199:
432:, 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.
5734:
10852:
5439:
7984:, 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
5230:
4590:
3844:
7980:. 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
4995:
6848:
5729:{\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 }
4360:
7952:
3638:
4274:
2963:
3558:
9891:
8642:
298:
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
8114:
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
299:
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
6645:
9075:
7820:
4034:
7685:
5908:
82:. 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,
8101:
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
2714:
879:
3351:
9689:
8440:
7258:
10265:
6001:
4703:
6359:
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
5225:{\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}}}
8775:
4585:{\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}
3839:{\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}}}.}
2657:
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
10139:
8933:
7136:
7364:
7582:
9224:
4780:
1024:
7009:
10397:
5808:
5304:
10008:
8944:
7787:
1917:
2170:
373:
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
7589:
5812:
4347:
3633:
9439:
1381:
5395:
3940:
2082:
9377:
8194:
2561:
751:
8049:
10816:
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
9526:, 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
7145:
1447:
10150:
6417:
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
7504:
2474:
5912:
5444:
5000:
4619:
1846:
6318:
1085:
10533:
8678:
2409:
417:. 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).
6843:{\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 )}
1535:
31:
7416:
9279:
10802:
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.
1993:
228:
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
2262:
1584:
10703:
7947:{\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}}}}
6425:. 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
1708:
738:
413:
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
4269:{\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).}
10054:
8848:
7051:
1653:
1210:
1149:
6888:
10582:
4961:
3297:
1265:
10451:
9468:
7265:
4827:
8349:
4914:
3250:
2958:{\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}}
1753:
8223:
326:
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
7511:
6067:
3553:{\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).}
660:
11623:
9922:
9886:{\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 )}
9105:
8673:
8637:{\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 )}
4990:
4714:
8295:
6040:
912:
607:
10609:
9495:
9306:
5434:
6096:
3875:
1307:
506:
8250:
7443:
2641:
At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on
8273:
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
6917:
6343:
6253:
5324:
9661:
8412:
7809:
6617:
6172:
4006:
3163:
3028:
10478:
10044:
9598:
8840:
8802:
7712:
7043:
6554:
6501:
6230:
2705:
2590:
1479:
564:
537:
10295:
5746:
6527:
3100:
10290:
9927:
9684:
9621:
9100:
8435:
8372:
6640:
6577:
4873:
4850:
4029:
3966:
3346:
3323:
3209:
3186:
3123:
3074:
3051:
2988:
477:
357:. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.
9641:
9564:
9544:
8392:
8315:
8091:
8071:
7974:
6908:
6597:
6463:
6443:
6273:
6196:
4610:
3986:
3143:
3008:
2013:
1937:
1777:
378:, 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.
12415:
7717:
449:
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.
1315:
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,
905:
6392:
1851:
9522:
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
9388:
1320:
6396:
388:
The rapidly moving particles constantly collide among themselves and with the walls of the container, and all these collisions are perfectly elastic.
12444:
11368:"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,"
9385:
yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as
7812:
381:
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
9311:
8131:
6102:
5235:
2491:
12408:
11138:
6388:
8351:
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
12352:
7989:
11422:
Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium".
1385:
6400:
12377:, (first edition 1939, second edition 1952), third edition 1970 prepared in co-operation with D. Burnett, Cambridge University Press, London
11654:
187:
proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This
7448:
2418:
6379:
In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of
Kinetic gas theory to dense gas mixtures,
12457:
10932:
2108:
1791:
887:, of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed,
6278:
105:
and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.
4279:
3565:
12497:
12375:
The
Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases
12479:
10483:
6421:
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
5329:
3880:
2346:
2023:
4791:
To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with
12030:"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen"
9600:
be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area
9070:{\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)}
11620:
6154:
is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision. Let
1484:
8277:. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy
7374:
353:
in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as
9234:
6175:
428:
model, relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with
260:
10402:
278:
developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also
11747:"The Enskog theory for multicomponent mixtures. III. Transport properties of dense binary mixtures with one tracer component"
11280:
10292:
direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
1540:
392:
10665:
673:
8098:
6380:
10832:(matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.
7680:{\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}=2{\sqrt {{\frac {2}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}}}
5903:{\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}
1153:
1092:
11131:
746:, which states that the force experienced by a particle is related to the time rate of change of its momentum, such that
10611:
is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation
10544:
9102:
direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
1214:
12485:
11886:
11197:
10966:
10659:
10407:
308:
12024:
de Groot, S. R., W. A. van
Leeuwen and Ch. G. van Weert (1980), Relativistic Kinetic Theory, North-Holland, Amsterdam.
11354:"Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"
2322:= 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that
1031:
12336:
12224:
11972:
11398:
2623:
874:{\displaystyle F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.}
614:
12412:
10974:
10894:
2326:= 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like
406:
Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.
10829:
3848:
This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed
10643:
10629:
9497:
is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.
8252:
is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.
7253:{\displaystyle \tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}l\,{du \over dy}\right)}
153:
12503:
12500:
allowing high school students to experiment and discover how various factors affect rates of chemical reactions.
10260:{\displaystyle J_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}\cdot \left(n_{0}\pm {\frac {2}{3}}l\,{dn \over dy}\right)}
8111:
1270:
12366:
11124:
9567:
8318:
7011:
where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient
6466:
2708:
350:
108:
The theory's introduction allowed many principal concepts of thermodynamics to be established. It explains the
17:
10012:
Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient
8808:. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient
8110:
which have a negative part that attracts the other molecule from distances longer than the hard core radius.
5996:{\displaystyle v_{\text{rms}}={\sqrt {\frac {3}{2}}}v_{p}={\sqrt {{3}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}
2218:
1942:
12176:
11609:
4698:{\displaystyle \Phi _{\text{effusion}}=J_{\text{collision}}A=nA{\sqrt {\frac {k_{\mathrm {B} }T}{2\pi m}}}.}
3635:
yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
1666:
12529:
10899:
10817:
10811:
7369:
2634:
2615:
161:
4963:; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of
243:
Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were
10889:
10884:
9506:
6855:
1607:
8770:{\displaystyle \varepsilon ^{\pm }=\left(\varepsilon _{0}\pm mc_{v}l\cos \theta \,{dT \over dy}\right),}
4919:
3255:
10904:
9446:
4794:
8327:
8225:
is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and
4878:
3214:
1729:
11881:(3 ed.). Reading, Massachusetts, USA: Addison-Wesley Publishing Company, Inc. pp. 286–291.
11567:
11162:
10821:
10635:
9470:
is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and
8201:
1780:
354:
145:
10063:
8857:
7060:
6045:
10958:
9898:
8649:
7791:
and insert the velocity in the viscosity equation above. This gives the well known equation (with
4966:
369:
The gas consists of very small particles. This smallness of their size is such that the sum of the
338:, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.
225:
12113:
8280:
6018:
569:
395:, meaning that there are no three-body (or higher) interactions, and the particles have no memory.
12524:
11020:
10942:
10587:
9473:
9284:
8116:
8103:
8097:
of a molecule in a monomolecular gas. There are no simple general relation between the collision
2667:
1848:
which leads to a simplified expression of the average translational kinetic energy per molecule,
12491:
12268:
7262:
The net rate of momentum per unit area that is transported across the imaginary surface is thus
6072:
5400:
3851:
2663:
2196:
The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature
482:
232:
determines the temperature of the gas. The theory was not immediately accepted, in part because
12380:
11605:
8805:
8228:
7421:
6118:
743:
398:
Except during collisions, the interactions among molecules are negligible. They exert no other
271:
233:
59:
12454:
11962:
6911:
6325:
6235:
6151:
5309:
1588:
This is an important, non-trivial result of the kinetic theory because it relates pressure, a
83:
12449:
12168:
12096:
11367:
11353:
11187:
11037:
10843:
10538:
10134:{\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}
9646:
9382:
8928:{\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}
8397:
8123:
7981:
7794:
7131:{\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}
6602:
6157:
3991:
3148:
3013:
2646:
2607:
Thus, the kinetic energy per unit kelvin of an ideal monatomic gas can be calculated easily:
2338:
2307:
2300:
509:
425:
421:
248:
168:
12301:
12280:
12244:
12147:
12101:
12044:
12004:
11915:
11836:
11797:
11758:
11688:
11441:
11192:
11172:
11080:
10778:
10456:
10015:
9576:
8811:
8780:
8267:
8261:
7690:
7014:
6532:
6479:
6365:
6201:
2676:
2568:
1457:
542:
515:
345:, a framework for its use in developing transport equations was developed independently by
295:
256:
133:
12172:
12135:
11992:
7359:{\displaystyle \tau =\tau ^{+}-\tau ^{-}={\frac {1}{3}}{\bar {v}}nm\cdot l\,{du \over dy}}
8:
11075:
11065:
10877:
8126:
reduces to the above expression in the limit of infinite dilution, and can be written as
6506:
6354:
3079:
429:
382:
331:
291:
125:
71:
58:
of pressure. The atoms have an average speed relative to their size slowed down here two
12284:
12248:
12197:
12151:
12048:
12008:
11919:
11840:
11801:
11762:
11692:
11445:
10272:
9666:
9603:
9082:
8417:
8354:
6622:
6559:
6412:
4855:
4832:
4011:
3948:
3328:
3305:
3191:
3168:
3105:
3056:
3033:
2970:
12346:
12325:
12201:
11457:
11431:
11253:
11202:
11157:
10950:
10766:
10612:
9626:
9549:
9529:
8377:
8300:
8076:
8056:
7959:
7577:{\displaystyle \eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m\cdot {\bar {v}}}{\sigma }}}
6893:
6582:
6448:
6428:
6258:
6181:
6178:
of one molecule colliding with another. As in the previous section, the number density
4613:
4595:
3971:
3128:
2993:
2659:
2601:
2311:
1998:
1922:
1762:
1756:
455:
342:
167:
The theory was historically significant as the first explicit exercise of the ideas of
141:
11938:
11903:
10828:(matter flow due to temperature gradient, and heat flow due to pressure gradient) and
9219:{\displaystyle q=q_{y}^{+}-q_{y}^{-}=-{\frac {1}{3}}{\bar {v}}nmc_{v}l\,{dT \over dy}}
8675:
above and below the gas layer, and each will contribute a molecular kinetic energy of
4775:{\displaystyle \Phi _{\text{effusion}}={\frac {PA}{\sqrt {2\pi mk_{\mathrm {B} }T}}}.}
2015:
is related to the translational kinetic energy by the description above, resulting in
12482:
to the kinetic molecular theory of gases, from The Upper Canada
District School Board
12388:
12332:
12220:
12193:
11968:
11943:
11882:
11848:
11587:
11583:
11517:
11404:
11394:
11276:
11100:
11042:
10727:
8274:
6133:
2593:
1019:{\displaystyle v^{2}={{\vec {v}}_{x}^{2}}={{\vec {v}}_{y}^{2}}={{\vec {v}}_{z}^{2}}.}
244:
180:
109:
102:
86:
between its temperature, pressure, and volume. The particles are now known to be the
55:
11521:
11461:
11257:
198:
12288:
12252:
12155:
12079:
12052:
12012:
11933:
11923:
11852:
11844:
11805:
11766:
11727:
11696:
11579:
11533:
11449:
11245:
11167:
11095:
11070:
10919:
10715:
10639:
9512:
9229:
8094:
6369:
2413:
Thus, the energy added to the system per gas particle kinetic degree of freedom is
890:
304:
207:
149:
137:
12131:
7586:
Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
365:
The application of kinetic theory to ideal gases makes the following assumptions:
282:
and vibrational molecular motions. In this same work he introduced the concept of
267:
12467:
12461:
12419:
12029:
11627:
11217:
11152:
11085:
11047:
11027:
10647:
8107:
7004:{\displaystyle p_{x}^{\pm }=m\left(u_{0}\pm l\cos \theta \,{du \over dy}\right),}
6123:
4785:
2341:
requires that kinetic energy is partitioned equally, the total kinetic energy is
414:
335:
327:
275:
157:
11453:
7138:
yields the forward momentum transfer per unit time per unit area (also known as
6556:
is along the horizontal direction. The number of molecules arriving at the area
409:
As a simplifying assumption, the particles are usually assumed to have the same
191:
atomistic point of view was rarely considered in the subsequent centuries, when
12519:
12398:
12370:
12269:"Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"
11824:
11207:
11090:
10997:
10909:
10861:
10730:
10655:
10638:
of the gas particles' detailed dynamics, the system must obey the principle of
10144:
9566:
above the lower plate. The non-equilibrium molecular flow is superimposed on a
9523:
6384:
6373:
6232:. The collision cross section per volume or collision cross section density is
6145:
1452:
283:
229:
217:
75:
43:
12513:
12504:
https://www.youtube.com/watch?v=47bF13o8pb8&list=UUXrJjdDeqLgGjJbP1sMnH8A
12292:
12159:
12092:
12057:
12016:
11964:
Molecular
Driving Forces: Statistical Thermodynamics in Chemistry and Biology
11591:
11537:
10392:{\displaystyle J=J_{y}^{+}-J_{y}^{-}=-{\frac {1}{3}}{\bar {v}}l{dn \over dy}}
8073:
is called collision cross section radius or kinetic radius, and the diameter
5803:{\displaystyle v_{\text{p}}={\sqrt {2\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}
5299:{\textstyle v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }
4708:
4354:
2673:
Assume that in the container, the number density (number per unit volume) is
1656:
375:
252:
212:
12384:
12189:
Liboff, R. L. (1990), Kinetic Theory, Prentice-Hall, Englewood Cliffs, N. J.
11928:
11408:
8317:
above the lower plate. The non-equilibrium energy flow is superimposed on a
6015:
is the mass of one molecule of gas in kg. The most probable (or mode) speed
3945:
The momentum transfer to the container wall from particles hitting the area
1028:
Further, assume that the volume is symmetrical about its three dimensions,
12083:
11947:
11677:"The Enskog theory for multicomponent mixtures. I. Linear transport theory"
11249:
11212:
11110:
10003:{\displaystyle n^{\pm }=\left(n_{0}\pm l\cos \theta \,{dn \over dy}\right)}
9519:
7139:
6473:
6418:
3348:
also depends on the velocity distribution; All in all, it calculates to be:
1592:
property, to the translational kinetic energy of the molecules, which is a
346:
188:
12423:
6503:
be the forward velocity of the gas at a horizontal flat layer (labeled as
12067:
11105:
11007:
11002:
10992:
10143:
yields the molecular transfer per unit time per unit area (also known as
9511:
Following a similar logic as above, one can derive the kinetic model for
8266:
Following a similar logic as above, one can derive the kinetic model for
2281:
1783:
1593:
1589:
391:
Interactions (i.e. collisions) between particles are strictly binary and
320:
121:
47:
35:
7782:{\displaystyle k_{B}\cdot N_{A}=R\quad {\text{and}}\quad M=m\cdot N_{A}}
2318:. A monatomic gas is axially symmetric about each spatial axis, so that
2310:
requires that kinetic energy is partitioned equally between all kinetic
10825:
7977:
4276:
Integrating this over all appropriate velocities within the constraint
3562:
Integrating this over all appropriate velocities within the constraint
2299:) are called the "classical results", which could also be derived from
2284:
is proportional to the average translational molecular kinetic energy.
30:
11857:
11786:"The Enskog theory for multicomponent mixtures. IV. Thermal diffusion"
10851:
9281:
gives the equation for thermal conductivity, which is usually denoted
8937:
yields the energy transfer per unit time per unit area (also known as
12257:
11809:
11770:
11731:
11716:"The Enskog theory for multicomponent mixtures. II. Mutual diffusion"
11700:
10651:
8938:
6465:
above the lower plate. The non-equilibrium flow is superimposed on a
6361:
2479:
2334:= 6, comprising 3 translational and 3 rotational degrees of freedom.
1912:{\displaystyle {\frac {1}{2}}mv^{2}={\frac {3}{2}}k_{\mathrm {B} }T.}
312:
287:
237:
192:
184:
129:
98:
39:
10046:
can be considered to be constant over a distance of mean free path.
8842:
can be considered to be constant over a distance of mean free path.
7045:
can be considered to be constant over a distance of mean free path.
1267:
The total surface area on which the gas particles act is therefore
11785:
11746:
11715:
11676:
11436:
11177:
11032:
8845:
Integrating over all appropriate velocities within the constraint
7508:
Combining this equation with the equation for mean free path gives
7048:
Integrating over all appropriate velocities within the constraint
6099:
4350:
2165:{\displaystyle T={\frac {2}{3}}{\frac {K_{t}}{Nk_{\mathrm {B} }}}.}
665:
609:. The gas particle encounters a boundary after characteristic time
446:
300:
279:
117:
91:
10453:
gives the equation for mass diffusivity, which is usually denoted
10049:
Integrating over all appropriate velocities within the constraint
6198:
is defined as the number of molecules per (extensive) volume, or
2478:
Therefore, the kinetic energy per kelvin of one mole of monatomic
12403:
Kinetic Theory: Classical, Quantum, and
Relativistic Descriptions
9924:
above and below the gas layer, where the local number density is
7418:
gives the equation for shear viscosity, which is usually denoted
6112:
4342:{\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }
3877:
of the
Maxwell's velocity distribution, one has to integrate over
3628:{\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }
316:
240:
how the collisions between molecules could be perfectly elastic.
12217:
The Man Who
Changed Everything – the Life of James Clerk Maxwell
11391:
The Man Who Changed Everything – the Life of James Clerk Maxwell
9434:{\displaystyle \kappa =\alpha _{\kappa }\kappa _{0}+\kappa _{c}}
1376:{\displaystyle P={\frac {N{\overline {F}}}{A}}={\frac {NLF}{V}}}
12469:
Temperature and Pressure of an Ideal Gas: The Equation of State
12114:"On the Causes, Laws and Phenomena of Heat, Gases, Gravitation"
6322:
Notice that the unit of the collision cross section per volume
5390:{\displaystyle \int f(v,\theta ,\phi )\,dv\,d\theta \,d\phi =1}
3935:{\displaystyle v>0,0<\theta <\pi ,0<\phi <2\pi }
2077:{\displaystyle T={\frac {1}{3}}{mv^{2} \over k_{\mathrm {B} }}}
370:
259:(1843), which connected their research with the development of
113:
51:
12424:
Macroscopic and kinetic modelling of rarefied polyatomic gases
11879:
Thermodynamics, Kinetic Theory, and Statistical Thermodynamics
270:
created a simple gas-kinetic model, which only considered the
54:
atoms relative to their spacing is shown to scale under 1,950
12506:
A demonstration apparatus for the thermal agitation in gases.
11604:
The average kinetic energy of a fluid is proportional to the
10914:
10734:
2327:
399:
9372:{\displaystyle \kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}l}
8189:{\displaystyle \eta =(1+\alpha _{\eta })\eta _{0}+\eta _{c}}
2556:{\displaystyle K={\frac {D}{2}}k_{B}N_{A}T={\frac {3}{2}}R,}
148:), the kinetic theory is also connected to the principle of
27:
Understanding of gas properties in terms of molecular motion
12494:
linking together kinetic theory concepts, from HyperPhysics
12488:
illustrating the kinetic theory from University of Arkansas
12395:, revised edition (Wiley-Interscience), ISBN 978-0471400653
11784:
Kincaid, J. M.; Cohen, E. G. D.; Lòpez de Haro, M. (1987).
11714:
Kincaid, J. M.; Lòpez de Haro, M.; Cohen, E. G. D. (1983).
11675:
Lòpez de Haro, M.; Cohen, E. G. D.; Kincaid, J. M. (1983).
11236:
Maxwell, J. C. (1867). "On the Dynamical Theory of Gases".
11182:
10127:
9500:
8921:
7124:
6128:
410:
87:
11270:
11153:
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations
12473:
11783:
11713:
11674:
11238:
Philosophical Transactions of the Royal Society of London
8044:{\displaystyle \sigma =\pi \left(2r\right)^{2}=\pi d^{2}}
2330:, is not radially symmetric about any axis, resulting in
221:
79:
9895:
These molecules made their last collision at a distance
8646:
These molecules made their last collision at a distance
1442:{\displaystyle \Rightarrow PV=NLF={\frac {N}{3}}mv^{2}.}
286:
of a particle. In 1859, after reading a paper about the
236:
had not yet been established, and it was not obvious to
12208:, Cambridge: Harvard University Press, pp. 224–233
12206:
Mikhail Vasil'evich Lomonosov on the Corpuscular Theory
11522:"Inelastic collisions and gas-kinetic effects of light"
12198:"On the Relation of the Amount of Material and Weight"
10269:
Note that the molecular transfer from above is in the
6914:. Each molecule will contribute a forward momentum of
5403:
5238:
1945:
1610:
893:
458:
11993:"Ueber die Art der Bewegung, welche wir Wärme nennen"
11643:. New York: University Science Books. pp. 56–61.
11572:
Physica A: Statistical Mechanics and its Applications
11506:. New York, NY: University Science Books. p. 37.
11264:
10668:
10590:
10547:
10486:
10459:
10410:
10298:
10275:
10153:
10057:
10018:
9930:
9901:
9692:
9669:
9649:
9629:
9606:
9579:
9552:
9532:
9476:
9449:
9391:
9314:
9287:
9237:
9108:
9085:
8947:
8851:
8814:
8783:
8681:
8652:
8443:
8420:
8400:
8380:
8357:
8330:
8303:
8283:
8255:
8231:
8204:
8134:
8079:
8059:
7992:
7962:
7823:
7797:
7720:
7693:
7592:
7514:
7499:{\displaystyle \eta _{0}={\frac {1}{3}}{\bar {v}}nml}
7451:
7424:
7377:
7268:
7148:
7054:
7017:
6920:
6896:
6858:
6648:
6625:
6605:
6585:
6562:
6535:
6509:
6482:
6451:
6431:
6328:
6281:
6261:
6238:
6204:
6184:
6160:
6075:
6048:
6021:
5915:
5815:
5749:
5743:
From the kinetic energy formula it can be shown that
5442:
5332:
5312:
4998:
4969:
4922:
4881:
4858:
4835:
4797:
4717:
4622:
4598:
4363:
4282:
4037:
4014:
3994:
3974:
3951:
3883:
3854:
3641:
3568:
3354:
3331:
3308:
3258:
3217:
3194:
3171:
3151:
3131:
3108:
3082:
3059:
3036:
3016:
2996:
2973:
2717:
2679:
2649:
is needed to accurately compute these contributions.
2626:(273.15 K), the kinetic energy can also be obtained:
2571:
2494:
2469:{\displaystyle {\frac {K}{ND}}={\frac {1}{2}}k_{B}T.}
2421:
2349:
2221:
2111:
2026:
2001:
1925:
1854:
1794:
1765:
1732:
1669:
1543:
1487:
1460:
1388:
1323:
1273:
1217:
1156:
1095:
1034:
915:
754:
676:
617:
572:
545:
518:
485:
307:
generalized Maxwell's achievement and formulated the
12070:(1949), "On the Kinetic Theory of Rarefied Gases.",
10634:
The kinetic theory of gases entails that due to the
12183:, Geneva & Paris: J.J. Paschoud, pp. 1–186
11424:
Studies in History and Philosophy of Modern Physics
9079:Note that the energy transfer from above is in the
6069:, and the mean (arithmetic mean, or average) speed
1841:{\displaystyle k_{\mathrm {B} }T={mv^{2} \over 3},}
12405:, third edition (Springer), ISBN 978-0-387-21775-8
12324:
12322:
11744:
10726:is the "mobility", or the ratio of the particle's
10697:
10603:
10576:
10527:
10472:
10445:
10391:
10284:
10259:
10133:
10038:
10002:
9916:
9885:
9678:
9655:
9635:
9615:
9592:
9558:
9538:
9489:
9462:
9433:
9371:
9300:
9273:
9218:
9094:
9069:
8927:
8834:
8796:
8769:
8667:
8636:
8429:
8406:
8386:
8366:
8343:
8309:
8289:
8244:
8217:
8188:
8085:
8065:
8043:
7968:
7946:
7803:
7781:
7706:
7679:
7576:
7498:
7437:
7410:
7358:
7252:
7130:
7037:
7003:
6902:
6882:
6842:
6634:
6611:
6591:
6571:
6548:
6521:
6495:
6457:
6437:
6337:
6313:{\displaystyle l={\frac {1}{n\sigma {\sqrt {2}}}}}
6312:
6267:
6247:
6224:
6190:
6166:
6090:
6061:
6034:
5995:
5902:
5802:
5728:
5428:
5389:
5318:
5298:
5224:
4984:
4955:
4908:
4875:are contained in the tilted pipe with a height of
4867:
4844:
4821:
4774:
4697:
4604:
4584:
4341:
4268:
4023:
4000:
3980:
3960:
3934:
3869:
3838:
3627:
3552:
3340:
3317:
3291:
3244:
3211:are contained in the tilted pipe with a height of
3203:
3180:
3157:
3137:
3117:
3094:
3068:
3053:, will collide with the area within time interval
3045:
3022:
3002:
2982:
2957:
2699:
2584:
2555:
2468:
2403:
2256:
2164:
2076:
2007:
1987:
1931:
1919:The translational kinetic energy of the system is
1911:
1840:
1771:
1747:
1702:
1647:
1578:
1529:
1473:
1441:
1375:
1301:
1259:
1204:
1143:
1079:
1018:
899:
873:
732:
654:
601:
558:
531:
500:
471:
12327:Mathematical Methods in Particle Transport Theory
11822:
6406:
5326:can be determined by the normalization condition
2652:
2194:) is one important result of the kinetic theory:
1599:
12511:
1080:{\displaystyle {\hat {i}},{\hat {j}},{\hat {k}}}
11516:
11271:L.I Ponomarev; I.V Kurchatov (1 January 1993).
10805:
10528:{\displaystyle D_{0}={\frac {1}{3}}{\bar {v}}l}
9546:in the layer increases uniformly with distance
2662:s, which is useful in applications such as the
2404:{\displaystyle K=DK_{t}={\frac {D}{2}}Nmv^{2}.}
1604:Rewriting the above result for the pressure as
452:Consider a gas particle traveling at velocity,
12173:"Physique Mécanique des Georges-Louis Le Sage"
12072:Communications on Pure and Applied Mathematics
11895:
11621:Configuration integral (statistical mechanics)
10623:
8093:is called collision cross section diameter or
3302:The total number of particles that reach area
1311:The pressure exerted by the collisions of the
440:
290:of molecules by Clausius, Scottish physicist
11876:
11132:
6852:These molecules made their last collision at
2990:on the container wall, a particle with speed
2280:Thus, the product of pressure and volume per
668:of the gas particle can then be described as
11960:
11872:
11870:
11868:
11641:Physical Chemistry for the Chemical Sciences
11639:Chang, Raymond; Thoman, John W. Jr. (2014).
11504:Physical Chemistry for the Chemical Sciences
11502:Chang, Raymond; Thoman, John W. Jr. (2014).
11388:
8112:The radius for zero Lennard-Jones potential
4558:
4539:
97:The basic version of the model describes an
11745:Lòpez de Haro, M.; Cohen, E. G. D. (1984).
11638:
11608:, which always exceeds the mean velocity -
11501:
10537:The corresponding expression obtained from
8122:The expression for viscosity obtained from
1530:{\displaystyle K_{t}={\frac {N}{2}}mv^{2},}
12351:: CS1 maint: location missing publisher (
12235:Maxwell, James Clerk (1873), "Molecules",
11901:
11877:Sears, F.W.; Salinger, G.L. (1975). "10".
11825:"The non-linear Enskog-Boltzmann equation"
11139:
11125:
10850:
10401:Combining the above kinetic equation with
9568:Maxwell-Boltzmann equilibrium distribution
9228:Combining the above kinetic equation with
8319:Maxwell-Boltzmann equilibrium distribution
7411:{\displaystyle \tau =\eta \,{du \over dy}}
7368:Combining the above kinetic equation with
6467:Maxwell-Boltzmann equilibrium distribution
6255:, and it is related to the mean free path
3125:. Therefore, all the particles with speed
12256:
12056:
11937:
11927:
11865:
11856:
11555:. New York, NY: University Science Press.
11550:
11435:
10678:
10231:
9974:
9873:
9866:
9859:
9721:
9714:
9623:on one side of the gas layer, with speed
9274:{\displaystyle q=-\kappa \,{dT \over dy}}
9250:
9195:
9041:
8738:
8624:
8617:
8610:
8472:
8465:
8374:on one side of the gas layer, with speed
7714:is the most probable speed. We note that
7387:
7335:
7224:
6972:
6830:
6823:
6816:
6742:
6679:
6672:
6579:on one side of the gas layer, with speed
5707:
5681:
5651:
5644:
5637:
5485:
5478:
5471:
5374:
5367:
5360:
5277:
5251:
5208:
5201:
5194:
5041:
5034:
5027:
4251:
4244:
4237:
4099:
4092:
3535:
3528:
3521:
3383:
3376:
2941:
2927:
2913:
2798:
2784:
2770:
435:
9501:Diffusion Coefficient and diffusion flux
8297:which increases uniformly with distance
4784:The above expression is consistent with
2645:and the total molecular energy. Quantum
1988:{\textstyle K_{t}={\frac {1}{2}}Nmv^{2}}
1786:defined by the ideal gas law, to obtain
197:
29:
11961:Dill, Ken A.; Bromberg, Sarina (2003).
11823:van Beijeren, H.; Ernst, M. H. (1973).
11235:
10824:of the particles), when applied to the
6445:which increase uniformly with distance
6348:
6042:is 81.6% of the root-mean-square speed
4612:is punched to become a small hole, the
2257:{\displaystyle PV={\frac {2}{3}}K_{t}.}
1579:{\displaystyle PV={\frac {2}{3}}K_{t}.}
14:
12512:
11904:"Viscosity of dilute gases and vapors"
11421:
10698:{\displaystyle D=\mu \,k_{\text{B}}T,}
1703:{\displaystyle PV=Nk_{\mathrm {B} }T,}
733:{\displaystyle p_{i}=mv_{i}=mL_{i}/t.}
508:-direction in an enclosed volume with
261:mechanical explanations of gravitation
12393:Molecular Theory of Gases and Liquids
12097:"On the physical properties of gases"
11565:
5738:
12124:, Baldwin, Cradock, and Joy: 273–293
11048:Grand potential / Landau free energy
3165:from the normal that can reach area
2212:
2102:
2017:
1660:
1648:{\textstyle PV={\frac {Nmv^{2}}{3}}}
1205:{\displaystyle F=F_{i}=F_{j}=F_{k},}
1144:{\displaystyle v=v_{i}=v_{j}=v_{k},}
445:In the kinetic theory of gases, the
323:was also first stated by Boltzmann.
12464:- a chapter from an online textbook
12434:, 437–505, DOI 10.1017/jfm.2016.604
11647:
10618:
6883:{\displaystyle y=\pm l\cos \theta }
6413:Viscosity § Momentum transport
62:fold from that at room temperature.
24:
12360:
12181:Deux Traites de Physique Mécanique
12136:"Grundzüge einer Theorie der Gase"
10820:(i.e. the detailed balance of the
10577:{\displaystyle D=\alpha _{D}D_{0}}
9754:
9663:from the normal, in time interval
8505:
8414:from the normal, in time interval
8256:Thermal conductivity and heat flux
7871:
7811:subsequently estimated below) for
7660:
6778:
6712:
6619:from the normal, in time interval
5973:
5880:
5780:
5588:
5531:
5156:
4956:{\displaystyle v\cos(\theta )dAdt}
4757:
4719:
4668:
4624:
4573:
4008:from the normal, in time interval
3812:
3292:{\displaystyle v\cos(\theta )dAdt}
2150:
2066:
1897:
1801:
1739:
1688:
1260:{\displaystyle A_{i}=A_{j}=A_{k}.}
907:, in every direction is identical
789:
772:
420:More modern developments, such as
303:of air and other gases." In 1871,
25:
12541:
12438:
11520:; Nienhuis, Gerard (1991-12-01).
10446:{\displaystyle J=-D{dn \over dy}}
9463:{\displaystyle \alpha _{\kappa }}
6139:
4822:{\displaystyle (v,\theta ,\phi )}
1939:times that of a molecule, namely
415:Dalton's Law of partial pressures
341:Following the development of the
12306:Thoughts on the Mental Functions
11967:. Garland Science. p. 327.
8344:{\displaystyle \varepsilon _{0}}
7813:shear viscosity for dilute gases
6383:, was developed in 1983-1987 by
6150:In kinetic theory of gases, the
4909:{\displaystyle v\cos(\theta )dt}
3245:{\displaystyle v\cos(\theta )dt}
1748:{\displaystyle k_{\mathrm {B} }}
11954:
11816:
11790:The Journal of Chemical Physics
11777:
11751:The Journal of Chemical Physics
11738:
11720:The Journal of Chemical Physics
11707:
11681:The Journal of Chemical Physics
11668:
11632:
11614:
11598:
11568:"Fifty years of kinetic theory"
11559:
11544:
11510:
11495:
11486:
11477:
11468:
11415:
11382:
10644:fluctuation-dissipation theorem
10630:Fluctuation-dissipation theorem
8218:{\displaystyle \alpha _{\eta }}
7756:
7750:
5668:
3076:, if it is within the distance
2709:Maxwell's velocity distribution
216:, which laid the basis for the
154:fluctuation-dissipation theorem
42:is proportional to the average
12219:, Hoboken, New Jersey: Wiley,
11566:Cohen, E. G. D. (1993-03-15).
11343:
11334:
11325:
11316:
11307:
11298:
11289:
11229:
11198:Maxwell–Boltzmann distribution
10660:Einstein–Smoluchowski equation
10516:
10360:
10191:
9880:
9856:
9850:
9831:
9711:
9705:
9344:
9170:
8985:
8631:
8607:
8601:
8582:
8462:
8456:
8160:
8141:
7599:
7562:
7481:
7317:
7178:
6837:
6792:
6669:
6661:
6407:Viscosity and kinetic momentum
6082:
6062:{\displaystyle v_{\text{rms}}}
5822:
5658:
5602:
5468:
5450:
5420:
5357:
5339:
5215:
5170:
5024:
5006:
4938:
4932:
4897:
4891:
4816:
4798:
4234:
4228:
4089:
4083:
4065:
4062:
4056:
4038:
3861:
3780:
3752:
3518:
3512:
3373:
3367:
3274:
3268:
3233:
3227:
2767:
2728:
2653:Collisions with container wall
1600:Temperature and kinetic energy
1389:
1071:
1056:
1041:
995:
966:
937:
655:{\displaystyle t=L_{i}/v_{i}.}
492:
360:
309:Maxwell–Boltzmann distribution
101:. It treats the collisions as
13:
1:
11984:
11551:McQuarrie, Donald A. (1976).
10403:Fick's first law of diffusion
9917:{\displaystyle l\cos \theta }
8668:{\displaystyle l\cos \theta }
4985:{\displaystyle v\cos \theta }
1655:, we may combine it with the
883:Now consider a large number,
112:properties of gases, such as
11849:10.1016/0375-9601(73)90346-0
11655:"5.62 Physical Chemistry II"
11584:10.1016/0378-4371(93)90357-A
10818:Onsager reciprocal relations
10812:Onsager reciprocal relations
10806:Onsager reciprocal relations
8290:{\displaystyle \varepsilon }
6035:{\displaystyle v_{\text{p}}}
3030:from the normal of the area
2707:and that the particles obey
1341:
602:{\displaystyle V=A_{i}L_{i}}
251:(ca. 1780, published 1818),
162:Onsager reciprocal relations
7:
12381:Joseph Oakland Hirschfelder
12323:Williams, M. M. R. (1971).
11454:10.1016/j.shpsb.2017.01.001
10890:Indistinguishable particles
10835:
10624:Fluctuation and dissipation
10604:{\displaystyle \alpha _{D}}
9490:{\displaystyle \kappa _{c}}
9301:{\displaystyle \kappa _{0}}
6098:is 92.1% of the rms speed (
5429:{\textstyle {4}/{\bar {v}}}
2295:
2289:
2270:
2206:
2200:
2190:
2178:
2090:
1716:
441:Pressure and kinetic energy
430:internal degrees of freedom
10:
12546:
12445:PHYSICAL CHEMISTRY – Gases
12428:Journal of Fluid Mechanics
10809:
10627:
9504:
8259:
6410:
6352:
6143:
6091:{\displaystyle {\bar {v}}}
3870:{\displaystyle {\bar {v}}}
1302:{\displaystyle A=3*A_{i}.}
742:We combine the above with
501:{\displaystyle {\hat {i}}}
274:of the particles. In 1857
174:
12267:Smoluchowski, M. (1906),
11902:Hildebrand, J.H. (1976).
11606:root mean-square velocity
10636:microscopic reversibility
10480:when it is a dilute gas:
9308:when it is a dilute gas:
8245:{\displaystyle \eta _{c}}
7445:when it is a dilute gas:
7438:{\displaystyle \eta _{0}}
7370:Newton's law of viscosity
6472:Inside a dilute gas in a
6345:is reciprocal of length.
4852:within the time interval
2303:; for more details, see:
1481:of the gas is defined as
202:Hydrodynamica front cover
146:microscopic reversibility
144:of microscopic dynamics (
12498:Interactive Java Applets
12331:. Butterworths, London.
12293:10.1002/andp.19063261405
12160:10.1002/andp.18561751008
12058:10.1002/andp.19053220806
12017:10.1002/andp.18571760302
11908:Proc Natl Acad Sci U S A
11610:Kinetic Molecular Theory
11538:10.1103/PhysRevA.44.7615
11223:
9507:Fick's laws of diffusion
9381:Similarly to viscosity,
6338:{\displaystyle n\sigma }
6248:{\displaystyle n\sigma }
5319:{\displaystyle \lambda }
1451:The total translational
539:, cross-sectional area,
183:, the Roman philosopher
12450:Early Theories of Gases
12399:Richard Lawrence Liboff
12385:Charles Francis Curtiss
12105:, Robert Baldwin: 56–60
11929:10.1073/pnas.73.12.4302
10933:Thermodynamic ensembles
10885:Spin–statistics theorem
10785:Note that the mobility
9656:{\displaystyle \theta }
8407:{\displaystyle \theta }
8104:Lennard-Jones potential
7804:{\displaystyle \sigma }
6612:{\displaystyle \theta }
6167:{\displaystyle \sigma }
4001:{\displaystyle \theta }
3158:{\displaystyle \theta }
3023:{\displaystyle \theta }
68:kinetic theory of gases
12084:10.1002/cpa.3160020403
11393:. Hoboken, NJ: Wiley.
11371:Philosophical Magazine
11357:Philosophical Magazine
11250:10.1098/rstl.1867.0004
10699:
10605:
10578:
10529:
10474:
10447:
10393:
10286:
10261:
10135:
10040:
10004:
9918:
9887:
9680:
9657:
9637:
9617:
9594:
9570:of molecular motions.
9560:
9540:
9491:
9464:
9435:
9373:
9302:
9275:
9220:
9096:
9071:
8929:
8836:
8806:specific heat capacity
8798:
8771:
8669:
8638:
8431:
8408:
8388:
8368:
8345:
8321:of molecular motions.
8311:
8291:
8246:
8219:
8190:
8087:
8067:
8045:
7970:
7948:
7805:
7783:
7708:
7681:
7578:
7500:
7439:
7412:
7360:
7254:
7132:
7039:
7005:
6904:
6884:
6844:
6636:
6613:
6593:
6573:
6550:
6523:
6497:
6469:of molecular motions.
6459:
6439:
6395:, building on work by
6339:
6314:
6269:
6249:
6226:
6192:
6168:
6119:Root-mean-square speed
6103:distribution of speeds
6092:
6063:
6036:
5997:
5904:
5804:
5730:
5430:
5391:
5320:
5300:
5226:
4986:
4957:
4910:
4869:
4846:
4823:
4776:
4699:
4606:
4586:
4343:
4270:
4025:
4002:
3982:
3962:
3936:
3871:
3840:
3629:
3554:
3342:
3319:
3293:
3246:
3205:
3182:
3159:
3139:
3119:
3096:
3070:
3047:
3024:
3004:
2984:
2967:Then for a small area
2959:
2701:
2586:
2557:
2470:
2405:
2258:
2166:
2078:
2009:
1989:
1933:
1913:
1842:
1773:
1749:
1704:
1649:
1580:
1531:
1475:
1443:
1377:
1303:
1261:
1206:
1145:
1081:
1020:
901:
875:
734:
656:
603:
560:
533:
502:
473:
436:Equilibrium properties
234:conservation of energy
203:
63:
46:of its particles. The
12371:Thomas George Cowling
12302:Waterston, John James
12215:Mahon, Basil (2003),
12112:Herapath, J. (1821),
12028:Einstein, A. (1905),
11991:Clausius, R. (1857),
11553:Statistical Mechanics
11389:Mahon, Basil (2003).
11366:Maxwell, J.C. (1860)
11352:Maxwell, J.C. (1860)
11188:Interatomic potential
11163:Chapman–Enskog theory
11038:Helmholtz free energy
10967:Isoenthalpic–isobaric
10844:Statistical mechanics
10700:
10658:, which leads to the
10606:
10579:
10539:Revised Enskog Theory
10530:
10475:
10473:{\displaystyle D_{0}}
10448:
10394:
10287:
10262:
10136:
10041:
10039:{\displaystyle dn/dy}
10005:
9919:
9888:
9681:
9658:
9638:
9618:
9595:
9593:{\displaystyle n_{0}}
9561:
9541:
9492:
9465:
9436:
9383:Revised Enskog Theory
9374:
9303:
9276:
9221:
9097:
9072:
8930:
8837:
8835:{\displaystyle dT/dy}
8799:
8797:{\displaystyle c_{v}}
8772:
8670:
8639:
8432:
8409:
8389:
8369:
8346:
8312:
8292:
8247:
8220:
8191:
8124:Revised Enskog Theory
8088:
8068:
8046:
7982:Revised Enskog theory
7971:
7949:
7806:
7784:
7709:
7707:{\displaystyle v_{p}}
7682:
7579:
7501:
7440:
7413:
7361:
7255:
7133:
7040:
7038:{\displaystyle du/dy}
7006:
6905:
6885:
6845:
6637:
6614:
6594:
6574:
6551:
6549:{\displaystyle u_{0}}
6524:
6498:
6496:{\displaystyle u_{0}}
6460:
6440:
6381:Revised Enskog Theory
6340:
6315:
6270:
6250:
6227:
6225:{\displaystyle n=N/V}
6193:
6169:
6093:
6064:
6037:
5998:
5905:
5805:
5731:
5431:
5392:
5321:
5301:
5227:
4987:
4958:
4911:
4870:
4847:
4824:
4777:
4700:
4607:
4587:
4344:
4271:
4026:
4003:
3983:
3963:
3937:
3872:
3841:
3630:
3555:
3343:
3325:within time interval
3320:
3294:
3247:
3206:
3188:within time interval
3183:
3160:
3140:
3120:
3097:
3071:
3048:
3025:
3005:
2985:
2960:
2702:
2700:{\displaystyle n=N/V}
2647:statistical mechanics
2611:per mole: 12.47 J / K
2587:
2585:{\displaystyle N_{A}}
2558:
2471:
2406:
2339:equipartition theorem
2308:equipartition theorem
2301:statistical mechanics
2259:
2167:
2079:
2010:
1990:
1934:
1914:
1843:
1774:
1750:
1705:
1650:
1581:
1537:providing the result
1532:
1476:
1474:{\displaystyle K_{t}}
1444:
1378:
1304:
1262:
1207:
1146:
1082:
1021:
902:
876:
735:
657:
604:
561:
559:{\displaystyle A_{i}}
534:
532:{\displaystyle L_{i}}
510:characteristic length
503:
474:
422:Revised Enskog Theory
355:Chapman–Enskog theory
249:Georges-Louis Le Sage
201:
195:ideas were dominant.
169:statistical mechanics
33:
12118:Annals of Philosophy
12102:Annals of Philosophy
11193:Magnetohydrodynamics
11173:Critical temperature
10779:absolute temperature
10666:
10642:. Specifically, the
10588:
10545:
10484:
10457:
10408:
10296:
10273:
10151:
10055:
10016:
9928:
9899:
9690:
9667:
9647:
9627:
9604:
9577:
9550:
9530:
9474:
9447:
9389:
9312:
9285:
9235:
9106:
9083:
8945:
8849:
8812:
8781:
8679:
8650:
8441:
8418:
8398:
8378:
8355:
8328:
8301:
8281:
8268:thermal conductivity
8262:Thermal conductivity
8229:
8202:
8132:
8077:
8057:
7990:
7960:
7821:
7795:
7718:
7691:
7590:
7512:
7449:
7422:
7375:
7266:
7146:
7052:
7015:
6918:
6894:
6856:
6646:
6623:
6603:
6583:
6560:
6533:
6507:
6480:
6449:
6429:
6366:thermal conductivity
6349:Transport properties
6326:
6279:
6259:
6236:
6202:
6182:
6158:
6073:
6046:
6019:
5913:
5813:
5747:
5440:
5401:
5330:
5310:
5236:
5232:with the constraint
4996:
4967:
4920:
4879:
4856:
4833:
4795:
4715:
4620:
4596:
4361:
4280:
4035:
4012:
3992:
3972:
3949:
3881:
3852:
3639:
3566:
3352:
3329:
3306:
3256:
3215:
3192:
3169:
3149:
3129:
3106:
3080:
3057:
3034:
3014:
2994:
2971:
2715:
2677:
2624:standard temperature
2569:
2492:
2419:
2347:
2219:
2109:
2024:
1999:
1943:
1923:
1852:
1792:
1763:
1730:
1667:
1608:
1541:
1485:
1458:
1386:
1321:
1271:
1215:
1154:
1093:
1032:
913:
891:
752:
674:
615:
570:
543:
516:
483:
456:
334:'s (1906) papers on
296:Maxwell distribution
272:translational motion
257:John James Waterston
134:thermal conductivity
126:transport properties
12530:Classical mechanics
12285:1906AnP...326..756V
12249:1873Natur...8..437.
12152:1856AnP...175..315K
12049:1905AnP...322..549E
12009:1857AnP...176..353C
11920:1976PNAS...73.4302H
11841:1973PhLA...43..367V
11802:1987JChPh..86..963K
11763:1984JChPh..80..408L
11693:1983JChPh..78.2746L
11446:2017SHPMP..57...53G
11313:Herapath 1816, 1821
10975:Isothermal–isobaric
10878:Particle statistics
10822:reversible dynamics
10613:chemical potentials
10337:
10319:
10168:
9147:
9129:
8962:
8117:collision integrals
6935:
6522:{\displaystyle y=0}
6355:Transport phenomena
4592:If this small area
4522:
4488:
4446:
4398:
3721:
3680:
3095:{\displaystyle vdt}
2633:per molecule: 5.65
2614:per molecule: 20.7
1995:. The temperature,
1011:
982:
953:
854:
744:Newton's second law
383:thermodynamic limit
332:Marian Smoluchowski
315:connection between
292:James Clerk Maxwell
12460:2017-02-28 at the
12418:2021-07-25 at the
12413:Henning Struchtrup
12309:(reprinted in his
12273:Annalen der Physik
12202:Henry M. Leicester
12140:Annalen der Physik
12037:Annalen der Physik
11997:Annalen der Physik
11662:MIT OpenCourseWare
11626:2012-04-28 at the
11518:van Enk, Steven J.
11203:Mixmaster universe
11158:Boltzmann equation
10915:Anyonic statistics
10767:Boltzmann constant
10695:
10601:
10574:
10541:may be written as
10525:
10470:
10443:
10389:
10323:
10305:
10285:{\displaystyle -y}
10282:
10257:
10154:
10131:
10126:
10036:
10000:
9914:
9883:
9679:{\displaystyle dt}
9676:
9653:
9633:
9616:{\displaystyle dA}
9613:
9590:
9556:
9536:
9487:
9460:
9431:
9369:
9298:
9271:
9216:
9133:
9115:
9095:{\displaystyle -y}
9092:
9067:
8948:
8925:
8920:
8832:
8794:
8767:
8665:
8634:
8430:{\displaystyle dt}
8427:
8404:
8384:
8367:{\displaystyle dA}
8364:
8341:
8307:
8287:
8275:thermal reservoirs
8242:
8215:
8186:
8083:
8063:
8041:
7966:
7944:
7801:
7779:
7704:
7677:
7574:
7496:
7435:
7408:
7356:
7250:
7128:
7123:
7035:
7001:
6921:
6900:
6880:
6840:
6635:{\displaystyle dt}
6632:
6609:
6589:
6572:{\displaystyle dA}
6569:
6546:
6519:
6493:
6455:
6435:
6335:
6310:
6265:
6245:
6222:
6188:
6164:
6088:
6059:
6032:
6011:is in kelvin, and
5993:
5900:
5800:
5739:Speed of molecules
5726:
5663:
5426:
5387:
5316:
5296:
5222:
5220:
4982:
4953:
4906:
4868:{\displaystyle dt}
4865:
4845:{\displaystyle dA}
4842:
4829:that hit the area
4819:
4772:
4707:Combined with the
4695:
4614:effusive flow rate
4602:
4582:
4508:
4474:
4432:
4376:
4339:
4266:
4024:{\displaystyle dt}
4021:
3998:
3978:
3961:{\displaystyle dA}
3958:
3932:
3867:
3836:
3707:
3658:
3625:
3550:
3341:{\displaystyle dt}
3338:
3318:{\displaystyle dA}
3315:
3289:
3242:
3204:{\displaystyle dt}
3201:
3181:{\displaystyle dA}
3178:
3155:
3135:
3118:{\displaystyle dA}
3115:
3092:
3069:{\displaystyle dt}
3066:
3046:{\displaystyle dA}
3043:
3020:
3000:
2983:{\displaystyle dA}
2980:
2955:
2697:
2668:isotope separation
2660:effusive flow rate
2602:ideal gas constant
2582:
2553:
2466:
2401:
2312:degrees of freedom
2254:
2198:. From equations (
2162:
2074:
2005:
1985:
1929:
1909:
1838:
1769:
1757:Boltzmann constant
1745:
1700:
1645:
1576:
1527:
1471:
1439:
1373:
1299:
1257:
1202:
1141:
1077:
1016:
988:
959:
930:
897:
871:
840:
730:
652:
599:
556:
529:
498:
472:{\textstyle v_{i}}
469:
343:Boltzmann equation
204:
152:, in terms of the
142:time reversibility
64:
12389:Robert Byron Bird
11914:(12): 4302–4303.
11829:Physics Letters A
11532:(11): 7615–7625.
11526:Physical Review A
11492:Smoluchowski 1906
11304:Le Sage 1780/1818
11282:978-0-7503-0251-7
11149:
11148:
11043:Gibbs free energy
10895:Maxwell–Boltzmann
10686:
10519:
10508:
10441:
10387:
10363:
10352:
10250:
10226:
10194:
10183:
9993:
9827:
9764:
9636:{\displaystyle v}
9559:{\displaystyle y}
9539:{\displaystyle n}
9515:of a dilute gas:
9347:
9336:
9269:
9214:
9173:
9162:
9060:
9023:
8988:
8977:
8757:
8578:
8515:
8387:{\displaystyle v}
8310:{\displaystyle y}
8270:of a dilute gas:
8086:{\displaystyle d}
8066:{\displaystyle r}
7969:{\displaystyle M}
7942:
7923:
7906:
7903:
7884:
7880:
7854:
7851:
7754:
7675:
7673:
7646:
7618:
7617:
7602:
7572:
7565:
7545:
7542:
7484:
7473:
7406:
7354:
7320:
7309:
7243:
7219:
7181:
7170:
6991:
6903:{\displaystyle l}
6788:
6722:
6592:{\displaystyle v}
6458:{\displaystyle y}
6438:{\displaystyle u}
6374:thermal diffusion
6308:
6305:
6268:{\displaystyle l}
6191:{\displaystyle n}
6174:be the collision
6134:Mode (statistics)
6085:
6056:
6029:
5988:
5986:
5939:
5938:
5923:
5895:
5893:
5866:
5841:
5840:
5825:
5795:
5793:
5757:
5702:
5598:
5541:
5512:
5423:
5272:
5166:
5101:
4767:
4766:
4725:
4690:
4689:
4643:
4630:
4605:{\displaystyle A}
4534:
4515:
4500:
4481:
4463:
4353:(consistent with
4203:
4140:
3981:{\displaystyle v}
3864:
3831:
3830:
3797:
3783:
3769:
3755:
3738:
3649:
3487:
3424:
3138:{\displaystyle v}
3003:{\displaystyle v}
2909:
2846:
2725:
2664:gaseous diffusion
2618:/ K = 129 μeV / K
2594:Avogadro constant
2545:
2509:
2448:
2435:
2380:
2278:
2277:
2239:
2186:
2185:
2157:
2126:
2098:
2097:
2072:
2041:
2008:{\displaystyle T}
1967:
1932:{\displaystyle N}
1889:
1863:
1833:
1772:{\displaystyle T}
1724:
1723:
1643:
1561:
1509:
1421:
1371:
1350:
1344:
1074:
1059:
1044:
998:
969:
940:
866:
829:
797:
495:
424:and the Extended
245:Mikhail Lomonosov
224:. In this work,
103:perfectly elastic
16:(Redirected from
12537:
12356:
12350:
12342:
12330:
12308:
12295:
12261:
12260:
12258:10.1038/008437a0
12243:(204): 437–441,
12229:
12209:
12184:
12162:
12125:
12106:
12086:
12061:
12060:
12034:
12019:
11979:
11978:
11958:
11952:
11951:
11941:
11931:
11899:
11893:
11892:
11874:
11863:
11862:
11860:
11820:
11814:
11813:
11810:10.1063/1.452243
11781:
11775:
11774:
11771:10.1063/1.446463
11742:
11736:
11735:
11732:10.1063/1.446388
11726:(9): 4509–4521.
11711:
11705:
11704:
11701:10.1063/1.444985
11687:(5): 2746–2759.
11672:
11666:
11665:
11659:
11651:
11645:
11644:
11636:
11630:
11618:
11612:
11602:
11596:
11595:
11563:
11557:
11556:
11548:
11542:
11541:
11514:
11508:
11507:
11499:
11493:
11490:
11484:
11481:
11475:
11472:
11466:
11465:
11439:
11419:
11413:
11412:
11386:
11380:
11347:
11341:
11338:
11332:
11329:
11323:
11320:
11314:
11311:
11305:
11302:
11296:
11293:
11287:
11286:
11273:The Quantum Dice
11268:
11262:
11261:
11233:
11168:Collision theory
11141:
11134:
11127:
10920:Braid statistics
10854:
10840:
10839:
10801:
10775:
10764:
10753:
10724:
10716:mass diffusivity
10712:
10704:
10702:
10701:
10696:
10688:
10687:
10684:
10640:detailed balance
10619:Detailed balance
10610:
10608:
10607:
10602:
10600:
10599:
10583:
10581:
10580:
10575:
10573:
10572:
10563:
10562:
10534:
10532:
10531:
10526:
10521:
10520:
10512:
10509:
10501:
10496:
10495:
10479:
10477:
10476:
10471:
10469:
10468:
10452:
10450:
10449:
10444:
10442:
10440:
10432:
10424:
10398:
10396:
10395:
10390:
10388:
10386:
10378:
10370:
10365:
10364:
10356:
10353:
10345:
10336:
10331:
10318:
10313:
10291:
10289:
10288:
10283:
10266:
10264:
10263:
10258:
10256:
10252:
10251:
10249:
10241:
10233:
10227:
10219:
10214:
10213:
10196:
10195:
10187:
10184:
10176:
10167:
10162:
10140:
10138:
10137:
10132:
10130:
10129:
10098:
10045:
10043:
10042:
10037:
10029:
10009:
10007:
10006:
10001:
9999:
9995:
9994:
9992:
9984:
9976:
9958:
9957:
9940:
9939:
9923:
9921:
9920:
9915:
9892:
9890:
9889:
9884:
9843:
9842:
9830:
9829:
9828:
9826:
9822:
9821:
9808:
9807:
9806:
9793:
9783:
9782:
9778:
9769:
9765:
9763:
9759:
9758:
9757:
9737:
9685:
9683:
9682:
9677:
9662:
9660:
9659:
9654:
9642:
9640:
9639:
9634:
9622:
9620:
9619:
9614:
9599:
9597:
9596:
9591:
9589:
9588:
9565:
9563:
9562:
9557:
9545:
9543:
9542:
9537:
9524:number densities
9513:mass diffusivity
9496:
9494:
9493:
9488:
9486:
9485:
9469:
9467:
9466:
9461:
9459:
9458:
9440:
9438:
9437:
9432:
9430:
9429:
9417:
9416:
9407:
9406:
9378:
9376:
9375:
9370:
9365:
9364:
9349:
9348:
9340:
9337:
9329:
9324:
9323:
9307:
9305:
9304:
9299:
9297:
9296:
9280:
9278:
9277:
9272:
9270:
9268:
9260:
9252:
9225:
9223:
9222:
9217:
9215:
9213:
9205:
9197:
9191:
9190:
9175:
9174:
9166:
9163:
9155:
9146:
9141:
9128:
9123:
9101:
9099:
9098:
9093:
9076:
9074:
9073:
9068:
9066:
9062:
9061:
9059:
9051:
9043:
9037:
9036:
9024:
9016:
9011:
9010:
8990:
8989:
8981:
8978:
8970:
8961:
8956:
8934:
8932:
8931:
8926:
8924:
8923:
8892:
8841:
8839:
8838:
8833:
8825:
8803:
8801:
8800:
8795:
8793:
8792:
8776:
8774:
8773:
8768:
8763:
8759:
8758:
8756:
8748:
8740:
8725:
8724:
8709:
8708:
8691:
8690:
8674:
8672:
8671:
8666:
8643:
8641:
8640:
8635:
8594:
8593:
8581:
8580:
8579:
8577:
8573:
8572:
8559:
8558:
8557:
8544:
8534:
8533:
8529:
8520:
8516:
8514:
8510:
8509:
8508:
8488:
8436:
8434:
8433:
8428:
8413:
8411:
8410:
8405:
8393:
8391:
8390:
8385:
8373:
8371:
8370:
8365:
8350:
8348:
8347:
8342:
8340:
8339:
8316:
8314:
8313:
8308:
8296:
8294:
8293:
8288:
8251:
8249:
8248:
8243:
8241:
8240:
8224:
8222:
8221:
8216:
8214:
8213:
8195:
8193:
8192:
8187:
8185:
8184:
8172:
8171:
8159:
8158:
8095:kinetic diameter
8092:
8090:
8089:
8084:
8072:
8070:
8069:
8064:
8050:
8048:
8047:
8042:
8040:
8039:
8024:
8023:
8018:
8014:
7975:
7973:
7972:
7967:
7953:
7951:
7950:
7945:
7943:
7941:
7940:
7939:
7913:
7912:
7907:
7905:
7904:
7899:
7890:
7885:
7876:
7875:
7874:
7861:
7860:
7855:
7853:
7852:
7847:
7838:
7833:
7832:
7810:
7808:
7807:
7802:
7788:
7786:
7785:
7780:
7778:
7777:
7755:
7752:
7743:
7742:
7730:
7729:
7713:
7711:
7710:
7705:
7703:
7702:
7686:
7684:
7683:
7678:
7676:
7674:
7669:
7665:
7664:
7663:
7652:
7647:
7639:
7637:
7629:
7628:
7619:
7613:
7609:
7604:
7603:
7595:
7583:
7581:
7580:
7575:
7573:
7568:
7567:
7566:
7558:
7548:
7546:
7544:
7543:
7538:
7529:
7524:
7523:
7505:
7503:
7502:
7497:
7486:
7485:
7477:
7474:
7466:
7461:
7460:
7444:
7442:
7441:
7436:
7434:
7433:
7417:
7415:
7414:
7409:
7407:
7405:
7397:
7389:
7365:
7363:
7362:
7357:
7355:
7353:
7345:
7337:
7322:
7321:
7313:
7310:
7302:
7297:
7296:
7284:
7283:
7259:
7257:
7256:
7251:
7249:
7245:
7244:
7242:
7234:
7226:
7220:
7212:
7207:
7206:
7183:
7182:
7174:
7171:
7163:
7158:
7157:
7137:
7135:
7134:
7129:
7127:
7126:
7095:
7044:
7042:
7041:
7036:
7028:
7010:
7008:
7007:
7002:
6997:
6993:
6992:
6990:
6982:
6974:
6956:
6955:
6934:
6929:
6909:
6907:
6906:
6901:
6889:
6887:
6886:
6881:
6849:
6847:
6846:
6841:
6815:
6804:
6803:
6791:
6790:
6789:
6787:
6783:
6782:
6781:
6767:
6766:
6765:
6752:
6741:
6740:
6736:
6727:
6723:
6721:
6717:
6716:
6715:
6695:
6668:
6641:
6639:
6638:
6633:
6618:
6616:
6615:
6610:
6598:
6596:
6595:
6590:
6578:
6576:
6575:
6570:
6555:
6553:
6552:
6547:
6545:
6544:
6528:
6526:
6525:
6520:
6502:
6500:
6499:
6494:
6492:
6491:
6464:
6462:
6461:
6456:
6444:
6442:
6441:
6436:
6393:M. Lòpez de Haro
6370:mass diffusivity
6344:
6342:
6341:
6336:
6319:
6317:
6316:
6311:
6309:
6307:
6306:
6301:
6289:
6274:
6272:
6271:
6266:
6254:
6252:
6251:
6246:
6231:
6229:
6228:
6223:
6218:
6197:
6195:
6194:
6189:
6173:
6171:
6170:
6165:
6097:
6095:
6094:
6089:
6087:
6086:
6078:
6068:
6066:
6065:
6060:
6058:
6057:
6054:
6041:
6039:
6038:
6033:
6031:
6030:
6027:
6002:
6000:
5999:
5994:
5989:
5987:
5982:
5978:
5977:
5976:
5965:
5960:
5955:
5950:
5949:
5940:
5931:
5930:
5925:
5924:
5921:
5909:
5907:
5906:
5901:
5896:
5894:
5889:
5885:
5884:
5883:
5872:
5867:
5859:
5857:
5852:
5851:
5842:
5836:
5832:
5827:
5826:
5818:
5809:
5807:
5806:
5801:
5796:
5794:
5789:
5785:
5784:
5783:
5772:
5764:
5759:
5758:
5755:
5735:
5733:
5732:
5727:
5703:
5695:
5664:
5636:
5625:
5614:
5613:
5601:
5600:
5599:
5597:
5593:
5592:
5591:
5577:
5576:
5575:
5562:
5552:
5551:
5546:
5542:
5540:
5536:
5535:
5534:
5520:
5513:
5511:
5500:
5435:
5433:
5432:
5427:
5425:
5424:
5416:
5413:
5408:
5396:
5394:
5393:
5388:
5325:
5323:
5322:
5317:
5305:
5303:
5302:
5297:
5273:
5265:
5231:
5229:
5228:
5223:
5221:
5193:
5182:
5181:
5169:
5168:
5167:
5165:
5161:
5160:
5159:
5145:
5144:
5143:
5130:
5120:
5119:
5115:
5106:
5102:
5100:
5083:
5076:
5071:
4991:
4989:
4988:
4983:
4962:
4960:
4959:
4954:
4916:and a volume of
4915:
4913:
4912:
4907:
4874:
4872:
4871:
4866:
4851:
4849:
4848:
4843:
4828:
4826:
4825:
4820:
4781:
4779:
4778:
4773:
4768:
4762:
4761:
4760:
4741:
4740:
4732:
4727:
4726:
4723:
4704:
4702:
4701:
4696:
4691:
4688:
4677:
4673:
4672:
4671:
4660:
4659:
4645:
4644:
4641:
4632:
4631:
4628:
4611:
4609:
4608:
4603:
4591:
4589:
4588:
4583:
4578:
4577:
4576:
4557:
4556:
4535:
4527:
4521:
4516:
4513:
4501:
4493:
4487:
4482:
4479:
4464:
4462:
4445:
4440:
4430:
4408:
4407:
4397:
4393:
4384:
4371:
4348:
4346:
4345:
4340:
4314:
4275:
4273:
4272:
4267:
4262:
4258:
4221:
4220:
4206:
4205:
4204:
4202:
4198:
4197:
4184:
4183:
4182:
4169:
4159:
4158:
4154:
4145:
4141:
4139:
4135:
4134:
4115:
4030:
4028:
4027:
4022:
4007:
4005:
4004:
3999:
3987:
3985:
3984:
3979:
3967:
3965:
3964:
3959:
3941:
3939:
3938:
3933:
3876:
3874:
3873:
3868:
3866:
3865:
3857:
3845:
3843:
3842:
3837:
3832:
3829:
3821:
3817:
3816:
3815:
3801:
3800:
3798:
3790:
3785:
3784:
3776:
3770:
3762:
3757:
3756:
3748:
3739:
3737:
3720:
3715:
3705:
3679:
3675:
3666:
3656:
3651:
3650:
3647:
3634:
3632:
3631:
3626:
3600:
3559:
3557:
3556:
3551:
3546:
3542:
3505:
3504:
3490:
3489:
3488:
3486:
3482:
3481:
3468:
3467:
3466:
3453:
3443:
3442:
3438:
3429:
3425:
3423:
3419:
3418:
3399:
3347:
3345:
3344:
3339:
3324:
3322:
3321:
3316:
3298:
3296:
3295:
3290:
3252:and a volume of
3251:
3249:
3248:
3243:
3210:
3208:
3207:
3202:
3187:
3185:
3184:
3179:
3164:
3162:
3161:
3156:
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3101:
3099:
3098:
3093:
3075:
3073:
3072:
3067:
3052:
3050:
3049:
3044:
3029:
3027:
3026:
3021:
3009:
3007:
3006:
3001:
2989:
2987:
2986:
2981:
2964:
2962:
2961:
2956:
2954:
2953:
2940:
2939:
2926:
2925:
2912:
2911:
2910:
2908:
2904:
2903:
2890:
2889:
2888:
2875:
2865:
2864:
2860:
2851:
2847:
2845:
2841:
2840:
2821:
2811:
2810:
2797:
2796:
2783:
2782:
2766:
2765:
2753:
2752:
2740:
2739:
2727:
2726:
2723:
2706:
2704:
2703:
2698:
2693:
2630:per mole: 3406 J
2591:
2589:
2588:
2583:
2581:
2580:
2562:
2560:
2559:
2554:
2546:
2538:
2530:
2529:
2520:
2519:
2510:
2502:
2475:
2473:
2472:
2467:
2459:
2458:
2449:
2441:
2436:
2434:
2423:
2410:
2408:
2407:
2402:
2397:
2396:
2381:
2373:
2368:
2367:
2272:
2263:
2261:
2260:
2255:
2250:
2249:
2240:
2232:
2213:
2180:
2171:
2169:
2168:
2163:
2158:
2156:
2155:
2154:
2153:
2139:
2138:
2129:
2127:
2119:
2103:
2092:
2083:
2081:
2080:
2075:
2073:
2071:
2070:
2069:
2059:
2058:
2057:
2044:
2042:
2034:
2018:
2014:
2012:
2011:
2006:
1994:
1992:
1991:
1986:
1984:
1983:
1968:
1960:
1955:
1954:
1938:
1936:
1935:
1930:
1918:
1916:
1915:
1910:
1902:
1901:
1900:
1890:
1882:
1877:
1876:
1864:
1856:
1847:
1845:
1844:
1839:
1834:
1829:
1828:
1827:
1814:
1806:
1805:
1804:
1778:
1776:
1775:
1770:
1754:
1752:
1751:
1746:
1744:
1743:
1742:
1718:
1709:
1707:
1706:
1701:
1693:
1692:
1691:
1661:
1654:
1652:
1651:
1646:
1644:
1639:
1638:
1637:
1621:
1585:
1583:
1582:
1577:
1572:
1571:
1562:
1554:
1536:
1534:
1533:
1528:
1523:
1522:
1510:
1502:
1497:
1496:
1480:
1478:
1477:
1472:
1470:
1469:
1448:
1446:
1445:
1440:
1435:
1434:
1422:
1414:
1382:
1380:
1379:
1374:
1372:
1367:
1356:
1351:
1346:
1345:
1337:
1331:
1308:
1306:
1305:
1300:
1295:
1294:
1266:
1264:
1263:
1258:
1253:
1252:
1240:
1239:
1227:
1226:
1211:
1209:
1208:
1203:
1198:
1197:
1185:
1184:
1172:
1171:
1150:
1148:
1147:
1142:
1137:
1136:
1124:
1123:
1111:
1110:
1086:
1084:
1083:
1078:
1076:
1075:
1067:
1061:
1060:
1052:
1046:
1045:
1037:
1025:
1023:
1022:
1017:
1012:
1010:
1005:
1000:
999:
991:
983:
981:
976:
971:
970:
962:
954:
952:
947:
942:
941:
933:
925:
924:
906:
904:
903:
898:
880:
878:
877:
872:
867:
865:
864:
855:
853:
848:
835:
830:
828:
827:
818:
817:
816:
803:
798:
796:
792:
786:
785:
784:
775:
769:
764:
763:
739:
737:
736:
731:
723:
718:
717:
702:
701:
686:
685:
661:
659:
658:
653:
648:
647:
638:
633:
632:
608:
606:
605:
600:
598:
597:
588:
587:
565:
563:
562:
557:
555:
554:
538:
536:
535:
530:
528:
527:
507:
505:
504:
499:
497:
496:
488:
478:
476:
475:
470:
468:
467:
305:Ludwig Boltzmann
208:Daniel Bernoulli
150:detailed balance
138:mass diffusivity
84:the relationship
21:
12545:
12544:
12540:
12539:
12538:
12536:
12535:
12534:
12510:
12509:
12474:Project PHYSNET
12462:Wayback Machine
12441:
12420:Wayback Machine
12363:
12361:Further reading
12344:
12343:
12339:
12300:
12279:(14): 756–780,
12266:
12234:
12227:
12214:
12192:
12177:Prévost, Pierre
12167:
12146:(10): 315–322,
12130:
12111:
12091:
12066:
12032:
12027:
11990:
11987:
11982:
11975:
11959:
11955:
11900:
11896:
11889:
11875:
11866:
11821:
11817:
11782:
11778:
11743:
11739:
11712:
11708:
11673:
11669:
11657:
11653:
11652:
11648:
11637:
11633:
11628:Wayback Machine
11619:
11615:
11603:
11599:
11564:
11560:
11549:
11545:
11515:
11511:
11500:
11496:
11491:
11487:
11482:
11478:
11473:
11469:
11420:
11416:
11401:
11387:
11383:
11377: : 21–37.
11363: : 19–32.
11348:
11344:
11339:
11335:
11330:
11326:
11321:
11317:
11312:
11308:
11303:
11299:
11294:
11290:
11283:
11269:
11265:
11234:
11230:
11226:
11218:Vlasov equation
11145:
11116:
11115:
11061:
11053:
11052:
11028:Internal energy
11023:
11013:
11012:
10988:
10980:
10979:
10959:Grand canonical
10935:
10925:
10924:
10880:
10838:
10814:
10808:
10796:
10786:
10773:
10763:
10757:
10748:
10738:
10722:
10710:
10683:
10679:
10667:
10664:
10663:
10648:Brownian motion
10646:applies to the
10632:
10626:
10621:
10595:
10591:
10589:
10586:
10585:
10568:
10564:
10558:
10554:
10546:
10543:
10542:
10511:
10510:
10500:
10491:
10487:
10485:
10482:
10481:
10464:
10460:
10458:
10455:
10454:
10433:
10425:
10423:
10409:
10406:
10405:
10379:
10371:
10369:
10355:
10354:
10344:
10332:
10327:
10314:
10309:
10297:
10294:
10293:
10274:
10271:
10270:
10242:
10234:
10232:
10218:
10209:
10205:
10204:
10200:
10186:
10185:
10175:
10163:
10158:
10152:
10149:
10148:
10125:
10124:
10103:
10102:
10094:
10076:
10075:
10059:
10058:
10056:
10053:
10052:
10025:
10017:
10014:
10013:
9985:
9977:
9975:
9953:
9949:
9948:
9944:
9935:
9931:
9929:
9926:
9925:
9900:
9897:
9896:
9838:
9834:
9817:
9813:
9809:
9802:
9798:
9794:
9792:
9788:
9784:
9774:
9770:
9753:
9752:
9748:
9741:
9736:
9732:
9731:
9691:
9688:
9687:
9668:
9665:
9664:
9648:
9645:
9644:
9628:
9625:
9624:
9605:
9602:
9601:
9584:
9580:
9578:
9575:
9574:
9551:
9548:
9547:
9531:
9528:
9527:
9509:
9503:
9481:
9477:
9475:
9472:
9471:
9454:
9450:
9448:
9445:
9444:
9425:
9421:
9412:
9408:
9402:
9398:
9390:
9387:
9386:
9360:
9356:
9339:
9338:
9328:
9319:
9315:
9313:
9310:
9309:
9292:
9288:
9286:
9283:
9282:
9261:
9253:
9251:
9236:
9233:
9232:
9206:
9198:
9196:
9186:
9182:
9165:
9164:
9154:
9142:
9137:
9124:
9119:
9107:
9104:
9103:
9084:
9081:
9080:
9052:
9044:
9042:
9032:
9028:
9015:
9006:
9002:
9001:
8997:
8980:
8979:
8969:
8957:
8952:
8946:
8943:
8942:
8919:
8918:
8897:
8896:
8888:
8870:
8869:
8853:
8852:
8850:
8847:
8846:
8821:
8813:
8810:
8809:
8788:
8784:
8782:
8779:
8778:
8749:
8741:
8739:
8720:
8716:
8704:
8700:
8699:
8695:
8686:
8682:
8680:
8677:
8676:
8651:
8648:
8647:
8589:
8585:
8568:
8564:
8560:
8553:
8549:
8545:
8543:
8539:
8535:
8525:
8521:
8504:
8503:
8499:
8492:
8487:
8483:
8482:
8442:
8439:
8438:
8419:
8416:
8415:
8399:
8396:
8395:
8379:
8376:
8375:
8356:
8353:
8352:
8335:
8331:
8329:
8326:
8325:
8302:
8299:
8298:
8282:
8279:
8278:
8264:
8258:
8236:
8232:
8230:
8227:
8226:
8209:
8205:
8203:
8200:
8199:
8180:
8176:
8167:
8163:
8154:
8150:
8133:
8130:
8129:
8108:Morse potential
8078:
8075:
8074:
8058:
8055:
8054:
8035:
8031:
8019:
8007:
8003:
8002:
7991:
7988:
7987:
7961:
7958:
7957:
7935:
7931:
7924:
7911:
7898:
7894:
7889:
7870:
7869:
7865:
7859:
7846:
7842:
7837:
7828:
7824:
7822:
7819:
7818:
7796:
7793:
7792:
7773:
7769:
7751:
7738:
7734:
7725:
7721:
7719:
7716:
7715:
7698:
7694:
7692:
7689:
7688:
7659:
7658:
7654:
7653:
7651:
7638:
7636:
7624:
7620:
7608:
7594:
7593:
7591:
7588:
7587:
7557:
7556:
7549:
7547:
7537:
7533:
7528:
7519:
7515:
7513:
7510:
7509:
7476:
7475:
7465:
7456:
7452:
7450:
7447:
7446:
7429:
7425:
7423:
7420:
7419:
7398:
7390:
7388:
7376:
7373:
7372:
7346:
7338:
7336:
7312:
7311:
7301:
7292:
7288:
7279:
7275:
7267:
7264:
7263:
7235:
7227:
7225:
7211:
7202:
7198:
7197:
7193:
7173:
7172:
7162:
7153:
7149:
7147:
7144:
7143:
7122:
7121:
7100:
7099:
7091:
7073:
7072:
7056:
7055:
7053:
7050:
7049:
7024:
7016:
7013:
7012:
6983:
6975:
6973:
6951:
6947:
6946:
6942:
6930:
6925:
6919:
6916:
6915:
6895:
6892:
6891:
6857:
6854:
6853:
6811:
6799:
6795:
6777:
6776:
6772:
6768:
6761:
6757:
6753:
6751:
6747:
6743:
6732:
6728:
6711:
6710:
6706:
6699:
6694:
6690:
6689:
6664:
6647:
6644:
6643:
6624:
6621:
6620:
6604:
6601:
6600:
6584:
6581:
6580:
6561:
6558:
6557:
6540:
6536:
6534:
6531:
6530:
6508:
6505:
6504:
6487:
6483:
6481:
6478:
6477:
6450:
6447:
6446:
6430:
6427:
6426:
6415:
6409:
6397:H. van Beijeren
6357:
6351:
6327:
6324:
6323:
6300:
6293:
6288:
6280:
6277:
6276:
6260:
6257:
6256:
6237:
6234:
6233:
6214:
6203:
6200:
6199:
6183:
6180:
6179:
6159:
6156:
6155:
6148:
6142:
6124:Arithmetic mean
6077:
6076:
6074:
6071:
6070:
6053:
6049:
6047:
6044:
6043:
6026:
6022:
6020:
6017:
6016:
5972:
5971:
5967:
5966:
5964:
5956:
5954:
5945:
5941:
5929:
5920:
5916:
5914:
5911:
5910:
5879:
5878:
5874:
5873:
5871:
5858:
5856:
5847:
5843:
5831:
5817:
5816:
5814:
5811:
5810:
5779:
5778:
5774:
5773:
5771:
5763:
5754:
5750:
5748:
5745:
5744:
5741:
5694:
5662:
5661:
5632:
5621:
5609:
5605:
5587:
5586:
5582:
5578:
5571:
5567:
5563:
5561:
5557:
5553:
5547:
5530:
5529:
5525:
5524:
5519:
5515:
5514:
5504:
5499:
5492:
5443:
5441:
5438:
5437:
5415:
5414:
5409:
5404:
5402:
5399:
5398:
5331:
5328:
5327:
5311:
5308:
5307:
5306:. The constant
5264:
5237:
5234:
5233:
5219:
5218:
5189:
5177:
5173:
5155:
5154:
5150:
5146:
5139:
5135:
5131:
5129:
5125:
5121:
5111:
5107:
5087:
5082:
5078:
5077:
5072:
5067:
5048:
4999:
4997:
4994:
4993:
4968:
4965:
4964:
4921:
4918:
4917:
4880:
4877:
4876:
4857:
4854:
4853:
4834:
4831:
4830:
4796:
4793:
4792:
4756:
4755:
4751:
4733:
4731:
4722:
4718:
4716:
4713:
4712:
4678:
4667:
4666:
4662:
4661:
4658:
4640:
4636:
4627:
4623:
4621:
4618:
4617:
4597:
4594:
4593:
4572:
4571:
4567:
4546:
4542:
4526:
4517:
4512:
4492:
4483:
4478:
4441:
4436:
4431:
4403:
4399:
4389:
4385:
4380:
4372:
4370:
4362:
4359:
4358:
4310:
4281:
4278:
4277:
4216:
4212:
4211:
4207:
4193:
4189:
4185:
4178:
4174:
4170:
4168:
4164:
4160:
4150:
4146:
4130:
4126:
4119:
4114:
4110:
4109:
4036:
4033:
4032:
4013:
4010:
4009:
3993:
3990:
3989:
3973:
3970:
3969:
3950:
3947:
3946:
3882:
3879:
3878:
3856:
3855:
3853:
3850:
3849:
3822:
3811:
3810:
3806:
3802:
3799:
3789:
3775:
3774:
3761:
3747:
3746:
3716:
3711:
3706:
3671:
3667:
3662:
3657:
3655:
3646:
3642:
3640:
3637:
3636:
3596:
3567:
3564:
3563:
3500:
3496:
3495:
3491:
3477:
3473:
3469:
3462:
3458:
3454:
3452:
3448:
3444:
3434:
3430:
3414:
3410:
3403:
3398:
3394:
3393:
3353:
3350:
3349:
3330:
3327:
3326:
3307:
3304:
3303:
3257:
3254:
3253:
3216:
3213:
3212:
3193:
3190:
3189:
3170:
3167:
3166:
3150:
3147:
3146:
3130:
3127:
3126:
3107:
3104:
3103:
3081:
3078:
3077:
3058:
3055:
3054:
3035:
3032:
3031:
3015:
3012:
3011:
2995:
2992:
2991:
2972:
2969:
2968:
2949:
2945:
2935:
2931:
2921:
2917:
2899:
2895:
2891:
2884:
2880:
2876:
2874:
2870:
2866:
2856:
2852:
2836:
2832:
2825:
2820:
2816:
2815:
2806:
2802:
2792:
2788:
2778:
2774:
2761:
2757:
2748:
2744:
2735:
2731:
2722:
2718:
2716:
2713:
2712:
2689:
2678:
2675:
2674:
2655:
2576:
2572:
2570:
2567:
2566:
2537:
2525:
2521:
2515:
2511:
2501:
2493:
2490:
2489:
2454:
2450:
2440:
2427:
2422:
2420:
2417:
2416:
2392:
2388:
2372:
2363:
2359:
2348:
2345:
2344:
2245:
2241:
2231:
2220:
2217:
2216:
2149:
2148:
2144:
2140:
2134:
2130:
2128:
2118:
2110:
2107:
2106:
2065:
2064:
2060:
2053:
2049:
2045:
2043:
2033:
2025:
2022:
2021:
2000:
1997:
1996:
1979:
1975:
1959:
1950:
1946:
1944:
1941:
1940:
1924:
1921:
1920:
1896:
1895:
1891:
1881:
1872:
1868:
1855:
1853:
1850:
1849:
1823:
1819:
1815:
1813:
1800:
1799:
1795:
1793:
1790:
1789:
1764:
1761:
1760:
1738:
1737:
1733:
1731:
1728:
1727:
1687:
1686:
1682:
1668:
1665:
1664:
1633:
1629:
1622:
1620:
1609:
1606:
1605:
1602:
1567:
1563:
1553:
1542:
1539:
1538:
1518:
1514:
1501:
1492:
1488:
1486:
1483:
1482:
1465:
1461:
1459:
1456:
1455:
1430:
1426:
1413:
1387:
1384:
1383:
1357:
1355:
1336:
1332:
1330:
1322:
1319:
1318:
1290:
1286:
1272:
1269:
1268:
1248:
1244:
1235:
1231:
1222:
1218:
1216:
1213:
1212:
1193:
1189:
1180:
1176:
1167:
1163:
1155:
1152:
1151:
1132:
1128:
1119:
1115:
1106:
1102:
1094:
1091:
1090:
1066:
1065:
1051:
1050:
1036:
1035:
1033:
1030:
1029:
1006:
1001:
990:
989:
987:
977:
972:
961:
960:
958:
948:
943:
932:
931:
929:
920:
916:
914:
911:
910:
892:
889:
888:
860:
856:
849:
844:
836:
834:
823:
819:
812:
808:
804:
802:
788:
787:
780:
776:
771:
770:
768:
759:
755:
753:
750:
749:
719:
713:
709:
697:
693:
681:
677:
675:
672:
671:
643:
639:
634:
628:
624:
616:
613:
612:
593:
589:
583:
579:
571:
568:
567:
550:
546:
544:
541:
540:
523:
519:
517:
514:
513:
487:
486:
484:
481:
480:
463:
459:
457:
454:
453:
443:
438:
402:on one another.
363:
336:Brownian motion
328:Albert Einstein
294:formulated the
276:Rudolf Clausius
177:
158:Brownian motion
28:
23:
22:
15:
12:
11:
5:
12543:
12533:
12532:
12527:
12525:Thermodynamics
12522:
12508:
12507:
12501:
12495:
12489:
12486:Java animation
12483:
12477:
12465:
12455:Thermodynamics
12452:
12447:
12440:
12439:External links
12437:
12436:
12435:
12406:
12396:
12378:
12367:Sydney Chapman
12362:
12359:
12358:
12357:
12337:
12319:
12318:
12297:
12296:
12263:
12262:
12231:
12230:
12225:
12211:
12210:
12190:
12186:
12185:
12169:Le Sage, G.-L.
12164:
12163:
12127:
12126:
12108:
12107:
12088:
12087:
12078:(4): 331–407,
12063:
12062:
12043:(8): 549–560,
12025:
12021:
12020:
12003:(3): 353–379,
11986:
11983:
11981:
11980:
11973:
11953:
11894:
11888:978-0201068948
11887:
11864:
11835:(4): 367–368.
11815:
11796:(2): 963–975.
11776:
11757:(1): 408–415.
11737:
11706:
11667:
11646:
11631:
11613:
11597:
11578:(1): 229–257.
11558:
11543:
11509:
11494:
11485:
11476:
11467:
11414:
11399:
11381:
11379:
11378:
11373:, 4th series,
11364:
11359:, 4th series,
11342:
11333:
11324:
11322:Waterston 1843
11315:
11306:
11297:
11295:Lomonosov 1758
11288:
11281:
11263:
11227:
11225:
11222:
11221:
11220:
11215:
11210:
11208:Thermodynamics
11205:
11200:
11195:
11190:
11185:
11180:
11175:
11170:
11165:
11160:
11155:
11147:
11146:
11144:
11143:
11136:
11129:
11121:
11118:
11117:
11114:
11113:
11108:
11103:
11098:
11093:
11088:
11083:
11078:
11073:
11068:
11062:
11059:
11058:
11055:
11054:
11051:
11050:
11045:
11040:
11035:
11030:
11024:
11019:
11018:
11015:
11014:
11011:
11010:
11005:
11000:
10995:
10989:
10986:
10985:
10982:
10981:
10978:
10977:
10969:
10961:
10953:
10945:
10943:Microcanonical
10936:
10931:
10930:
10927:
10926:
10923:
10922:
10917:
10912:
10910:Parastatistics
10907:
10902:
10897:
10892:
10887:
10881:
10876:
10875:
10872:
10871:
10870:
10869:
10867:Kinetic theory
10864:
10862:Thermodynamics
10856:
10855:
10847:
10846:
10837:
10834:
10810:Main article:
10807:
10804:
10794:
10783:
10782:
10770:
10761:
10755:
10746:
10733:to an applied
10731:drift velocity
10719:
10694:
10691:
10682:
10677:
10674:
10671:
10628:Main article:
10625:
10622:
10620:
10617:
10615:with density.
10598:
10594:
10571:
10567:
10561:
10557:
10553:
10550:
10524:
10518:
10515:
10507:
10504:
10499:
10494:
10490:
10467:
10463:
10439:
10436:
10431:
10428:
10422:
10419:
10416:
10413:
10385:
10382:
10377:
10374:
10368:
10362:
10359:
10351:
10348:
10343:
10340:
10335:
10330:
10326:
10322:
10317:
10312:
10308:
10304:
10301:
10281:
10278:
10255:
10248:
10245:
10240:
10237:
10230:
10225:
10222:
10217:
10212:
10208:
10203:
10199:
10193:
10190:
10182:
10179:
10174:
10171:
10166:
10161:
10157:
10145:diffusion flux
10128:
10123:
10120:
10117:
10114:
10111:
10108:
10105:
10104:
10101:
10097:
10093:
10090:
10087:
10084:
10081:
10078:
10077:
10074:
10071:
10068:
10065:
10064:
10062:
10035:
10032:
10028:
10024:
10021:
9998:
9991:
9988:
9983:
9980:
9973:
9970:
9967:
9964:
9961:
9956:
9952:
9947:
9943:
9938:
9934:
9913:
9910:
9907:
9904:
9882:
9879:
9876:
9872:
9869:
9865:
9862:
9858:
9855:
9852:
9849:
9846:
9841:
9837:
9833:
9825:
9820:
9816:
9812:
9805:
9801:
9797:
9791:
9787:
9781:
9777:
9773:
9768:
9762:
9756:
9751:
9747:
9744:
9740:
9735:
9730:
9727:
9724:
9720:
9717:
9713:
9710:
9707:
9704:
9701:
9698:
9695:
9675:
9672:
9652:
9632:
9612:
9609:
9587:
9583:
9555:
9535:
9502:
9499:
9484:
9480:
9457:
9453:
9428:
9424:
9420:
9415:
9411:
9405:
9401:
9397:
9394:
9368:
9363:
9359:
9355:
9352:
9346:
9343:
9335:
9332:
9327:
9322:
9318:
9295:
9291:
9267:
9264:
9259:
9256:
9249:
9246:
9243:
9240:
9212:
9209:
9204:
9201:
9194:
9189:
9185:
9181:
9178:
9172:
9169:
9161:
9158:
9153:
9150:
9145:
9140:
9136:
9132:
9127:
9122:
9118:
9114:
9111:
9091:
9088:
9065:
9058:
9055:
9050:
9047:
9040:
9035:
9031:
9027:
9022:
9019:
9014:
9009:
9005:
9000:
8996:
8993:
8987:
8984:
8976:
8973:
8968:
8965:
8960:
8955:
8951:
8922:
8917:
8914:
8911:
8908:
8905:
8902:
8899:
8898:
8895:
8891:
8887:
8884:
8881:
8878:
8875:
8872:
8871:
8868:
8865:
8862:
8859:
8858:
8856:
8831:
8828:
8824:
8820:
8817:
8791:
8787:
8766:
8762:
8755:
8752:
8747:
8744:
8737:
8734:
8731:
8728:
8723:
8719:
8715:
8712:
8707:
8703:
8698:
8694:
8689:
8685:
8664:
8661:
8658:
8655:
8633:
8630:
8627:
8623:
8620:
8616:
8613:
8609:
8606:
8603:
8600:
8597:
8592:
8588:
8584:
8576:
8571:
8567:
8563:
8556:
8552:
8548:
8542:
8538:
8532:
8528:
8524:
8519:
8513:
8507:
8502:
8498:
8495:
8491:
8486:
8481:
8478:
8475:
8471:
8468:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8426:
8423:
8403:
8383:
8363:
8360:
8338:
8334:
8306:
8286:
8257:
8254:
8239:
8235:
8212:
8208:
8183:
8179:
8175:
8170:
8166:
8162:
8157:
8153:
8149:
8146:
8143:
8140:
8137:
8082:
8062:
8038:
8034:
8030:
8027:
8022:
8017:
8013:
8010:
8006:
8001:
7998:
7995:
7965:
7938:
7934:
7930:
7927:
7922:
7919:
7916:
7910:
7902:
7897:
7893:
7888:
7883:
7879:
7873:
7868:
7864:
7858:
7850:
7845:
7841:
7836:
7831:
7827:
7800:
7776:
7772:
7768:
7765:
7762:
7759:
7749:
7746:
7741:
7737:
7733:
7728:
7724:
7701:
7697:
7672:
7668:
7662:
7657:
7650:
7645:
7642:
7635:
7632:
7627:
7623:
7616:
7612:
7607:
7601:
7598:
7571:
7564:
7561:
7555:
7552:
7541:
7536:
7532:
7527:
7522:
7518:
7495:
7492:
7489:
7483:
7480:
7472:
7469:
7464:
7459:
7455:
7432:
7428:
7404:
7401:
7396:
7393:
7386:
7383:
7380:
7352:
7349:
7344:
7341:
7334:
7331:
7328:
7325:
7319:
7316:
7308:
7305:
7300:
7295:
7291:
7287:
7282:
7278:
7274:
7271:
7248:
7241:
7238:
7233:
7230:
7223:
7218:
7215:
7210:
7205:
7201:
7196:
7192:
7189:
7186:
7180:
7177:
7169:
7166:
7161:
7156:
7152:
7125:
7120:
7117:
7114:
7111:
7108:
7105:
7102:
7101:
7098:
7094:
7090:
7087:
7084:
7081:
7078:
7075:
7074:
7071:
7068:
7065:
7062:
7061:
7059:
7034:
7031:
7027:
7023:
7020:
7000:
6996:
6989:
6986:
6981:
6978:
6971:
6968:
6965:
6962:
6959:
6954:
6950:
6945:
6941:
6938:
6933:
6928:
6924:
6912:mean free path
6899:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6839:
6836:
6833:
6829:
6826:
6822:
6819:
6814:
6810:
6807:
6802:
6798:
6794:
6786:
6780:
6775:
6771:
6764:
6760:
6756:
6750:
6746:
6739:
6735:
6731:
6726:
6720:
6714:
6709:
6705:
6702:
6698:
6693:
6688:
6685:
6682:
6678:
6675:
6671:
6667:
6663:
6660:
6657:
6654:
6651:
6631:
6628:
6608:
6588:
6568:
6565:
6543:
6539:
6518:
6515:
6512:
6490:
6486:
6454:
6434:
6408:
6405:
6385:E. G. D. Cohen
6350:
6347:
6334:
6331:
6304:
6299:
6296:
6292:
6287:
6284:
6264:
6244:
6241:
6221:
6217:
6213:
6210:
6207:
6187:
6163:
6152:mean free path
6146:Mean free path
6144:Main article:
6141:
6140:Mean free path
6138:
6137:
6136:
6131:
6126:
6121:
6116:
6084:
6081:
6052:
6025:
5992:
5985:
5981:
5975:
5970:
5963:
5959:
5953:
5948:
5944:
5937:
5934:
5928:
5919:
5899:
5892:
5888:
5882:
5877:
5870:
5865:
5862:
5855:
5850:
5846:
5839:
5835:
5830:
5824:
5821:
5799:
5792:
5788:
5782:
5777:
5770:
5767:
5762:
5753:
5740:
5737:
5725:
5722:
5719:
5716:
5713:
5710:
5706:
5701:
5698:
5693:
5690:
5687:
5684:
5680:
5677:
5674:
5671:
5667:
5660:
5657:
5654:
5650:
5647:
5643:
5640:
5635:
5631:
5628:
5624:
5620:
5617:
5612:
5608:
5604:
5596:
5590:
5585:
5581:
5574:
5570:
5566:
5560:
5556:
5550:
5545:
5539:
5533:
5528:
5523:
5518:
5510:
5507:
5503:
5498:
5495:
5493:
5491:
5488:
5484:
5481:
5477:
5474:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5445:
5436:, and overall:
5422:
5419:
5412:
5407:
5386:
5383:
5380:
5377:
5373:
5370:
5366:
5363:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5315:
5295:
5292:
5289:
5286:
5283:
5280:
5276:
5271:
5268:
5263:
5260:
5257:
5254:
5250:
5247:
5244:
5241:
5217:
5214:
5211:
5207:
5204:
5200:
5197:
5192:
5188:
5185:
5180:
5176:
5172:
5164:
5158:
5153:
5149:
5142:
5138:
5134:
5128:
5124:
5118:
5114:
5110:
5105:
5099:
5096:
5093:
5090:
5086:
5081:
5075:
5070:
5066:
5063:
5060:
5057:
5054:
5051:
5049:
5047:
5044:
5040:
5037:
5033:
5030:
5026:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
5002:
5001:
4981:
4978:
4975:
4972:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4864:
4861:
4841:
4838:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4771:
4765:
4759:
4754:
4750:
4747:
4744:
4739:
4736:
4730:
4721:
4711:, this yields
4694:
4687:
4684:
4681:
4676:
4670:
4665:
4657:
4654:
4651:
4648:
4639:
4635:
4626:
4601:
4581:
4575:
4570:
4566:
4563:
4560:
4555:
4552:
4549:
4545:
4541:
4538:
4533:
4530:
4525:
4520:
4511:
4507:
4504:
4499:
4496:
4491:
4486:
4477:
4473:
4470:
4467:
4461:
4458:
4455:
4452:
4449:
4444:
4439:
4435:
4429:
4426:
4423:
4420:
4417:
4414:
4411:
4406:
4402:
4396:
4392:
4388:
4383:
4379:
4375:
4369:
4366:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4313:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4265:
4261:
4257:
4254:
4250:
4247:
4243:
4240:
4236:
4233:
4230:
4227:
4224:
4219:
4215:
4210:
4201:
4196:
4192:
4188:
4181:
4177:
4173:
4167:
4163:
4157:
4153:
4149:
4144:
4138:
4133:
4129:
4125:
4122:
4118:
4113:
4108:
4105:
4102:
4098:
4095:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4020:
4017:
3997:
3977:
3957:
3954:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3863:
3860:
3835:
3828:
3825:
3820:
3814:
3809:
3805:
3796:
3793:
3788:
3782:
3779:
3773:
3768:
3765:
3760:
3754:
3751:
3745:
3742:
3736:
3733:
3730:
3727:
3724:
3719:
3714:
3710:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3678:
3674:
3670:
3665:
3661:
3654:
3645:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3599:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3549:
3545:
3541:
3538:
3534:
3531:
3527:
3524:
3520:
3517:
3514:
3511:
3508:
3503:
3499:
3494:
3485:
3480:
3476:
3472:
3465:
3461:
3457:
3451:
3447:
3441:
3437:
3433:
3428:
3422:
3417:
3413:
3409:
3406:
3402:
3397:
3392:
3389:
3386:
3382:
3379:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3337:
3334:
3314:
3311:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3200:
3197:
3177:
3174:
3154:
3134:
3114:
3111:
3102:from the area
3091:
3088:
3085:
3065:
3062:
3042:
3039:
3019:
2999:
2979:
2976:
2952:
2948:
2944:
2938:
2934:
2930:
2924:
2920:
2916:
2907:
2902:
2898:
2894:
2887:
2883:
2879:
2873:
2869:
2863:
2859:
2855:
2850:
2844:
2839:
2835:
2831:
2828:
2824:
2819:
2814:
2809:
2805:
2801:
2795:
2791:
2787:
2781:
2777:
2773:
2769:
2764:
2760:
2756:
2751:
2747:
2743:
2738:
2734:
2730:
2721:
2696:
2692:
2688:
2685:
2682:
2654:
2651:
2639:
2638:
2631:
2620:
2619:
2612:
2579:
2575:
2552:
2549:
2544:
2541:
2536:
2533:
2528:
2524:
2518:
2514:
2508:
2505:
2500:
2497:
2465:
2462:
2457:
2453:
2447:
2444:
2439:
2433:
2430:
2426:
2400:
2395:
2391:
2387:
2384:
2379:
2376:
2371:
2366:
2362:
2358:
2355:
2352:
2276:
2275:
2266:
2264:
2253:
2248:
2244:
2238:
2235:
2230:
2227:
2224:
2184:
2183:
2174:
2172:
2161:
2152:
2147:
2143:
2137:
2133:
2125:
2122:
2117:
2114:
2100:which becomes
2096:
2095:
2086:
2084:
2068:
2063:
2056:
2052:
2048:
2040:
2037:
2032:
2029:
2004:
1982:
1978:
1974:
1971:
1966:
1963:
1958:
1953:
1949:
1928:
1908:
1905:
1899:
1894:
1888:
1885:
1880:
1875:
1871:
1867:
1862:
1859:
1837:
1832:
1826:
1822:
1818:
1812:
1809:
1803:
1798:
1768:
1741:
1736:
1722:
1721:
1712:
1710:
1699:
1696:
1690:
1685:
1681:
1678:
1675:
1672:
1642:
1636:
1632:
1628:
1625:
1619:
1616:
1613:
1601:
1598:
1575:
1570:
1566:
1560:
1557:
1552:
1549:
1546:
1526:
1521:
1517:
1513:
1508:
1505:
1500:
1495:
1491:
1468:
1464:
1453:kinetic energy
1438:
1433:
1429:
1425:
1420:
1417:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1370:
1366:
1363:
1360:
1354:
1349:
1343:
1340:
1335:
1329:
1326:
1298:
1293:
1289:
1285:
1282:
1279:
1276:
1256:
1251:
1247:
1243:
1238:
1234:
1230:
1225:
1221:
1201:
1196:
1192:
1188:
1183:
1179:
1175:
1170:
1166:
1162:
1159:
1140:
1135:
1131:
1127:
1122:
1118:
1114:
1109:
1105:
1101:
1098:
1073:
1070:
1064:
1058:
1055:
1049:
1043:
1040:
1015:
1009:
1004:
997:
994:
986:
980:
975:
968:
965:
957:
951:
946:
939:
936:
928:
923:
919:
900:{\textstyle v}
896:
870:
863:
859:
852:
847:
843:
839:
833:
826:
822:
815:
811:
807:
801:
795:
791:
783:
779:
774:
767:
762:
758:
729:
726:
722:
716:
712:
708:
705:
700:
696:
692:
689:
684:
680:
651:
646:
642:
637:
631:
627:
623:
620:
596:
592:
586:
582:
578:
575:
566:, and volume,
553:
549:
526:
522:
494:
491:
466:
462:
442:
439:
437:
434:
404:
403:
396:
389:
386:
379:
362:
359:
351:Sydney Chapman
330:'s (1905) and
284:mean free path
230:kinetic energy
176:
173:
44:kinetic energy
26:
18:Thermal motion
9:
6:
4:
3:
2:
12542:
12531:
12528:
12526:
12523:
12521:
12518:
12517:
12515:
12505:
12502:
12499:
12496:
12493:
12490:
12487:
12484:
12481:
12478:
12475:
12471:
12470:
12466:
12463:
12459:
12456:
12453:
12451:
12448:
12446:
12443:
12442:
12433:
12429:
12425:
12421:
12417:
12414:
12410:
12409:Behnam Rahimi
12407:
12404:
12400:
12397:
12394:
12390:
12386:
12382:
12379:
12376:
12373:(1939/1970),
12372:
12368:
12365:
12364:
12354:
12348:
12340:
12338:9780408700696
12334:
12329:
12328:
12321:
12320:
12316:
12312:
12307:
12303:
12299:
12298:
12294:
12290:
12286:
12282:
12278:
12274:
12270:
12265:
12264:
12259:
12254:
12250:
12246:
12242:
12238:
12233:
12232:
12228:
12226:0-470-86171-1
12222:
12218:
12213:
12212:
12207:
12203:
12199:
12195:
12194:Lomonosov, M.
12191:
12188:
12187:
12182:
12178:
12174:
12170:
12166:
12165:
12161:
12157:
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12115:
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12109:
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12098:
12094:
12090:
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12081:
12077:
12073:
12069:
12065:
12064:
12059:
12054:
12050:
12046:
12042:
12038:
12031:
12026:
12023:
12022:
12018:
12014:
12010:
12006:
12002:
11998:
11994:
11989:
11988:
11976:
11974:9780815320517
11970:
11966:
11965:
11957:
11949:
11945:
11940:
11935:
11930:
11925:
11921:
11917:
11913:
11909:
11905:
11898:
11890:
11884:
11880:
11873:
11871:
11869:
11859:
11854:
11850:
11846:
11842:
11838:
11834:
11830:
11826:
11819:
11811:
11807:
11803:
11799:
11795:
11791:
11787:
11780:
11772:
11768:
11764:
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11756:
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11748:
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11733:
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11725:
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11717:
11710:
11702:
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11656:
11650:
11642:
11635:
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11625:
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11607:
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11593:
11589:
11585:
11581:
11577:
11573:
11569:
11562:
11554:
11547:
11539:
11535:
11531:
11527:
11523:
11519:
11513:
11505:
11498:
11489:
11483:Einstein 1905
11480:
11471:
11463:
11459:
11455:
11451:
11447:
11443:
11438:
11433:
11429:
11425:
11418:
11410:
11406:
11402:
11400:0-470-86171-1
11396:
11392:
11385:
11376:
11372:
11369:
11365:
11362:
11358:
11355:
11351:
11350:
11346:
11340:Clausius 1857
11337:
11328:
11319:
11310:
11301:
11292:
11284:
11278:
11275:. CRC Press.
11274:
11267:
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11017:
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11001:
10999:
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10994:
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10990:
10984:
10983:
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10968:
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10957:
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10946:
10944:
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10938:
10937:
10934:
10929:
10928:
10921:
10918:
10916:
10913:
10911:
10908:
10906:
10903:
10901:
10900:Bose–Einstein
10898:
10896:
10893:
10891:
10888:
10886:
10883:
10882:
10879:
10874:
10873:
10868:
10865:
10863:
10860:
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10858:
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10853:
10849:
10848:
10845:
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10833:
10831:
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10813:
10803:
10800:
10793:
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10776:
10771:
10768:
10760:
10756:
10752:
10745:
10741:
10736:
10732:
10729:
10725:
10720:
10717:
10713:
10708:
10707:
10706:
10692:
10689:
10680:
10675:
10672:
10669:
10661:
10657:
10653:
10649:
10645:
10641:
10637:
10631:
10616:
10614:
10596:
10592:
10569:
10565:
10559:
10555:
10551:
10548:
10540:
10535:
10522:
10513:
10505:
10502:
10497:
10492:
10488:
10465:
10461:
10437:
10434:
10429:
10426:
10420:
10417:
10414:
10411:
10404:
10399:
10383:
10380:
10375:
10372:
10366:
10357:
10349:
10346:
10341:
10338:
10333:
10328:
10324:
10320:
10315:
10310:
10306:
10302:
10299:
10279:
10276:
10267:
10253:
10246:
10243:
10238:
10235:
10228:
10223:
10220:
10215:
10210:
10206:
10201:
10197:
10188:
10180:
10177:
10172:
10169:
10164:
10159:
10155:
10146:
10141:
10121:
10118:
10115:
10112:
10109:
10106:
10099:
10095:
10091:
10088:
10085:
10082:
10079:
10072:
10069:
10066:
10060:
10050:
10047:
10033:
10030:
10026:
10022:
10019:
10010:
9996:
9989:
9986:
9981:
9978:
9971:
9968:
9965:
9962:
9959:
9954:
9950:
9945:
9941:
9936:
9932:
9911:
9908:
9905:
9902:
9893:
9877:
9874:
9870:
9867:
9863:
9860:
9853:
9847:
9844:
9839:
9835:
9823:
9818:
9814:
9810:
9803:
9799:
9795:
9789:
9785:
9779:
9775:
9771:
9766:
9760:
9749:
9745:
9742:
9738:
9733:
9728:
9725:
9722:
9718:
9715:
9708:
9702:
9699:
9696:
9693:
9673:
9670:
9650:
9630:
9610:
9607:
9585:
9581:
9571:
9569:
9553:
9533:
9525:
9521:
9516:
9514:
9508:
9498:
9482:
9478:
9455:
9451:
9441:
9426:
9422:
9418:
9413:
9409:
9403:
9399:
9395:
9392:
9384:
9379:
9366:
9361:
9357:
9353:
9350:
9341:
9333:
9330:
9325:
9320:
9316:
9293:
9289:
9265:
9262:
9257:
9254:
9247:
9244:
9241:
9238:
9231:
9230:Fourier's law
9226:
9210:
9207:
9202:
9199:
9192:
9187:
9183:
9179:
9176:
9167:
9159:
9156:
9151:
9148:
9143:
9138:
9134:
9130:
9125:
9120:
9116:
9112:
9109:
9089:
9086:
9077:
9063:
9056:
9053:
9048:
9045:
9038:
9033:
9029:
9025:
9020:
9017:
9012:
9007:
9003:
8998:
8994:
8991:
8982:
8974:
8971:
8966:
8963:
8958:
8953:
8949:
8940:
8935:
8915:
8912:
8909:
8906:
8903:
8900:
8893:
8889:
8885:
8882:
8879:
8876:
8873:
8866:
8863:
8860:
8854:
8843:
8829:
8826:
8822:
8818:
8815:
8807:
8789:
8785:
8764:
8760:
8753:
8750:
8745:
8742:
8735:
8732:
8729:
8726:
8721:
8717:
8713:
8710:
8705:
8701:
8696:
8692:
8687:
8683:
8662:
8659:
8656:
8653:
8644:
8628:
8625:
8621:
8618:
8614:
8611:
8604:
8598:
8595:
8590:
8586:
8574:
8569:
8565:
8561:
8554:
8550:
8546:
8540:
8536:
8530:
8526:
8522:
8517:
8511:
8500:
8496:
8493:
8489:
8484:
8479:
8476:
8473:
8469:
8466:
8459:
8453:
8450:
8447:
8444:
8424:
8421:
8401:
8381:
8361:
8358:
8336:
8332:
8322:
8320:
8304:
8284:
8276:
8271:
8269:
8263:
8253:
8237:
8233:
8210:
8206:
8196:
8181:
8177:
8173:
8168:
8164:
8155:
8151:
8147:
8144:
8138:
8135:
8127:
8125:
8120:
8118:
8113:
8109:
8105:
8100:
8099:cross section
8096:
8080:
8060:
8051:
8036:
8032:
8028:
8025:
8020:
8015:
8011:
8008:
8004:
7999:
7996:
7993:
7985:
7983:
7979:
7963:
7954:
7936:
7932:
7928:
7925:
7920:
7917:
7914:
7908:
7900:
7895:
7891:
7886:
7881:
7877:
7866:
7862:
7856:
7848:
7843:
7839:
7834:
7829:
7825:
7816:
7814:
7798:
7789:
7774:
7770:
7766:
7763:
7760:
7757:
7747:
7744:
7739:
7735:
7731:
7726:
7722:
7699:
7695:
7670:
7666:
7655:
7648:
7643:
7640:
7633:
7630:
7625:
7621:
7614:
7610:
7605:
7596:
7584:
7569:
7559:
7553:
7550:
7539:
7534:
7530:
7525:
7520:
7516:
7506:
7493:
7490:
7487:
7478:
7470:
7467:
7462:
7457:
7453:
7430:
7426:
7402:
7399:
7394:
7391:
7384:
7381:
7378:
7371:
7366:
7350:
7347:
7342:
7339:
7332:
7329:
7326:
7323:
7314:
7306:
7303:
7298:
7293:
7289:
7285:
7280:
7276:
7272:
7269:
7260:
7246:
7239:
7236:
7231:
7228:
7221:
7216:
7213:
7208:
7203:
7199:
7194:
7190:
7187:
7184:
7175:
7167:
7164:
7159:
7154:
7150:
7141:
7118:
7115:
7112:
7109:
7106:
7103:
7096:
7092:
7088:
7085:
7082:
7079:
7076:
7069:
7066:
7063:
7057:
7046:
7032:
7029:
7025:
7021:
7018:
6998:
6994:
6987:
6984:
6979:
6976:
6969:
6966:
6963:
6960:
6957:
6952:
6948:
6943:
6939:
6936:
6931:
6926:
6922:
6913:
6897:
6877:
6874:
6871:
6868:
6865:
6862:
6859:
6850:
6834:
6831:
6827:
6824:
6820:
6817:
6812:
6808:
6805:
6800:
6796:
6784:
6773:
6769:
6762:
6758:
6754:
6748:
6744:
6737:
6733:
6729:
6724:
6718:
6707:
6703:
6700:
6696:
6691:
6686:
6683:
6680:
6676:
6673:
6665:
6658:
6655:
6652:
6649:
6629:
6626:
6606:
6586:
6566:
6563:
6541:
6537:
6516:
6513:
6510:
6488:
6484:
6475:
6470:
6468:
6452:
6432:
6424:
6420:
6414:
6404:
6402:
6398:
6394:
6390:
6389:J. M. Kincaid
6386:
6382:
6377:
6375:
6371:
6367:
6363:
6356:
6346:
6332:
6329:
6320:
6302:
6297:
6294:
6290:
6285:
6282:
6262:
6242:
6239:
6219:
6215:
6211:
6208:
6205:
6185:
6177:
6176:cross section
6161:
6153:
6147:
6135:
6132:
6130:
6127:
6125:
6122:
6120:
6117:
6114:
6111:
6110:
6109:
6106:
6104:
6101:
6079:
6050:
6023:
6014:
6010:
6006:
5990:
5983:
5979:
5968:
5961:
5957:
5951:
5946:
5942:
5935:
5932:
5926:
5917:
5897:
5890:
5886:
5875:
5868:
5863:
5860:
5853:
5848:
5844:
5837:
5833:
5828:
5819:
5797:
5790:
5786:
5775:
5768:
5765:
5760:
5751:
5736:
5723:
5720:
5717:
5714:
5711:
5708:
5704:
5699:
5696:
5691:
5688:
5685:
5682:
5678:
5675:
5672:
5669:
5665:
5655:
5652:
5648:
5645:
5641:
5638:
5633:
5629:
5626:
5622:
5618:
5615:
5610:
5606:
5594:
5583:
5579:
5572:
5568:
5564:
5558:
5554:
5548:
5543:
5537:
5526:
5521:
5516:
5508:
5505:
5501:
5496:
5494:
5489:
5486:
5482:
5479:
5475:
5472:
5465:
5462:
5459:
5456:
5453:
5447:
5417:
5410:
5405:
5384:
5381:
5378:
5375:
5371:
5368:
5364:
5361:
5354:
5351:
5348:
5345:
5342:
5336:
5333:
5313:
5293:
5290:
5287:
5284:
5281:
5278:
5274:
5269:
5266:
5261:
5258:
5255:
5252:
5248:
5245:
5242:
5239:
5212:
5209:
5205:
5202:
5198:
5195:
5190:
5186:
5183:
5178:
5174:
5162:
5151:
5147:
5140:
5136:
5132:
5126:
5122:
5116:
5112:
5108:
5103:
5097:
5094:
5091:
5088:
5084:
5079:
5073:
5068:
5064:
5061:
5058:
5055:
5052:
5050:
5045:
5042:
5038:
5035:
5031:
5028:
5021:
5018:
5015:
5012:
5009:
5003:
4979:
4976:
4973:
4970:
4950:
4947:
4944:
4941:
4935:
4929:
4926:
4923:
4903:
4900:
4894:
4888:
4885:
4882:
4862:
4859:
4839:
4836:
4813:
4810:
4807:
4804:
4801:
4789:
4787:
4782:
4769:
4763:
4752:
4748:
4745:
4742:
4737:
4734:
4728:
4710:
4709:ideal gas law
4705:
4692:
4685:
4682:
4679:
4674:
4663:
4655:
4652:
4649:
4646:
4637:
4633:
4615:
4599:
4579:
4568:
4564:
4561:
4553:
4550:
4547:
4543:
4536:
4531:
4528:
4523:
4518:
4509:
4505:
4502:
4497:
4494:
4489:
4484:
4475:
4471:
4468:
4465:
4459:
4456:
4453:
4450:
4447:
4442:
4437:
4433:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4404:
4400:
4394:
4390:
4386:
4381:
4377:
4373:
4367:
4364:
4356:
4355:Ideal gas law
4352:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4311:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4263:
4259:
4255:
4252:
4248:
4245:
4241:
4238:
4231:
4225:
4222:
4217:
4213:
4208:
4199:
4194:
4190:
4186:
4179:
4175:
4171:
4165:
4161:
4155:
4151:
4147:
4142:
4136:
4131:
4127:
4123:
4120:
4116:
4111:
4106:
4103:
4100:
4096:
4093:
4086:
4080:
4077:
4074:
4071:
4068:
4059:
4053:
4050:
4047:
4044:
4041:
4018:
4015:
3995:
3975:
3955:
3952:
3943:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3858:
3846:
3833:
3826:
3823:
3818:
3807:
3803:
3794:
3791:
3786:
3777:
3771:
3766:
3763:
3758:
3749:
3743:
3740:
3734:
3731:
3728:
3725:
3722:
3717:
3712:
3708:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3676:
3672:
3668:
3663:
3659:
3652:
3643:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3597:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3560:
3547:
3543:
3539:
3536:
3532:
3529:
3525:
3522:
3515:
3509:
3506:
3501:
3497:
3492:
3483:
3478:
3474:
3470:
3463:
3459:
3455:
3449:
3445:
3439:
3435:
3431:
3426:
3420:
3415:
3411:
3407:
3404:
3400:
3395:
3390:
3387:
3384:
3380:
3377:
3370:
3364:
3361:
3358:
3355:
3335:
3332:
3312:
3309:
3300:
3286:
3283:
3280:
3277:
3271:
3265:
3262:
3259:
3239:
3236:
3230:
3224:
3221:
3218:
3198:
3195:
3175:
3172:
3152:
3132:
3112:
3109:
3089:
3086:
3083:
3063:
3060:
3040:
3037:
3017:
2997:
2977:
2974:
2965:
2950:
2946:
2942:
2936:
2932:
2928:
2922:
2918:
2914:
2905:
2900:
2896:
2892:
2885:
2881:
2877:
2871:
2867:
2861:
2857:
2853:
2848:
2842:
2837:
2833:
2829:
2826:
2822:
2817:
2812:
2807:
2803:
2799:
2793:
2789:
2785:
2779:
2775:
2771:
2762:
2758:
2754:
2749:
2745:
2741:
2736:
2732:
2719:
2710:
2694:
2690:
2686:
2683:
2680:
2671:
2669:
2665:
2661:
2650:
2648:
2644:
2636:
2632:
2629:
2628:
2627:
2625:
2617:
2613:
2610:
2609:
2608:
2605:
2603:
2599:
2595:
2577:
2573:
2563:
2550:
2547:
2542:
2539:
2534:
2531:
2526:
2522:
2516:
2512:
2506:
2503:
2498:
2495:
2487:
2485:
2481:
2476:
2463:
2460:
2455:
2451:
2445:
2442:
2437:
2431:
2428:
2424:
2414:
2411:
2398:
2393:
2389:
2385:
2382:
2377:
2374:
2369:
2364:
2360:
2356:
2353:
2350:
2342:
2340:
2335:
2333:
2329:
2325:
2321:
2317:
2313:
2309:
2304:
2302:
2298:
2297:
2292:
2291:
2285:
2283:
2274:
2267:
2265:
2251:
2246:
2242:
2236:
2233:
2228:
2225:
2222:
2215:
2214:
2211:
2209:
2208:
2203:
2202:
2197:
2193:
2192:
2182:
2175:
2173:
2159:
2145:
2141:
2135:
2131:
2123:
2120:
2115:
2112:
2105:
2104:
2101:
2094:
2087:
2085:
2061:
2054:
2050:
2046:
2038:
2035:
2030:
2027:
2020:
2019:
2016:
2002:
1980:
1976:
1972:
1969:
1964:
1961:
1956:
1951:
1947:
1926:
1906:
1903:
1892:
1886:
1883:
1878:
1873:
1869:
1865:
1860:
1857:
1835:
1830:
1824:
1820:
1816:
1810:
1807:
1796:
1787:
1785:
1782:
1766:
1758:
1734:
1720:
1713:
1711:
1697:
1694:
1683:
1679:
1676:
1673:
1670:
1663:
1662:
1659:
1658:
1657:ideal gas law
1640:
1634:
1630:
1626:
1623:
1617:
1614:
1611:
1597:
1595:
1591:
1586:
1573:
1568:
1564:
1558:
1555:
1550:
1547:
1544:
1524:
1519:
1515:
1511:
1506:
1503:
1498:
1493:
1489:
1466:
1462:
1454:
1449:
1436:
1431:
1427:
1423:
1418:
1415:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1368:
1364:
1361:
1358:
1352:
1347:
1338:
1333:
1327:
1324:
1316:
1314:
1309:
1296:
1291:
1287:
1283:
1280:
1277:
1274:
1254:
1249:
1245:
1241:
1236:
1232:
1228:
1223:
1219:
1199:
1194:
1190:
1186:
1181:
1177:
1173:
1168:
1164:
1160:
1157:
1138:
1133:
1129:
1125:
1120:
1116:
1112:
1107:
1103:
1099:
1096:
1088:
1068:
1062:
1053:
1047:
1038:
1026:
1013:
1007:
1002:
992:
984:
978:
973:
963:
955:
949:
944:
934:
926:
921:
917:
908:
894:
886:
881:
868:
861:
857:
850:
845:
841:
837:
831:
824:
820:
813:
809:
805:
799:
793:
781:
777:
765:
760:
756:
747:
745:
740:
727:
724:
720:
714:
710:
706:
703:
698:
694:
690:
687:
682:
678:
669:
667:
662:
649:
644:
640:
635:
629:
625:
621:
618:
610:
594:
590:
584:
580:
576:
573:
551:
547:
524:
520:
511:
489:
464:
460:
450:
448:
433:
431:
427:
423:
418:
416:
412:
407:
401:
397:
394:
390:
387:
384:
380:
377:
372:
368:
367:
366:
358:
356:
352:
348:
344:
339:
337:
333:
329:
324:
322:
318:
314:
310:
306:
302:
297:
293:
289:
285:
281:
277:
273:
269:
268:August Krönig
264:
262:
258:
254:
253:John Herapath
250:
246:
241:
239:
235:
231:
227:
223:
219:
215:
214:
213:Hydrodynamica
209:
200:
196:
194:
190:
186:
182:
172:
170:
165:
163:
159:
155:
151:
147:
143:
140:. Due to the
139:
135:
131:
127:
124:, as well as
123:
119:
115:
111:
106:
104:
100:
95:
93:
89:
85:
81:
77:
76:thermodynamic
74:model of the
73:
69:
61:
57:
53:
49:
45:
41:
37:
32:
19:
12480:Introduction
12468:
12431:
12427:
12402:
12392:
12374:
12326:
12317:, 167, 183.)
12314:
12310:
12305:
12276:
12272:
12240:
12236:
12216:
12205:
12180:
12143:
12139:
12121:
12117:
12100:
12093:Herapath, J.
12075:
12071:
12068:Grad, Harold
12040:
12036:
12000:
11996:
11963:
11956:
11911:
11907:
11897:
11878:
11832:
11828:
11818:
11793:
11789:
11779:
11754:
11750:
11740:
11723:
11719:
11709:
11684:
11680:
11670:
11661:
11649:
11640:
11634:
11616:
11600:
11575:
11571:
11561:
11552:
11546:
11529:
11525:
11512:
11503:
11497:
11488:
11479:
11474:Maxwell 1873
11470:
11427:
11423:
11417:
11390:
11384:
11374:
11370:
11360:
11356:
11345:
11336:
11327:
11318:
11309:
11300:
11291:
11272:
11266:
11241:
11237:
11231:
11213:Vicsek model
10971:
10963:
10955:
10947:
10939:
10866:
10815:
10798:
10791:
10787:
10784:
10772:
10758:
10750:
10743:
10739:
10721:
10709:
10633:
10536:
10400:
10268:
10142:
10051:
10048:
10011:
9894:
9572:
9517:
9510:
9442:
9380:
9227:
9078:
8936:
8844:
8645:
8323:
8272:
8265:
8197:
8128:
8121:
8052:
7986:
7955:
7817:
7790:
7585:
7507:
7367:
7261:
7140:shear stress
7047:
6851:
6474:Couette flow
6471:
6422:
6419:Couette flow
6416:
6378:
6358:
6321:
6149:
6107:
6012:
6008:
6004:
5742:
4790:
4786:Graham's law
4783:
4706:
3944:
3847:
3561:
3301:
2966:
2672:
2656:
2642:
2640:
2621:
2606:
2597:
2564:
2488:
2483:
2477:
2415:
2412:
2343:
2337:Because the
2336:
2331:
2323:
2319:
2315:
2305:
2294:
2288:
2286:
2279:
2268:
2205:
2199:
2195:
2189:
2187:
2176:
2099:
2088:
1788:
1725:
1714:
1603:
1587:
1450:
1317:
1312:
1310:
1089:
1087:, such that
1027:
909:
884:
882:
748:
741:
670:
663:
611:
479:, along the
451:
444:
419:
408:
405:
393:uncorrelated
364:
347:David Enskog
340:
325:
265:
242:
211:
205:
179:In about 50
178:
166:
107:
96:
94:of the gas.
78:behavior of
70:is a simple
67:
65:
11331:Krönig 1856
11096:von Neumann
10905:Fermi–Dirac
9518:Consider a
8053:The radius
6476:setup, let
6401:M. H. Ernst
6007:is in m/s,
4349:yields the
3968:with speed
2666:method for
2637:= 35.2 meV.
2287:Equations (
2210:), we have
1784:temperature
1594:microscopic
1590:macroscopic
361:Assumptions
321:probability
313:logarithmic
255:(1816) and
193:Aristotlean
122:temperature
110:macroscopic
56:atmospheres
36:temperature
12514:Categories
12132:Krönig, A.
11985:References
11858:1874/36979
11437:1702.01411
11060:Scientists
11021:Potentials
10826:convection
10656:drag force
10654:) and the
9505:See also:
8260:See also:
7978:molar mass
6411:See also:
6353:See also:
2188:Equation (
1596:property.
280:rotational
238:physicists
220:theory of
210:published
160:) and the
12492:Flowchart
12422:(2016), "
12347:cite book
12196:(1970) ,
11592:0378-4371
11430:: 53–65.
11244:: 49–88.
11091:Ehrenfest
11071:Boltzmann
10951:Canonical
10830:advection
10676:μ
10652:diffusion
10593:α
10556:α
10517:¯
10418:−
10361:¯
10342:−
10334:−
10321:−
10277:−
10216:±
10198:⋅
10192:¯
10173:−
10165:±
10122:π
10113:ϕ
10092:π
10086:θ
9972:θ
9969:
9960:±
9937:±
9912:θ
9909:
9878:ϕ
9871:θ
9854:θ
9848:
9790:−
9746:π
9729:×
9709:θ
9703:
9651:θ
9643:at angle
9479:κ
9456:κ
9452:α
9423:κ
9410:κ
9404:κ
9400:α
9393:κ
9345:¯
9317:κ
9290:κ
9248:κ
9245:−
9171:¯
9152:−
9144:−
9131:−
9087:−
9013:±
9004:ε
8995:⋅
8986:¯
8967:−
8959:±
8939:heat flux
8916:π
8907:ϕ
8886:π
8880:θ
8736:θ
8733:
8711:±
8702:ε
8688:±
8684:ε
8663:θ
8660:
8629:ϕ
8622:θ
8605:θ
8599:
8541:−
8497:π
8480:×
8460:θ
8454:
8402:θ
8394:at angle
8333:ε
8285:ε
8234:η
8211:η
8207:α
8178:η
8165:η
8156:η
8152:α
8136:η
8029:π
8000:π
7994:σ
7929:⋅
7926:σ
7909:⋅
7901:π
7882:σ
7857:⋅
7849:π
7826:η
7799:σ
7767:⋅
7732:⋅
7649:⋅
7644:π
7615:π
7600:¯
7570:σ
7563:¯
7554:⋅
7517:η
7482:¯
7454:η
7427:η
7385:η
7379:τ
7330:⋅
7318:¯
7294:−
7290:τ
7286:−
7277:τ
7270:τ
7209:±
7188:⋅
7179:¯
7155:±
7151:τ
7119:π
7110:ϕ
7089:π
7083:θ
6970:θ
6967:
6958:±
6932:±
6878:θ
6875:
6866:±
6835:ϕ
6828:θ
6813:θ
6809:
6749:−
6704:π
6687:×
6666:θ
6659:
6607:θ
6599:at angle
6362:viscosity
6333:σ
6298:σ
6243:σ
6162:σ
6100:isotropic
6083:¯
5962:⋅
5869:⋅
5864:π
5838:π
5823:¯
5769:⋅
5724:π
5715:ϕ
5697:π
5689:θ
5656:ϕ
5649:θ
5634:θ
5630:
5623:θ
5619:
5559:−
5509:π
5490:ϕ
5483:θ
5466:ϕ
5460:θ
5421:¯
5379:ϕ
5372:θ
5355:ϕ
5349:θ
5334:∫
5314:λ
5294:π
5285:ϕ
5267:π
5259:θ
5213:ϕ
5206:θ
5191:θ
5187:
5127:−
5092:π
5074:×
5069:θ
5065:
5056:λ
5046:ϕ
5039:θ
5022:ϕ
5016:θ
4980:θ
4977:
4936:θ
4930:
4895:θ
4889:
4814:ϕ
4808:θ
4746:π
4720:Φ
4683:π
4642:collision
4625:Φ
4616:will be:
4559:⟩
4540:⟨
4466:×
4460:θ
4454:θ
4451:
4443:π
4434:∫
4428:θ
4422:θ
4419:
4413:θ
4410:
4387:π
4378:∫
4337:π
4328:ϕ
4308:π
4302:θ
4256:ϕ
4249:θ
4232:θ
4226:
4166:−
4124:π
4107:×
4087:θ
4081:
4069:×
4060:θ
4054:
3996:θ
3988:at angle
3930:π
3921:ϕ
3909:π
3903:θ
3862:¯
3824:π
3781:¯
3753:¯
3741:×
3735:θ
3729:θ
3726:
3718:π
3709:∫
3703:θ
3697:θ
3694:
3688:θ
3685:
3669:π
3660:∫
3648:collision
3623:π
3614:ϕ
3594:π
3588:θ
3540:ϕ
3533:θ
3516:θ
3510:
3450:−
3408:π
3391:×
3371:θ
3365:
3272:θ
3266:
3231:θ
3225:
3153:θ
3145:at angle
3018:θ
3010:at angle
2872:−
2830:π
2480:ideal gas
1390:⇒
1342:¯
1284:∗
1072:^
1057:^
1042:^
996:→
967:→
938:→
493:^
288:diffusion
226:Bernoulli
189:Epicurean
185:Lucretius
130:viscosity
99:ideal gas
92:molecules
72:classical
40:ideal gas
12458:Archived
12416:Archived
12401:(2003),
12391:(1964),
12304:(1843),
12171:(1818),
12134:(1856),
12095:(1816),
11948:16592372
11624:Archived
11462:38272381
11409:52358254
11258:96568430
11178:Gas laws
11086:Einstein
11033:Enthalpy
10998:Einstein
10836:See also
10728:terminal
6890:, where
4724:effusion
4629:effusion
4351:pressure
2486:= 3) is
1781:absolute
666:momentum
447:pressure
301:pressure
266:In 1856
247:(1747),
206:In 1738
128:such as
118:pressure
60:trillion
12281:Bibcode
12245:Bibcode
12204:(ed.),
12179:(ed.),
12148:Bibcode
12045:Bibcode
12005:Bibcode
11916:Bibcode
11837:Bibcode
11798:Bibcode
11759:Bibcode
11689:Bibcode
11442:Bibcode
11066:Maxwell
10777:is the
10765:is the
10714:is the
8804:is the
7976:is the
6910:is the
6113:Average
2724:Maxwell
2600:is the
2592:is the
2293:) and (
2204:) and (
1755:is the
317:entropy
218:kinetic
175:History
38:of the
12387:, and
12335:
12311:Papers
12237:Nature
12223:
11971:
11946:
11939:431439
11936:
11885:
11590:
11460:
11407:
11397:
11279:
11256:
11101:Tolman
10987:Models
10705:where
10584:where
9520:steady
9443:where
8777:where
8198:where
7687:where
6108:See:
6003:where
5397:to be
2596:, and
2565:where
1726:where
400:forces
371:volume
311:. The
120:, and
114:volume
52:helium
12520:Gases
12200:, in
12175:, in
12033:(PDF)
11658:(PDF)
11458:S2CID
11432:arXiv
11349:See:
11254:S2CID
11224:Notes
11111:Fermi
11106:Debye
11081:Gibbs
11008:Potts
11003:Ising
10993:Debye
10735:force
2328:water
222:gases
156:(for
88:atoms
80:gases
12411:and
12369:and
12353:link
12333:ISBN
12221:ISBN
11969:ISBN
11944:PMID
11883:ISBN
11588:ISSN
11405:OCLC
11395:ISBN
11277:ISBN
11183:Heat
11076:Bose
10650:(or
10116:<
10110:<
10089:<
10083:<
10070:>
9686:is
9573:Let
8910:<
8904:<
8883:<
8877:<
8864:>
8437:is
8324:Let
7956:and
7113:<
7107:<
7086:<
7080:<
7067:>
6642:is
6399:and
6391:and
6372:and
6129:Mean
5718:<
5712:<
5692:<
5686:<
5673:>
5288:<
5282:<
5262:<
5256:<
5243:>
4331:<
4325:<
4305:<
4299:<
4287:>
3924:<
3918:<
3906:<
3900:<
3888:>
3617:<
3611:<
3591:<
3585:<
3573:>
2306:The
2282:mole
1779:the
1759:and
664:The
411:mass
376:size
349:and
319:and
136:and
66:The
48:size
34:The
12472:on
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851:2
846:i
842:v
838:m
832:=
825:2
821:t
814:i
810:L
806:m
800:=
794:t
790:d
782:i
778:p
773:d
766:=
761:i
757:F
728:.
725:t
721:/
715:i
711:L
707:m
704:=
699:i
695:v
691:m
688:=
683:i
679:p
650:.
645:i
641:v
636:/
630:i
626:L
622:=
619:t
595:i
591:L
585:i
581:A
577:=
574:V
552:i
548:A
525:i
521:L
490:i
465:i
461:v
385:.
20:)
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