature
10768:absolute temperature
10655:
10631:. Specifically, the
10577:
10534:
10473:
10446:
10397:
10285:
10262:
10140:
10044:
10005:
9917:
9888:
9679:
9656:
9636:
9616:
9593:
9566:
9539:
9519:
9463:
9436:
9378:
9301:
9274:
9224:
9095:
9072:
8934:
8838:
8801:
8770:
8668:
8639:
8430:
8407:
8387:
8367:
8344:
8317:
8290:
8270:
8257:thermal conductivity
8251:Thermal conductivity
8218:
8191:
8121:
8066:
8046:
7979:
7949:
7810:
7784:
7707:
7680:
7579:
7501:
7438:
7411:
7364:
7255:
7135:
7041:
7004:
6907:
6883:
6845:
6635:
6612:
6592:
6572:
6549:
6522:
6496:
6469:
6438:
6418:
6355:thermal conductivity
6338:Transport properties
6315:
6268:
6248:
6225:
6191:
6171:
6147:
6062:
6035:
6008:
5902:
5802:
5736:
5429:
5390:
5319:
5299:
5225:
5221:with the constraint
4985:
4956:
4909:
4868:
4845:
4822:
4784:
4704:
4609:
4585:
4350:
4269:
4024:
4001:
3981:
3961:
3938:
3870:
3841:
3628:
3555:
3341:
3318:
3295:
3245:
3204:
3181:
3158:
3138:
3118:
3095:
3069:
3046:
3023:
3003:
2983:
2960:
2704:
2666:
2613:standard temperature
2558:
2481:
2408:
2336:
2208:
2098:
2013:
1988:
1932:
1912:
1841:
1781:
1752:
1719:
1656:
1597:
1530:
1474:
1447:
1375:
1310:
1260:
1204:
1143:
1082:
1021:
902:
880:
741:
663:
604:
559:
532:
505:
472:
445:
323:'s (1906) papers on
285:Maxwell distribution
261:translational motion
246:John James Waterston
123:thermal conductivity
115:transport properties
12519:Classical mechanics
12274:1906AnP...326..756V
12238:1873Natur...8..437.
12141:1856AnP...175..315K
12038:1905AnP...322..549E
11998:1857AnP...176..353C
11909:1976PNAS...73.4302H
11830:1973PhLA...43..367V
11791:1987JChPh..86..963K
11752:1984JChPh..80..408L
11682:1983JChPh..78.2746L
11435:2017SHPMP..57...53G
11302:Herapath 1816, 1821
10964:Isothermal–isobaric
10867:Particle statistics
10811:reversible dynamics
10602:chemical potentials
10326:
10308:
10157:
9136:
9118:
8951:
8106:collision integrals
6924:
6511:{\displaystyle y=0}
6344:Transport phenomena
4581:If this small area
4511:
4477:
4435:
4387:
3710:
3669:
3084:{\displaystyle vdt}
2622:per molecule: 5.65
2603:per molecule: 20.7
1984:. The temperature,
1000:
971:
942:
843:
733:Newton's second law
372:thermodynamic limit
321:Marian Smoluchowski
304:connection between
281:James Clerk Maxwell
12449:2017-02-28 at the
12407:2021-07-25 at the
12402:Henning Struchtrup
12298:(reprinted in his
12262:Annalen der Physik
12191:Henry M. Leicester
12129:Annalen der Physik
12026:Annalen der Physik
11986:Annalen der Physik
11651:MIT OpenCourseWare
11615:2012-04-28 at the
11507:van Enk, Steven J.
11192:Mixmaster universe
11147:Boltzmann equation
10904:Anyonic statistics
10756:Boltzmann constant
10684:
10590:
10563:
10530:may be written as
10514:
10459:
10432:
10378:
10312:
10294:
10274:{\displaystyle -y}
10271:
10246:
10143:
10120:
10115:
10025:
9989:
9903:
9872:
9668:{\displaystyle dt}
9665:
9642:
9622:
9605:{\displaystyle dA}
9602:
9579:
9545:
9525:
9476:
9449:
9420:
9358:
9287:
9260:
9205:
9122:
9104:
9084:{\displaystyle -y}
9081:
9056:
8937:
8914:
8909:
8821:
8783:
8756:
8654:
8623:
8419:{\displaystyle dt}
8416:
8393:
8373:
8356:{\displaystyle dA}
8353:
8330:
8296:
8276:
8264:thermal reservoirs
8231:
8204:
8175:
8072:
8052:
8030:
7955:
7933:
7790:
7768:
7693:
7666:
7563:
7485:
7424:
7397:
7345:
7239:
7117:
7112:
7024:
6990:
6910:
6889:
6869:
6829:
6624:{\displaystyle dt}
6621:
6598:
6578:
6561:{\displaystyle dA}
6558:
6535:
6508:
6482:
6444:
6424:
6324:
6299:
6254:
6234:
6211:
6177:
6153:
6077:
6048:
6021:
6000:is in kelvin, and
5982:
5889:
5789:
5728:Speed of molecules
5715:
5652:
5415:
5376:
5305:
5285:
5211:
5209:
4971:
4942:
4895:
4857:{\displaystyle dt}
4854:
4834:{\displaystyle dA}
4831:
4818:that hit the area
4808:
4761:
4696:Combined with the
4684:
4603:effusive flow rate
4591:
4571:
4497:
4463:
4421:
4365:
4328:
4255:
4013:{\displaystyle dt}
4010:
3987:
3967:
3950:{\displaystyle dA}
3947:
3921:
3856:
3825:
3696:
3647:
3614:
3539:
3330:{\displaystyle dt}
3327:
3307:{\displaystyle dA}
3304:
3278:
3231:
3193:{\displaystyle dt}
3190:
3170:{\displaystyle dA}
3167:
3144:
3124:
3107:{\displaystyle dA}
3104:
3081:
3058:{\displaystyle dt}
3055:
3035:{\displaystyle dA}
3032:
3009:
2989:
2972:{\displaystyle dA}
2969:
2944:
2686:
2657:isotope separation
2649:effusive flow rate
2591:ideal gas constant
2571:
2542:
2455:
2390:
2301:degrees of freedom
2243:
2187:. From equations (
2151:
2063:
1994:
1974:
1918:
1898:
1827:
1758:
1746:Boltzmann constant
1734:
1689:
1634:
1565:
1516:
1460:
1428:
1362:
1288:
1246:
1191:
1130:
1066:
1005:
977:
948:
919:
886:
860:
829:
719:
641:
588:
545:
518:
487:
461:{\textstyle v_{i}}
458:
332:Boltzmann equation
193:
141:, in terms of the
131:time reversibility
53:
12378:Robert Byron Bird
11903:(12): 4302–4303.
11818:Physics Letters A
11521:(11): 7615–7625.
11515:Physical Review A
11481:Smoluchowski 1906
11293:Le Sage 1780/1818
11271:978-0-7503-0251-7
11138:
11137:
11032:Gibbs free energy
10884:Maxwell–Boltzmann
10675:
10508:
10497:
10430:
10376:
10352:
10341:
10239:
10215:
10183:
10172:
9982:
9816:
9753:
9625:{\displaystyle v}
9548:{\displaystyle y}
9528:{\displaystyle n}
9504:of a dilute gas:
9336:
9325:
9258:
9203:
9162:
9151:
9049:
9012:
8977:
8966:
8746:
8567:
8504:
8376:{\displaystyle v}
8299:{\displaystyle y}
8259:of a dilute gas:
8075:{\displaystyle d}
8055:{\displaystyle r}
7958:{\displaystyle M}
7931:
7912:
7895:
7892:
7873:
7869:
7843:
7840:
7743:
7664:
7662:
7635:
7607:
7606:
7591:
7561:
7554:
7534:
7531:
7473:
7462:
7395:
7343:
7309:
7298:
7232:
7208:
7170:
7159:
6980:
6892:{\displaystyle l}
6777:
6711:
6581:{\displaystyle v}
6447:{\displaystyle y}
6427:{\displaystyle u}
6363:thermal diffusion
6297:
6294:
6257:{\displaystyle l}
6180:{\displaystyle n}
6163:be the collision
6123:Mode (statistics)
6074:
6045:
6018:
5977:
5975:
5928:
5927:
5912:
5884:
5882:
5855:
5830:
5829:
5814:
5784:
5782:
5746:
5691:
5587:
5530:
5501:
5412:
5261:
5155:
5090:
4756:
4755:
4714:
4679:
4678:
4632:
4619:
4594:{\displaystyle A}
4523:
4504:
4489:
4470:
4452:
4342:(consistent with
4192:
4129:
3970:{\displaystyle v}
3853:
3820:
3819:
3786:
3772:
3758:
3744:
3727:
3638:
3476:
3413:
3127:{\displaystyle v}
2992:{\displaystyle v}
2898:
2835:
2714:
2653:gaseous diffusion
2607:/ K = 129 μeV / K
2583:Avogadro constant
2534:
2498:
2437:
2424:
2369:
2267:
2266:
2228:
2175:
2174:
2146:
2115:
2087:
2086:
2061:
2030:
1997:{\displaystyle T}
1956:
1921:{\displaystyle N}
1878:
1852:
1822:
1761:{\displaystyle T}
1713:
1712:
1632:
1550:
1498:
1410:
1360:
1339:
1333:
1063:
1048:
1033:
987:
958:
929:
855:
818:
786:
484:
413:and the Extended
234:Mikhail Lomonosov
213:. In this work,
92:perfectly elastic
12526:
12345:
12339:
12331:
12319:
12297:
12284:
12250:
12249:
12247:10.1038/008437a0
12232:(204): 437–441,
12218:
12198:
12173:
12151:
12114:
12095:
12075:
12050:
12049:
12023:
12008:
11968:
11967:
11947:
11941:
11940:
11930:
11920:
11888:
11882:
11881:
11863:
11852:
11851:
11849:
11809:
11803:
11802:
11799:10.1063/1.452243
11770:
11764:
11763:
11760:10.1063/1.446463
11731:
11725:
11724:
11721:10.1063/1.446388
11715:(9): 4509–4521.
11700:
11694:
11693:
11690:10.1063/1.444985
11676:(5): 2746–2759.
11661:
11655:
11654:
11648:
11640:
11634:
11633:
11625:
11619:
11607:
11601:
11591:
11585:
11584:
11552:
11546:
11545:
11537:
11531:
11530:
11503:
11497:
11496:
11488:
11482:
11479:
11473:
11470:
11464:
11461:
11455:
11454:
11428:
11408:
11402:
11401:
11375:
11369:
11336:
11330:
11327:
11321:
11318:
11312:
11309:
11303:
11300:
11294:
11291:
11285:
11282:
11276:
11275:
11262:The Quantum Dice
11257:
11251:
11250:
11222:
11157:Collision theory
11130:
11123:
11116:
10909:Braid statistics
10843:
10829:
10828:
10790:
10764:
10753:
10742:
10713:
10705:mass diffusivity
10701:
10693:
10691:
10690:
10685:
10677:
10676:
10673:
10629:detailed balance
10608:Detailed balance
10599:
10597:
10596:
10591:
10589:
10588:
10572:
10570:
10569:
10564:
10562:
10561:
10552:
10551:
10523:
10521:
10520:
10515:
10510:
10509:
10501:
10498:
10490:
10485:
10484:
10468:
10466:
10465:
10460:
10458:
10457:
10441:
10439:
10438:
10433:
10431:
10429:
10421:
10413:
10387:
10385:
10384:
10379:
10377:
10375:
10367:
10359:
10354:
10353:
10345:
10342:
10334:
10325:
10320:
10307:
10302:
10280:
10278:
10277:
10272:
10255:
10253:
10252:
10247:
10245:
10241:
10240:
10238:
10230:
10222:
10216:
10208:
10203:
10202:
10185:
10184:
10176:
10173:
10165:
10156:
10151:
10129:
10127:
10126:
10121:
10119:
10118:
10087:
10034:
10032:
10031:
10026:
10018:
9998:
9996:
9995:
9990:
9988:
9984:
9983:
9981:
9973:
9965:
9947:
9946:
9929:
9928:
9912:
9910:
9909:
9904:
9881:
9879:
9878:
9873:
9832:
9831:
9819:
9818:
9817:
9815:
9811:
9810:
9797:
9796:
9795:
9782:
9772:
9771:
9767:
9758:
9754:
9752:
9748:
9747:
9746:
9726:
9674:
9672:
9671:
9666:
9651:
9649:
9648:
9643:
9631:
9629:
9628:
9623:
9611:
9609:
9608:
9603:
9588:
9586:
9585:
9580:
9578:
9577:
9554:
9552:
9551:
9546:
9534:
9532:
9531:
9526:
9513:number densities
9502:mass diffusivity
9485:
9483:
9482:
9477:
9475:
9474:
9458:
9456:
9455:
9450:
9448:
9447:
9429:
9427:
9426:
9421:
9419:
9418:
9406:
9405:
9396:
9395:
9367:
9365:
9364:
9359:
9354:
9353:
9338:
9337:
9329:
9326:
9318:
9313:
9312:
9296:
9294:
9293:
9288:
9286:
9285:
9269:
9267:
9266:
9261:
9259:
9257:
9249:
9241:
9214:
9212:
9211:
9206:
9204:
9202:
9194:
9186:
9180:
9179:
9164:
9163:
9155:
9152:
9144:
9135:
9130:
9117:
9112:
9090:
9088:
9087:
9082:
9065:
9063:
9062:
9057:
9055:
9051:
9050:
9048:
9040:
9032:
9026:
9025:
9013:
9005:
9000:
8999:
8979:
8978:
8970:
8967:
8959:
8950:
8945:
8923:
8921:
8920:
8915:
8913:
8912:
8881:
8830:
8828:
8827:
8822:
8814:
8792:
8790:
8789:
8784:
8782:
8781:
8765:
8763:
8762:
8757:
8752:
8748:
8747:
8745:
8737:
8729:
8714:
8713:
8698:
8697:
8680:
8679:
8663:
8661:
8660:
8655:
8632:
8630:
8629:
8624:
8583:
8582:
8570:
8569:
8568:
8566:
8562:
8561:
8548:
8547:
8546:
8533:
8523:
8522:
8518:
8509:
8505:
8503:
8499:
8498:
8497:
8477:
8425:
8423:
8422:
8417:
8402:
8400:
8399:
8394:
8382:
8380:
8379:
8374:
8362:
8360:
8359:
8354:
8339:
8337:
8336:
8331:
8329:
8328:
8305:
8303:
8302:
8297:
8285:
8283:
8282:
8277:
8240:
8238:
8237:
8232:
8230:
8229:
8213:
8211:
8210:
8205:
8203:
8202:
8184:
8182:
8181:
8176:
8174:
8173:
8161:
8160:
8148:
8147:
8084:kinetic diameter
8081:
8079:
8078:
8073:
8061:
8059:
8058:
8053:
8039:
8037:
8036:
8031:
8029:
8028:
8013:
8012:
8007:
8003:
7964:
7962:
7961:
7956:
7942:
7940:
7939:
7934:
7932:
7930:
7929:
7928:
7902:
7901:
7896:
7894:
7893:
7888:
7879:
7874:
7865:
7864:
7863:
7850:
7849:
7844:
7842:
7841:
7836:
7827:
7822:
7821:
7799:
7797:
7796:
7791:
7777:
7775:
7774:
7769:
7767:
7766:
7744:
7741:
7732:
7731:
7719:
7718:
7702:
7700:
7699:
7694:
7692:
7691:
7675:
7673:
7672:
7667:
7665:
7663:
7658:
7654:
7653:
7652:
7641:
7636:
7628:
7626:
7618:
7617:
7608:
7602:
7598:
7593:
7592:
7584:
7572:
7570:
7569:
7564:
7562:
7557:
7556:
7555:
7547:
7537:
7535:
7533:
7532:
7527:
7518:
7513:
7512:
7494:
7492:
7491:
7486:
7475:
7474:
7466:
7463:
7455:
7450:
7449:
7433:
7431:
7430:
7425:
7423:
7422:
7406:
7404:
7403:
7398:
7396:
7394:
7386:
7378:
7354:
7352:
7351:
7346:
7344:
7342:
7334:
7326:
7311:
7310:
7302:
7299:
7291:
7286:
7285:
7273:
7272:
7248:
7246:
7245:
7240:
7238:
7234:
7233:
7231:
7223:
7215:
7209:
7201:
7196:
7195:
7172:
7171:
7163:
7160:
7152:
7147:
7146:
7126:
7124:
7123:
7118:
7116:
7115:
7084:
7033:
7031:
7030:
7025:
7017:
6999:
6997:
6996:
6991:
6986:
6982:
6981:
6979:
6971:
6963:
6945:
6944:
6923:
6918:
6898:
6896:
6895:
6890:
6878:
6876:
6875:
6870:
6838:
6836:
6835:
6830:
6804:
6793:
6792:
6780:
6779:
6778:
6776:
6772:
6771:
6770:
6756:
6755:
6754:
6741:
6730:
6729:
6725:
6716:
6712:
6710:
6706:
6705:
6704:
6684:
6657:
6630:
6628:
6627:
6622:
6607:
6605:
6604:
6599:
6587:
6585:
6584:
6579:
6567:
6565:
6564:
6559:
6544:
6542:
6541:
6536:
6534:
6533:
6517:
6515:
6514:
6509:
6491:
6489:
6488:
6483:
6481:
6480:
6453:
6451:
6450:
6445:
6433:
6431:
6430:
6425:
6382:M. Lòpez de Haro
6359:mass diffusivity
6333:
6331:
6330:
6325:
6308:
6306:
6305:
6300:
6298:
6296:
6295:
6290:
6278:
6263:
6261:
6260:
6255:
6243:
6241:
6240:
6235:
6220:
6218:
6217:
6212:
6207:
6186:
6184:
6183:
6178:
6162:
6160:
6159:
6154:
6086:
6084:
6083:
6078:
6076:
6075:
6067:
6057:
6055:
6054:
6049:
6047:
6046:
6043:
6030:
6028:
6027:
6022:
6020:
6019:
6016:
5991:
5989:
5988:
5983:
5978:
5976:
5971:
5967:
5966:
5965:
5954:
5949:
5944:
5939:
5938:
5929:
5920:
5919:
5914:
5913:
5910:
5898:
5896:
5895:
5890:
5885:
5883:
5878:
5874:
5873:
5872:
5861:
5856:
5848:
5846:
5841:
5840:
5831:
5825:
5821:
5816:
5815:
5807:
5798:
5796:
5795:
5790:
5785:
5783:
5778:
5774:
5773:
5772:
5761:
5753:
5748:
5747:
5744:
5724:
5722:
5721:
5716:
5692:
5684:
5653:
5625:
5614:
5603:
5602:
5590:
5589:
5588:
5586:
5582:
5581:
5580:
5566:
5565:
5564:
5551:
5541:
5540:
5535:
5531:
5529:
5525:
5524:
5523:
5509:
5502:
5500:
5489:
5424:
5422:
5421:
5416:
5414:
5413:
5405:
5402:
5397:
5385:
5383:
5382:
5377:
5314:
5312:
5311:
5306:
5294:
5292:
5291:
5286:
5262:
5254:
5220:
5218:
5217:
5212:
5210:
5182:
5171:
5170:
5158:
5157:
5156:
5154:
5150:
5149:
5148:
5134:
5133:
5132:
5119:
5109:
5108:
5104:
5095:
5091:
5089:
5072:
5065:
5060:
4980:
4978:
4977:
4972:
4951:
4949:
4948:
4943:
4905:and a volume of
4904:
4902:
4901:
4896:
4863:
4861:
4860:
4855:
4840:
4838:
4837:
4832:
4817:
4815:
4814:
4809:
4770:
4768:
4767:
4762:
4757:
4751:
4750:
4749:
4730:
4729:
4721:
4716:
4715:
4712:
4693:
4691:
4690:
4685:
4680:
4677:
4666:
4662:
4661:
4660:
4649:
4648:
4634:
4633:
4630:
4621:
4620:
4617:
4600:
4598:
4597:
4592:
4580:
4578:
4577:
4572:
4567:
4566:
4565:
4546:
4545:
4524:
4516:
4510:
4505:
4502:
4490:
4482:
4476:
4471:
4468:
4453:
4451:
4434:
4429:
4419:
4397:
4396:
4386:
4382:
4373:
4360:
4337:
4335:
4334:
4329:
4303:
4264:
4262:
4261:
4256:
4251:
4247:
4210:
4209:
4195:
4194:
4193:
4191:
4187:
4186:
4173:
4172:
4171:
4158:
4148:
4147:
4143:
4134:
4130:
4128:
4124:
4123:
4104:
4019:
4017:
4016:
4011:
3996:
3994:
3993:
3988:
3976:
3974:
3973:
3968:
3956:
3954:
3953:
3948:
3930:
3928:
3927:
3922:
3865:
3863:
3862:
3857:
3855:
3854:
3846:
3834:
3832:
3831:
3826:
3821:
3818:
3810:
3806:
3805:
3804:
3790:
3789:
3787:
3779:
3774:
3773:
3765:
3759:
3751:
3746:
3745:
3737:
3728:
3726:
3709:
3704:
3694:
3668:
3664:
3655:
3645:
3640:
3639:
3636:
3623:
3621:
3620:
3615:
3589:
3548:
3546:
3545:
3540:
3535:
3531:
3494:
3493:
3479:
3478:
3477:
3475:
3471:
3470:
3457:
3456:
3455:
3442:
3432:
3431:
3427:
3418:
3414:
3412:
3408:
3407:
3388:
3336:
3334:
3333:
3328:
3313:
3311:
3310:
3305:
3287:
3285:
3284:
3279:
3241:and a volume of
3240:
3238:
3237:
3232:
3199:
3197:
3196:
3191:
3176:
3174:
3173:
3168:
3153:
3151:
3150:
3145:
3133:
3131:
3130:
3125:
3113:
3111:
3110:
3105:
3090:
3088:
3087:
3082:
3064:
3062:
3061:
3056:
3041:
3039:
3038:
3033:
3018:
3016:
3015:
3010:
2998:
2996:
2995:
2990:
2978:
2976:
2975:
2970:
2953:
2951:
2950:
2945:
2943:
2942:
2929:
2928:
2915:
2914:
2901:
2900:
2899:
2897:
2893:
2892:
2879:
2878:
2877:
2864:
2854:
2853:
2849:
2840:
2836:
2834:
2830:
2829:
2810:
2800:
2799:
2786:
2785:
2772:
2771:
2755:
2754:
2742:
2741:
2729:
2728:
2716:
2715:
2712:
2695:
2693:
2692:
2687:
2682:
2619:per mole: 3406 J
2580:
2578:
2577:
2572:
2570:
2569:
2551:
2549:
2548:
2543:
2535:
2527:
2519:
2518:
2509:
2508:
2499:
2491:
2464:
2462:
2461:
2456:
2448:
2447:
2438:
2430:
2425:
2423:
2412:
2399:
2397:
2396:
2391:
2386:
2385:
2370:
2362:
2357:
2356:
2261:
2252:
2250:
2249:
2244:
2239:
2238:
2229:
2221:
2202:
2169:
2160:
2158:
2157:
2152:
2147:
2145:
2144:
2143:
2142:
2128:
2127:
2118:
2116:
2108:
2092:
2081:
2072:
2070:
2069:
2064:
2062:
2060:
2059:
2058:
2048:
2047:
2046:
2033:
2031:
2023:
2007:
2003:
2001:
2000:
1995:
1983:
1981:
1980:
1975:
1973:
1972:
1957:
1949:
1944:
1943:
1927:
1925:
1924:
1919:
1907:
1905:
1904:
1899:
1891:
1890:
1889:
1879:
1871:
1866:
1865:
1853:
1845:
1836:
1834:
1833:
1828:
1823:
1818:
1817:
1816:
1803:
1795:
1794:
1793:
1767:
1765:
1764:
1759:
1743:
1741:
1740:
1735:
1733:
1732:
1731:
1707:
1698:
1696:
1695:
1690:
1682:
1681:
1680:
1650:
1643:
1641:
1640:
1635:
1633:
1628:
1627:
1626:
1610:
1574:
1572:
1571:
1566:
1561:
1560:
1551:
1543:
1525:
1523:
1522:
1517:
1512:
1511:
1499:
1491:
1486:
1485:
1469:
1467:
1466:
1461:
1459:
1458:
1437:
1435:
1434:
1429:
1424:
1423:
1411:
1403:
1371:
1369:
1368:
1363:
1361:
1356:
1345:
1340:
1335:
1334:
1326:
1320:
1297:
1295:
1294:
1289:
1284:
1283:
1255:
1253:
1252:
1247:
1242:
1241:
1229:
1228:
1216:
1215:
1200:
1198:
1197:
1192:
1187:
1186:
1174:
1173:
1161:
1160:
1139:
1137:
1136:
1131:
1126:
1125:
1113:
1112:
1100:
1099:
1075:
1073:
1072:
1067:
1065:
1064:
1056:
1050:
1049:
1041:
1035:
1034:
1026:
1014:
1012:
1011:
1006:
1001:
999:
994:
989:
988:
980:
972:
970:
965:
960:
959:
951:
943:
941:
936:
931:
930:
922:
914:
913:
895:
893:
892:
887:
869:
867:
866:
861:
856:
854:
853:
844:
842:
837:
824:
819:
817:
816:
807:
806:
805:
792:
787:
785:
781:
775:
774:
773:
764:
758:
753:
752:
728:
726:
725:
720:
712:
707:
706:
691:
690:
675:
674:
650:
648:
647:
642:
637:
636:
627:
622:
621:
597:
595:
594:
589:
587:
586:
577:
576:
554:
552:
551:
546:
544:
543:
527:
525:
524:
519:
517:
516:
496:
494:
493:
488:
486:
485:
477:
467:
465:
464:
459:
457:
456:
294:Ludwig Boltzmann
197:Daniel Bernoulli
139:detailed balance
127:mass diffusivity
73:the relationship
12534:
12533:
12529:
12528:
12527:
12525:
12524:
12523:
12499:
12498:
12463:Project PHYSNET
12451:Wayback Machine
12430:
12409:Wayback Machine
12352:
12350:Further reading
12333:
12332:
12328:
12289:
12268:(14): 756–780,
12255:
12223:
12216:
12203:
12181:
12166:Prévost, Pierre
12156:
12135:(10): 315–322,
12119:
12100:
12080:
12055:
12021:
12016:
11979:
11976:
11971:
11964:
11948:
11944:
11889:
11885:
11878:
11864:
11855:
11810:
11806:
11771:
11767:
11732:
11728:
11701:
11697:
11662:
11658:
11646:
11642:
11641:
11637:
11626:
11622:
11617:Wayback Machine
11608:
11604:
11592:
11588:
11553:
11549:
11538:
11534:
11504:
11500:
11489:
11485:
11480:
11476:
11471:
11467:
11462:
11458:
11409:
11405:
11390:
11376:
11372:
11366: : 21–37.
11352: : 19–32.
11337:
11333:
11328:
11324:
11319:
11315:
11310:
11306:
11301:
11297:
11292:
11288:
11283:
11279:
11272:
11258:
11254:
11223:
11219:
11215:
11207:Vlasov equation
11134:
11105:
11104:
11050:
11042:
11041:
11017:Internal energy
11012:
11002:
11001:
10977:
10969:
10968:
10948:Grand canonical
10924:
10914:
10913:
10869:
10827:
10803:
10797:
10785:
10775:
10762:
10752:
10746:
10737:
10727:
10711:
10699:
10672:
10668:
10656:
10653:
10652:
10637:Brownian motion
10635:applies to the
10621:
10615:
10610:
10584:
10580:
10578:
10575:
10574:
10557:
10553:
10547:
10543:
10535:
10532:
10531:
10500:
10499:
10489:
10480:
10476:
10474:
10471:
10470:
10453:
10449:
10447:
10444:
10443:
10422:
10414:
10412:
10398:
10395:
10394:
10368:
10360:
10358:
10344:
10343:
10333:
10321:
10316:
10303:
10298:
10286:
10283:
10282:
10263:
10260:
10259:
10231:
10223:
10221:
10207:
10198:
10194:
10193:
10189:
10175:
10174:
10164:
10152:
10147:
10141:
10138:
10137:
10114:
10113:
10092:
10091:
10083:
10065:
10064:
10048:
10047:
10045:
10042:
10041:
10014:
10006:
10003:
10002:
9974:
9966:
9964:
9942:
9938:
9937:
9933:
9924:
9920:
9918:
9915:
9914:
9889:
9886:
9885:
9827:
9823:
9806:
9802:
9798:
9791:
9787:
9783:
9781:
9777:
9773:
9763:
9759:
9742:
9741:
9737:
9730:
9725:
9721:
9720:
9680:
9677:
9676:
9657:
9654:
9653:
9637:
9634:
9633:
9617:
9614:
9613:
9594:
9591:
9590:
9573:
9569:
9567:
9564:
9563:
9540:
9537:
9536:
9520:
9517:
9516:
9498:
9492:
9470:
9466:
9464:
9461:
9460:
9443:
9439:
9437:
9434:
9433:
9414:
9410:
9401:
9397:
9391:
9387:
9379:
9376:
9375:
9349:
9345:
9328:
9327:
9317:
9308:
9304:
9302:
9299:
9298:
9281:
9277:
9275:
9272:
9271:
9250:
9242:
9240:
9225:
9222:
9221:
9195:
9187:
9185:
9175:
9171:
9154:
9153:
9143:
9131:
9126:
9113:
9108:
9096:
9093:
9092:
9073:
9070:
9069:
9041:
9033:
9031:
9021:
9017:
9004:
8995:
8991:
8990:
8986:
8969:
8968:
8958:
8946:
8941:
8935:
8932:
8931:
8908:
8907:
8886:
8885:
8877:
8859:
8858:
8842:
8841:
8839:
8836:
8835:
8810:
8802:
8799:
8798:
8777:
8773:
8771:
8768:
8767:
8738:
8730:
8728:
8709:
8705:
8693:
8689:
8688:
8684:
8675:
8671:
8669:
8666:
8665:
8640:
8637:
8636:
8578:
8574:
8557:
8553:
8549:
8542:
8538:
8534:
8532:
8528:
8524:
8514:
8510:
8493:
8492:
8488:
8481:
8476:
8472:
8471:
8431:
8428:
8427:
8408:
8405:
8404:
8388:
8385:
8384:
8368:
8365:
8364:
8345:
8342:
8341:
8324:
8320:
8318:
8315:
8314:
8291:
8288:
8287:
8271:
8268:
8267:
8253:
8247:
8225:
8221:
8219:
8216:
8215:
8198:
8194:
8192:
8189:
8188:
8169:
8165:
8156:
8152:
8143:
8139:
8122:
8119:
8118:
8097:Morse potential
8067:
8064:
8063:
8047:
8044:
8043:
8024:
8020:
8008:
7996:
7992:
7991:
7980:
7977:
7976:
7950:
7947:
7946:
7924:
7920:
7913:
7900:
7887:
7883:
7878:
7859:
7858:
7854:
7848:
7835:
7831:
7826:
7817:
7813:
7811:
7808:
7807:
7785:
7782:
7781:
7762:
7758:
7740:
7727:
7723:
7714:
7710:
7708:
7705:
7704:
7687:
7683:
7681:
7678:
7677:
7648:
7647:
7643:
7642:
7640:
7627:
7625:
7613:
7609:
7597:
7583:
7582:
7580:
7577:
7576:
7546:
7545:
7538:
7536:
7526:
7522:
7517:
7508:
7504:
7502:
7499:
7498:
7465:
7464:
7454:
7445:
7441:
7439:
7436:
7435:
7418:
7414:
7412:
7409:
7408:
7387:
7379:
7377:
7365:
7362:
7361:
7335:
7327:
7325:
7301:
7300:
7290:
7281:
7277:
7268:
7264:
7256:
7253:
7252:
7224:
7216:
7214:
7200:
7191:
7187:
7186:
7182:
7162:
7161:
7151:
7142:
7138:
7136:
7133:
7132:
7111:
7110:
7089:
7088:
7080:
7062:
7061:
7045:
7044:
7042:
7039:
7038:
7013:
7005:
7002:
7001:
6972:
6964:
6962:
6940:
6936:
6935:
6931:
6919:
6914:
6908:
6905:
6904:
6884:
6881:
6880:
6846:
6843:
6842:
6800:
6788:
6784:
6766:
6765:
6761:
6757:
6750:
6746:
6742:
6740:
6736:
6732:
6721:
6717:
6700:
6699:
6695:
6688:
6683:
6679:
6678:
6653:
6636:
6633:
6632:
6613:
6610:
6609:
6593:
6590:
6589:
6573:
6570:
6569:
6550:
6547:
6546:
6529:
6525:
6523:
6520:
6519:
6497:
6494:
6493:
6476:
6472:
6470:
6467:
6466:
6439:
6436:
6435:
6419:
6416:
6415:
6404:
6398:
6386:H. van Beijeren
6346:
6340:
6316:
6313:
6312:
6289:
6282:
6277:
6269:
6266:
6265:
6249:
6246:
6245:
6226:
6223:
6222:
6203:
6192:
6189:
6188:
6172:
6169:
6168:
6148:
6145:
6144:
6137:
6131:
6113:Arithmetic mean
6066:
6065:
6063:
6060:
6059:
6042:
6038:
6036:
6033:
6032:
6015:
6011:
6009:
6006:
6005:
5961:
5960:
5956:
5955:
5953:
5945:
5943:
5934:
5930:
5918:
5909:
5905:
5903:
5900:
5899:
5868:
5867:
5863:
5862:
5860:
5847:
5845:
5836:
5832:
5820:
5806:
5805:
5803:
5800:
5799:
5768:
5767:
5763:
5762:
5760:
5752:
5743:
5739:
5737:
5734:
5733:
5730:
5683:
5651:
5650:
5621:
5610:
5598:
5594:
5576:
5575:
5571:
5567:
5560:
5556:
5552:
5550:
5546:
5542:
5536:
5519:
5518:
5514:
5513:
5508:
5504:
5503:
5493:
5488:
5481:
5432:
5430:
5427:
5426:
5404:
5403:
5398:
5393:
5391:
5388:
5387:
5320:
5317:
5316:
5300:
5297:
5296:
5295:. The constant
5253:
5226:
5223:
5222:
5208:
5207:
5178:
5166:
5162:
5144:
5143:
5139:
5135:
5128:
5124:
5120:
5118:
5114:
5110:
5100:
5096:
5076:
5071:
5067:
5066:
5061:
5056:
5037:
4988:
4986:
4983:
4982:
4957:
4954:
4953:
4910:
4907:
4906:
4869:
4866:
4865:
4846:
4843:
4842:
4823:
4820:
4819:
4785:
4782:
4781:
4745:
4744:
4740:
4722:
4720:
4711:
4707:
4705:
4702:
4701:
4667:
4656:
4655:
4651:
4650:
4647:
4629:
4625:
4616:
4612:
4610:
4607:
4606:
4586:
4583:
4582:
4561:
4560:
4556:
4535:
4531:
4515:
4506:
4501:
4481:
4472:
4467:
4430:
4425:
4420:
4392:
4388:
4378:
4374:
4369:
4361:
4359:
4351:
4348:
4347:
4299:
4270:
4267:
4266:
4205:
4201:
4200:
4196:
4182:
4178:
4174:
4167:
4163:
4159:
4157:
4153:
4149:
4139:
4135:
4119:
4115:
4108:
4103:
4099:
4098:
4025:
4022:
4021:
4002:
3999:
3998:
3982:
3979:
3978:
3962:
3959:
3958:
3939:
3936:
3935:
3871:
3868:
3867:
3845:
3844:
3842:
3839:
3838:
3811:
3800:
3799:
3795:
3791:
3788:
3778:
3764:
3763:
3750:
3736:
3735:
3705:
3700:
3695:
3660:
3656:
3651:
3646:
3644:
3635:
3631:
3629:
3626:
3625:
3585:
3556:
3553:
3552:
3489:
3485:
3484:
3480:
3466:
3462:
3458:
3451:
3447:
3443:
3441:
3437:
3433:
3423:
3419:
3403:
3399:
3392:
3387:
3383:
3382:
3342:
3339:
3338:
3319:
3316:
3315:
3296:
3293:
3292:
3246:
3243:
3242:
3205:
3202:
3201:
3182:
3179:
3178:
3159:
3156:
3155:
3139:
3136:
3135:
3119:
3116:
3115:
3096:
3093:
3092:
3070:
3067:
3066:
3047:
3044:
3043:
3024:
3021:
3020:
3004:
3001:
3000:
2984:
2981:
2980:
2961:
2958:
2957:
2938:
2934:
2924:
2920:
2910:
2906:
2888:
2884:
2880:
2873:
2869:
2865:
2863:
2859:
2855:
2845:
2841:
2825:
2821:
2814:
2809:
2805:
2804:
2795:
2791:
2781:
2777:
2767:
2763:
2750:
2746:
2737:
2733:
2724:
2720:
2711:
2707:
2705:
2702:
2701:
2678:
2667:
2664:
2663:
2644:
2565:
2561:
2559:
2556:
2555:
2526:
2514:
2510:
2504:
2500:
2490:
2482:
2479:
2478:
2443:
2439:
2429:
2416:
2411:
2409:
2406:
2405:
2381:
2377:
2361:
2352:
2348:
2337:
2334:
2333:
2234:
2230:
2220:
2209:
2206:
2205:
2138:
2137:
2133:
2129:
2123:
2119:
2117:
2107:
2099:
2096:
2095:
2054:
2053:
2049:
2042:
2038:
2034:
2032:
2022:
2014:
2011:
2010:
1989:
1986:
1985:
1968:
1964:
1948:
1939:
1935:
1933:
1930:
1929:
1913:
1910:
1909:
1885:
1884:
1880:
1870:
1861:
1857:
1844:
1842:
1839:
1838:
1812:
1808:
1804:
1802:
1789:
1788:
1784:
1782:
1779:
1778:
1753:
1750:
1749:
1727:
1726:
1722:
1720:
1717:
1716:
1676:
1675:
1671:
1657:
1654:
1653:
1622:
1618:
1611:
1609:
1598:
1595:
1594:
1591:
1556:
1552:
1542:
1531:
1528:
1527:
1507:
1503:
1490:
1481:
1477:
1475:
1472:
1471:
1454:
1450:
1448:
1445:
1444:
1419:
1415:
1402:
1376:
1373:
1372:
1346:
1344:
1325:
1321:
1319:
1311:
1308:
1307:
1279:
1275:
1261:
1258:
1257:
1237:
1233:
1224:
1220:
1211:
1207:
1205:
1202:
1201:
1182:
1178:
1169:
1165:
1156:
1152:
1144:
1141:
1140:
1121:
1117:
1108:
1104:
1095:
1091:
1083:
1080:
1079:
1055:
1054:
1040:
1039:
1025:
1024:
1022:
1019:
1018:
995:
990:
979:
978:
976:
966:
961:
950:
949:
947:
937:
932:
921:
920:
918:
909:
905:
903:
900:
899:
881:
878:
877:
849:
845:
838:
833:
825:
823:
812:
808:
801:
797:
793:
791:
777:
776:
769:
765:
760:
759:
757:
748:
744:
742:
739:
738:
708:
702:
698:
686:
682:
670:
666:
664:
661:
660:
632:
628:
623:
617:
613:
605:
602:
601:
582:
578:
572:
568:
560:
557:
556:
539:
535:
533:
530:
529:
512:
508:
506:
503:
502:
476:
475:
473:
470:
469:
452:
448:
446:
443:
442:
432:
427:
391:on one another.
352:
325:Brownian motion
317:Albert Einstein
283:formulated the
265:Rudolf Clausius
166:
147:Brownian motion
17:
12:
11:
5:
12532:
12522:
12521:
12516:
12514:Thermodynamics
12511:
12497:
12496:
12490:
12484:
12478:
12475:Java animation
12472:
12466:
12454:
12444:Thermodynamics
12441:
12436:
12429:
12428:External links
12426:
12425:
12424:
12395:
12385:
12367:
12356:Sydney Chapman
12351:
12348:
12347:
12346:
12326:
12308:
12307:
12286:
12285:
12252:
12251:
12220:
12219:
12214:
12200:
12199:
12179:
12175:
12174:
12158:Le Sage, G.-L.
12153:
12152:
12116:
12115:
12097:
12096:
12077:
12076:
12067:(4): 331–407,
12052:
12051:
12032:(8): 549–560,
12014:
12010:
12009:
11992:(3): 353–379,
11975:
11972:
11970:
11969:
11962:
11942:
11883:
11877:978-0201068948
11876:
11853:
11824:(4): 367–368.
11804:
11785:(2): 963–975.
11765:
11746:(1): 408–415.
11726:
11695:
11656:
11635:
11620:
11602:
11586:
11567:(1): 229–257.
11547:
11532:
11498:
11483:
11474:
11465:
11456:
11403:
11388:
11370:
11368:
11367:
11362:, 4th series,
11353:
11348:, 4th series,
11331:
11322:
11313:
11311:Waterston 1843
11304:
11295:
11286:
11284:Lomonosov 1758
11277:
11270:
11252:
11216:
11214:
11211:
11210:
11209:
11204:
11199:
11197:Thermodynamics
11194:
11189:
11184:
11179:
11174:
11169:
11164:
11159:
11154:
11149:
11144:
11136:
11135:
11133:
11132:
11125:
11118:
11110:
11107:
11106:
11103:
11102:
11097:
11092:
11087:
11082:
11077:
11072:
11067:
11062:
11057:
11051:
11048:
11047:
11044:
11043:
11040:
11039:
11034:
11029:
11024:
11019:
11013:
11008:
11007:
11004:
11003:
11000:
10999:
10994:
10989:
10984:
10978:
10975:
10974:
10971:
10970:
10967:
10966:
10958:
10950:
10942:
10934:
10932:Microcanonical
10925:
10920:
10919:
10916:
10915:
10912:
10911:
10906:
10901:
10899:Parastatistics
10896:
10891:
10886:
10881:
10876:
10870:
10865:
10864:
10861:
10860:
10859:
10858:
10856:Kinetic theory
10853:
10851:Thermodynamics
10845:
10844:
10836:
10835:
10826:
10823:
10799:Main article:
10796:
10793:
10783:
10772:
10771:
10759:
10750:
10744:
10735:
10722:to an applied
10720:drift velocity
10708:
10683:
10680:
10671:
10666:
10663:
10660:
10617:Main article:
10614:
10611:
10609:
10606:
10604:with density.
10587:
10583:
10560:
10556:
10550:
10546:
10542:
10539:
10513:
10507:
10504:
10496:
10493:
10488:
10483:
10479:
10456:
10452:
10428:
10425:
10420:
10417:
10411:
10408:
10405:
10402:
10374:
10371:
10366:
10363:
10357:
10351:
10348:
10340:
10337:
10332:
10329:
10324:
10319:
10315:
10311:
10306:
10301:
10297:
10293:
10290:
10270:
10267:
10244:
10237:
10234:
10229:
10226:
10219:
10214:
10211:
10206:
10201:
10197:
10192:
10188:
10182:
10179:
10171:
10168:
10163:
10160:
10155:
10150:
10146:
10134:diffusion flux
10117:
10112:
10109:
10106:
10103:
10100:
10097:
10094:
10093:
10090:
10086:
10082:
10079:
10076:
10073:
10070:
10067:
10066:
10063:
10060:
10057:
10054:
10053:
10051:
10024:
10021:
10017:
10013:
10010:
9987:
9980:
9977:
9972:
9969:
9962:
9959:
9956:
9953:
9950:
9945:
9941:
9936:
9932:
9927:
9923:
9902:
9899:
9896:
9893:
9871:
9868:
9865:
9861:
9858:
9854:
9851:
9847:
9844:
9841:
9838:
9835:
9830:
9826:
9822:
9814:
9809:
9805:
9801:
9794:
9790:
9786:
9780:
9776:
9770:
9766:
9762:
9757:
9751:
9745:
9740:
9736:
9733:
9729:
9724:
9719:
9716:
9713:
9709:
9706:
9702:
9699:
9696:
9693:
9690:
9687:
9684:
9664:
9661:
9641:
9621:
9601:
9598:
9576:
9572:
9544:
9524:
9491:
9488:
9473:
9469:
9446:
9442:
9417:
9413:
9409:
9404:
9400:
9394:
9390:
9386:
9383:
9357:
9352:
9348:
9344:
9341:
9335:
9332:
9324:
9321:
9316:
9311:
9307:
9284:
9280:
9256:
9253:
9248:
9245:
9238:
9235:
9232:
9229:
9201:
9198:
9193:
9190:
9183:
9178:
9174:
9170:
9167:
9161:
9158:
9150:
9147:
9142:
9139:
9134:
9129:
9125:
9121:
9116:
9111:
9107:
9103:
9100:
9080:
9077:
9054:
9047:
9044:
9039:
9036:
9029:
9024:
9020:
9016:
9011:
9008:
9003:
8998:
8994:
8989:
8985:
8982:
8976:
8973:
8965:
8962:
8957:
8954:
8949:
8944:
8940:
8911:
8906:
8903:
8900:
8897:
8894:
8891:
8888:
8887:
8884:
8880:
8876:
8873:
8870:
8867:
8864:
8861:
8860:
8857:
8854:
8851:
8848:
8847:
8845:
8820:
8817:
8813:
8809:
8806:
8780:
8776:
8755:
8751:
8744:
8741:
8736:
8733:
8726:
8723:
8720:
8717:
8712:
8708:
8704:
8701:
8696:
8692:
8687:
8683:
8678:
8674:
8653:
8650:
8647:
8644:
8622:
8619:
8616:
8612:
8609:
8605:
8602:
8598:
8595:
8592:
8589:
8586:
8581:
8577:
8573:
8565:
8560:
8556:
8552:
8545:
8541:
8537:
8531:
8527:
8521:
8517:
8513:
8508:
8502:
8496:
8491:
8487:
8484:
8480:
8475:
8470:
8467:
8464:
8460:
8457:
8453:
8450:
8447:
8444:
8441:
8438:
8435:
8415:
8412:
8392:
8372:
8352:
8349:
8327:
8323:
8295:
8275:
8246:
8243:
8228:
8224:
8201:
8197:
8172:
8168:
8164:
8159:
8155:
8151:
8146:
8142:
8138:
8135:
8132:
8129:
8126:
8071:
8051:
8027:
8023:
8019:
8016:
8011:
8006:
8002:
7999:
7995:
7990:
7987:
7984:
7954:
7927:
7923:
7919:
7916:
7911:
7908:
7905:
7899:
7891:
7886:
7882:
7877:
7872:
7868:
7862:
7857:
7853:
7847:
7839:
7834:
7830:
7825:
7820:
7816:
7789:
7765:
7761:
7757:
7754:
7751:
7748:
7738:
7735:
7730:
7726:
7722:
7717:
7713:
7690:
7686:
7661:
7657:
7651:
7646:
7639:
7634:
7631:
7624:
7621:
7616:
7612:
7605:
7601:
7596:
7590:
7587:
7560:
7553:
7550:
7544:
7541:
7530:
7525:
7521:
7516:
7511:
7507:
7484:
7481:
7478:
7472:
7469:
7461:
7458:
7453:
7448:
7444:
7421:
7417:
7393:
7390:
7385:
7382:
7375:
7372:
7369:
7341:
7338:
7333:
7330:
7323:
7320:
7317:
7314:
7308:
7305:
7297:
7294:
7289:
7284:
7280:
7276:
7271:
7267:
7263:
7260:
7237:
7230:
7227:
7222:
7219:
7212:
7207:
7204:
7199:
7194:
7190:
7185:
7181:
7178:
7175:
7169:
7166:
7158:
7155:
7150:
7145:
7141:
7114:
7109:
7106:
7103:
7100:
7097:
7094:
7091:
7090:
7087:
7083:
7079:
7076:
7073:
7070:
7067:
7064:
7063:
7060:
7057:
7054:
7051:
7050:
7048:
7023:
7020:
7016:
7012:
7009:
6989:
6985:
6978:
6975:
6970:
6967:
6960:
6957:
6954:
6951:
6948:
6943:
6939:
6934:
6930:
6927:
6922:
6917:
6913:
6901:mean free path
6888:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6828:
6825:
6822:
6818:
6815:
6811:
6808:
6803:
6799:
6796:
6791:
6787:
6783:
6775:
6769:
6764:
6760:
6753:
6749:
6745:
6739:
6735:
6728:
6724:
6720:
6715:
6709:
6703:
6698:
6694:
6691:
6687:
6682:
6677:
6674:
6671:
6667:
6664:
6660:
6656:
6652:
6649:
6646:
6643:
6640:
6620:
6617:
6597:
6577:
6557:
6554:
6532:
6528:
6507:
6504:
6501:
6479:
6475:
6443:
6423:
6397:
6394:
6374:E. G. D. Cohen
6339:
6336:
6323:
6320:
6293:
6288:
6285:
6281:
6276:
6273:
6253:
6233:
6230:
6210:
6206:
6202:
6199:
6196:
6176:
6152:
6141:mean free path
6135:Mean free path
6133:Main article:
6130:
6129:Mean free path
6127:
6126:
6125:
6120:
6115:
6110:
6105:
6073:
6070:
6041:
6014:
5981:
5974:
5970:
5964:
5959:
5952:
5948:
5942:
5937:
5933:
5926:
5923:
5917:
5908:
5888:
5881:
5877:
5871:
5866:
5859:
5854:
5851:
5844:
5839:
5835:
5828:
5824:
5819:
5813:
5810:
5788:
5781:
5777:
5771:
5766:
5759:
5756:
5751:
5742:
5729:
5726:
5714:
5711:
5708:
5705:
5702:
5699:
5695:
5690:
5687:
5682:
5679:
5676:
5673:
5669:
5666:
5663:
5660:
5656:
5649:
5646:
5643:
5639:
5636:
5632:
5629:
5624:
5620:
5617:
5613:
5609:
5606:
5601:
5597:
5593:
5585:
5579:
5574:
5570:
5563:
5559:
5555:
5549:
5545:
5539:
5534:
5528:
5522:
5517:
5512:
5507:
5499:
5496:
5492:
5487:
5484:
5482:
5480:
5477:
5473:
5470:
5466:
5463:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5434:
5425:, and overall:
5411:
5408:
5401:
5396:
5375:
5372:
5369:
5366:
5362:
5359:
5355:
5352:
5348:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5304:
5284:
5281:
5278:
5275:
5272:
5269:
5265:
5260:
5257:
5252:
5249:
5246:
5243:
5239:
5236:
5233:
5230:
5206:
5203:
5200:
5196:
5193:
5189:
5186:
5181:
5177:
5174:
5169:
5165:
5161:
5153:
5147:
5142:
5138:
5131:
5127:
5123:
5117:
5113:
5107:
5103:
5099:
5094:
5088:
5085:
5082:
5079:
5075:
5070:
5064:
5059:
5055:
5052:
5049:
5046:
5043:
5040:
5038:
5036:
5033:
5029:
5026:
5022:
5019:
5015:
5012:
5009:
5006:
5003:
5000:
4997:
4994:
4991:
4990:
4970:
4967:
4964:
4961:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4853:
4850:
4830:
4827:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4760:
4754:
4748:
4743:
4739:
4736:
4733:
4728:
4725:
4719:
4710:
4700:, this yields
4683:
4676:
4673:
4670:
4665:
4659:
4654:
4646:
4643:
4640:
4637:
4628:
4624:
4615:
4590:
4570:
4564:
4559:
4555:
4552:
4549:
4544:
4541:
4538:
4534:
4530:
4527:
4522:
4519:
4514:
4509:
4500:
4496:
4493:
4488:
4485:
4480:
4475:
4466:
4462:
4459:
4456:
4450:
4447:
4444:
4441:
4438:
4433:
4428:
4424:
4418:
4415:
4412:
4409:
4406:
4403:
4400:
4395:
4391:
4385:
4381:
4377:
4372:
4368:
4364:
4358:
4355:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4302:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4254:
4250:
4246:
4243:
4239:
4236:
4232:
4229:
4225:
4222:
4219:
4216:
4213:
4208:
4204:
4199:
4190:
4185:
4181:
4177:
4170:
4166:
4162:
4156:
4152:
4146:
4142:
4138:
4133:
4127:
4122:
4118:
4114:
4111:
4107:
4102:
4097:
4094:
4091:
4087:
4084:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4009:
4006:
3986:
3966:
3946:
3943:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3852:
3849:
3824:
3817:
3814:
3809:
3803:
3798:
3794:
3785:
3782:
3777:
3771:
3768:
3762:
3757:
3754:
3749:
3743:
3740:
3734:
3731:
3725:
3722:
3719:
3716:
3713:
3708:
3703:
3699:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3667:
3663:
3659:
3654:
3650:
3643:
3634:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3588:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3538:
3534:
3530:
3527:
3523:
3520:
3516:
3513:
3509:
3506:
3503:
3500:
3497:
3492:
3488:
3483:
3474:
3469:
3465:
3461:
3454:
3450:
3446:
3440:
3436:
3430:
3426:
3422:
3417:
3411:
3406:
3402:
3398:
3395:
3391:
3386:
3381:
3378:
3375:
3371:
3368:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3326:
3323:
3303:
3300:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3189:
3186:
3166:
3163:
3143:
3123:
3103:
3100:
3091:from the area
3080:
3077:
3074:
3054:
3051:
3031:
3028:
3008:
2988:
2968:
2965:
2941:
2937:
2933:
2927:
2923:
2919:
2913:
2909:
2905:
2896:
2891:
2887:
2883:
2876:
2872:
2868:
2862:
2858:
2852:
2848:
2844:
2839:
2833:
2828:
2824:
2820:
2817:
2813:
2808:
2803:
2798:
2794:
2790:
2784:
2780:
2776:
2770:
2766:
2762:
2758:
2753:
2749:
2745:
2740:
2736:
2732:
2727:
2723:
2719:
2710:
2685:
2681:
2677:
2674:
2671:
2643:
2640:
2628:
2627:
2620:
2609:
2608:
2601:
2568:
2564:
2541:
2538:
2533:
2530:
2525:
2522:
2517:
2513:
2507:
2503:
2497:
2494:
2489:
2486:
2454:
2451:
2446:
2442:
2436:
2433:
2428:
2422:
2419:
2415:
2389:
2384:
2380:
2376:
2373:
2368:
2365:
2360:
2355:
2351:
2347:
2344:
2341:
2265:
2264:
2255:
2253:
2242:
2237:
2233:
2227:
2224:
2219:
2216:
2213:
2173:
2172:
2163:
2161:
2150:
2141:
2136:
2132:
2126:
2122:
2114:
2111:
2106:
2103:
2089:which becomes
2085:
2084:
2075:
2073:
2057:
2052:
2045:
2041:
2037:
2029:
2026:
2021:
2018:
1993:
1971:
1967:
1963:
1960:
1955:
1952:
1947:
1942:
1938:
1917:
1897:
1894:
1888:
1883:
1877:
1874:
1869:
1864:
1860:
1856:
1851:
1848:
1826:
1821:
1815:
1811:
1807:
1801:
1798:
1792:
1787:
1757:
1730:
1725:
1711:
1710:
1701:
1699:
1688:
1685:
1679:
1674:
1670:
1667:
1664:
1661:
1631:
1625:
1621:
1617:
1614:
1608:
1605:
1602:
1590:
1587:
1564:
1559:
1555:
1549:
1546:
1541:
1538:
1535:
1515:
1510:
1506:
1502:
1497:
1494:
1489:
1484:
1480:
1457:
1453:
1442:kinetic energy
1427:
1422:
1418:
1414:
1409:
1406:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1359:
1355:
1352:
1349:
1343:
1338:
1332:
1329:
1324:
1318:
1315:
1287:
1282:
1278:
1274:
1271:
1268:
1265:
1245:
1240:
1236:
1232:
1227:
1223:
1219:
1214:
1210:
1190:
1185:
1181:
1177:
1172:
1168:
1164:
1159:
1155:
1151:
1148:
1129:
1124:
1120:
1116:
1111:
1107:
1103:
1098:
1094:
1090:
1087:
1062:
1059:
1053:
1047:
1044:
1038:
1032:
1029:
1004:
998:
993:
986:
983:
975:
969:
964:
957:
954:
946:
940:
935:
928:
925:
917:
912:
908:
889:{\textstyle v}
885:
859:
852:
848:
841:
836:
832:
828:
822:
815:
811:
804:
800:
796:
790:
784:
780:
772:
768:
763:
756:
751:
747:
718:
715:
711:
705:
701:
697:
694:
689:
685:
681:
678:
673:
669:
640:
635:
631:
626:
620:
616:
612:
609:
585:
581:
575:
571:
567:
564:
555:, and volume,
542:
538:
515:
511:
483:
480:
455:
451:
431:
428:
426:
423:
393:
392:
385:
378:
375:
368:
351:
348:
340:Sydney Chapman
319:'s (1905) and
273:mean free path
219:kinetic energy
165:
162:
33:kinetic energy
15:
9:
6:
4:
3:
2:
12531:
12520:
12517:
12515:
12512:
12510:
12507:
12506:
12504:
12494:
12491:
12488:
12485:
12482:
12479:
12476:
12473:
12470:
12467:
12464:
12460:
12459:
12455:
12452:
12448:
12445:
12442:
12440:
12437:
12435:
12432:
12431:
12422:
12418:
12414:
12410:
12406:
12403:
12399:
12398:Behnam Rahimi
12396:
12393:
12389:
12386:
12383:
12379:
12375:
12371:
12368:
12365:
12362:(1939/1970),
12361:
12357:
12354:
12353:
12343:
12337:
12329:
12327:9780408700696
12323:
12318:
12317:
12310:
12309:
12305:
12301:
12296:
12292:
12288:
12287:
12283:
12279:
12275:
12271:
12267:
12263:
12259:
12254:
12253:
12248:
12243:
12239:
12235:
12231:
12227:
12222:
12221:
12217:
12215:0-470-86171-1
12211:
12207:
12202:
12201:
12196:
12192:
12188:
12184:
12183:Lomonosov, M.
12180:
12177:
12176:
12171:
12167:
12163:
12159:
12155:
12154:
12150:
12146:
12142:
12138:
12134:
12130:
12126:
12122:
12118:
12117:
12112:
12108:
12104:
12099:
12098:
12093:
12092:
12087:
12083:
12079:
12078:
12074:
12070:
12066:
12062:
12058:
12054:
12053:
12048:
12043:
12039:
12035:
12031:
12027:
12020:
12015:
12012:
12011:
12007:
12003:
11999:
11995:
11991:
11987:
11983:
11978:
11977:
11965:
11963:9780815320517
11959:
11955:
11954:
11946:
11938:
11934:
11929:
11924:
11919:
11914:
11910:
11906:
11902:
11898:
11894:
11887:
11879:
11873:
11869:
11862:
11860:
11858:
11848:
11843:
11839:
11835:
11831:
11827:
11823:
11819:
11815:
11808:
11800:
11796:
11792:
11788:
11784:
11780:
11776:
11769:
11761:
11757:
11753:
11749:
11745:
11741:
11737:
11730:
11722:
11718:
11714:
11710:
11706:
11699:
11691:
11687:
11683:
11679:
11675:
11671:
11667:
11660:
11652:
11645:
11639:
11631:
11624:
11618:
11614:
11611:
11606:
11600:
11596:
11590:
11582:
11578:
11574:
11570:
11566:
11562:
11558:
11551:
11543:
11536:
11528:
11524:
11520:
11516:
11512:
11508:
11502:
11494:
11487:
11478:
11472:Einstein 1905
11469:
11460:
11452:
11448:
11444:
11440:
11436:
11432:
11427:
11422:
11418:
11414:
11407:
11399:
11395:
11391:
11389:0-470-86171-1
11385:
11381:
11374:
11365:
11361:
11358:
11354:
11351:
11347:
11344:
11340:
11339:
11335:
11329:Clausius 1857
11326:
11317:
11308:
11299:
11290:
11281:
11273:
11267:
11264:. CRC Press.
11263:
11256:
11248:
11244:
11240:
11236:
11232:
11228:
11221:
11217:
11208:
11205:
11203:
11200:
11198:
11195:
11193:
11190:
11188:
11185:
11183:
11180:
11178:
11175:
11173:
11170:
11168:
11165:
11163:
11160:
11158:
11155:
11153:
11150:
11148:
11145:
11143:
11140:
11139:
11131:
11126:
11124:
11119:
11117:
11112:
11111:
11109:
11108:
11101:
11098:
11096:
11093:
11091:
11088:
11086:
11083:
11081:
11078:
11076:
11073:
11071:
11068:
11066:
11063:
11061:
11058:
11056:
11053:
11052:
11046:
11045:
11038:
11035:
11033:
11030:
11028:
11025:
11023:
11020:
11018:
11015:
11014:
11011:
11006:
11005:
10998:
10995:
10993:
10990:
10988:
10985:
10983:
10980:
10979:
10973:
10972:
10965:
10962:
10959:
10957:
10954:
10951:
10949:
10946:
10943:
10941:
10938:
10935:
10933:
10930:
10927:
10926:
10923:
10918:
10917:
10910:
10907:
10905:
10902:
10900:
10897:
10895:
10892:
10890:
10889:Bose–Einstein
10887:
10885:
10882:
10880:
10877:
10875:
10872:
10871:
10868:
10863:
10862:
10857:
10854:
10852:
10849:
10848:
10847:
10846:
10842:
10838:
10837:
10834:
10831:
10830:
10822:
10820:
10816:
10812:
10808:
10802:
10792:
10789:
10782:
10778:
10769:
10765:
10760:
10757:
10749:
10745:
10741:
10734:
10730:
10725:
10721:
10718:
10714:
10709:
10706:
10702:
10697:
10696:
10695:
10681:
10678:
10669:
10664:
10661:
10658:
10650:
10646:
10642:
10638:
10634:
10630:
10626:
10620:
10605:
10603:
10585:
10581:
10558:
10554:
10548:
10544:
10540:
10537:
10529:
10524:
10511:
10502:
10494:
10491:
10486:
10481:
10477:
10454:
10450:
10426:
10423:
10418:
10415:
10409:
10406:
10403:
10400:
10393:
10388:
10372:
10369:
10364:
10361:
10355:
10346:
10338:
10335:
10330:
10327:
10322:
10317:
10313:
10309:
10304:
10299:
10295:
10291:
10288:
10268:
10265:
10256:
10242:
10235:
10232:
10227:
10224:
10217:
10212:
10209:
10204:
10199:
10195:
10190:
10186:
10177:
10169:
10166:
10161:
10158:
10153:
10148:
10144:
10135:
10130:
10110:
10107:
10104:
10101:
10098:
10095:
10088:
10084:
10080:
10077:
10074:
10071:
10068:
10061:
10058:
10055:
10049:
10039:
10036:
10022:
10019:
10015:
10011:
10008:
9999:
9985:
9978:
9975:
9970:
9967:
9960:
9957:
9954:
9951:
9948:
9943:
9939:
9934:
9930:
9925:
9921:
9900:
9897:
9894:
9891:
9882:
9866:
9863:
9859:
9856:
9852:
9849:
9842:
9836:
9833:
9828:
9824:
9812:
9807:
9803:
9799:
9792:
9788:
9784:
9778:
9774:
9768:
9764:
9760:
9755:
9749:
9738:
9734:
9731:
9727:
9722:
9717:
9714:
9711:
9707:
9704:
9697:
9691:
9688:
9685:
9682:
9662:
9659:
9639:
9619:
9599:
9596:
9574:
9570:
9560:
9558:
9542:
9522:
9514:
9510:
9505:
9503:
9497:
9487:
9471:
9467:
9444:
9440:
9430:
9415:
9411:
9407:
9402:
9398:
9392:
9388:
9384:
9381:
9373:
9368:
9355:
9350:
9346:
9342:
9339:
9330:
9322:
9319:
9314:
9309:
9305:
9282:
9278:
9254:
9251:
9246:
9243:
9236:
9233:
9230:
9227:
9220:
9219:Fourier's law
9215:
9199:
9196:
9191:
9188:
9181:
9176:
9172:
9168:
9165:
9156:
9148:
9145:
9140:
9137:
9132:
9127:
9123:
9119:
9114:
9109:
9105:
9101:
9098:
9078:
9075:
9066:
9052:
9045:
9042:
9037:
9034:
9027:
9022:
9018:
9014:
9009:
9006:
9001:
8996:
8992:
8987:
8983:
8980:
8971:
8963:
8960:
8955:
8952:
8947:
8942:
8938:
8929:
8924:
8904:
8901:
8898:
8895:
8892:
8889:
8882:
8878:
8874:
8871:
8868:
8865:
8862:
8855:
8852:
8849:
8843:
8832:
8818:
8815:
8811:
8807:
8804:
8796:
8778:
8774:
8753:
8749:
8742:
8739:
8734:
8731:
8724:
8721:
8718:
8715:
8710:
8706:
8702:
8699:
8694:
8690:
8685:
8681:
8676:
8672:
8651:
8648:
8645:
8642:
8633:
8617:
8614:
8610:
8607:
8603:
8600:
8593:
8587:
8584:
8579:
8575:
8563:
8558:
8554:
8550:
8543:
8539:
8535:
8529:
8525:
8519:
8515:
8511:
8506:
8500:
8489:
8485:
8482:
8478:
8473:
8468:
8465:
8462:
8458:
8455:
8448:
8442:
8439:
8436:
8433:
8413:
8410:
8390:
8370:
8350:
8347:
8325:
8321:
8311:
8309:
8293:
8273:
8265:
8260:
8258:
8252:
8242:
8226:
8222:
8199:
8195:
8185:
8170:
8166:
8162:
8157:
8153:
8144:
8140:
8136:
8133:
8127:
8124:
8116:
8114:
8109:
8107:
8102:
8098:
8094:
8089:
8088:cross section
8085:
8069:
8049:
8040:
8025:
8021:
8017:
8014:
8009:
8004:
8000:
7997:
7993:
7988:
7985:
7982:
7974:
7972:
7968:
7952:
7943:
7925:
7921:
7917:
7914:
7909:
7906:
7903:
7897:
7889:
7884:
7880:
7875:
7870:
7866:
7855:
7851:
7845:
7837:
7832:
7828:
7823:
7818:
7814:
7805:
7803:
7787:
7778:
7763:
7759:
7755:
7752:
7749:
7746:
7736:
7733:
7728:
7724:
7720:
7715:
7711:
7688:
7684:
7659:
7655:
7644:
7637:
7632:
7629:
7622:
7619:
7614:
7610:
7603:
7599:
7594:
7585:
7573:
7558:
7548:
7542:
7539:
7528:
7523:
7519:
7514:
7509:
7505:
7495:
7482:
7479:
7476:
7467:
7459:
7456:
7451:
7446:
7442:
7419:
7415:
7391:
7388:
7383:
7380:
7373:
7370:
7367:
7360:
7355:
7339:
7336:
7331:
7328:
7321:
7318:
7315:
7312:
7303:
7295:
7292:
7287:
7282:
7278:
7274:
7269:
7265:
7261:
7258:
7249:
7235:
7228:
7225:
7220:
7217:
7210:
7205:
7202:
7197:
7192:
7188:
7183:
7179:
7176:
7173:
7164:
7156:
7153:
7148:
7143:
7139:
7130:
7107:
7104:
7101:
7098:
7095:
7092:
7085:
7081:
7077:
7074:
7071:
7068:
7065:
7058:
7055:
7052:
7046:
7035:
7021:
7018:
7014:
7010:
7007:
6987:
6983:
6976:
6973:
6968:
6965:
6958:
6955:
6952:
6949:
6946:
6941:
6937:
6932:
6928:
6925:
6920:
6915:
6911:
6902:
6886:
6866:
6863:
6860:
6857:
6854:
6851:
6848:
6839:
6823:
6820:
6816:
6813:
6809:
6806:
6801:
6797:
6794:
6789:
6785:
6773:
6762:
6758:
6751:
6747:
6743:
6737:
6733:
6726:
6722:
6718:
6713:
6707:
6696:
6692:
6689:
6685:
6680:
6675:
6672:
6669:
6665:
6662:
6654:
6647:
6644:
6641:
6638:
6618:
6615:
6595:
6575:
6555:
6552:
6530:
6526:
6505:
6502:
6499:
6477:
6473:
6464:
6459:
6457:
6441:
6421:
6413:
6409:
6403:
6393:
6391:
6387:
6383:
6379:
6378:J. M. Kincaid
6375:
6371:
6366:
6364:
6360:
6356:
6352:
6345:
6335:
6321:
6318:
6309:
6291:
6286:
6283:
6279:
6274:
6271:
6251:
6231:
6228:
6208:
6204:
6200:
6197:
6194:
6174:
6166:
6165:cross section
6150:
6142:
6136:
6124:
6121:
6119:
6116:
6114:
6111:
6109:
6106:
6103:
6100:
6099:
6098:
6095:
6093:
6090:
6068:
6039:
6012:
6003:
5999:
5995:
5979:
5972:
5968:
5957:
5950:
5946:
5940:
5935:
5931:
5924:
5921:
5915:
5906:
5886:
5879:
5875:
5864:
5857:
5852:
5849:
5842:
5837:
5833:
5826:
5822:
5817:
5808:
5786:
5779:
5775:
5764:
5757:
5754:
5749:
5740:
5725:
5712:
5709:
5706:
5703:
5700:
5697:
5693:
5688:
5685:
5680:
5677:
5674:
5671:
5667:
5664:
5661:
5658:
5654:
5644:
5641:
5637:
5634:
5630:
5627:
5622:
5618:
5615:
5611:
5607:
5604:
5599:
5595:
5583:
5572:
5568:
5561:
5557:
5553:
5547:
5543:
5537:
5532:
5526:
5515:
5510:
5505:
5497:
5494:
5490:
5485:
5483:
5478:
5475:
5471:
5468:
5464:
5461:
5454:
5451:
5448:
5445:
5442:
5436:
5406:
5399:
5394:
5373:
5370:
5367:
5364:
5360:
5357:
5353:
5350:
5343:
5340:
5337:
5334:
5331:
5325:
5322:
5302:
5282:
5279:
5276:
5273:
5270:
5267:
5263:
5258:
5255:
5250:
5247:
5244:
5241:
5237:
5234:
5231:
5228:
5201:
5198:
5194:
5191:
5187:
5184:
5179:
5175:
5172:
5167:
5163:
5151:
5140:
5136:
5129:
5125:
5121:
5115:
5111:
5105:
5101:
5097:
5092:
5086:
5083:
5080:
5077:
5073:
5068:
5062:
5057:
5053:
5050:
5047:
5044:
5041:
5039:
5034:
5031:
5027:
5024:
5020:
5017:
5010:
5007:
5004:
5001:
4998:
4992:
4968:
4965:
4962:
4959:
4939:
4936:
4933:
4930:
4924:
4918:
4915:
4912:
4892:
4889:
4883:
4877:
4874:
4871:
4851:
4848:
4828:
4825:
4802:
4799:
4796:
4793:
4790:
4778:
4776:
4771:
4758:
4752:
4741:
4737:
4734:
4731:
4726:
4723:
4717:
4699:
4698:ideal gas law
4694:
4681:
4674:
4671:
4668:
4663:
4652:
4644:
4641:
4638:
4635:
4626:
4622:
4604:
4588:
4568:
4557:
4553:
4550:
4542:
4539:
4536:
4532:
4525:
4520:
4517:
4512:
4507:
4498:
4494:
4491:
4486:
4483:
4478:
4473:
4464:
4460:
4457:
4454:
4448:
4445:
4442:
4439:
4436:
4431:
4426:
4422:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4393:
4389:
4383:
4379:
4375:
4370:
4366:
4362:
4356:
4353:
4345:
4344:Ideal gas law
4341:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4300:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4252:
4248:
4244:
4241:
4237:
4234:
4230:
4227:
4220:
4214:
4211:
4206:
4202:
4197:
4188:
4183:
4179:
4175:
4168:
4164:
4160:
4154:
4150:
4144:
4140:
4136:
4131:
4125:
4120:
4116:
4112:
4109:
4105:
4100:
4095:
4092:
4089:
4085:
4082:
4075:
4069:
4066:
4063:
4060:
4057:
4048:
4042:
4039:
4036:
4033:
4030:
4007:
4004:
3984:
3964:
3944:
3941:
3932:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3847:
3835:
3822:
3815:
3812:
3807:
3796:
3792:
3783:
3780:
3775:
3766:
3760:
3755:
3752:
3747:
3738:
3732:
3729:
3723:
3720:
3717:
3714:
3711:
3706:
3701:
3697:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3665:
3661:
3657:
3652:
3648:
3641:
3632:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3586:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3549:
3536:
3532:
3528:
3525:
3521:
3518:
3514:
3511:
3504:
3498:
3495:
3490:
3486:
3481:
3472:
3467:
3463:
3459:
3452:
3448:
3444:
3438:
3434:
3428:
3424:
3420:
3415:
3409:
3404:
3400:
3396:
3393:
3389:
3384:
3379:
3376:
3373:
3369:
3366:
3359:
3353:
3350:
3347:
3344:
3324:
3321:
3301:
3298:
3289:
3275:
3272:
3269:
3266:
3260:
3254:
3251:
3248:
3228:
3225:
3219:
3213:
3210:
3207:
3187:
3184:
3164:
3161:
3141:
3121:
3101:
3098:
3078:
3075:
3072:
3052:
3049:
3029:
3026:
3006:
2986:
2966:
2963:
2954:
2939:
2935:
2931:
2925:
2921:
2917:
2911:
2907:
2903:
2894:
2889:
2885:
2881:
2874:
2870:
2866:
2860:
2856:
2850:
2846:
2842:
2837:
2831:
2826:
2822:
2818:
2815:
2811:
2806:
2801:
2796:
2792:
2788:
2782:
2778:
2774:
2768:
2764:
2760:
2751:
2747:
2743:
2738:
2734:
2730:
2725:
2721:
2708:
2699:
2683:
2679:
2675:
2672:
2669:
2660:
2658:
2654:
2650:
2639:
2637:
2633:
2625:
2621:
2618:
2617:
2616:
2614:
2606:
2602:
2599:
2598:
2597:
2594:
2592:
2588:
2584:
2566:
2562:
2552:
2539:
2536:
2531:
2528:
2523:
2520:
2515:
2511:
2505:
2501:
2495:
2492:
2487:
2484:
2476:
2474:
2470:
2465:
2452:
2449:
2444:
2440:
2434:
2431:
2426:
2420:
2417:
2413:
2403:
2400:
2387:
2382:
2378:
2374:
2371:
2366:
2363:
2358:
2353:
2349:
2345:
2342:
2339:
2331:
2329:
2324:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2293:
2291:
2287:
2286:
2281:
2280:
2274:
2272:
2263:
2256:
2254:
2240:
2235:
2231:
2225:
2222:
2217:
2214:
2211:
2204:
2203:
2200:
2198:
2197:
2192:
2191:
2186:
2182:
2181:
2171:
2164:
2162:
2148:
2134:
2130:
2124:
2120:
2112:
2109:
2104:
2101:
2094:
2093:
2090:
2083:
2076:
2074:
2050:
2043:
2039:
2035:
2027:
2024:
2019:
2016:
2009:
2008:
2005:
1991:
1969:
1965:
1961:
1958:
1953:
1950:
1945:
1940:
1936:
1915:
1895:
1892:
1881:
1875:
1872:
1867:
1862:
1858:
1854:
1849:
1846:
1824:
1819:
1813:
1809:
1805:
1799:
1796:
1785:
1776:
1774:
1771:
1755:
1747:
1723:
1709:
1702:
1700:
1686:
1683:
1672:
1668:
1665:
1662:
1659:
1652:
1651:
1648:
1647:
1646:ideal gas law
1629:
1623:
1619:
1615:
1612:
1606:
1603:
1600:
1586:
1584:
1580:
1575:
1562:
1557:
1553:
1547:
1544:
1539:
1536:
1533:
1513:
1508:
1504:
1500:
1495:
1492:
1487:
1482:
1478:
1455:
1451:
1443:
1438:
1425:
1420:
1416:
1412:
1407:
1404:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1357:
1353:
1350:
1347:
1341:
1336:
1327:
1322:
1316:
1313:
1305:
1303:
1298:
1285:
1280:
1276:
1272:
1269:
1266:
1263:
1243:
1238:
1234:
1230:
1225:
1221:
1217:
1212:
1208:
1188:
1183:
1179:
1175:
1170:
1166:
1162:
1157:
1153:
1149:
1146:
1127:
1122:
1118:
1114:
1109:
1105:
1101:
1096:
1092:
1088:
1085:
1077:
1057:
1051:
1042:
1036:
1027:
1015:
1002:
996:
991:
981:
973:
967:
962:
952:
944:
938:
933:
923:
915:
910:
906:
897:
883:
875:
870:
857:
850:
846:
839:
834:
830:
826:
820:
813:
809:
802:
798:
794:
788:
782:
770:
766:
754:
749:
745:
736:
734:
729:
716:
713:
709:
703:
699:
695:
692:
687:
683:
679:
676:
671:
667:
658:
656:
651:
638:
633:
629:
624:
618:
614:
610:
607:
599:
583:
579:
573:
569:
565:
562:
540:
536:
513:
509:
500:
478:
453:
449:
439:
437:
422:
420:
416:
412:
407:
405:
401:
396:
390:
386:
383:
379:
376:
373:
369:
366:
361:
357:
356:
355:
347:
345:
341:
337:
333:
328:
326:
322:
318:
313:
311:
307:
303:
299:
295:
291:
286:
282:
278:
274:
270:
266:
262:
258:
257:August Krönig
253:
251:
247:
243:
242:John Herapath
239:
235:
230:
228:
224:
220:
216:
212:
208:
204:
203:
202:Hydrodynamica
198:
189:
185:
183:
179:
175:
171:
161:
159:
154:
152:
148:
144:
140:
136:
132:
129:. Due to the
128:
124:
120:
116:
113:, as well as
112:
108:
104:
100:
95:
93:
89:
84:
82:
78:
74:
70:
66:
65:thermodynamic
63:model of the
62:
58:
50:
46:
42:
38:
34:
30:
26:
21:
12469:Introduction
12457:
12420:
12416:
12391:
12381:
12363:
12315:
12306:, 167, 183.)
12303:
12299:
12294:
12265:
12261:
12229:
12225:
12205:
12194:
12169:
12132:
12128:
12110:
12106:
12089:
12082:Herapath, J.
12064:
12060:
12057:Grad, Harold
12029:
12025:
11989:
11985:
11952:
11945:
11900:
11896:
11886:
11867:
11821:
11817:
11807:
11782:
11778:
11768:
11743:
11739:
11729:
11712:
11708:
11698:
11673:
11669:
11659:
11650:
11638:
11629:
11623:
11605:
11589:
11564:
11560:
11550:
11541:
11535:
11518:
11514:
11501:
11492:
11486:
11477:
11468:
11463:Maxwell 1873
11459:
11416:
11412:
11406:
11379:
11373:
11363:
11359:
11349:
11345:
11334:
11325:
11316:
11307:
11298:
11289:
11280:
11261:
11255:
11230:
11226:
11220:
11202:Vicsek model
10960:
10952:
10944:
10936:
10928:
10855:
10804:
10787:
10780:
10776:
10773:
10761:
10747:
10739:
10732:
10728:
10710:
10698:
10622:
10525:
10389:
10257:
10131:
10040:
10037:
10000:
9883:
9561:
9506:
9499:
9431:
9369:
9216:
9067:
8925:
8833:
8634:
8312:
8261:
8254:
8186:
8117:
8110:
8041:
7975:
7944:
7806:
7779:
7574:
7496:
7356:
7250:
7129:shear stress
7036:
6840:
6463:Couette flow
6460:
6411:
6408:Couette flow
6405:
6367:
6347:
6310:
6138:
6096:
6001:
5997:
5993:
5731:
4779:
4775:Graham's law
4772:
4695:
3933:
3836:
3550:
3290:
2955:
2661:
2645:
2631:
2629:
2610:
2595:
2586:
2553:
2477:
2472:
2466:
2404:
2401:
2332:
2326:Because the
2325:
2320:
2312:
2308:
2304:
2294:
2283:
2277:
2275:
2268:
2257:
2194:
2188:
2184:
2178:
2176:
2165:
2088:
2077:
1777:
1714:
1703:
1592:
1576:
1439:
1306:
1301:
1299:
1078:
1076:, such that
1016:
898:
873:
871:
737:
730:
659:
652:
600:
468:, along the
440:
433:
408:
397:
394:
382:uncorrelated
353:
336:David Enskog
329:
314:
254:
231:
200:
194:
168:In about 50
167:
155:
96:
85:
83:of the gas.
67:behavior of
59:is a simple
56:
54:
11320:Krönig 1856
11085:von Neumann
10894:Fermi–Dirac
9507:Consider a
8042:The radius
6465:setup, let
6390:M. H. Ernst
5996:is in m/s,
4338:yields the
3957:with speed
2655:method for
2626:= 35.2 meV.
2276:Equations (
2199:), we have
1773:temperature
1583:microscopic
1579:macroscopic
350:Assumptions
310:probability
302:logarithmic
244:(1816) and
182:Aristotlean
111:temperature
99:macroscopic
45:atmospheres
25:temperature
12503:Categories
12121:Krönig, A.
11974:References
11847:1874/36979
11426:1702.01411
11049:Scientists
11010:Potentials
10815:convection
10645:drag force
10643:) and the
9494:See also:
8249:See also:
7967:molar mass
6400:See also:
6342:See also:
2177:Equation (
1585:property.
269:rotational
227:physicists
209:theory of
199:published
149:) and the
12481:Flowchart
12411:(2016), "
12336:cite book
12185:(1970) ,
11581:0378-4371
11419:: 53–65.
11233:: 49–88.
11080:Ehrenfest
11060:Boltzmann
10940:Canonical
10819:advection
10665:μ
10641:diffusion
10582:α
10545:α
10506:¯
10407:−
10350:¯
10331:−
10323:−
10310:−
10266:−
10205:±
10187:⋅
10181:¯
10162:−
10154:±
10111:π
10102:ϕ
10081:π
10075:θ
9961:θ
9958:
9949:±
9926:±
9901:θ
9898:
9867:ϕ
9860:θ
9843:θ
9837:
9779:−
9735:π
9718:×
9698:θ
9692:
9640:θ
9632:at angle
9468:κ
9445:κ
9441:α
9412:κ
9399:κ
9393:κ
9389:α
9382:κ
9334:¯
9306:κ
9279:κ
9237:κ
9234:−
9160:¯
9141:−
9133:−
9120:−
9076:−
9002:±
8993:ε
8984:⋅
8975:¯
8956:−
8948:±
8928:heat flux
8905:π
8896:ϕ
8875:π
8869:θ
8725:θ
8722:
8700:±
8691:ε
8677:±
8673:ε
8652:θ
8649:
8618:ϕ
8611:θ
8594:θ
8588:
8530:−
8486:π
8469:×
8449:θ
8443:
8391:θ
8383:at angle
8322:ε
8274:ε
8223:η
8200:η
8196:α
8167:η
8154:η
8145:η
8141:α
8125:η
8018:π
7989:π
7983:σ
7918:⋅
7915:σ
7898:⋅
7890:π
7871:σ
7846:⋅
7838:π
7815:η
7788:σ
7756:⋅
7721:⋅
7638:⋅
7633:π
7604:π
7589:¯
7559:σ
7552:¯
7543:⋅
7506:η
7471:¯
7443:η
7416:η
7374:η
7368:τ
7319:⋅
7307:¯
7283:−
7279:τ
7275:−
7266:τ
7259:τ
7198:±
7177:⋅
7168:¯
7144:±
7140:τ
7108:π
7099:ϕ
7078:π
7072:θ
6959:θ
6956:
6947:±
6921:±
6867:θ
6864:
6855:±
6824:ϕ
6817:θ
6802:θ
6798:
6738:−
6693:π
6676:×
6655:θ
6648:
6596:θ
6588:at angle
6351:viscosity
6322:σ
6287:σ
6232:σ
6151:σ
6089:isotropic
6072:¯
5951:⋅
5858:⋅
5853:π
5827:π
5812:¯
5758:⋅
5713:π
5704:ϕ
5686:π
5678:θ
5645:ϕ
5638:θ
5623:θ
5619:
5612:θ
5608:
5548:−
5498:π
5479:ϕ
5472:θ
5455:ϕ
5449:θ
5410:¯
5368:ϕ
5361:θ
5344:ϕ
5338:θ
5323:∫
5303:λ
5283:π
5274:ϕ
5256:π
5248:θ
5202:ϕ
5195:θ
5180:θ
5176:
5116:−
5081:π
5063:×
5058:θ
5054:
5045:λ
5035:ϕ
5028:θ
5011:ϕ
5005:θ
4969:θ
4966:
4925:θ
4919:
4884:θ
4878:
4803:ϕ
4797:θ
4735:π
4709:Φ
4672:π
4631:collision
4614:Φ
4605:will be:
4548:⟩
4529:⟨
4455:×
4449:θ
4443:θ
4440:
4432:π
4423:∫
4417:θ
4411:θ
4408:
4402:θ
4399:
4376:π
4367:∫
4326:π
4317:ϕ
4297:π
4291:θ
4245:ϕ
4238:θ
4221:θ
4215:
4155:−
4113:π
4096:×
4076:θ
4070:
4058:×
4049:θ
4043:
3985:θ
3977:at angle
3919:π
3910:ϕ
3898:π
3892:θ
3851:¯
3813:π
3770:¯
3742:¯
3730:×
3724:θ
3718:θ
3715:
3707:π
3698:∫
3692:θ
3686:θ
3683:
3677:θ
3674:
3658:π
3649:∫
3637:collision
3612:π
3603:ϕ
3583:π
3577:θ
3529:ϕ
3522:θ
3505:θ
3499:
3439:−
3397:π
3380:×
3360:θ
3354:
3261:θ
3255:
3220:θ
3214:
3142:θ
3134:at angle
3007:θ
2999:at angle
2861:−
2819:π
2469:ideal gas
1379:⇒
1331:¯
1273:∗
1061:^
1046:^
1031:^
985:→
956:→
927:→
482:^
277:diffusion
215:Bernoulli
178:Epicurean
174:Lucretius
119:viscosity
88:ideal gas
81:molecules
61:classical
29:ideal gas
12447:Archived
12405:Archived
12390:(2003),
12380:(1964),
12293:(1843),
12160:(1818),
12123:(1856),
12084:(1816),
11937:16592372
11613:Archived
11451:38272381
11398:52358254
11247:96568430
11167:Gas laws
11075:Einstein
11022:Enthalpy
10987:Einstein
10825:See also
10717:terminal
6879:, where
4713:effusion
4618:effusion
4340:pressure
2475:= 3) is
1770:absolute
655:momentum
436:pressure
290:pressure
255:In 1856
236:(1747),
195:In 1738
117:such as
107:pressure
49:trillion
12270:Bibcode
12234:Bibcode
12193:(ed.),
12168:(ed.),
12137:Bibcode
12034:Bibcode
11994:Bibcode
11905:Bibcode
11826:Bibcode
11787:Bibcode
11748:Bibcode
11678:Bibcode
11431:Bibcode
11055:Maxwell
10766:is the
10754:is the
10703:is the
8793:is the
7965:is the
6899:is the
6102:Average
2713:Maxwell
2589:is the
2581:is the
2282:) and (
2193:) and (
1744:is the
306:entropy
207:kinetic
164:History
27:of the
12376:, and
12324:
12300:Papers
12226:Nature
12212:
11960:
11935:
11928:431439
11925:
11874:
11579:
11449:
11396:
11386:
11268:
11245:
11090:Tolman
10976:Models
10694:where
10573:where
9509:steady
9432:where
8766:where
8187:where
7676:where
6097:See:
5992:where
5386:to be
2585:, and
2554:where
1715:where
389:forces
360:volume
300:. The
109:, and
103:volume
41:helium
12509:Gases
12189:, in
12164:, in
12022:(PDF)
11647:(PDF)
11447:S2CID
11421:arXiv
11338:See:
11243:S2CID
11213:Notes
11100:Fermi
11095:Debye
11070:Gibbs
10997:Potts
10992:Ising
10982:Debye
10724:force
2317:water
211:gases
145:(for
77:atoms
69:gases
12400:and
12358:and
12342:link
12322:ISBN
12210:ISBN
11958:ISBN
11933:PMID
11872:ISBN
11577:ISSN
11394:OCLC
11384:ISBN
11266:ISBN
11172:Heat
11065:Bose
10639:(or
10105:<
10099:<
10078:<
10072:<
10059:>
9675:is
9562:Let
8899:<
8893:<
8872:<
8866:<
8853:>
8426:is
8313:Let
7945:and
7102:<
7096:<
7075:<
7069:<
7056:>
6631:is
6388:and
6380:and
6361:and
6118:Mean
5707:<
5701:<
5681:<
5675:<
5662:>
5277:<
5271:<
5251:<
5245:<
5232:>
4320:<
4314:<
4294:<
4288:<
4276:>
3913:<
3907:<
3895:<
3889:<
3877:>
3606:<
3600:<
3580:<
3574:<
3562:>
2295:The
2271:mole
1768:the
1748:and
653:The
400:mass
365:size
338:and
308:and
125:and
55:The
37:size
23:The
12461:on
12421:806
12415:",
12278:doi
12242:doi
12145:doi
12069:doi
12042:doi
12002:doi
11990:176
11923:PMC
11913:doi
11842:hdl
11834:doi
11795:doi
11756:doi
11717:doi
11686:doi
11569:doi
11565:194
11523:doi
11439:doi
11235:doi
11231:157
10961:NPT
10953:NPH
10945:µVT
10937:NVT
10929:NVE
10136:):
9955:cos
9895:cos
9834:sin
9689:cos
8930:):
8719:cos
8646:cos
8585:sin
8440:cos
8095:or
7742:and
7131:):
6953:cos
6861:cos
6795:sin
6645:cos
6518:);
6094:).
6044:rms
5911:rms
5616:cos
5605:sin
5173:sin
5051:cos
4963:cos
4916:cos
4875:cos
4503:rms
4469:rms
4437:sin
4405:sin
4390:cos
4212:sin
4067:cos
4040:cos
4020:is:
3712:sin
3680:sin
3671:cos
3496:sin
3351:cos
3252:cos
3211:cos
2611:At
415:BGK
170:BCE
79:or
39:of
12505::
12419:,
12372:,
12338:}}
12334:{{
12302:,
12276:,
12266:21
12264:,
12260:,
12240:,
12228:,
12143:,
12133:99
12131:,
12127:,
12109:,
12105:,
12088:,
12063:,
12040:,
12030:17
12028:,
12024:,
12000:,
11988:,
11984:,
11931:.
11921:.
11911:.
11901:76
11899:.
11895:.
11856:^
11840:.
11832:.
11822:43
11820:.
11816:.
11793:.
11783:86
11781:.
11777:.
11754:.
11744:80
11742:.
11738:.
11713:79
11711:.
11707:.
11684:.
11674:78
11672:.
11668:.
11649:.
11575:.
11563:.
11559:.
11519:44
11517:.
11513:.
11445:.
11437:.
11429:.
11417:57
11415:.
11392:.
11364:20
11350:19
11241:.
11229:.
10779:=
10731:=
10726:,
8108:.
7804::
6392:.
6376:,
6365:.
6357:,
6353:,
6264:by
4981::
4777:.
4346:):
3931:.
3288:.
2700::
2659:.
2624:zJ
2605:yJ
2593:.
2303:,
501:,
252:.
160:.
153:.
121:,
105:,
12465:.
12344:)
12330:.
12304:3
12280::
12272::
12244::
12236::
12230:8
12147::
12139::
12111:9
12071::
12065:2
12044::
12036::
12004::
11996::
11966:.
11939:.
11915::
11907::
11880:.
11850:.
11844::
11836::
11828::
11801:.
11797::
11789::
11762:.
11758::
11750::
11723:.
11719::
11692:.
11688::
11680::
11653:.
11583:.
11571::
11529:.
11525::
11453:.
11441::
11433::
11423::
11400:.
11274:.
11249:.
11237::
11129:e
11122:t
11115:v
10788:F
10786:/
10784:d
10781:v
10777:μ
10770:.
10763:T
10758:;
10751:B
10748:k
10743:;
10740:F
10738:/
10736:d
10733:v
10729:μ
10712:μ
10707:;
10700:D
10682:,
10679:T
10674:B
10670:k
10662:=
10659:D
10651::
10586:D
10559:0
10555:D
10549:D
10541:=
10538:D
10512:l
10503:v
10495:3
10492:1
10487:=
10482:0
10478:D
10455:0
10451:D
10427:y
10424:d
10419:n
10416:d
10410:D
10404:=
10401:J
10373:y
10370:d
10365:n
10362:d
10356:l
10347:v
10339:3
10336:1
10328:=
10318:y
10314:J
10305:+
10300:y
10296:J
10292:=
10289:J
10269:y
10243:)
10236:y
10233:d
10228:n
10225:d
10218:l
10213:3
10210:2
10200:0
10196:n
10191:(
10178:v
10170:4
10167:1
10159:=
10149:y
10145:J
10108:2
10096:0
10089:2
10085:/
10069:0
10062:0
10056:v
10050:{
10023:y
10020:d
10016:/
10012:n
10009:d
9986:)
9979:y
9976:d
9971:n
9968:d
9952:l
9944:0
9940:n
9935:(
9931:=
9922:n
9892:l
9870:)
9864:d
9857:d
9853:v
9850:d
9846:)
9840:(
9829:2
9825:v
9821:(
9813:T
9808:B
9804:k
9800:2
9793:2
9789:v
9785:m
9775:e
9769:2
9765:/
9761:3
9756:)
9750:T
9744:B
9739:k
9732:2
9728:m
9723:(
9715:t
9712:d
9708:A
9705:d
9701:)
9695:(
9686:v
9683:n
9663:t
9660:d
9620:v
9600:A
9597:d
9575:0
9571:n
9543:y
9523:n
9472:c
9416:c
9408:+
9403:0
9385:=
9356:l
9351:v
9347:c
9343:m
9340:n
9331:v
9323:3
9320:1
9315:=
9310:0
9283:0
9255:y
9252:d
9247:T
9244:d
9231:=
9228:q
9200:y
9197:d
9192:T
9189:d
9182:l
9177:v
9173:c
9169:m
9166:n
9157:v
9149:3
9146:1
9138:=
9128:y
9124:q
9115:+
9110:y
9106:q
9102:=
9099:q
9079:y
9053:)
9046:y
9043:d
9038:T
9035:d
9028:l
9023:v
9019:c
9015:m
9010:3
9007:2
8997:0
8988:(
8981:n
8972:v
8964:4
8961:1
8953:=
8943:y
8939:q
8902:2
8890:0
8883:2
8879:/
8863:0
8856:0
8850:v
8844:{
8819:y
8816:d
8812:/
8808:T
8805:d
8779:v
8775:c
8754:,
8750:)
8743:y
8740:d
8735:T
8732:d
8716:l
8711:v
8707:c
8703:m
8695:0
8686:(
8682:=
8643:l
8621:)
8615:d
8608:d
8604:v
8601:d
8597:)
8591:(
8580:2
8576:v
8572:(
8564:T
8559:B
8555:k
8551:2
8544:2
8540:v
8536:m
8526:e
8520:2
8516:/
8512:3
8507:)
8501:T
8495:B
8490:k
8483:2
8479:m
8474:(
8466:t
8463:d
8459:A
8456:d
8452:)
8446:(
8437:v
8434:n
8414:t
8411:d
8371:v
8351:A
8348:d
8326:0
8294:y
8227:c
8171:c
8163:+
8158:0
8150:)
8137:+
8134:1
8131:(
8128:=
8070:d
8050:r
8026:2
8022:d
8015:=
8010:2
8005:)
8001:r
7998:2
7994:(
7986:=
7953:M
7926:A
7922:N
7910:T
7907:R
7904:M
7885:3
7881:2
7876:=
7867:T
7861:B
7856:k
7852:m
7833:3
7829:2
7824:=
7819:0
7764:A
7760:N
7753:m
7750:=
7747:M
7737:R
7734:=
7729:A
7725:N
7716:B
7712:k
7689:p
7685:v
7660:m
7656:T
7650:B
7645:k
7630:2
7623:2
7620:=
7615:p
7611:v
7600:2
7595:=
7586:v
7549:v
7540:m
7529:2
7524:3
7520:1
7515:=
7510:0
7483:l
7480:m
7477:n
7468:v
7460:3
7457:1
7452:=
7447:0
7420:0
7392:y
7389:d
7384:u
7381:d
7371:=
7340:y
7337:d
7332:u
7329:d
7322:l
7316:m
7313:n
7304:v
7296:3
7293:1
7288:=
7270:+
7262:=
7236:)
7229:y
7226:d
7221:u
7218:d
7211:l
7206:3
7203:2
7193:0
7189:u
7184:(
7180:m
7174:n
7165:v
7157:4
7154:1
7149:=
7105:2
7093:0
7086:2
7082:/
7066:0
7059:0
7053:v
7047:{
7022:y
7019:d
7015:/
7011:u
7008:d
6988:,
6984:)
6977:y
6974:d
6969:u
6966:d
6950:l
6942:0
6938:u
6933:(
6929:m
6926:=
6916:x
6912:p
6887:l
6858:l
6852:=
6849:y
6827:)
6821:d
6814:d
6810:v
6807:d
6790:2
6786:v
6782:(
6774:T
6768:B
6763:k
6759:2
6752:2
6748:v
6744:m
6734:e
6727:2
6723:/
6719:3
6714:)
6708:T
6702:B
6697:k
6690:2
6686:m
6681:(
6673:t
6670:d
6666:A
6663:d
6659:)
6651:(
6642:v
6639:n
6619:t
6616:d
6576:v
6556:A
6553:d
6531:0
6527:u
6506:0
6503:=
6500:y
6478:0
6474:u
6442:y
6422:u
6412:F
6319:n
6292:2
6284:n
6280:1
6275:=
6272:l
6252:l
6229:n
6209:V
6205:/
6201:N
6198:=
6195:n
6175:n
6104:,
6069:v
6040:v
6017:p
6013:v
6002:m
5998:T
5994:v
5980:,
5973:m
5969:T
5963:B
5958:k
5947:3
5941:=
5936:p
5932:v
5925:2
5922:3
5916:=
5907:v
5887:,
5880:m
5876:T
5870:B
5865:k
5850:8
5843:=
5838:p
5834:v
5823:2
5818:=
5809:v
5787:,
5780:m
5776:T
5770:B
5765:k
5755:2
5750:=
5745:p
5741:v
5710:2
5698:0
5694:,
5689:2
5672:0
5668:,
5665:0
5659:v
5655:;
5648:)
5642:d
5635:d
5631:v
5628:d
5600:3
5596:v
5592:(
5584:T
5578:B
5573:k
5569:2
5562:2
5558:v
5554:m
5544:e
5538:2
5533:)
5527:T
5521:B
5516:k
5511:m
5506:(
5495:2
5491:1
5486:=
5476:d
5469:d
5465:v
5462:d
5458:)
5452:,
5446:,
5443:v
5440:(
5437:f
5407:v
5400:/
5395:4
5374:1
5371:=
5365:d
5358:d
5354:v
5351:d
5347:)
5341:,
5335:,
5332:v
5329:(
5326:f
5280:2
5268:0
5264:,
5259:2
5242:0
5238:,
5235:0
5229:v
5205:)
5199:d
5192:d
5188:v
5185:d
5168:2
5164:v
5160:(
5152:T
5146:B
5141:k
5137:2
5130:2
5126:v
5122:m
5112:e
5106:2
5102:/
5098:3
5093:)
5087:T
5084:k
5078:2
5074:m
5069:(
5048:v
5042:=
5032:d
5025:d
5021:v
5018:d
5014:)
5008:,
5002:,
4999:v
4996:(
4993:f
4960:v
4940:t
4937:d
4934:A
4931:d
4928:)
4922:(
4913:v
4893:t
4890:d
4887:)
4881:(
4872:v
4852:t
4849:d
4829:A
4826:d
4806:)
4800:,
4794:,
4791:v
4788:(
4759:.
4753:T
4747:B
4742:k
4738:m
4732:2
4727:A
4724:P
4718:=
4682:.
4675:m
4669:2
4664:T
4658:B
4653:k
4645:A
4642:n
4639:=
4636:A
4627:J
4623:=
4589:A
4569:T
4563:B
4558:k
4554:n
4551:=
4543:n
4540:i
4537:k
4533:E
4526:n
4521:3
4518:2
4513:=
4508:2
4499:v
4495:m
4492:n
4487:3
4484:1
4479:=
4474:2
4465:v
4461:m
4458:n
4446:d
4427:0
4414:d
4394:2
4384:2
4380:/
4371:0
4363:2
4357:=
4354:P
4323:2
4311:0
4308:,
4305:2
4301:/
4285:0
4282:,
4279:0
4273:v
4253:.
4249:)
4242:d
4235:d
4231:v
4228:d
4224:)
4218:(
4207:2
4203:v
4198:(
4189:T
4184:B
4180:k
4176:2
4169:2
4165:v
4161:m
4151:e
4145:2
4141:/
4137:3
4132:)
4126:T
4121:B
4117:k
4110:2
4106:m
4101:(
4093:t
4090:d
4086:A
4083:d
4079:)
4073:(
4064:v
4061:n
4055:]
4052:)
4046:(
4037:v
4034:m
4031:2
4028:[
4008:t
4005:d
3965:v
3945:A
3942:d
3916:2
3904:0
3901:,
3886:0
3883:,
3880:0
3874:v
3848:v
3823:.
3816:m
3808:T
3802:B
3797:k
3793:8
3784:4
3781:n
3776:=
3767:v
3761:n
3756:4
3753:1
3748:=
3739:v
3733:n
3721:d
3702:0
3689:d
3666:2
3662:/
3653:0
3642:=
3633:J
3609:2
3597:0
3594:,
3591:2
3587:/
3571:0
3568:,
3565:0
3559:v
3537:.
3533:)
3526:d
3519:d
3515:v
3512:d
3508:)
3502:(
3491:2
3487:v
3482:(
3473:T
3468:B
3464:k
3460:2
3453:2
3449:v
3445:m
3435:e
3429:2
3425:/
3421:3
3416:)
3410:T
3405:B
3401:k
3394:2
3390:m
3385:(
3377:t
3374:d
3370:A
3367:d
3363:)
3357:(
3348:v
3345:n
3325:t
3322:d
3302:A
3299:d
3276:t
3273:d
3270:A
3267:d
3264:)
3258:(
3249:v
3229:t
3226:d
3223:)
3217:(
3208:v
3188:t
3185:d
3165:A
3162:d
3122:v
3102:A
3099:d
3079:t
3076:d
3073:v
3053:t
3050:d
3030:A
3027:d
2987:v
2967:A
2964:d
2940:z
2936:v
2932:d
2926:y
2922:v
2918:d
2912:x
2908:v
2904:d
2895:T
2890:B
2886:k
2882:2
2875:2
2871:v
2867:m
2857:e
2851:2
2847:/
2843:3
2838:)
2832:T
2827:B
2823:k
2816:2
2812:m
2807:(
2802:=
2797:z
2793:v
2789:d
2783:y
2779:v
2775:d
2769:x
2765:v
2761:d
2757:)
2752:z
2748:v
2744:,
2739:y
2735:v
2731:,
2726:x
2722:v
2718:(
2709:f
2684:V
2680:/
2676:N
2673:=
2670:n
2632:D
2587:R
2567:A
2563:N
2540:,
2537:R
2532:2
2529:3
2524:=
2521:T
2516:A
2512:N
2506:B
2502:k
2496:2
2493:D
2488:=
2485:K
2473:D
2471:(
2453:.
2450:T
2445:B
2441:k
2435:2
2432:1
2427:=
2421:D
2418:N
2414:K
2388:.
2383:2
2379:v
2375:m
2372:N
2367:2
2364:D
2359:=
2354:t
2350:K
2346:D
2343:=
2340:K
2321:D
2313:D
2309:D
2305:D
2285:4
2279:1
2262:)
2260:4
2258:(
2241:.
2236:t
2232:K
2226:3
2223:2
2218:=
2215:V
2212:P
2196:3
2190:1
2180:3
2170:)
2168:3
2166:(
2149:.
2140:B
2135:k
2131:N
2125:t
2121:K
2113:3
2110:2
2105:=
2102:T
2082:)
2080:2
2078:(
2056:B
2051:k
2044:2
2040:v
2036:m
2028:3
2025:1
2020:=
2017:T
1992:T
1970:2
1966:v
1962:m
1959:N
1954:2
1951:1
1946:=
1941:t
1937:K
1916:N
1896:.
1893:T
1887:B
1882:k
1876:2
1873:3
1868:=
1863:2
1859:v
1855:m
1850:2
1847:1
1825:,
1820:3
1814:2
1810:v
1806:m
1800:=
1797:T
1791:B
1786:k
1756:T
1729:B
1724:k
1708:)
1706:1
1704:(
1687:,
1684:T
1678:B
1673:k
1669:N
1666:=
1663:V
1660:P
1630:3
1624:2
1620:v
1616:m
1613:N
1607:=
1604:V
1601:P
1563:.
1558:t
1554:K
1548:3
1545:2
1540:=
1537:V
1534:P
1514:,
1509:2
1505:v
1501:m
1496:2
1493:N
1488:=
1483:t
1479:K
1456:t
1452:K
1426:.
1421:2
1417:v
1413:m
1408:3
1405:N
1400:=
1397:F
1394:L
1391:N
1388:=
1385:V
1382:P
1358:V
1354:F
1351:L
1348:N
1342:=
1337:A
1328:F
1323:N
1317:=
1314:P
1302:N
1286:.
1281:i
1277:A
1270:3
1267:=
1264:A
1244:.
1239:k
1235:A
1231:=
1226:j
1222:A
1218:=
1213:i
1209:A
1189:,
1184:k
1180:F
1176:=
1171:j
1167:F
1163:=
1158:i
1154:F
1150:=
1147:F
1128:,
1123:k
1119:v
1115:=
1110:j
1106:v
1102:=
1097:i
1093:v
1089:=
1086:v
1058:k
1052:,
1043:j
1037:,
1028:i
1003:.
997:2
992:z
982:v
974:=
968:2
963:y
953:v
945:=
939:2
934:x
924:v
916:=
911:2
907:v
884:v
874:N
858:.
851:i
847:L
840:2
835:i
831:v
827:m
821:=
814:2
810:t
803:i
799:L
795:m
789:=
783:t
779:d
771:i
767:p
762:d
755:=
750:i
746:F
717:.
714:t
710:/
704:i
700:L
696:m
693:=
688:i
684:v
680:m
677:=
672:i
668:p
639:.
634:i
630:v
625:/
619:i
615:L
611:=
608:t
584:i
580:L
574:i
570:A
566:=
563:V
541:i
537:A
514:i
510:L
479:i
454:i
450:v
374:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
