Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 22
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But, once again, that determines r only, if µ is known and viceversa. Thus the game seemed to be destined to go on and on in an viciousspiral: Each new thought added a new quantity which could not bedetermined unless one of the previous quantities was known.However, now the end of the spiral was near, because Maxwell didcalculate the viscosity Ș of a gas. He obtained K 13 P VN lc , cf. Insert 4.3,and Ș could be measured. Thus now we have 5 equations for the 5unknowns k, N, µ, r, l viz.pVk,NTmN 2lπ rVNP , ȡliq r3 = µ,1,1 Nµ lc .3 VηIt was Josef Loschmidt14 (1821–1895) who recognized that Maxwell’sformula for the viscosity could be used to close the argument, and hecalculated the missing values.
I have repeated Loschmidt’s calculation for 1litre of air at p = 1atm, T = 298K, m = 1.210–3kg, c = 503m/s, cf. Insert4.1, and with ȡliq§103kg/m3 and Ș = 1.810–5Ns/m2 – all measurable, orcalculable, or reasonably estimable values – and have obtained-2322-27-10-7k = 1.110 J/K, N = 3.210 , µ = 37.810 kg, r = 3.410 m, l = 0.910 m.Since air is a mixture of particles with an average relative molecularmass Mr = 29, the mass of the hydrogen atom comes out as µ0 = 1.310-27kgand therefore the number of particles in a mol is L = 7.71023. That numberis officially called the Avogadro number, although in Austria and Germany,where Loschmidt lived, it is also known as the Loschmidt number. None ofthese values is really good by modern standards, due to the coarseness ofthe assumptions and of the input values. Even so, the orders of magnitudeare fine,15 and that was all physicists could do in the mid-nineteenth century.It was Kelvin who emphasized the enormous size of the number mostpoignantly, when he suggested to dilute a glass full of marked watermolecules with all the water of the seven seas.
Afterwards each glass of seawater would still contain about 100 marked molecules!“„14J. Loschmidt: Zur Grösse der Luftmolecüle. [On the size of air molecules] Zeitschrift fürmathematische Physik 10, (1865), p. 511.15 The modern value of the Avogadro constant is L = 6.02213671023, so that we have–27–23µ0 = 1.66054010 kg. The proper value of the Boltzmann constant is k = 1.3804410 J/K.James Clerk Maxwell (1831–1879)87James Clerk Maxwell (1831–1879)None of the scientists before Maxwell had recognized in his calculationsthat the atoms of a gas move with different speeds. This was not becausethey thought that the speeds were all equal.
Rather they did not know howto account for different speeds mathematically. This changed whenMaxwell took up the question.In a recent biography16 we read that there are no anecdotes to tell aboutMaxwell because he led a quiet life, devoted to his family and science. Andhe died prematurely of cancer at the age of only 48 years. The interest in hisperson is based on the admiration of his scientific work. Indeed, Maxwellwas a genius – both as a mathematician and as a physicist – who is bestknown for his formulation of the equations governing electro-magnetism, atheory of vast scientific and technical importance, without which modernlife would be inconceivable, see Chap. 2 above.
Boltzmann, Maxwell’scongenial contemporary, and occasional correspondent was so much movedto enthusiasm over the Maxwell equations – that is what they are called tothis day – that he exclaimed: War es ein Gott, der diese Zeichen schrieb?17Maxwell had put the keystone on Faraday’s collection of the phenomena ofelectro-magnetism and suggested that light were an electro-magneticphenomenon, – truly a revolutionary discovery. We have discussed this inChap.
2.However, here it is not Maxwell’s electro-magnetism that is of interest.Rather it is his equally profound – albeit, perhaps, less momentous – contribution to the kinetic theory of gases.18 In an early work Maxwell –stimulated by the offer of an award in an open competition – had studied therings of Saturn.19 He proved that they could not consist of flat, hollow disks.Such rigid disks would be broken up by tidal forces. Therefore, whatappeared to be solid rings, had to consist of numerous small solid rocks andicy lumps that travel around Saturn on elliptical orbits like so manysatellites, which have different orbital speeds. Occasionally the lumps mightcollide and thus be kicked inwards or outwards, thereby carrying theirorbital momentum into the faster or slower adjacent ellipses.
Maxwellbecame well-known by winning the competition. Also the work found him16Giulio Peruzzi: “Maxwell, der Begründer der Elektrodynamik” [Maxwell the founder ofelectrodynamics] Spektrum der Wissenschaften, German edition of Scientific American.Biografie 2 (2000).17 Was it a God who wrote these marks? This is a quotation from Goethe’s Faust.18 Maxwell wrote several papers on the kinetic theory. Here we are concerned with the firstone: J.C.
Maxwell: “Illustrations of the dynamical theory of gases.” PhilosophicalMagazine 19 and 20, both (1860).19 J.C. Maxwell: “On theories of the constitution of Saturn’s rings.” Proceedings of theRoyal Society of Edinburgh IV (1859).J.C. Maxwell: “On the Stability of the Motion of Saturn’s Rings.” An Essay whichobtained the Adams Prize for the year 1856, in the University of Cambridge.88 4 Entropie as S = k ln Wwell-acquainted with the properties of swarms of particles when he turnedhis attention to gases.Maxwell introduced the function ij(ci)dci (i = 1,2,3) for the fraction ofatoms in a gas that have velocity components in the i-direction between ciand ci + dci, and he proved, cf. Insert 4.2, that the form of the function ij(ci)in equilibrium is given by a Gaussian, whose peak lies at zero velocity andwhose height and width is determined by temperatureÈ µ ci 2 Øϕ equ ( ci )exp É Ù,kkT2ÊÚ2π µ T1( i = 1,2,3).The fraction of atoms with velocities between (c1, c2, c3) and (c1 + dc1,c2 + dc2, c3 + dc3) is then a function of the speed cfequ (c1 , c2 , c3 )dc1dc2 dc31kT2π µÈ µc 2 Ødc1dc2 dc3 .Ê 2kT ÙÚexp É 3Accordingly the fraction of atoms Fequ(c)dc with speeds between c andc + dc is given byFequ (c)dc4π c 2kT2π µ3È µc 2 Ødc .Ê 2kT ÙÚexp É Thus most atoms have a small velocities and only few are moving fast.But small speeds are also rare, since only few velocities represent smallspeeds.
The mean speed follows as c3 Pk T . 20All three of these equilibrium distribution functions are often calledMaxwell distributions, or simply Maxwellians.The Maxwell distributionConsider a gas of N atoms in the volume V which is at rest as a whole and3possesses the internal energy U = N /2 kT, because the atoms have velocities(c1,c2,c3).
The gas is in equilibrium, and therefore homogeneous with an isotropicdistribution of atomic velocities.Let ijequ(ci)dci be the fraction of atoms with the velocity component i between ciand ci + dci,, such that ijequ(c1)ijequ(c2)ijequ(c3) determines the fraction of atoms with the20Actually this is the root mean square velocity. There are slight differences between c , andthe mean speed, and the most probable speed which we ignore.James Clerk Maxwell (1831–1879)89velocity (c1,c2,c3). Because of isotropy that product can only depend on the speedc222c1 c2 c3 .
We thus haven(c) = ijequ(c1) ijequ(c2) ijequ(c3).Logarithmic differentiation with respect to ci provides1 d ln M equ ,c i dc i1 d ln nc dcsuch that both sides must be constants. Hence follows by integrationM equ ( c i )2A exp( Bc i ) .The two constants A and B may be calculated fromff1³ A exp( Bc2i)d c iand12kT³P222ci A exp( Bc i )dciffso that we obtainM equ ( c i )§ Pci 2exp ¨¨ 2S Pk T© 2 kT1·.¸¸¹Insert 4.2Brush21 says that Maxwell’s derivation of the equilibrium distribution –replayed in Insert 4.2 – mystified his contemporaries because of its noveltyand originality and he suggests.
... that the proof may have been simplycopied from a book on statistics by Quételet 22 or from a review[of that book] by Herschel in the Edinburgh Review.23 In that reviewJohn Herschel, – the son of the eminent astronomer Friedrich WilhelmHerschel (1738–1822) – calculates the probability of the deviation of a balldropped from a height in order to hit a mark; his analysis is very similar toMaxwell’s.Then, still in the same paper, Maxwell proceeded to propose aningenious interpretation for the friction in a gas, cf.
Insert 4.3.24 Thatinterpretation could have been motivated by the investigation of Saturn’srings, although they are not mentioned in the paper. Newton had assumedthat the force needed to maintain a velocity gradient in a fluid or a gas – orin the ether for that matter – is proportional to the value of the velocity21S.G. Brush: (1976) loc.cit. p. 342.A. Quételet: “La théorie des probabilités appliquées aux sciences morales et politiques”.23 J. Herschel: Edinburgh Review 92 (1850).24Insert 4.3 presents a caricature of Maxwell’s argument, which I have found useful whenexplaining the mechanism of gaseous friction to students.2290 4 Entropie as S = k ln Wgradient. The factor of proportionality is the viscosity Ș and it is a commonexperience that the viscosity, or shear resistance of water (say), or honeydecreases with increasing temperature. The same behaviour was expectedfor gases.Viscous friction in a gas, a caricatureThe mechanics of viscous friction can be appreciated from the consideration of twotrains of equal masses M with velocities V1 and V2 passing each other on paralleladjacent tracks.
People change the momentum of the trains by stepping from one tothe other at the equal mass rate µ in both directions. Upon arrival in the new train, aperson must support himself against either the forward or the backward wall inorder to stay on his feet. Thereby he accelerates or brakes the new train, and thusthe two trains eventually equalize their velocities.
The equations of motion for thetrains readMdV1dtP (V2 V1 )henceMd (V1 V2 )dtMdV2dt2P (V1 V2 ).P (V1 V2 )It follows that the velocity difference of the trains decreases exponentially due to a“shear force” proportional to the actual value of the difference. The jumping rate µis the factor of proportionality; if it increases, the braking is more efficient.Basically the same argument was used by Maxwell to calculate the shear force IJbetween two gas layers moving with a y-dependent flow velocity V(y) in the xdirection. The result reads1dV ,WUlc3dywhere ȡ is the mass density and l the mean free path, c is the mean speed of theatoms which jump between the layers, much like the passengers of the train modeldo.Insert 4.3We have anticipated Maxwell’s result for the viscosity K 13 P VN lcabove, when we reported Loschmidt’s calculation of the size of the airmolecules.
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