john sydney dugdale entropy and its physical meaning 1996

5/26/2018 John Sydney Dugdale Entropy and Its Physical Meaning 1996

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Entropy and itsPhysical Meaning


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Entropy and itsPhysical Meaning

J . S. DUGDALEEmeritus Professor, University of Leeds

T aylor & FrancisPublishers since 1798


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UK Tay lor & F ranc is l td , 1 Gunpowder Square, Lon don EC4A 3DEUSA Tay lor & F ranc is Inc . , 325 Chestnut Street, 8th Floo r, Philadelphia , PA 19106Copyright 0 J . S. Dugda le 1996Reprinted 1998

All rights reserved. No part of this publication ma y be reproduced, stored ina retrieval system, or transmitted, in any form or by any means, electronic,electrostatic, m agnetic tape, mechanical, photocopying, recording or otherwise,without the prior permission of the cop yright owner.British Library Cata loguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 0-7484-0568-2 (cased)ISBN 0-7484-0569-0 (paperback)Library of Con gress Cata loguing Publication Data are avai lableCover design by Jim WilkieTypeset in Times 10/12pt by Keyset Composition, Colchester, EssexPrinted in Great Britain by T . J . International Ltd.


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Contents

Preface xPART ONE Entropy in thermodynamics

1 Temperature and heat: some historical developmentsThe nature of heat

2 Temperature and heat: a different approach 1The zeroth law o f thermody na mics 3Equations of state 4T h e wor k don e in thermod yn am ic proc esses: quasi-static changes 14Work in irreversible p ro cesses 5Exercises 63 The first law of thermodynamics 9

The interna l energy an ana logy 1Atomic interpretation of intern a l energy 2T he in terna l energy as a function of temperature and volume 3Exercises 44 The second law of thermodynamics 7The behaviour of idealised heat engines 8The abso lute sca le of temperature 5

The Carnot engine as a refrigerator 6Exercises 6

5 Entropy: how it is measured and how it is used 9T h e definition of entropy 9


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ContentsThe measurement of entropy 3Entropytemperature diagrams 4The entropy of a perfect gas 4Entropy change and latent heat 6The central equation of thermodynamics 8T hermodyna mic function s an d their ap prop riate variables 9The ClausiusClapeyron equation 1The Jo ule Tho mson effect 3Exercises 5

6 Entropy in irreversible changes 7Hea t fl o w 7Adiaba tic pro cesses which are irreversible 9Exercises 2PART TWO The statistical interpretation of entropy 5

7 The statistical approach: a specific example 7A specific exa mple 0Exercises 8

8 General ideas and development: the definition of a microstate 9M ore general treatment of loc alised assembly 2T h e Boltzmann distribution 4A solid of mo lecules with two ener gy levels each 7Exercises 2

9 Temperature and entropy in statistical mechanics 3T wo assemblies that ca n share en ergy 3Entropy a nd absolute temperature 5A different point of view 00Entropy a nd disorder 01Exercises 02

10 Application to solids 03T h e entropy of a solid-like assembly 03The Einstein solid 05A simple magnetic solid 09Exercises 12

11 Applicat ion to gases: (1) the classical approximat ion 15Gases 15T h e entropy of an ideal gas 19T h e evaluation of the par tition function , Z, for an ideal gas 20The SackurTetrode equation for the entropy of an ideal gas 21vi


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ContentsGas mixtures 22The distribution of velocities in the gas 23T h e Helmholtz free energy 24The equation of state 25Exercises 25

12 Application to gases: (2) BoseEinstein and FermiDirac gases 27A FermiDirac assembly 27T he different distribution functions 29Some pro perties of a Bose Einstein gas at low temperatures 31Some p ro perties of an ideal F ermi Dirac gas 33Exercises 35

13 Fluctuations and Maxwell's demon 39The recurrence paradox 41Exercises 42PART THREE Entropy at low temperatures 43

14 The third law of thermodynamics 45Entropy at low temperatures 45Superconductivity 47Liquid H e 3 49Liquid H e 4 51Mixtures 52Entropy differen ces 54Hea t capa cities near the absolute zero 58Gases and the third law 59Non -equilibrium states 62The third la w an d chemica l equilibrium 63The third law a summary 63

15 Absolute temperature and low temperatures 65Measurement of absolute temperature 65Measurement of high tempera tures 65Low temperatures 66T h e liquefaction of gases 67T h e H e 3 He 4 dilution refrigerator 70The magnetic method of cool ing 71The measuremen t of low temperatures 73Exercises 75

16 The third law of thermodynamics and the unattainabil ity ofabsolute zero 77

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ContentsNotes 81An swers to exercises 85Reading list 93Index 95

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Preface

This book is a revised version of my ear l ier book Entropy a nd Lo wTemperature Physics, with here r ather less emphasis on the low temperatureaspects of the subject. The concept of entropy lies at the heart ofthermodynamics and is often thought of as obscur e, even m ysterious. Theaim of this book is thus the same as befor e, na mely, to ma ke accessible theidea o f entropy and to enc our age an intuitive appreciation of its nature anduse. In this n ew version, I have ad ded exercises for the reader; mo stly theyare mean t to be stra ightfor wa rd tests of understanding but sometimes theyare used to extend the coverage of the text.

I am very grateful to Bryan Coles for reading the manuscript, for hissuggestion s and for his encoura gement. May I also express my thanks to T on yGunault for valuable discussions, and the staff of T ay lor & Francis for theirhelp in preparing the manuscript for publication.I am indebted to the Editor of A nnalen der Physik for permission toreproduce Figure 10b.F inal ly I wish to record my great debt to my teachers: at school Mr G.

R. Noakes; as an undergraduate Dr C. Hurst; as a research student SirFr anc is Simon.

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PART ONE

Entropy in thermodynamics


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1

Temperature and heat: somehistorical developments

Entropy, as we shall see, is defined in terms of temperature and heat, sowe sha l l begin with a brief study o f these qua n tit ies . W e shal l do this byhaving a look at wha t some of the great experimenters and thinkers on thesetopics have done and written.Fahrenheit (1686-1736) produced the first thermometer which wasaccur ately reproducible and this ma de possible a systematic quantitative studyo f temperature and hea t. It seem s to have been soon recogn ised that , whenbodies at different temper atures wer e brou ght into contac t and when a l lchanges had ceased, a thermometer plac ed in c ontact with eac h body in turngave the same rea ding.This is a most important o bservation, so important indeed tha t it has sinc ebeen given the status of a law o f thermody na mics it is often called the zero thl aw s inc e the first and s econd laws of thermody na mics had a lready beenform ulated before this fact of observation wa s 'ca no nised'. At the time,however, this 'equilibrium o f heat' or thermal equilibrium as we shouldnow call it gave rise to considerable confusion. For example, it wasinterpreted by some to mea n tha t at equilibrium there wa s an equal amountof heat per un it volume throughout the different bodies. As we can no w see,this interpretat ion a ro se fro m a failure to distinguish between heat andtemperature.J oseph Black (1728-17 99) did muc h to c la rify this and other questionsrelating to the nature of heat and temperature. His work was publishedpo sthumo usly (1803) in his Lectures on the Elements of Chemistry edited byhis friend , who ha d a lso been a pupil and colleague, John Robison. ' Blackdistinguished clearly between quantity and intensity o f heat, that is, betweenheat and temperature. H e also introduced the con cepts of heat capac ity and

'Superior figures refer to Notes on pp. 181-3.3


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Entropy and i ts physical mean inglatent heat and by his exper imen ts laid the foun dations of the science ofcalorimetry.A quotation fro m Ro bison's notes which are appen ded to his edition ofBlack's Lectures read s 'Before the year 1765 , Dr Black had made manyexperiments on the heats comm un icated to w ater by differen t solid bodies,and had co mpletely establ ished their regular and stead y d ifferen ces in thisrespect. . . .' Already, therefore, by 1765 Black had demonstrated thatdifferent substances ha d different hea t capa cities. H is metho d of measuringthem was by what we now call 'the method of mixtures'. Here isRo bison 's description o f the method an d of the precautions taken to en sureaccuracy:

Dr Black estimated the [heat] cap ac ities, by m ixing the two bodies in equalmasses, but of differen t tempera tures; and then stated their capacities asinversely prop ortional to the changes of temperature of each by the mixture. Thus,a pound of gold, of the temperature 150, being sudden ly mixed with a poundof water, of the temperature 50, ra ises it to 55 near ly: therefore the capacityof gold is to that of an equal weight of water as 5 to 95, or as 1 to 19; for thegold lo ses 95 0 , and the water gains 5. . . .

These experiments require the most scrupulous attention to many cir-cumstan ces which may affect the result.1 T he mixture must be made in a very extended surface, that it may quickly

attain the medium temperature.2 The stuff which is poured into the other should have the temperature of theroom, that no change may happen in the pou ring it out of its containing

vessel.3 The effect of the vessel in which the mixture is made must be con-sidered.4 Less chance of error will be incurred when the substances are of equalbulk.5 The change of temperature of the mixture, during a few successive m omen ts,must be observed, in order to obtain the real temperature at the begin-

ning.6 No substanc es should be mixed which produce any chan ge of temperatureby their chemical a ct ion , or w hich change their temperature, if mixed whenof the same tempera ture.7 Each substance must be compa red in a variety of temperatures, lest the ratioof the ca pa cities should be d ifferent in different temperatures.

W hen a l l of these circumstanc es have been duly a ttended to, we obtain themeasure of the capacities of different substances for heat.2

T o this day ca lorimetry requires 'the most scrup ulous attention to man ycircumstanc es which may affect the result'. A great deal of scientific effortstill goes in to the accurate measurement o f heat ca pac ities over a wide rangeof temperatures and ind eed this is still one of the primary mea surements inthermod yn amics. F ro m it is derived the great bulk of o ur information about4


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Temperature and heat: som e historical developm entsthe therma l properties of substan ces ( in par t icula r, as we shall see, theirinterna l energy and entropy).The nature of heatW e see therefore that ca lorimetry had d eveloped into a quan titative scienceby the end o f the eighteenth cen tury . Ho wever, there wa s still a great divisionof opinion a bout the nature o f heat. In 178 3, for example, Henry Cavendish(1731-1810), writing about 'the co ld generated by the melting of ice' and 'theheat produced by the freezing of water' , makes the fol lowing observationin a footnote:

I am informed that Dr Black explains the above mention ed phenom ena inthe same man ner; on ly , instead of using the expression , heat is genera ted orpro duced, he say s, latent heat is evolved o r set free; but as this expression relatesto an hypo thesis depending on the suppo sition , that the heat of bodies is owingto their c on taining more or less of a substanc e called the matter of heat; andas I think Sir Isaac Newton's opinion, that heat consists in the internal motionof the particles of bodies, much the most probable, I chose to use theexpression, hea t is gener ated. . . .In fac t, it is probable that Black never held any theory of heat with greatco nviction s inc e on the whole he seems to have felt that all theories were

a waste of time. However, it does seem clear that while some scientists atthat time tho ught o f heat as 'a substance called the matter of heat' or a s 'anigneous fluid' ( later it wa s cal led c alo ric) others thought this an unnecessaryhypothesis. (Incidentally, although Cavendish gives Newton the credit forhaving thought of heat as a form of motion, Newton was by no means thefirst to h ave this idea .)At about this time, Count Rumford (Benjamin T hompson , 1753-1814)began his important experiments on the nature of heat. Here is an extractfrom his acco unt of the experiments: 3Being engaged lately, in superintending the boring of cannon, in theworkshops of the military arsenal at Munich, I was struck with the very

con siderable degree of heat which a brass gun ac quires, in a sho rt time, in beingbored; and with the still more intense heat (much greater than that of boilingwa ter, as I found by experiment,) of the metallic chips separated f rom i t bythe borer.T he more I meditated on these pheno mena , the mor e they appea red to meto be curious and interesting. A thorough investigation of them seemed evento bid fair to give a farther insight into the hidden na ture of heat; and to ena bleus to form some reasonable conjectures respecting the existence, or n o n -existence, o f an igneous fluid: a subject o n which the opinion s of philosophershave, in all ages, been much divided. . . .Fro m whence comes the heat actual ly produc ed in the mechanical operationabove mention ed?

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Entropy and i ts physical mea ningRum ford set out to an swer this question by exper iment . From a c a n n o nhe made a bra ss cy lind er which wo uld just fit over a blunt steel borer. Thisborer w as forc ed against the bottom of the cy l inder an d the cyl inder was madeto turn on its axis by means of the boring machine driven by ho rses. Of the

four experiments Rumford describes, one was concerned with the heatcapacity of the metal chips produced in the boring pro cess. T his he show edwas the same as that of an equal ma ss of the or igina l metal. Of the otherexperiments, the third w as the mo st striking. In this experiment Rumfordsurrounded the cylinder being bored by a wooden water-tight box. T ocon t inue in his own wor ds:T he box wa s fi lled with co ld wa ter (viz, at the temperature of 60) an d themachine was put in mo tion .The result of this beautiful experimen t was very striking, an d the pleasure

it afforded me a mply repaid me for all the trouble I had had, in contriving andarranging the complicated machinery used in making it.The cylind er, revolving at the rate of about 32 times in a minute, had been

in mo tion but a shor t time, when I perceived, by putting my ha nd into the water,and touching the outside of the cy l inder , that heat wa s generated; and it wasno t l ong before the water which surro unded the cy l in der began to be sensiblywarm.

At the end of 1 hour I found, by plunging a thermometer into thewater in the box, . . . that its temperature had been raised n o less than47 degrees. . . .At the end of 2 hours , reckoning from the beginning of the experiment, thetemperature of the water was found to be raised to 178.

At 2 hours 20 minutes it wa s 200; and at 2 hours 30 minutes it ACT UALLYBOILEDIt wou ld be d ifficult to d escribe the surprise and astonishment expressed inthe countenances of the by-standers, on seeing so large a quantity of cold waterheated, and actually made to boil, without any fire.

Rumford then com putes the quantity of heat produ ced in the experiment:he estima tes it to be equivalen t to the heat needed to ra ise the temperatureof 26.58 lb of wa ter by 180F .

As the machinery used in th is exper imen t could eas i ly be c arr ied ar oun dby the for ce of on e horse, (thou gh, to rend er the wo rk lighter, two hor ses wereactual ly employed in doing it,) these com putation s show further how l arge aquantity of heat might be produc ed, by pro per mechan ical con trivan ce, merelyby the strength of a horse, without either fire, light, combustion, or chemicaldecomposition; and, in a case of necessity, the heat thus prod uced m ight beused in cooking victuals.In summa risin g his conclusions from these experiments Rumfordwrites:

And, in reasoning on this subject, we must no t for get to con sider that most6


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Temperature and heat: some historical developm entsremarkable circumstance, that the source of the heat genera ted by friction, inthese experimen ts, appea red evidently to be inexhaustible.It is hardly n ecessary to add, that an y th ing which an y insulated body, orsystem of bodies, can cont inue to furn ish without limitation, cannot possiblybe a material substance: and i t appea rs to me to be extremely difficult, if notquite impossible, to form a ny distinct idea of an y thing, capa ble of being excited,and communicated, in the manner the heat wa s excited and comm unicated inthese experiments, except it be MOTION.T hese experiments c lear ly show ed a c lo se con nect ion between heat andwo rk. Ho wever, they are on ly qualitative, although, as Joule later remarked,the data do in fact con ta in en ough information to yield rough quantitativeresults. Ma yer (1814-1878), a German physician and phy sicist, under stoo dclearly the relat ion ship between h eat and work and was able to make a

nu merical estimate of the con version factor. He is no w reco gnised, alon g withJ ou le , as the discoverer of the first law o f thermodynamics, but at the timehis wor k rec eived little recognition.The finally decisive, quantitative studies o n heat and work were thosemade by J ou le (1818 -1889) in the 1840s. Almost fifty y ear s after R umfor d'swork Joule presented a paper to the Royal Soc iety on this same subjectentitled 'T he Mecha nica l Equivalent of Hea t'. 4 He writes:F o r a long time it had been a favou rite hypo thesis that heat co nsists of 'a

force or power be lon ging to bod ies', but it wa s reserved for Count Rumfordto make the first experimen ts decidedly in favour of that view. That justlycelebrated natural philosopher demon strated by his ingen ious experiments thatthe very great qua ntity of heat excited by the boring o f c a n n o n could not beascr ibed to a change taking place in the calorific capacity of the metal; and hetherefore concluded that the motion of the borer wa s commun ica ted to theparticles of metal , thus pro ducing the phenomena of heat.Joule then po ints ou t that if, in Rum ford's third experiment, yo u assumethe rate of wo rking to be about one horsepower (as Rumford indicates) yo uca n estimate that the wo rk required to raise the temperature of 1 pound ofwa ter by 1F is abo ut 1000 foot-pounds which, as he observes, is not verydifferent fro m the valule derived from his own experiments, namely 772foot-pounds.

In 1843 I announced the fact that 'heat is evolved by the passaa ge of waterthrough narrow tubes', an d that eac h degree of heat per lb of water requiredfor its evolution in this way a mechanical force represented by 770 foo t-pound s.Subsequently, in 1845 and 1847, I employed a paddle-wheel to produce the fluidfriction, and obtained the equivalents 781.5, 782.1 an d 787 .6, respectively fromthe agitation of water, sperm-oil and mercu ry . Results so c losely co inciding witho n e another , and with those pr eviously derived from experimen ts with elasticfluids an d the electromagnetic machine, left no doubt on my mind as to theexistence of an equivalent relation between forc e and heat; but still it appea red

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Entropy and i ts physical mea ningof the highest importan ce to o btain that relation with sti ll greater accura cy. ThisI have attempted in the present paper.Joule no w goes on to describe the appa ratus with which he ca rried out

his experiments to determine 'the mechanical equivalent o f heat'. Theappa ratus c on sisted essential ly of a pad dle-wheel with eight sets of revolvingarms w orking between four sets of stationary vanes. The purpose of thepad dle-wheel wa s to stir a l iquid c on tained in the space between the van es.The pad dle-wheel , attac hed to a 'rol ler' , was made to rotate by means oftwo lea d w eights suitably co nn ected to the ro l ler by pulleys an d fine twine.W e can fo l low the course of experiment best in J oule's own wo rds:

The method of experimenting w as simply as follows: The temperature of thefrictiona l appar atus having been ascertained an d the weights wo un d up wi ththe assistance of the stand . . . the rol ler w as ref ixed to the axis. The preciseheight of the weights above the groun d having then been determined by mean sof the graduated slips of wood, . . ., the rol ler was set at l iberty an d al lowedto revo lve unti l the weights reached the f lagged f loor of the laboratory, afteraccomplishing a fall of abo ut 63 inches. The roller was then removed to thestan d, the weights wo un d up again , an d the friction renewed. After this hadbeen repeated twenty times, the experiment was concluded with anotherobservation o f the temperature of the apparatus. The mean temperature of thelabor atory w as determined by observation s mad e at the commen cement, middleand termina tion of each experiment.

Previously to, or immediately after each of the experiments, I made tr ia lof the effect of rad ia t ion an d con duct ion o f heat to or from the atmosphere,in depressing or raising the temperature of the frictional apparatus. In thesetrials, the position of the apparatus, the quantity of water con ta ined by it, thetime o ccu pied, the method of observing the thermometers, the position of theexperimenter, in short everything, with the exception of the appa ratus beingat rest, was the same as in the experiments in which the effect of friction wasobserved.Joule then d escribes in detail the results of these extremely carefulexperiments. At the end of the paper he wr ites:

I will therefore conclude by considering it as demonstrated by theexperiments co ntained in this pa per . . . .1st. That the quantity of heat produced by the friction of bo dies, whether solid

or liquid, is always proportional to the quantity of force expended.And2nd. T hat the quantity of heat capa ble of increasing the temperature of a pound

of water (weighed in vacuo, and taken at between 55 0 and 60) by?Fa hrenheit requires for its evolution the expenditure of a mechanicalforce represented by the fall of 772 lb through the space of one foot.

This brief account gives some idea of how the subject o f heat haddeveloped up to about the middle of the nineteenth century. As we shal l8


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Temperature and heat: some historical developmentssee later, a great deal o f wo rk was going on a t this time o n a different aspectof the subjec t, that relating to the secon d law of thermodynamics . Beforegoing on to this , how ever , I wo uld l ike to look at the co nc epts of temperatureand of heat from a different point of view.

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2

Temperature and heat: adifferent approach

Once we recognise that thermodynamics and mechanics are intimatelyrela ted, it is then lo gical to begin w ith qua n tities alrea dy familiar and welldefined from such bran ches of physics as mechanics and electromagnetism,and to use them to d efin e, with the help o f experimental information, thespecifical ly thermo dy na mic quan tities such as temperature, internal energyand entropy.Such 'mecha n ical' pro perties, derived fro m other bran ches of physics orchemistry, are pressure, volume, chemical composition, magnetisation,dielectric constant, refractive index an d so on. In simple systems., suc h asare usually d ealt with in phy sics , on ly a few o f these variables need to bespecified in or der to describe the thermody n amic state of the system. Forexample, suppose that we have one gram o f helium gas and tha t we fix itsvolume an d pressure. Suppose that we n ow measure some pro perty of thegas such as its viscosity, refractive index, heat conductivity or dielectricconstant. W e then f ind that pro vided the volume and the pressure of thisone gram of helium are always the same, these o ther pr oper ties are alsoalways the same n o matter by w hom they are measured or when o r where.W e may thus think of the mass of helium , its pressure an d volume as thethermodynamic 'co-ordinates' of the system. It is cha ra cteristic of thermo-dynamics that it deals on ly w ith ma cro scopic , large-scale quan tit ies of thiskind and no t with the variables which chara cterise individua l atoms ormolecules. A com plete a tomic description o f the hel ium gas wo uld specifythe mass of the helium atoms, their mo men tum or kinetic energy, theirpositions and mutual po tent ia l energy . The atomic c o-ordina tes of onegram-atom of helium wo uld number a bout 10 2 4 par ameters; by con trast thethermo dyn amic co-o rdina tes number on ly three (the mass of gas, its pressureand volume). Later on we sha ll see how these two ver y different descriptionsof the gas are linked together.

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Entropy and i ts physical mea ningI t is c lear that on ly un der certa in con ditions do the mass, pressure an dvolume o f the gas suffice to d escr ibe its state. For example, if a gas sudden lyexplodes from a co ntain er, the system is far too com plicated to be describedsimply by pressure an d volume, which are, in any case, no lo nger wel l enough

defin ed to be very u seful pa ra meters. Fo r a s imple thermodynamic descrip-tion (in terms of , say , the pressure an d volum e) to be a dequa te it is eviden tthat the pressure must be uniform and the volume well defined. Thisrestriction can be relaxed if the systems can be divided into a set ofsub-systems in each of which these con ditions are satisfied. At present,how ever, we shal l assume that our systems are everywhere uniform.T he concept of thermo dyn amic equilibrium ca n be defined in terms o f thethermodynamic co-ordinates of the system. If these co-ordinates (e.g.pressure , vo lume an d m ass for a s imple system) do not change with time,and pro vided there are no large-scale flow pro cesses going on , then thesystem is said to be in thermodynamic equilibrium.In most of our discussions we shal l assume that the mass and chemicalcomposition of the systems we deal with remain unchanged. Nonethelessthere are important examples in physics wherein the system is 'open', i.e.the mass is not fixed, for example, where electrons can f low from o neconductor to another. Moreover, although it is outside the scope o f thisbook, we must bear in mind that thermodynamics has had some of itsgreatest triumphs and mo st fruitful ap plications in dealing w ith chem icalreactions.Born 5 has shown how with the help of these 'mechan ical ' co-or dina tes(whose definition co mes entirely fro m o ther branches of p hy sics) it is possibleto define the essentially thermal con cepts of 'insulating' and 'conducting'.Basical ly the idea is simple en ough: if an insulating wa ll separa tes two systemsthe thermodyn amic co -ordinates of one can be altered at will withoutinfluencing those of the other. If the wa ll between them is co n ducting, thisis no lon ger true. (If we knew a l l abo ut the co nc ept of temperature, we wo ulddescribe these situations by say ing: if the systems are separa ted by aninsulating wa ll , the temperature of on e does not influence the other, whereasif they are l inked by a conducting wal l , it does. For our present purpo ses,however, we wish to describe these situation without explicit ly using thecon cept o r the wo rd 'tempera ture'.)T he not ion s of insulating and con ducting wal ls ar e, of cour se, idealisation salthough a silvered Dewar vessel with a vacuum space between the wallsprovides a good a pproximation to the for mer (especially a t low temper atur es)and a thin pure copper sheet to the latter. Even tho ugh the full detaileddevelopment of the ar gument outlined here is ra ther cumberso me, the mainthing is to recognise that nevertheless it is possible to f in d sat isfactor ydefinitions of 'insulating' and 'con ducting' wal ls without usin g the words'heat' or 'temperature'.The idea o f an insulat in g wa l l then leads to the important no tion of an'adiabatic' pro cess: if a system is entirely surr oun ded by an insulating wall12


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Temperature and heat: a different approachthe system is an adiaba tic system and an y processes that it can undergo aread iabatic pr oc esses.

The zeroth law of thermodynamicsHaving in this way introd uced the co n cepts o f 'conducting' and 'insulating'walls we can go on to defin e thermal equilibrium. Con sider two systemssurroun ded by insulating w al ls except where they have a common con ductingwa l l . T hen, wha tever states the systems are in when they are first broughtinto con tact, their thermody na mic co-o rdinates will, if the systems are leftto themselves, even tually stop changing. The systems are then said to be inthermal equilibrium.

The zeroth law of thermody na mics may then be s tated as fo l lows: twosystems which are in thermal equilibrium with a third are in thermalequilibrium w ith ea ch o ther.This law is the basis for the con cept of temperature . Any con venient body,who se state can readily be varied and which has a con venient thermodyna micco -ord ina te (e.g. the length of a mercury column etc.), can be used tomea sure tempera ture, i.e. be a thermometer. If the thermo dyn amic state ofthe thermometer, after having come into thermal equilibrium with onesystem, A, is uncha nged when brought into thermal co ntact with a secondsystem, B, then A and B are, acco rding to the zeroth law, in thermalequilibrium, and are said to have the same temper atur e. It is foun d that inthe simplest thermometer on ly a s ingle co -o rdina te has to be measured inor der to indicate its thermo dy na mic state. A measure of this co-o rdina te isthen a measure of the temperature on this par ticula r tempera ture scale.T o a v o id a tota l arbitrariness in the temperature sca le we shal l for thepresent use the gas scale of temperature. W e can do this co nveniently in o neo f two ways. In the constant volume gas thermometer we define thetemperature O v by the relationship:

O v ocp 1)where p is the pressure at this temperature of a fixed ma ss of gas kept atco n stan t volume, V.On the constant pressure gas scale, we define the temperature O p by therelationship:

O p IX 2)where now V is the volume of a fixed ma ss of gas under con stan t pressureat the temperature Op. So far these definitions a llow us to measure on ly ratiosof temperatures; the size of the degree is fixed by defin ing the n or mal meltin gpoint of ice as having the value 273.15 (cf. p. 36).If the measurements of O v and O p are made with an y gas in the limit ofvery low pressures , the two tempera ture scales are found to be the same

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Entropy and i ts physical m eaningand, in addition, are found to be independent of the pa rticular gas used.Beca use this tempera ture scale is in depend ent of the particular gas chosenwe shall make use of it until we are able to define a truly 'abso lute'temperature scale. It is important to remember that the gas scale oftempera ture is an experimental one; temperatures o n it are found by makingmeasurements on real gases at low pressures. Temperatures on this gas sca lewill be denoted by 0 and we shall refer to the scale as the 'gas scale' oftemperature.

Equations of stateAs we have a lready seen , on ly a few macroscopic parameters are needed tospecify the thermody na mic state of a system: in the simplest ca se the ma ss,the pressure an d the volume o f the system. If we fix the ma ss of the systemand measur e its tempera ture as a fun ction of pressure an d volume we obtainin this way a relationship between p, V and O . (Note: 0 here is measuredon the gas scale of temperature.) This relation ship is cal led an equation o fstate. An example of such an equation of state is one due to Dieterici w hichdescr ibes quite well the behaviour of actual gases at modera te pressures. Itis as follows:

P = 1 RO - a I R O V 3where R per mo le o f gas is a con stan t for all gases, and a and b are con stan tsfor a particular gas.Although such equation s strictly apply on ly to systems in thermodynamicequilibrium, nevertheless they ca n be used to c alc ulate the work done whenthe system changes its state, pro vided tha t in these cha n ges the system isalway s very c lose to equilibrium. Such cha nges in which the system reallypasses through a succession o f equilibrium states are cal led quasi-taticprocesses.The work done in thermodynamic processes: quasi-static changesI have empha sised tha t the fundamen tal measurements in thermodynamicsare taken directly from mecha nics or other bran ches of physics. In this way,for example , we kn ow that in a quasi-static chan ge of volume, the work doneon the system is

W=-1-V 2 p dV

VI(4)

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Temperature and heat: a different approachwhere V2 is the volume o f the system at the end o f the cha nge an d V1 thatat the begin ning. (The wo rk don e by the system wo uld have the op positesign.) p is, of course, the pressure at any instant during the change.Expressions can also be derived for the work done in other qua si-staticchanges, such as, for example, the magnetisation of a body in a magneticfield. Since, how ever, these expression s have to be defined and used w ithcon siderable ca re , an d since the details are irrelevant to the main a rgumen t,I will not go into these questions here. 6In the expression (4) for the work done in a quasi-static chan ge of volumetwo para meters are involved. One of these, the pr essure, does no t dependon the mass of the system; the o ther one, V, is (other thin gs bein g equa l)directly pr opor t ion a l to the mass. For this reason , the pressure is oftenreferred to as the 'intensive' an d the volume as the 'extensive' var iable. Asimilar distinction is possible in other expressions for the w o r k do ne inquasi-static processes.The work done in quasi-static processes can always be representedgraphically o n a suitable diagra m of state. For example, i f on a pV diagramthe pressure o f a fluid is plotted a gain st its volume for some particular path,the work done by the fluid for that path is represen ted by the area underthe curve, since this area is equal to

V 2 p dViv,

Since there are an un limited num ber o f paths having the same starting po int,A, an d the same end po int, B, and since un der each o f these paths the areais in genera l different, the wor k don e in taking the fluid fro m state A to stateB depends no t on ly on the initial a nd f ina l states of the system, but on thepa rticular pa th between them.

Work in irreversible processesBefore discussing irreversible pr oc esses we must f irst define wha t we mea nby a reversible pro cess. T he test of a rever sible pr oc ess is suggested by itsname but some care is needed in app lying this test. W e ca l l a change in asystem 'reversible' if we can restore the system to its original state and atthe same time restore its enviro n men t to its original con dit ion . By 'env iron -ment' I mean the apparatus or equipment, o r an ything else, outside thesystem which is affected by the chan ge we are discussing. T he impor tant pointis this: after any change you c an a lways restore a given limited sy stem to itsor iginal co nd ition but usual ly you can not do so without causing a permanen tchan ge in it s environ men t . T o sum up then: the test of whether a chan geis reve rsible or n ot is to try to restore the status quo everywhere. If you cando this, the change is rever sible, otherw ise not. A reversible change is thus

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Entropy and its physical mea ningan ideal isat ion which ca n n ever be ful ly rea l ised in pra ctice. Nevertheless,the concept of reve rsibility is a vital one in thermodynamics and we shal lmeet it aga in in con nection with the secon d law of thermodynamics.An example of a simple reversible process is the slow compr ess ion of agas in an insulated cyl inder by a p iston which has negligible friction with thecyl inder wal ls . If the for ce o n the piston exceeds that exerted by the pressureof the gas by an infinitesimal amount, the piston moves inw ards andcompresses the gas. If the force on the piston is reduced so as to beinfinitesimally less than that exerted by the pressure of the gas the pistonmoves outwards an d the gas expan ds towa rds its original volume.Now we shall co ntra st this with two examples of irreversible pro cesses andindicate how yo u can ca lcu late the work done in such p ro cesses. If, in thereversible pro cess just descr ibed, the piston had appr eciable friction with thecylinder walls, the pro cess wo uld no longer be reversible. On the compr essionstroke more wo rk wou ld be required than if there w ere n o friction. On thereturn stro ke, how ever, this wo rk is n ot reco vered, but rather an additionalamount of energy is aga in expended in overcoming friction. W e thereforehave an irreversible proc ess. The wo rk don e against the frictional force, F(assumed constant), is just the product of F an d the distan ce moved by thepiston.Here is an other example. Suppose that you stir a fluid in a con tainer witha paddle-wheel . T o mo ve the wheel at a l l you must exert a co uple on i t, an dfrom the size of the co uple an d the number o f revolutions of the wheel yo ucan ca lcu late the w o r k do ne in the process. If the fluid is in a thermallyinsulated container the temperature of the fluid rises beca use o f the stirring.If you reverse the direction of stirr ing the temperature of the f luid does n otfall and you do no t recover the wo rk done .W e have no w show n h ow it is poss ib le to m easure wo rk in a number ofprocesses and our next step is to consider the relat ion ship between wo rk,heat and internal energy, which are related by the first law of ther-modynamics.

Exercises01 Give an example to illustrate the difference between a system inthermo dynam ic equilibrium an d a system in a steady state.02 Express as par tial differential coefficients the volume thermal expansionco efficient, the isotherma l bulk mod ulus, the isotherm al c om pressibility andthe isotherma l magn etic susceptibility.03 10-3 m 3 of lead is co mpr essed reversibly and i so thermal ly a t roo mtemperature from 1 to 1000 atmospheres pressure. How much work i s doneon the lead? The isotherm al c om pressibility of lead, pT = 1 7 -1 ( 19V Idp ) T ,16


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Temperature and heat: a different approa chis assumed independent of pressure with the value of 2.2 x 10-6 atm -1 . Takelatm= 105 Pa.Q4 W ater flows at constant temperature through a pipe at a ra te of 1 m 3 s -I .The input pre ssure is 2 x 106 Pa a nd the output pressure is one half this. Howmuch work is done in 1000 s?

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3

The first law of thermodynamics

The mechanica l theory o f heat, as it is sometimes called, was first firmlyestablished by the very c areful experiments o f Jo u le which we read aboutearlier. In these experiments the essential features were as follows. Workwas done on a kno w n mass of water (or other fluid), thus raising itstemperature; the change in temperature wa s then mea sured very acc urate ly .Joule ensur ed that as far a s possible no heat entered o r left the wa ter so thatthe temperature change was entirely a result of the wo rk. By per forming thework in a variety of different wa ys (for example, electrically, by stirring orby friction) he was able to show that to chan ge the temperature o f the samemass of wa ter through the same temperature interval always required thesame amount of work n o matter by what method the work was done. Hefound similar results with other liquids.We interpret these experiments and generalise from them as follows: ifa thermally isola ted system is taken fro m state A to state B, the work requireddepends on ly on the states A and B and not on the intervening states or themethod of doing the work.This statement allows us to define a new thermodynamic co-ordinate,ca l led the intern a l energy func t ion , U , with w hich to cha rac terise the stateof a thermody na mic system. If we choose some convenient state A as thereference state of a specif ied thermo dyn amic system, then the wor k don e intaking it by adiabatic processes to any other state B depends on B and noton the path. If this work is W we can thus write:

Ug-UA= W 5)In this way we define the difference in intern al energy between B and A asequal to the wor k don e in going from A to B by a diabatic mea ns. It is becauseW is independen t of the path (for a therma lly isolated system) tha t Ug hasa un ique value for eac h state of the system, pro vided of cour se that the samereference state is always used. (It is to be noted that it is not always possible

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Entropy and i ts physical m eaningto go from state A to state B by a diabatic pr oc esses a lon e; it is, however,always possible to go either fro m A to B or from B to A by such pro cessesand this is sufficient to define U.) We now imagine that by suitablemeasurements of the w o r k do ne in adiabatic processes a value of U isassociated with eac h possible state of the system o f interest. If we now relaxthe con dition that the cha nges are to be adiabat ic , we sha l l then gener a l lyfin d that in going from state A to state B

A U = U B - UA WW e now def ine the difference between AU and W (this differen ce is zero inadiabatic chan ges) as a measure of the heat Q which has en tered the systemin the cha nge. W e shal l trea t the heat entering a system as positive. Thus

Q = A U Wor

AU= Q+ W 6)which is a statement of the first law of thermodynamics. Notice that hereheat entering the system and w o r k do ne on the system are regarded aspositive.T o rec ap itulate this important ar gument: it is foun d experimental ly thatin a wide variety of processes the work done in taking a therma lly isolatedsystem from on e state to a no ther depen ds on ly on the two states and noton the path or the method of doing the wo rk. W e can thus associate witheac h state of a system a quantity U , cal led the intern al energy func tion , whosevalue for an y given state of a system is measured by the work done on thesystem in brin ging it to that state fro m a given r eferenc e state un der adiabaticconditions. W e n o w imagine that by these mean s we m easure the intern a lenergy relative to some referen ce state for all states of the system. In anyarbitrary process (no longer necessarily adiabatic) the difference between thechan ge in interna l energy , AU, an d the w o r k do ne on the system is nowdefined as the heat which has entered the system during the pro cess.

A number o f po int s concern ing the first law o f thermodynamics needcomment.i Sinc e we have no independent definition o f Q , the quantity o f heatinvolved in a thermodynamic process, it might appea r tha t the first law

of thermo dyn amics is a mere definition w ithou t physical con tent . Theessence of the law, however, lies in its reco gnition that in adiabaticprocesses W depends on ly on the initial an d final states of the system;this form s the basis for the definition of U , the intern al energy fun ction .This result concerning W is certainly based on experiment a lthough itis ap plied in circumstances far beyon d those for which it has beendirectly tested. The ultimate test of the first law, however, lies in thetesting of its predictions; these have been verified over and overagain.

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The first law of thermodynamics(ii) It is not at once eviden t that the quantity Q defined in equation (6)is the same as that determined in con vention al heat capac ity measure-ments. A detailed discussion of such methods in the light of equation(6) shows, ho we ver, that this is in fact so.

(iii) Let me emphasise again that in the approach outlined here (dueor iginal ly to Born) the thermodynamic concept of quantity of heat hasbeen introduced a nd defined in terms of purely mechanica l quan -tities.(iv) Notice, too, that the idea of heat arises on ly when the state of a system

changes. It is meaningless to talk about quantity of heat in a body.Quantity of heat is not a function of the state of a body liketemperature, pressure, volume or mass: hea t is a form of energy intransit an d cannot be said to exist except when changes of state areoccurring. In this respect it is of course similar to the thermodynamicconcept of wo rk which also is alwa ys associated with changes of state.It is equally meaningless to talk about the quantity of work in abody.

(v ) Note that the thermodynamic concept of heat conflicts, in someimportant wa ys, with the common usage of the wor d. W e say that heatis tra nsferred from one body to an other an d that, for example, the heatlost by one body is gained by the other. There is thus a strongimplication that the heat wa s or iginal ly inside the first body and endedup after the tran sfer inside the other body. It is prec isely this notionthat we have to get rid of in order to th ink c lear ly about heat, wo rkan d interna l energy.

(vi) Equation (6) app lies to an y kind of change, reversible o r irreversible.In or der that U be defin able the initial and final states must be statesof equilibrium, but the intermed iate states ma y be far from it.

The internal energy an analogyThe fo l low ing an alogy may help to make clear the significance of the internalenergy function . Let us suppose that a man has a bank accoun t in which acertain sum of mo ney is credited to him. He ma y a dd to this sum either bydepositing cash at the bank o r by pa ying into his acco unt cheques from otherpeople. Likewise he ma y diminish the amount of money in the acco unt bydrawing out cash or by issuin g cheques payable from the account . The pointis tha t the change in the amount of money in the account in any per iod oftime is the algebraic sum of all cash paid in o r taken out and of all the chequespaid into o r drawn on the account. The total (shown in his acco unt book)do es n ot distinguish between cheque payments and cash payments; on ly thesum of the two counts. So it is with the interna l energy function : heat o rwork do no t contribute separa tely; only the sum of the two ma tters.

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Entropy and i ts physical m eaningT his ana logy a lso makes c lear a no ther point . When the bank a ccoun t isstatic and n ot chan gin g, there is n o need to talk of cash or cheques. It ison l y for the process of transferring mon ey that we need say how the transfer

is to be made, i.e. by ca sh or cheque. So it is in thermodynamics. As longas there is n o exchange of energy with other bodies, the intern al energyfunction of a body is fixed and specifies the en ergy state of the bodycompletely. The use of the words 'heat' or 'work' in these circ umstan ces isentirely ina ppro priate. How ever, as soon as there is any change of energyit is then important to specify if this is by hea t or work. In brief, therefore,hea t refers to a particular kind o f energy transfer.

Atomic interpretation of internal energyTo see more vividly just wha t the internal energy of a body is, we need tochange our poin t of view and l ook at the body o n the atomic sca le. At thisstage it is simplest and n ot seriou sly misleadin g to descr ibe the atoms inclassical, as oppo sed to quantum, terms. In these terms we should expectto f ind , at an y f inite tempera ture of the body, that its atoms w ere mo vin gabout with velocities whose magn itude a n d distribution depend on thetemperature of the body . W e expec t, in fac t, that the higher the temperaturethe higher the mean square velocity of the atoms an d the greater the spreado f velocities abo ut the mean. This is a well -kno wn result from the kinetictheory of gases; it is equa lly tru e of solids and liquids at high tempera tures.In solids, of course, the atoms tend to be a ssoc iated with defin ite positionsin space an d in perfect cry stals the mean po sition s of the atoms for m a regularthree-dimensional array. In addition to the kinetic ener gy of the a toms, whichdepends on the temperature, the atoms also po ssess potentia l energy fromtheir mechan ical interaction with the other a toms. This potential energy toowil l usual ly depend on the temperature of the body an d also o n its volume.The internal energy of the body can no w be interpreted as the total of thekinetic and potential en ergy o f all the atom s that compose it. In this way,we ca n interpret a large-scale thermod yn amic quan tity, the intern al energy,whose chan ges are measured by calorimetry, in terms of the mechanica lproperties of atoms.Classical thermo dyn amics has n o need of this atom ic interpr etation ; itforms a co ns istent an d self-contained discipline in which al l the con ceptspertain to the large scale. On the other han d, if the internal energy of a bodyis just the tota l mechan ical energy of the atoms in it an d if this energy ca nbe changed by do ing work on the body or by heating it, then the first lawof thermody na mics is just a man ifestation o f the law of con servation of energyin mechanics. The conversion of work into internal energy is then to bethought of as the change of large-scale directed mechan ical mo tion ( in , forexample, the piston during the co mpression of a gas or in the paddle-wheelduring the stirring of a fluid) into the smal l -sca le ran dom m ovemen ts of22


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The first law of thermodynamicsatoms. As we have alrea dy implied, the kinetic theory o f gases is built upon the basis of these idea s; its suc cess in relating the lar ge-sca le proper tiesof gases to the mechanics o f atoms is un questioned . W e shal l come bac k tothe atomic point of view la ter when we try to un derstan d en tropy .

The internal energy as a function of temperature and volumeW e kn ow fr om the definition of heat that

AU = Q+W 6)Now apply this to a quasi-static infinitesimal chan ge in which the work

done is by a change of volume dV at pressure p . Let us denote the smallquantity o f heat entering the system by CI4 so that we have:dU = C t q p d V 7 )

(The symbo l dq is used to den ote an infinitesimal quantity which is not thedifferential 7 of a function.)If the chan ge takes place at con stan t volumedU = dg

If this chan ge causes a tempera ture rise (10, then the heat capa city at constantvolume mu st be given by:00 ) vv d O =

dg 7U 8)Consequently, if w e in tegrate this expression with V constant

02U2 - Ut = dO 9 )where U2 a n d U 1 are the values of the interna l energy at the same volumeat temperatures 02 an d 0 1 . Th is mean s that if we measure the heat capacityat con stan t volume over a temperature range we ca n then f ind the changein intern al energy of the system within that ra n ge. T his result is quitegeneral.The intern al energy, U, depends on ly on the thermod yn amic state of thebody; this, in turn, depends on ly on, say, the temperature, 0, an d vo lume,V , if the m ass is fixed. Con sequently, we ca n w rite for any arbitrary changein U brought about by chan ges dO and dV:

do+ ( aU) dv 10)U = dU \k 9This equation simply explo its the fact that an y chan ge in U depends on ly

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Entropy and its physical meaningon the initial an d final states and not on the path; thus we ca n first change0 with V con stan t and then change V with 0 con stant and this will describeany arbitrary infinitesimal chan ge in U we please.Ho wever , we have already seen that (dUlc10) v = Cv so that we can rewriteequation (10) as

dU = C v d0+ (7 1 1 ) dV 11)dVExperiments on gases at very low pressures show that in them the internalenergy at a given temperature does no t depend on the volume; this is kno wnas J oule's law . If we regard an ideal gas as the limit of an actual gas at veryl ow pressures, we can conclude that for an ideal gas ( dUldV) e = 0 an d so ,fo r any chan ge in such a ga s,dU = C v d0 12)This is NOT true for mo st systems; in solids, for example, the internalenergy depend s strongly on the volume and the secon d term in equa tion (11)

cannot be neglected.

ExercisesQI A gas, not an ideal gas, is con tained in a thermally insulated cylinder.It is quickly co mpr essed so tha t its tempera ture rises shar ply. H as there beena transfer of heat to the gas? Has work been done?Q2 A gas, not an ideal gas, is contained in a rigid therma lly in sulatedcontainer. It is then allowed to expand into a similar container initiallyevacuated. What is the chan ge in the interna l energy of the gas?03 10-3 m 3 of lead is compressed reversibly and adiabatically from 1 to1000 atmospheres pressure. W hat is the chan ge in the interna l energy of thelead? The ad iabatic com pressibil ity o f lead is assumed independent o fpressure with the value of 2.2 X 10 -6 atm - Take 1 atm = i0 5 Pa. (Cf.Exercise Q3 of Chapter 2.)Q4 A system, who se equation o f s ta te depends on ly on pressure p, volumeV and temperature 0, is taken qua si-statica lly from state A to state B in thefigure along the path A C B at the pressures indicated. In this proc ess 50 Jo f heat enter the system and 20 J of work are done by the system.(a ) How much heat enters the system a lo ng the path ADB?(b ) If the sy stem goes fro m B to A by the cur ved path indicated schematical lyo n the figur e, the work done on the system is 25 J . How much h eat enterso r leaves the system?24


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D

A

The first law of thermodynamics

5 P,

Pi

V

Figure Q4

(c) If the internal energy at A is denoted by UA etc., suppose thatUp -UA= 25 J . W hat then is the heat tra nsfer involved in the processesAD a nd DB?05 1 kg of water is slowly frozen at constant temperature and a pressureof 100 atm. The latent heat of melting of ice under these conditions is3.36 x 10 5 3 kg -1 . Ho w much w ork is don e a nd wha t is the change in internalenergy of the system? The density of ice relative to that of wa ter is 0.9.Q6 Show that in an ideal gas:

Cp = C v + Rwhere C p , the heat cap acity per mole at constant pressure, is defined asC p = (clii/d0) p = ) , t and R is the gas constant per mole, as in the equationof state of the gas: pV = RO.(Hint: use equations (12) an d (7) of the text.)07 Show that, when an ideal gas undergoes an ad iabatic qua si-staticchanage, pVY = constant, where y = C,,/C..

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Entropy and i ts physical m eaning(Hint: use equation (7) of the text, with dq = 0 for an adiabatic process, andequation (12) for dU in an ideal gas. Then use the differential form ofpV = RO, i.e. pdV + V dp = Rd() to eliminate dO. You will also need theresult derived in the previous question 06.)Show also that the result can be re-expressed a s 8V Y -1 = constant.

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4

The second law ofthermodynamics

In our discussion of the first law of thermodynamics we have seen how theinternal energy function is defined a nd from that how 'heat' is defined a ndmeasured. We have also noted that the internal energy can be identified withthe total mechanical energy (kinetic a nd potential) of the atoms in thesubstance.The second law of thermodynamics introduces a further state variable, theentropy. In addition, it makes possible the definit ion of an absolute scale oftemperature, that is, one which is independent of the properties of a n ysubstance or class of substances. (The size of the degree is, however, stilldefined in terms of the properties of a particular substance, usually water.See page 36.) The second law starts from the following fact of commonobservation: when a hot body and a co ld body are brought into thermalcontact, the hot body cools and the co ld body gets wa rm a nd never the otherway round. We may say, in brief, that heat always flows spontaneously fromhot to cold. The second law of thermodynamics attempts to express this factin such a way that it can be used to deduce certain important propertiescommon to all substances.

By the beginning of the nineteenth century, steam engines were widelyused in industry and a great deal of work was being done to improve theirperformance . It was this interest which led to the formulation and under-standing of the second law of thermodynamics. Carnot (1796-1832) was thefirst person to grasp the essentials of the problem an d we shall examine hisapproach to it in some detail.Before doing so, however, I would like to comment briefly on therelevance of heat engines to the wider topic of thermodynamics. Thermo-dynamics serves two quite different a nd at first sight unrelated purposes. O nthe one hand, it deals with the engineering problem of making heat engines

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Entropy and i ts physical m eaning(and refrigera tors) wo rk efficiently. On the other hand, it deals with theconditions of equilibrium in thermo dy na mic systems. Ho w is it that these twoapp ar ently differen t subjects are so clo sely l inked? The rea son is this (I quotefrom Carnot): 'W herever there is a difference of temperature, work ca n beproduc ed from it.' More generally, whenever two systems are not inthermod yn amic equil ibrium wo rk ca n be produced. The expression, 'not inthermody na mic equil ibrium', ca n inc lude a difference in temperature, adifference in pressure, a difference in chemical potential, and so on. So wesee tha t, just a s problems of stability an d equilibrium in mechanics are relatedto the ability of a system to do wo rk, so in thermodynamics the possibilityof work and the absence of equilibrium are two aspects of the same situation.So when w e study the behaviour of ideal ised heat en gin es, remember thatthe results will also be important in the study of the general thermodyn amicproperties of systems in equilibrium.W ith this in mind let us n ow look at Carnot 's work on the behaviour ofhea t en gines.

The behaviour of idealised heat enginesIn 1824 Carnot published his famous memoir entitled Rflexions sur lapuissance motrice du feu et sur les machines propres e t dvelopper cettepuissance. In this truly rema rkable wor k he was co nc erned w ith the generalquestion of how to produce mechan ica l wor k from sources of heat. He beginsa s follows: 8

Everybody knows that heat can cause movement, that it possesses greatmotive power: steam engines so common today are a vivid and familiar proofof it. The study of these engines is of the greatest interest, their importanceis enormous, a n d their use increases every day. They seem destined to producea great revolution in the civilised world.

Despite studies of all kinds devoted to steam engines, and in spite of thesatisfactory state they have reached today, the theory of them has advancedvery little and the attempts to improve them are still directed almost bychance.

The question has often been raised whether the motive power of heat islimited or if it is boundless; whether possible improvements in steam engineshave an assignable limit, a limit that, in the nature of things, cannot be exceededby any means whatever, or if on the contrary these improvements can beextended indefinitely.Carnot adds a foo tnote here to explain w hat he mean s by 'motive pow er'.He mea n s essentia l ly wha t we w ould ca l l 'wo rk' . In the next paragraph hecontinues:

To see in its full generality the principle of the production of work byheat we must think of it independently of any particular agent; we must

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The second law of thermodynamicsestablish arguments applicable not on ly to steam en gines but to an y imagina bleheat engine whatever the working substance and whatever its manner ofworking.After some further general remarks, Carnot now comes to a most

important point. Let me first give a literal translation of what he says an dthen comment on it.The production of motion in the steam en gine i s a lways a ccom pan ied by a

circumstance upon which we must fix attention. This circumstance is there-establishmen t of equilibrium in the ca loric , that is to say, its passage froma body where the temperature is more or less elevated to another where it islower.The expression, 're-establishment of equilibrium in the caloric', can betranslated quite accur ately by the phrase, 'return to thermal equilibrium'. So

the point he is making here is that the production of work by a heat enginerequires a temperature difference in the first place but that when the engineoperates it will reduce the temperature difference and tend to bring aboutthermal equilibrium.Again, later on, Carnot reiterates that the heat en gine requires both a hotbody and a co ld body to make it operate. In addition, and this is mostimpor tan t, 'W herever there exists a difference of tempera ture it is possibleto produce work from it.'Carnot then continues his argument as follows:

Sinc e an y return to thermal equilibrium can be used to produce w ork , anyreturn to equilibrium wh ich takes place without the product ion of this wo rkmust be con sidered as a rea l loss.The point here is that, given a tempera ture differen ce, this may either be

used to produce work o r it may be wastefully dissipated simply by a flowof heat fro m a high tempera ture to a low temperature without an y w ork beingproduced. As we a l l know, this spon taneo us flow of heat fro m a hot bodyto a co lder one wil l a lway s take place if it can. This mean s that a situationwhich might produce work can be irretrievably lost by natural heat flow.

Let me sum up the situation at this stage. (i) A heat engine requires atemperature difference in order to operate. (ii) W h e n the engine operatesit tak es in hea t at the high tempera ture an d gives out some heat at the l owtemperature so that it tends to reduce the temperature difference, i.e. torestore thermal equilibrium. (iii) Any temperature difference can, inprinciple, be used to produce work. (iv) T empera ture differen ces tend todisappear spontaneously by heat conduction without producing usefulwork.

These ideas form the basis of Carnot's thinking (although perhaps witha slightly different emphasis since at the time he wrote he was making useof the caloric theory, even though he had serious doubts about its validity).

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h

d

a

PESRE

Entropy and its physical meaning

ik d f h(a) b) O L U M E

Figure 1 (a) Copy of Carnot's original diagram. (b) The Carnot cycle on thep-diagram of a gas. O A i s the temperature of the heat source and O B that ofthe heat s ink. The letters on the V axis correspond to the positions of thepiston in Figure 1(a). X i s the starting point.

F ro m these idea s it is clear that an efficien t heat en gin e must be so designedthat there are no wasteful heat flows during its operation. When heat hasto be transferred from, say, the furnace to the working substance in theengine, this must be done while both bodies are very nearly at the sametemperature. Mor eover, there must be n o significa nt tempera ture gra dientsinside the working substance at an y t ime dur ing i ts oper at ion s inc e these toowould reduce the efficiency of the engine.Carnot devised a cycle of operations for a heat engine that would a chievethese purpo ses. T his cyc le (no w ca l led the Car no t cy c le) has been describedand considered so often since Carnot's time that it is of interest to see howhe himself described it in the first place. It is a very lucid description butit may perhaps be helped by showing the cycle of operations on the pVdiagram of a gas . Figure 1 shows such a diagram (Figure 1(b)) together w iththe original Figure 1 of Carnot's paper (Figure 1(a)). The lettering on theV axis in F igure 1(b) has been cho sen to correspond with that of the positionsof the piston in Figure 1(a). The cyc le shown in the figure consists of twoisothermal and two a diabatic proc esses in the sequence described by Carnot .Notice, how ever, that Carnot starts his description of the proc ess at the pointX, i.e. part way through an isothermal change. Now let us fol low Carnot:30


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The second law of thermodynamics. . . Let us imagine a gas, atmospheric air for example, enclosed in a cylindricalvessel abcd, Figure 1 [our Figure 1(a) ] , furnished with a movable diaphragmor piston cd; in addition let there be two bodies, AB, each held at a constanttemperature, that of A being higher than that of B; let us now imagine thefollowing series of operations:1 Contact of the body A with the air enclosed in the space abcd, or with the

wal l of this space, a wall which we will suppose to conduct heat easily.Through this contact the air assumes the same temperature as the body A;cd is the present position of the piston.

2 The piston rises gradually a n d takes the position ef. Contact is stillmaintained between the body A and the air, which is thus kept at a constanttemperature during the expansion. The body A furnishes the heat necessaryto keep the temperature constant.

3 The body A is taken a w a y and the air is no longer in contact with any bodycapable of giving it heat; the piston however continues to move and passesfrom the position ef to the position gh. The air expands without receivingheat, a n d its temperature falls. Let us imagine that it fails in this way untilit becomes equal to that of the body B: at this stage the piston stops andoccupies the position gh.

4 The air is put into contact with body B; it is compressed by the return ofthe piston as it is moved from the position gh to the position cd. The airhowever remains at a constant temperature because of its contact with thebody B to which it gives up its heat.

5 The body B is removed a n d we continue to compress the air which, beingnow isolated, rises in temperature. The compression is continued until theair has acquired the temperature of the body A . During this time the pistonpasses from the position cd to the position ik.

6 The air is put back into contact with body A ; the piston returns from theposition ik to the position ef; the temperature stays constant.

7 The operation described under No. 3 is repeated, then successively theoperations 4, 5, 6, 3, 4, 5, 6, 3, 4, 5 a n d so on. . . .

Carnot notes two points about this cycle. One is that the working substance(the air) produces a n et a m o unt of work in each cycle . We can see this readilyfrom the p V diagram. The work done by the air is fp dV which is thus thearea of the cycle on the p V diagram. If the representative point of theworking substance traces out the cyc le in the direction indicated by thearrows, this area is positive and a net amount of work is produced in eachcyc le .

The second point is that the work is produced in the most advantageousm a nne r possible. The two adiabatic processes (3) an d (5) change thetemperature of the air in the engine without adding or taking away any heat.In this way the air is always brought to the temperature of the heat sourcebefore being put into contact with i t, an d likewise the heat sink. This preventsany wasteful flow of heat between bodies at widely different tempera-tures.

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Entropy and i ts physical m eaningFrom our point of view we may say that all the pro cesses described inthe cy cle a re qua si-static; the gas is alwa ys effectively in equilibrium. For thisreason a l l the cha nges are rever sible which is importan t for the subsequentargument.Carnot continues:

All the operations described above can be carried out in the oppositedirection and in an inverse order. Let us imagine that a fter the sixth opera tion ,that is to say the piston having arr ived at the position ef, o n e makes it returnto the position ik and that at the same time on e keeps the air in con tact withbody A: the heat furnished by the body during the sixth operation w i ll returnto its source, that is to say to body A, and things w ill be in the state they w ereat the end of the fifth operation. If now we take away the body A and makethe piston move from cf to cd, the temperature of the air wil l decrea se by asmany degrees as it rose during the fifth opera tion and will become equal tothat of body B. W e can obv ious ly co nt inue a series of oper ations inverse tothose that w e first descr ibed: it is sufficient to rep ro duc e the same condit ionsand to carry out for each operation an expan sion instead of a compression an dvice versa.

The result of the first operation s was the pro duct ion of a certa in quantityof work an d the tran sport of heat from body A to body B ; the result of theinverse oper ation s is the consumption of the work pro duced an d the return ofthe heat from body B to body A: so that these two series of operations in away cancel or neutralise each other.Carnot then shows that n o heat engine can be more efficient than thereversible en gin e he has d escribed. He remarks:

W e have chosen atmospheric air as a n instrument for developing work fromheat; but it is obvious that the reasoning would have been the same for anyother gaseous substan ce a nd even for any other body whose temperature canbe changed by successive contractions and expansions which includes allsubstan ces in na ture or at least all those which are appropriate for obtainingwork from heat. Thus we are led to establish the fol lowing genera l propo si-tion:

The m otive pow er of heat is independent of the agents used to produ ce i t ; itsam oun t is f ixed uniquely by the temperatures of the bod ies between which thetransport of heat is ultimately made.W e sha l l not fo l low Carn ot 's own argument to esta blish this proposit ionbeca use it rests on the calor ic theory . T o make the argumen t con sistent withthe mechan ical theory of heat , we must cha nge it somew hat and also bringin a new principle: the second law of thermodynamics.As we have seen, the Carnot cycle consists of two adiaba tic and twoisotherm al pr oc esses (at tempera tures O A and O B ) so car ried out that at al l

stages the changes are reversible.Suppose no w that during one cycle the quantity of heat taken in by theworking substance at the high temper atur e is Q i , and that given out at the32


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The second law of thermodynamicslow temperature is Q2. Since the working substance returns exactly to itsinitial state at the end of one cycle, AU = O . If then the work done by theengine in o ne cyc l e i s W ,

W= Qi Q 2 13)by virtue of the first law o f thermodynamics . The ratio of the work doneper cycle to the heat taken in at the high temperature, i.e. W/Q i , is calledthe thermody na mic effic iency of the engine. We now wish to show that a l lreversible engines working between the same two temperatures have thesame efficien cy and further that n o engine working between these tempera-tures can be more efficient.In or der to pro ceed further, we require to state in a suitable form the axiomor postulate that heat always flows spontaneously from a high tempera tureto a lower on e. Clausius (1822-1888), in his statemen t of the secon d law o fthermodynamics, used the following form: 'It is impossible for a self-acting(cyclic) machine, unaided by any external agency, to convey heat from onebody at a given temperature to an other at a higher temperature.' 9 The formof the secon d law o f thermodynamics due to Kelvin (W illiam Thomson,1824-1907) is as fol lows: I 'It is impossible, by means o f inan imate materia lagenc y , to derive [con t inu ous] mechan ica l effect from an y port ion of matterby co oling it below the temperature of the coldest of the surro un ding o bjects.'These two statements can be shown to be completely equivalent but the firstis perhaps preferable as being more obvious ly a restatement of theobservation that heat flows spontaneously only from hot to cold.

We wish now to prove that n o engine using a given source and sink o fheat can be mor e efficient than a reversible engine (a Carn ot engine) workingbetween the same two temperatures. Let A an d B be two engines o f whichA is the Carn ot engine and B is the engin e which, by hyp othesis, ha s a greaterthermody na mic effic iency than A. Since A is perfectly reversible, we maysuppose that it is driven backwards absorbing completely the work, W,generated in each cycle by B. A is now no longer a heat engine but arefrigerator or heat pump whose function is to absorb heat at the l owtemperature and discharge a grea ter qua ntity at the higher temperature. IfB absor bs heat Qb at the high tempera ture and A restores heat Q a , then thethermod yn amic effic iency n a of A is W IQ a , and that of B(%) is W /Q b . N o wby hypothesis n b is greater than n a since we supposed B to be more efficientthan A . This implies that Q a is greater than Qb. That is to say that moreheat is delivered by A at the high temperature than is absorbed by B. Thisis no viola tion of the first law of thermod yn amics s ince A absor bs at the lowtemperature corr espon ding ly more heat than is discharged by B. It is,however , a violation of the second law of thermodynamics s ince the twoma chin es together co nstitute 'a self-ac tin g cy clic machine' w hich 'un aided byany external agency . . . conveys heat from o ne b o dy at a low temperatureto ano ther at a higher'. We conclude, therefore, that our original hypothesisis wrong, i.e. that n o heat engine with an eff ic iency greater than a Carno t

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Entropy and i ts physical m eaningengine can exist. From a similar argument it follo ws that a ll Carnot engineswo rking between the same two temperatures have the same efficienc y. If thiswere n ot so it wo uld be possible, by suitably coupling a pa ir of engines whoseefficienc ies were d ifferen t, to viola te the second law of thermodynamics.

W e have established, therefor e, that the thermod yn amic efficiency of aCarnot eng ine depends on ly on the tempera tures between w hich it wo rksand not on the working substance. This is essentially the proposition whichCarnot set out to prove; it is frequen t ly ca l led Car no t's theorem.Although w e have no t fo l lowed exact ly Carn ot 's origina l argumen t , wehave follow ed a lmo st all its essential featur es an d in particular the idea ofcoupling the reversible engine, wor king backwa rds, to the heat engine who seefficiency is un der study .

How did Carnot come upon such beautiful, general and powerfulargumen ts? Pa rt of the answer lies in the influence and achievements of hisfather, Lazare Carn ot, who wa s not on l y one of Nap oleo n 's mo st successfulgenerals, but was also a very capable mathematician. One of his mathematicalworks, Fundamental Principles of Equilibrium and Movement, was conc ernedwith the efficiency of machines such as pulleys and levers. In the preface,he w rites of mechan ical engines in genera l (I translate): . . to ma ke these[mac hines] pro duc e the grea test effect po ssible, it is essen tial that there aren o sudden jolts, that is to say that the movement must always change byimperceptible degrees.' In the text he a mp lifies this and i n discussing ho wto make an efficien t mecha n ical en gin e w rites: ` . . this prin ciple dema n dsthat one avoids . .. any collision or sudden change whatever that is notessential to the nature of the machine, sin ce ever y time there is a collisionther e is a loss of kinetic en ergy, and in consequence a part of the activemo vemen t is useless ly absor bed.' If you compare this with Sadi Ca rn ot'swords o n hea t engines (see p. 36), you see, I think, how closely the grainof Sadi's thinking matches that of his father. Non etheless his son 's wo rk onheat engines ca rr ies Lazar e's ideas in to quite new r ealms. Sadi Car no t's modeof thought is further revea led in the text of his trea tise. At one point he says(this is a fairly literal trans lation and no t very e legant):

Accor ding to our present ideas , one can , with sufficient accu rac y, compa re themotive pow er o f heat w ith that of wa ter descending: both have a maximum thatcan no t be exceeded o matter w hat machine is used to abso rb the act ionof the water or hat substance is used to absorb the action of the heat.T he motive power of the fall o f water depends on its height an d the quantityof liquid; the motive power of heat l ikewise depen ds on the quantity o f heatused and wha t we might ca l l , an d wha t ind eed we shal l call, the height of itsfall, that is to say , the difference in temperature of the bodies between whichthe exchange of heat takes place.

Carnot's ana logy turned out no t to be exact but it pro vided an astonishinglyclear insight into the working of heat en gines and fro m that to a profoundunderstanding of thermodynamics.34


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The second law of thermodynamicsWe now proceed to do two things: (i) to define the absolute tempera ture

scale; (ii) to define a new state variable, the entropy.

The absolute scale of temperatureW e have already shown that the effic iency of a Carnot engine whose workingtemperatures are t 1 and t 2 is a funct ion on ly of these temperatures, i.e.

n Qt Q2 = 1 Q2 = ( D o i t 2 )Q t tt 1 and t2 are not no w measured on the gas scale of tempera ture; they areat this point quite arbitra ry . T herefore Q 2 /Q is itself simply a function oft 1 and t 2 , i.e.

Q2 = F(t2, ti) 15)QiM or eover , it is no t difficult to show that the function F must separate in sucha way that Q 2 /Q = f(t2 )/f(t 1 ). This then pr ovides us with a gener al basis foran absolute scale of temperature. Note, however, that we may obviouslychoose the function f as we please.Thomson (later Lord Kelvin) recognised even before the general accept-ance of the present theory of hea t, that the fact expressed in Carnot's theoremcould provide the basis for an absolute scale of temperature. He firstproposed such a scale in 1848 defining it in the following way:"

The chara cteristic property of the scale which I now propose is, that al ldegrees have the same valu e; that is, tha t a un it of heat descending from a bodyA at the temperature T of this sca le, to a body B at the temperature ( T 1),wo uld give out the same mec han ical effect, wha tever be the number T . Thismay justly be termed an absolute scale, since its characteristic is quiteindependent of the physical pr operties o f an y specific substance.This turns out to be equivalent to choosing the function At) as the

exponential function so that in a Carnot cycleQ 1 l ? 16)Q2 i 2

where n d 73 are the temperatures of the sink and source on this firstThomson scale.In 1854 Thomson defined a second absolute scale , cho sen this t ime to agreewith the air thermometer scale, which is effectively the gas scale oftemperature." In this seco nd sca le , which is no w un iversa l ly used and cal led

14)

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Entropy and its physical meaningthe Kelvin temperature scale, the temperatures 7 i and T 2 of the source andsink of the Carnot engine are so defined that:

Q iQ2 T2 (17)This equation defines the ratio of two absolute temperatures. It does notdefine the size of the degree. To do this we must give a numerica l value tothe temperature of some convenient fixed point. For this purpose the triplepoint of water has been chosen as the most convenient a nd its temperatureis taken to have the value 273.16 K by definition. (This choice was made andrecommended by the Tenth General Conference on Weights an d Measuresin 1954.) The reason for choosing this particular number is that the size ofthe degree then remains essentially the same as the Centigrade degree usedpreviously in scientific work. Note, however, that whereas to use theCentigrade degree requires measurements at two fixed points (the ice pointa nd the normal boiling point of water) to use the degree defined in this newway requires measurements at one fixed point only.

From now on we shall always use this absolute temperature scale. It isusually denoted by K. By taking a perfect gas as the working substance ofa Carnot engine it is not difficult to show that the gas scale is then identicalwith the absolute scale. Over a wide range of temperatures the gas scale canthus be used as a means of determining the absolute temperature. At verylow temperatures, however, gases effectively cease to exist a nd then the gasscale can no longer be used. In this region, as we shall see, we have to goback to the definition of the absolute scale in order to measure thetemperature.

The Carnot engine as a refrigeratorA Carnot engine, being perfectly reversible, is both the most efficient heatengine an d also, when operating in reverse, the most efficient refrigeratoro r heat pump. It can thus be used as a stan dard with which to compare actualrefrigerating machines.

However, we must beware of thinking that the idealised efficiency definedabove has any close relationship to 'efficiency' as an engineering concept.To the engineer many other aspects of the engine, whether as a heat engineo r refrigerator, that are neglected in the idealisation are of importance,probably of overriding importance: speed of operation, cost, environmentaleffects an d so on.

Exercises01 An ideal gas with y = 1.5 is used as the working substance of a Carnotengine. The temperature of the source is 600 K an d that of the sink is 300 K.36


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The second law of thermod ynamicsThe volume of the gas changes from 4 to 1 litre at the low temperature andthe pressure at the volume of 4 l itres is 1 atmospher e. Ca lcula te the volum esan d pressures at the extreme points of the high tempera ture expans ion an dalso the heat absorbed there. How much heat is given out at the l owtemperature and what is the thermody na mic effic iency of the engine?Q 2 A house is heated by mea ns of a reversible heat engine act ing a s a heatpump. T he out s ide temper ature is 270 K and the inside tempera ture is 300 K.If when the house is heated by conventional electric space heaters 10 kW ofpower is used, how much power is consumed by the ideal heat pump tomaintain the same temperature?03 T wo ident ica l bodies of con stan t heat capac ity are initial ly a t tempera -tures T 1 and T2. T h e y are used as the sink and source of a Carnot engineand no other source of heat is available. Show that the final commontemperature when a l l possible work has been extracted from the system isTF= (T1 T2)" 2 . What is the maximum work obtainable?04 T he vo lume thermal expan sion coeff ic ient of helium gas at the ice pointand at low pressures is 0.00366 per C. T he temperatures were measured onthe perfect gas scale. What is the value of the ice point on the absolutetemperature scale?

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Entropy: how it is measured andhow it is used

The definition of entropySo far Q i a n d Q2 have been co nsidered as magnitudes on ly . Ho wever , if weadopt a sign convention by which Q is positive when heat is absorbed bythe working substance and n egative wh en hea t is given up, we may rewriteequation (17) in the form

Q 1 /T1 + Q2 1T 2 = 0 18)F igure 2 represents in a p V diagram the succession of states thro ughwhich a substan ce is taken in an ar bitrary reversible cycle: superimposed onit are drawn a succession of adiabatic and isothermal paths. The heavy l inein the diagram represents a path consisting only o f alterna ting ad iabatic andisotherma l steps, which appr oximate to the origina l arbitrary cyc l e and can ,in fac t, be made a s c lose to this as we wish by making the mesh of adiabaticand isother mal lines still finer. (This is no t obvious because the zig-zag pathis nec essar ily lon ger than the smooth on e; the first law of thermodynamics,ho wever , is sufficien t to show that the heat absorbed in any segmen t of thesmoo th path is the same as that absor bed in the cor respondin

john sydney dugdale entropy and its physical meaning 1996

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