Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 21
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10: “On the properties and motions of elastic fluids,particularly air”. In S. Brush: “Kinetic theory’’ Vol I, Pergamon Press, Oxford (1965).According to D. Lindley: “Boltzmann’s atom”. The Free Press, New York (2001), p. 1.S.G. Brush reviews, at some length, the efforts of both scientist to publish their works inhis comprehensive memoir: “The kind of motion we call heat, a history of the kinetictheory of gases in the 19th century” Vol I pp. 107–159. North Holland PublishingCompany, Amsterdam (1976).Brush reports that Lord Rayleigh found Waterston’s paper in the archives of the RoyalSociety, and published it in 1893, 48 years after it was submitted.
Rayleigh added anintroduction in which he gave good advise for junior scientists by saying thatElementary Kinetic Theory of Gases83However, in the 1850’s the kinetic theory gained some ground. Clausiuswrote his influential paper – later much praised by Maxwell 6 – “On thekind of motion we call heat” 7 which provided a clearly written and quiteconvincing kinetic interpretation x of temperature, x of the thermal equationof state of a gas, x of adiabatic heating upon compression, x of the liquidand solid state of matter in terms of molecular motion, and x ofcondensation and evaporation.
Stephen Brush has chosen Clausius’s title asthe motto and main title for his comprehensive two-volume history of thekinetic theory of gases.8 Actually Clausius had been anticipated by AugustKarl Krönig (1822—1879), 9 – at least in the derivation of the equation ofstate and in the interpretation of temperature.
Krönig had considered astrongly simplified caricature of a gas in which the particles all move withthe same speed and in only six directions – orthogonal to the six walls of arectangular box – see Insert 4.1. Even earlier, Joule had described asimilarly simple model in 1851.10 It seems that Joule was the first to showthat the mean speed c of the molecules of a gas at temperature T is suchthat P2 c 2 23 kT holds.
At room temperature c is thus of the order ofmagnitude of several hundred meters per second. That result met withconsiderable scepticism which was aired most poignantly by ChristophHendrik Diederik Buys-Ballot (1817-1890), a meteorologist with a butler.He argued that…. highly speculative investigations, especially by an unknown author, are bestbrought before the world through some other channel than a scientific society.Rayleigh praises the marvellous courage of the author, i.e. Waterston, and provides someadditional council:A young author, who believes himself capable of great things, would usually dowell to secure the favourable recognition of the scientific world by work whosescope is limited, and whose value is easily judged, before embarking on higherflights.I do not know whether Lord Raleigh was serious, when he wrote these sentences, orwhether, perhaps, he was being sarcastic.6 J.C.
Maxwell: “On the dynamical theory of gases”. Philosophical Transactions of the RoyalSociety of London 157 (1867).7 R. Clausius: “Über die Art der Bewegung, welche wir Wärme nennen,” Annalen derPhysik 100, (1857) pp. 353–380.8 S.G. Brush: “The kind of motion we call heat.” loc.cit.9 A.K. Krönig: “Grundzüge einer Theorie der Gase’’. [Basic theory of gases]Annalen derPhysik 99 (1856), p.
315.10 J.P. Joule: “Remarks on the heat and the constitution of elastic fluids”. Memoirs of theManchester Literary and Philosophical Society. November 1851. Reprinted inPhilosophical Magazine. Series IV, Vol. XIV (1857), p. 211.84 4 Entropie as S = k ln W… if he were sitting at one end of a long dining room and a butler broughtin dinner at the other end, it would be some moments before he couldsmell what he was about to eat. If atoms were flying at hundreds of metersper second … he should smell the dinner as soon as he saw it.Clausius was able to answer that objection by arguing that the molecules ofthe fragrant dinner fumes – like all gas molecules – do not fly on straightlines for any great length of time.
They collide with other molecules and areturned sideways and backwards, and those deflections lead to a zig-zag pathof an atom, which is much longer between two given points than thedistance of the points. Therefore, the atom needs more time between thosepoints along its path than it would, if it did not collide. In making thisargument Clausius developed the concept of the mean free path l of an atomor a molecule:11 If the particle is imagined to be a spherical billiard ball ofradius r, it sweeps out a cylinder of volume lʌr2 between two collisions – inthe mean – so that such a cylinder should contain just one other particle.Therefore, we must have lʌr2n§1, where n=N/V is the number density ofparticles. Hence follows the mean free path l and Clausius suspected itsvalue to be a small fraction of a millimetre for a gas under normalconditions of p=1atm and T=298 K (say).
But he could not be sure, since rand n were unknown. Therefore his argument remained somewhatinconclusive.Mean speed of molecules2We consider an ideal gas at rest in a rectangular box of volume V = LD with thenumber density N/V of atoms and assume that the inter-atomic forces arenegligible. The pressure p of the gas on the walls results from the bombardment2with gas atoms. –pD is the force of the right wall on the gas. By Newton’s law that2force is equal to the rate of change of momentum of the atoms that hit the area D .For simplicity we assume that all atoms have the same speed c and that one sixthof them are flying perpendicular to each of the six walls.
The change of momentum2of a single atom of mass µ in an elastic collision with the right wall of area D isequal to 2 Pc and the collision rate on that wall is c D 2 16 VN . Therefore the rateof change of momentum is11R. Clausius: “Über die mittlere Länge der Wege, welche bei Molekularbewegunggasförmiger Körper von den einzelnen Molekülen zurückgelegt werden, nebst einigenanderen Bemerkungen über die mechanische Wärmetheorie”. Annalen der Physik (2) 105(1858) pp. 239–258.
English translation: “On the mean free path of the molecular motionin gaseous bodies; also other remarks on the mechanical theory of heat.’’ In S. Brush:“Kinetic theory.” Vol I, Pergamon Press, Oxford (1965).Elementary Kinetic Theory of Gases 2Pc 21 N6 VD285 Vm 13 c 2 D 2 ,2This must be set equal to –pD and it follows that13pVc2.Fig. 4.2. On the pressure of a gasComparison with the thermal equation of state of an ideal gas provides13c2kPTTherefore the mean kinetic energy of the atoms equals 23 kT .33Consider air 12 in V =1m and with the mass m=1.2kg, the mass of air in 1m atp = 1atm and T = 298 K: We obtain c = 503m/s and that may be considered themean speed of the air molecules.Insert 4.1So, now it became imperative to determine how many particles there arein a given volume of a gas in some agreed-upon reference state of pressureand temperature.
To be sure, one mol contains L = 1g/µ0 particles but howbig is µ0, the mass of a hydrogen atom?Avogadro’s observation meant that the thermal equation of state of a gascontains a universal constant. We may put that equation into the form pV =NkT , where – according to Avogadro – k is a universal constant, now calledthe Boltzmann constant.13 The value of k was unknown, since N wasunknown and not measurable. Of course, one might try to be clever andreplace the particle number N by the mass m = Nµ of the gas. The mass canbe measured by weighing, but that does not help, because in terms of massthe formula reads pV = m k/µT and thus, while k/µ may now be calculated, kand µ individually are still unknown.Actually, that was the dilemma of the atomic hypothesis which delayedits acceptance among physicists: The whole idea had too much of ahypothetical flavour; neither the masses of the atoms were known, nor theirradii r, nor was it known how many there were in a given volume.
It is truethat the speeds of the particles were known, see above, but the distances andthe distances between collisions were again unknown.12Air is made up of molecules, not atoms. Moreover, in air there are at least two types ofmolecules, nitrogen and oxygen. But this does not matter for the present consideration oforders of magnitude.13 Boltzmann had just been born when Avogadro died. At that time the universal constantwas called the ideal gas constant, denoted by R which is k/µ0. I avoid that constant andtrust that the reader will tolerate the anachronism, perhaps.86 4 Entropie as S = k ln WOne could do something about the radius, because it seemed to be a goodassumption, that in the liquid phase, where the density ȡliq was known, theparticles lay close together so that ȡliqr3 § µ had to hold, at least by order ofmagnitude.
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