Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 20

a wall permeable to heat. TheClausius-Duhem inequality on the other hand implies that the normalqcomponent of Ti is also continuous, provided that no entropy is produced inthe wall. Therefore T must also be continuous. In this manner the zeroth law,cf. Chap. 1, may be said to represent a corollary of the Clausius-Duheminequality. Its continuity is the defining property of temperature, and byvirtue of the continuity, the temperature is measurable by contactthermometers. That is the reason why temperature plays a privileged roleamong thermodynamic variables. We shall review this role of temperaturein Chap. 8, cf.

Insert 8.3.What is Entropy ?A physicist likes to be able to grasp his concepts plausibly and on anintuitive level. In that respect, however, the entropy – for all its proven andrecognized importance – is a disappointment. The formula dS dTQ doesnot lend itself to a suggestive interpretation.What is needed for the modern student of physics, is an interpretation interms of atoms and molecules.

Like with temperature: It is all very well toexplain that temperature is defined by its continuity at a diathermic wall,but the ‘‘ahaa”-experience comes only after it is clear that temperaturemeasures the mean kinetic energy of the molecules, – and then it comesimmediately.Such a molecular interpretation of entropy was missing in the work ofClausius. It arrived, however, with Boltzmann, although one must admit,that the interpretation of entropy was considerable more subtle than that oftemperature. Let us consider this in the next chapter.60K.R. Popper: ‘‘Objective knowledge – an evolutionary approach.” Clarendon Press,Oxford (1972).4 Entropy as S = k ln WGreek and Roman philosophers had conceived of atoms, and theydeveloped the idea in more detail than we are usually led to believe.

In thethinking of Leukippus and Demokritus in the 5th and 4th century B.C., theatoms of air move in all directions, and only occasionally they change theirpaths when they hit each other. To the ancients this fairly modern viewimplied a kind of determinism, which was incompatible with the idea ofGod, or gods, playing out their pranks, benevolent or otherwise.

Thereforein later times, in the hands of Epicurus (341–270 B.C.) and Lucretius(95–55 B.C.), the atomistic philosophy of the “Natura Rerum” – this is thetitle of Lucretius’s long poem – adopted an anti-religious and even atheisticflavour, which rendered it politically and socially unacceptable. Thereforeatomism faded away, and in the end it came to represent no more than afootnote in ancient philosophy.In the Age of Reason, by the work of Pierre Simon Marquis de Laplace(1749–1827), determinism came back with a vengeance in the form ofLaplace’s demon: … an intelligent creature capable of knowing all forces… and all places of all things in the world, and equipped with theintelligence to analyse those data. Thus he can evaluate the motion of thegreatest celestial bodies as well as of the tiniest atom; nothing is hidden forthis demon: future and past lie open to his eyes.And just like in antiquity, this kind of determinism was considered asrunning counter to religion.

Laplace was a minister under Napoléon, towhom he presented a part of his voluminous “Traité de MécaniqueCéleste.” Napoléon is supposed to have remarked that he saw no mention ofGod in the book. I had no need of that hypothesis said Laplace. WhenLagrange1 – a colleague and frequent co-worker of Laplace – heard aboutthis exchange, he exclaimed: Ah, but it is a beautiful hypothesis just thesame. It explains so many things.Those enlightened men of post-revolutionary France clearly had their funat the expense of religion.1Joseph Louis Comte de Lagrange (1736–1813) was an eminent mechanician andmathematician.

Napoléon made him a count to reward him for his achievements.80 4 Entropie as S = k ln WRenaissance of the Atom in ChemistryAnd yet, when the concept of atoms was firmly established – at least inchemistry – that was the achievement of a devout Quaker, John Dalton(1766–1844). Dalton proposed that, in a chemical reaction, atoms combineto form molecules of a compound without losing their identity.

Using theevidence collected by others, notably by Joseph Louis Proust (1754–1826),Dalton was able to determine the relative atomic and molecular masses ofmany elements and compounds. Once conceived, the ideas is extremelysimple to explain and exploit: Carbon monoxide is made from carbon andoxygen in the definite proportion of 3 to 4 by mass, or weight. Thus, if webelieve that carbon monoxide molecules are made up of one atom each ofcarbon and oxygen, the oxygen atom must be 1.33 times as massive as thecarbon atom, which is correct.Occasionally this type of reasoning can go wrong, however, as it did forDalton with hydrogen and oxygen that form water in the proportion of 1 to8.

Thus Dalton concluded that the oxygen atom is 8 times as massive as thehydrogen atom. The proper number is 16, as we all know, because waterhas two hydrogen atoms for one oxygen atom. We shall soon see how thaterror was ironed out by Gay-Lussac and Avogadro.In 1808 Dalton published his results in a book “New System of ChemicalPhilosophy”, in which he gave relative atomic and molecular masses, mostof them correct.It became common practice to denote by Mr the ratio of the mass µ of any atom ormolecule to the mass µo of a hydrogen atom.2 And Mrg is defined as the mass ofwhat is called a “mol”. If a mol contains L molecules, so that its mass is Lµ, wehave1gMrg = Lµand hence L =.PoTherefore a mol of any element or compound has the same number of atoms ormolecules.The absolute mass – in kg (say) – of the atoms could not be had in thatway, and it took another half century before that was found.

We shall cometo this shortly.Dalton’s atoms were rather immediately accepted by chemists. However,some hard-nosed physicists waged a losing battle against the atomic hypothesis that lasted all through the 19th century.2Later the reference mass was based on the oxygen atom and still later – now – on thecarbon atom. The reasons do not concern us, and the changes of Mr are minute.Renaissance of the Atom in Chemistry81Dalton was colour-blind and he studiedthat condition, which is sometimes stillcalled Daltonism.When he was presented to KingWilliam IV, Dalton’s Quaker ethics did notallow him to put on the required colourfulcourt dress.

His friends had to convincehim that the dress was grey beforethe ceremony could go ahead.Fig. 4.1. John DaltonFor ideal gases there is a kind of corollary to Dalton’s law of definite proportions, and that helped to correct Dalton’s errors, e.g. the one on thecomposition of water. The chemist Gay-Lussac – pioneer of the thermalequation of state of ideal gases – was dealing with reactions, whosereactants are all gases; he found, that simple and definite proportions alsohold for volumes.

Thus one litre of hydrogen combines with one litre ofchlorine, both at the same pressure and temperature, and give hydrogenchloride. Or two litres of hydrogen and one litre of oxygen combine towater, or three litres of hydrogen and one litre of nitrogen provideammonia. These observations could most easily be understood by assumingthat equal volumes contain equal numbers of atoms or molecules. Thereforethe water molecule should contain two hydrogen atoms, – not one (!) – andammonia should contain three. That conclusion was drawn by the chemistJöns Jakob Berzelius (1779–1848) from Stockholm, and by AmadeoAvogadro (1776–1850), Conte di Quaregna. Avogadro was the physicistwho is responsible for the mantra still taught to schoolchildren:Equalvolumesofdifferentgasesatthesamepressureandtemperaturecontainequallymanyparticles.Also Avogadro was first to use the words atom and molecule in the sensewe are used to.

Berzelius, on the other hand, is the chemist who introducedthe now familiar nomenclature, like H2O for water, or NH3 for ammonia.So, chemistry – such as it was at those days – had been put into perfectshape in a short time by the use of the concept of atoms. But, alas, such ishuman nature that Dalton, who had started it all, was pretty much the onlychemist who could not bring himself to accept Gay-Lussac’s andAvogadro’s and Berzelius’s reasoning and nomenclature. He stuck to hisview that water contained only one hydrogen atom, and to a cumbersomenotation.82 4 Entropie as S = k ln WElementary Kinetic Theory of GasesIn physics it was Daniel Bernoulli (1700–1782), who first put to use theancient atomistic idea of the randomly flying molecules of a gas.

Heexplained the pressure of a gas on the wall of the container by the change ofmomentum of the molecules during their incessant bombardment of thewall. Bernoulli also related the temperature to the square of the (mean)speed of the molecules, and he was thus able to interpret the thermalequation of state of ideal gases, – the law found by Boyle, Mariotte,Amontons, Charles, and Gay-Lussac.Daniel Bernoulli came from a family of illustrious mathematicians.

Hisfather Johann (1667–1748) started variational calculus, and his uncle Jakob(1654–1705) progressed significantly in the calculus of probability; hediscovered the law of large numbers, and is the author of the Bernoullidistribution, whose limit for large numbers is the Gauss distribution or – ina gas – the Maxwell distribution, which is so important in the kinetic theoryof gases.

Also Jakob solved the non-linear ordinary differential equation,which carries his name and which we shall encounter in Chap. 8 in thecontext of acceleration waves. Daniel’s best-remembered theorem is theBernoulli equation, which states that the pressure of an incompressible fluiddrops when the speed increases. The theorem is part of Daniel Bernoulli’sbook on hydrodynamics – published in 1738 – in which the kinetic theoryof gases represents Sect. 10.3 That section was largely ignored by scientists,and it sank into oblivion for more than a century.Two other pioneers of the kinetic theory of gases fared no better. Theywere John Herapath (1790–1868), an engineer and amateur scientist andJohn James Waterston (1811–1883), a military instructor in the services ofthe East India Company in Bombay.

The former did a little less than whatBernoulli had done and the latter did a little more. Both sent their works tothe Royal Society of London for publication in the Philosophical Transactions and both found themselves rejected. Waterston received a less thancomplimentary evaluation to the effect that his work was nothing butnonsense.4 ,5345D. Bernoulli: “Hydrodynamica, sive de vivibus et motibus fluidorum commentarii. Sectiodecima: De affectionibus atque motibus fluidorum elasticorum, praecique autem aëris’’.English translation of Sect.