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ProgressinMathematicalPhysicsVolume57Editors-in-ChiefAnneBoutetdeMonvel(UniversitéParisVIIDenisDiderot,France)GeraldKaiser(CenterforSignalsandWaves,Austin,TX,USA)EditorialBoardC.Berenstein(UniversityofMaryland,CollegePark,USA)SirM.Berry(UniversityofBristol,UK)P.Blanchard(UniversityofBielefeld,Germany)M.Eastwood(UniversityofAdelaide,Australia)A.S.Fokas(UniversityofCambridge,UK)D.Sternheimer(UniversitédeBourgogne,Dijon,France)C.Tracy(UniversityofCalifornia,Davis,USA) MikhailV.VolkensteinEntropyandInformationTranslatedbyAbeShenitzerandRobertG.BurnsBirkhäuserBasel·Boston·Berlin Author:MikhailV.Volkenstein(1912–1992)Translators:AbeShenitzerRobertG.Burns4OakdaleStreetDept.ofMath.andStat.JamaicaPlainYorkUniversityMA021304700KeeleStreetUSATorontoM3J1P3Canada2000MathematicsSubjectClassification:94A17,68P30,33B10,11B73LibraryofCongressControlNumber:2009931264BibliographicinformationpublishedbyDieDeutscheBibliothek.DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie;detailedbibliographicdataisavailableintheInternetathttp://dnb.ddb.deISBN978-3-0346-0077-4BirkhäuserVerlagAG,Basel·Boston·BerlinThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,broadcasting,reproductiononmicrofilmsorinotherways,andstorageindatabanks.Foranykindofusewhatsoever,permissionfromthecopyrightownermustbeobtained.©2009BirkhäuserVerlagAGBasel·Boston·BerlinP.O.Box133,CH-4010Basel,SwitzerlandPartofSpringerScience+BusinessMediaPrintedonacid-freepaperproducedfromchlorine-freepulp.TCF∞PrintedinGermanyThisbookisatranslationofthebook“Знтропияиинформация”writtenbyMikhailV.Volkenstein(M.B.Волькенштейн)andwaspublishedinthebookseries“ProblemsofScienceandTechnicalProgress“byМоскваНаука(NaukaPublishers)in1986.ISBN978-3-0346-0077-4e-ISBN978-3-0346-0078-1987654321www.birkhauser.ch ContentsAbouttheAuthorixPreface11“Reflectionsonthemotivepoweroffire...”3SadiCarnot..................................3Caloric.....................................4Theirreversibilityofheatprocesses.....................6WhatdidCarnotdo?............................9TheCarnotcycle...............................11Thermodynamictemperature........................162Thelawsofthermodynamics19Lomonosovandtheconservationlaws...................19Thelawofconservationofenergy......................21Thesecondlaw................................23Thepressureoflight.............................24Entropy....................................25Thelogarithmandexponentialfunctions..................27Calculationofentropy............................30Measuringentropyexperimentally.....................34Irreversibleprocesses.............................373Entropyandfreeenergy41Obtainingusefulwork............................41Equilibriumconditions............................43Achemicalreaction.............................45Meltingofcrystalsandevaporationofliquids...............48Whydoesalcoholdissolveinwateryetgasolinenotdoso?........49Hydrophobicforcesandthealbuminglobule................51Whatdorubberandanidealgashaveincommon?............53Whydoweheataroom?..........................57 viContents“Themistressoftheworldandhershadow”................59WhywasEmdenright?...........................604Entropyandprobability63Boltzmann’sformula.............................63Stirling’sformula...............................66ThemeaningofBoltzmann’sformula....................68Thefusionofacrystalandtheevaporationofaliquid..........71Entropicforces................................72Entropyofequilibriumstates........................76Alittlequantummechanics.........................78Gibbs’paradox................................80Nernst’stheorem...............................815Statisticsandmechanics85Thedistributionofvelocities,andtemperature..............85Thebarometricformulaandthe“gravitationalperpetuummobile”...90Fluctuations..................................93Whyistheskyblue?.............................98TheageofDarwin..............................100Laplace’sdemonandSina˘ı’sbilliard....................103Thefateoftheuniverse...........................1076Opensystems113Theproductionandflowofentropy....................113Thedissipationfunction...........................116Anastronautlivesonnegativeentropy...................119Whydocellsdivide?.............................124Farfromequilibrium.............................125TheBelousov–Zhabotinski˘ıreaction....................131Organismsasdissipativesystems......................133Thethreestagesofthermodynamics....................1387Information141Informationandprobability.........................141Informationalentropy............................145Informationandentropy...........................151Maxwell’sdemon...............................154Obtainingandcreatinginformation.....................157Thevalueofinformation...........................160 Contentsvii8Entropy,information,life165Thethermodynamicsoflivingorganisms..................165Biologicalevolution,entropy,andinformation...............169Thevalueofbiologicalinformation.....................174Complexityandirreplaceability.......................181ComplexityandG¨odel’stheorem......................185Informationandartisticcreation......................186Index197 AbouttheAuthorEntropyformedacentralthemeofthescientificworkofMikhailVladimirovichVolkenstein,especiallyduringthelastperiodofhislife.Initially,fromabout1933to1948,hisscientificinterestslaymoreinspectroscopy;duringthisperiod,hismostsignificantworkconcernedtheintensitiesofmolecularoscillationspectra.Inthenextperiodofhislife,fromaround1948to1960,heturnedhisatten-tiontothenewfieldofmacromolecules,becomingoneofthemostimportantcontributorstothestatisticaltheoryofsuchmolecules.Inthe1960sand70shisintereststurnednaturallytoquestionsfrommolecularbiol-ogyandmolecularbiophysics.Itistohimthatweowemuchofourunderstandingofbiomolecularconformation;inparticular,hisworkshedlightontherelationsbetweenthechemicalstructureofpolymersandtheirplasticandelasticproperties,andonthereac-tivityofbiomolecules.Towardstheendofthe1970s,MikhailVladimirovichsetouttoexplorethedeepquestionsoftheoriginoflife,andsobeganhisinvesti-gationsofthecomplexofprob-Figure1:MikhailVladimirovichVolkensteinlemslinking“entropy,informa-duringavisittoBerlinin1981.(Pleasenotetion,andlife”.Itwasatthisthathewasnotaheavysmoker.)timethatIfirstmethim,inPoland;hispersonalityandhiseruditionmadeaprofoundimpressiononme.HesubsequentlyattendedseveralofourconferencesinBerlinonthetheme“Irre-versibleprocessesandself-organization”,wherehemetIlyaPrigogineandbecamefriendswithhim.Irememberclearlythewonderfulconversationsaboutscienceandculture,fullofcontroversy,betweenthesetwobrilliantpolyhistors,rangingfromthedancinggodShivatoAfricansculpture,carriedoutnowinonelanguage, xAbouttheAuthornowinsomeother.AtthattimeMikhailVladimirovichwasprofoundlyimpressedbyManfredEigen’sworkontheevolutionofbiologicalmacromoleculesandtheoriginsofbiologicalinformation,andbegancorrespondingwithhim.InMoscow,however,MikhailVladimirovichwasthenconsideredadissidentandnotpermit-tedtovisitWesterncountries.IrecallhowproudhewasofabirthdaypresentfromManfredEigen,atapeofapianorecitalperformedbyEigenhimself.Onlytowardstheendofthe1980swasMikhailVladimirovichabletoacceptinvitationsfromhisnumerousfriendstotravelwithhiswifetotheWest.Irememberinpar-ticularhismentioning,aproposofavisittoSpain,howhiswifehadfoughtthereahalf-centuryearlieronthebarricades.Inowturntothepresentbook,writtenbyMikhailVladimirovichinthe1980s.Thebookisabrilliantessaycoveringagreatmanytopics,rangingfromtheearlyhistoryofthermodynamicstothecomplexityandvalueofinformation.Iamcertainthatthereaderwillreceivegreatenjoymentfromthismasterpieceofpopularscience.WernerEbeling(EditoroftheGermantranslationofEntropyandInformation)Translators’noteOurspecialthanksgotothefollowing:EdwinF.Beschlerforarrangingpubli-cationandcheckingthetypescript,WernerEbelingforprovidingabiographicalforewordabouttheauthorandaphotographofhim,andalsoforcorrectingthespellingofsomenames,andMariaVolkenstein,theauthor’sdaughter,who,to-getherwithAleksandrY.MozdakowandVladimirSychevofthepublishinghouseNauka,helpedinobtainingthepublicationrights.WealsothankMarcHerbstrittofBirkh¨auserforhandlingthepublicationprocesssoefficientlyandamicably,andNinaIukhoveliforhelpwithRussian.Thepoeticalepigraphsatthebeginningofeachchapterwereoriginallyrhyming;forthesakeofaccuracy,noattemptwasmadetoproducerhymingEnglishtranslations(withtwoexceptions),althoughwedidtryforsomedegreeofrhythm.Thesecondtranslator(Burns)acknowledgeswithgratitudetheassistanceofstaffmembersofthedepartmentsofmathematics(andstatistics)atYorkUniversity,Toronto,andTheUniversityofQueensland,Brisbane.Thefootnoteshave,withrareexception,beenaddedbythesecondtranslator,whotakesfullresponsibilityforthem. PrefaceThisisjust...entropy,hesaid,thinkingthatthisexplainedeverything,andherepeatedthestrangewordafewtimes.KarelCapekˇ1,“Krakatit”This“strangeword”denotesoneofthemostbasicquantitiesofthephysicsofheatphenomena,thatis,ofthermodynamics.Althoughtheconceptofentropydidindeedoriginateinthermodynamics,itlaterbecameclearthatitwasamoreuniversalconcept,offundamentalsignificanceforchemistryandbiology,aswellasphysics.Althoughtheconceptofenergyisusuallyconsideredmoreimportantandeasiertograsp,itturnsout,asweshallsee,thattheideaofentropyisjustassubstantial—andmoreovernotallthatcomplicated.Wecancomputeormeasurethequantityofenergycontainedinthissheetofpaper,andthesameistrueofitsentropy.Furthermore,entropyhasremarkableproperties.Ourgalaxy,thesolarsystem,andthebiospherealltaketheirbeingfromentropy,asaresultofitstransferencetothesurroundingmedium.Thereisasurprisingconnectionbetweenentropyandinformation,thatis,thetotalintelligencecommunicatedbyamessage.Allofthisisexpoundedinthepresentbook,therebyconveyinginformationtothereaderanddecreasinghisentropy;butitisuptothereadertodecidehowvaluablethisinformationmightbe.Thesecondhalfofthe20thcenturyisnotableforthecreationanddevel-opmentofcomplexareasofscienceofthegreatestimportancenotonlyforthenaturalsciencesandtechnology,butalsoforthehumanities.Sucharecybernetics,informationtheory,andsynergetics.Althoughthesetermsdidnotexistfiftyyearsago2,theynowturnupconstantly.Inallthreeofthesedisciplinestheconceptsofentropyandinformationareabsolutelyindispensable,sothatwithoutthemitisnotpossibletograspthetrueessenceofmodernscience.Thefinalchaptersofthe1KarelCapek(1890–1938),Czechplaywrightandnovelist.Inventoroftheword“robot”inˇitspresentsense,inhisplayRUR.2Notethattheoriginalworkappearedin1986.Trans. 2Prefacebookcontainbrief,andofnecessityincomplete,expositionsofsynergeticsandin-formationtheory.Theaimofthepresentaccountistobringthesenewdisciplinestothereader’sattention,andintroducehimorhertothecircleofrelatedideas.IwishtothankM.I.Kaganov,Yu.L.Klimontovich,Ya.A.Smorodinski˘ı,andW.Ebelingfortheirusefulcommentsonthemanuscript.M.V.Volkenstein Chapter1“Reflectionsonthemotivepoweroffire...”Blackcoalburnsinthefirebox,Waterturnsintosteam,AndthepistonmovesstubbornlyBackandforth,backandforth.Thehotsteamcondenses,Turningtheheavywheels.Theengine-drivingfirePresentsseveralriddles.SadiCarnotAlittlebookofonly45pages,titled“Reflectionsonthemotivepoweroffire,andonmachinescapableofdevelopingthatpower”1,appearedinParisin1824.ItsauthorwasS.Carnot,accordingtothetitlepageaformerstudentatthePolytechnic.AtthattimeSadiCarnotwas28.HewasthesonofLazareCarnot,whohadbeengiventhehonorifictitleof“organizerofvictory”forhisservicesinthewarsofrevolutionaryFrance.L.Carnotwasaprominentmilitaryandpoliticalactivist,anengineer,andanexcellentmathematician;howevertheson’slittlememoir(asscientificpaperswerethencalled)wastooutweighinsignificancetheworksofthefather.ThiswasS.Carnot’sonlypublication.Hediedeightyearslater,in1832,attheageof36,acriticalageforgenius.Raphael,Mozart,andPushkindiedataboutthesameage.1SadiCarnot.R´eflexionssurlapuissancemotricedufeu.1824. 4Chapter1.“Reflectionsonthemotivepoweroffire...”Whatwasthestateofphysicsatthetime?Theoreticalmechanics,whosefoundationswerediscoveredbythegreatNewton,seemed—withgoodreason—tohavebeenperfectedbyLaplace,Lagrange,andotherscientists.Inopticstherewasacontinuingstrugglebetweenthecorpuscularandwavetheoriesoflight,thoughthefinalvictoryofthelatterwaswellunderway.Inelectricityandmagnetismimportantdiscoverieshadbeenmade(byAmp`ere,Oersted,andOhm),andthedecisivediscoveriesofFaradaywereimminent.Manypropertiesofgaseshadbeeninvestigatedindetail,and,thankstotheworkofDalton,atomictheoryheldswayinchemistry.Inshort,full-scalescience—inparticularphysics—wasadvancingrapidly.However,therewasoneimportantarea,thephysicsofheatphenomena,thatwasstillawaitingitsNewton.Thereweretworeasonsforthisdelay.Firstly,steamengines—heatmachines—hadalreadyinfiltratedtechnology,andsecondly,theessentialfeaturesofheatphenomenaturnedouttobefundamentallydifferentfromthoseofmechanics.Asisclearfromitstitle,Carnot’smemoirrelatesdirectlytoheatengines.TheactionofthesemachinespromptedCarnottothinkaboutheatanditstrans-formationintowork.Thisisarelativelyrarecaseinthehistoryofscience.Usuallysciencedevelopsindependentlyoftechnology,followingitsowninternallogic.Faraday,andafterhimMaxwell,werenotthinkingofthedynamowhentheycreatedthetheoryoftheelectromagneticfield.Maxwell,andthenHertz,werenotledtotheirdiscoveryofelectromagneticwaves—firsttheoreticallyandthenexperimentally—byexam-iningthefunctioningofradios.Soonerorlatersignificantphysicsfindsimportantpracticalapplications.CaloricItwasadifferentstoryforheat.Thesteamenginewasinventedindependentlyofanytheory.ThestoryhasitthatJamesWatt’snoticinghowthelidofaboilingkettlebobbedupanddownledhimtotheinventionofthesteamengine.Bethatasitmay,priortoCarnot’spaperthereexistednotheoryofheatwhatsoever,andtheimportantquestionofthemechanicalefficiencyofsteamenginesremainedopen.Atthattimeitwasdifficulttoconstructatheoryofheat;itsnaturewaslargelymysterious.Thereweretwohypotheses.Thefirstofthesewasbasedonthenotionofthe“caloric”,aweightless,invisiblefluidthatwhenaddedtoabodycauseditstemperaturetoriseandwascapableofchangingitsstate.ThuswastheEnglishphysicistJosephBlackledtowritedownthefollowingequations:ice+caloric=water,water+caloric=steam.Itwasthoughtthatcaloricwascontainedinagasmuchlikejuiceinanorange.Squeezeanorangeandyougetorangejuice.Compressagasandcaloric 5oozesout,thatis,itheatsup.Thecalorictheorywasstudied—withoutmuchsuccess—bythefamousrev-olutionaryJean-PaulMarat,whopublishedanarticleonitin1780.Thecalorichypothesisstoodinoppositiontothekinetictheory.Inthemid-18thcenturyM.V.LomonosovandD.Bernoulli,inthecourseofpropoundingvariousargumentsagainstthetheoryofthecaloric,suggestedakinetichypothesis.Lomonosovconjecturedthat“thecauseofheatconsistsintheinternalrotationalmotion”oftheparticlesofthebody,andthat“theparticlesofhotbodiesmustspinfasterandthoseofcoolerbodiesslower”.TheobservationsofBenjaminThompson(CountRumford)decisivelyrefutedthecalorichypothesis.In1798Rumfordmeasuredtheamountofheatproducedbydrillingmetal.Sincetheheatcapacityoftheshavingsremainedthesameasthatoftheoriginalmetal,whencecametheadditionalcaloric?Rumfordconcludedthatthesourceofheatmustindeedbekinetic.2Weremindthereaderthattheheatcapacityofabodyisdefinedtobetheratiooftheamountofheat3absorbedbythebodytotheresultingriseinitstemperature,orinotherwordstheamountofheatenergyneededtoraiseitstemperaturebyonedegree.NotethattheheatcapacityCVofabody,sayaquantityofgas,heldatconstantvolumeV,islessthanitsheatcapacityCpatconstantpressurep.For,ifthepressureisheldconstant,onlypartoftheheatΔQsuppliedtothegasisusedtoincreaseitsinternalenergy4ΔE,assomeisneededtodotheworkinvolvedinexpandingthegas,thatis,inincreasingitsvolumebyanamountΔV.ThuswehaveΔQ=CpΔT=ΔE+pΔV,(1.1)whereΔTistheincreaseintemperatureofthegas,andpΔVtheworkdoneinexpandingthegasagainstthefixedpressurep.5HenceΔEΔVCp=+p.ΔTΔTOntheotherhand,atconstantvolumeΔV=0,wehavebydefinitionΔECV=.ΔT2BenjaminThompson(CountRumford).“Heatisaformofmotion.Anexperimentinboringcannon.”PhilosophicalTransactions88,1798.3Heatisdefinedastheenergytransferredfromonebodyorsystemtoanotherduetoatemperaturedifference.Thethermalenergyofabodyisthetotalkineticenergyofitsatomsandmolecules.Trans.4Theinternalenergyofabodyorthermodynamicsystemisthetotalofthekineticenergyofitsmolecules(translational,rotational,andvibrational)andthepotentialenergyassociatedwiththevibrationalandelectricalenergyoftheatomsofitsmolecules.Trans.5Sincepressureisforceperunitsurfaceareaofthe(expansible)chambercontainingthegas,thisisindeedtheworkdoneinexpandingbyanamountΔV.Trans. 6Chapter1.“Reflectionsonthemotivepoweroffire...”HenceΔVCp=CV+p.ΔTForamoleofanidealgastheClapeyron-Mendeleevstateequation6holds:pV=RT,(1.2)whereR≈8.31joules/moleperdegree7istheideal-gasconstant,andTtheabsolutetemperature(seebelow).ItfollowsthatatconstantpressurepΔV=RΔT,whencetheaboveequationbecomesCp=CV+R.(1.3)InspiteofRumford’sexperiments,thecalorichypothesiscontinuedtodomi-nate.Therewereattemptstocombinebothpointsofview;forinstance,itwasheldthatthecaloricprovidedthemeansforthepropagationofmolecularvibrationsandrotationsinabody—anexceedinglyartificialconstruction.Weareoftenbaffledbywhatwelearnofthescienceofthepast.Forinstance,itisdifficulttocomprehendwhybeliefinthecaloricpersistedwhenphlogistonhadalreadybeenrejected.Chemistsofthe18thcenturypostulatedphlogistonasasubstanceconstitutingtheessenceofcombustibility;thatforinstanceametalisacompoundofashes(“slag”or“lime”)andphlogiston,andwhenitisburned,thephlogistonescapesandonlytheashesremain.TheworkofLomonosovandLavoisierdemonstratedthatinfactphlogistonwasafiction;combustionofametalisthesameasitsoxidation,asynthesisofthemetalandoxygen.Thusphlogistonwasasortofanti-oxygen,sotospeak.Thereweretworeasonsforthepersistenceofthecalorichypothesis.Firstly,therewereatthetimedifficultiesinexplainingtheheatofradiation.Howmightthisarisefrommolecularrotationsorvibrations?Secondly,thekinetictheoryasitstoodatthattimeofferedonlyqualitativeexplanations,andwasthereforeinthisrespectnotatallsuperiortothecalorictheory.Thekinetictheorywastotriumphatalaterstage.TheirreversibilityofheatprocessesWehavealreadymentionedthedissimilarityofheatandmechanicalphenomena.Thisdissimilarityinfactcomesdowntotheirreversibilityofheatprocessesas6InEnglishtextsusuallygivenasderivingfromacombinationofBoyle’slaw(1660)andCharles’law(1787).Trans.7Ajouleistheworkdonebyaforceofonenewton(theforceneededtoaccelerateamassofonekilogrambyonemetrepersecondpersecond)inmovinganobjectthroughonemetreinthedirectionoftheforce.Amoleofasubstancecontainsthesamenumberofparticlesas12gramsofcarbon-12atoms,namely6.022×1023,knownas“Avogadro’snumber”.Trans. 7opposedtothoseofordinarymechanics.Weshallnowconsiderthisthemebriefly,leavingthedetailedtreatmentforlater.AllphenomenadescribableintermsofNewtonianmechanics,thatis,purelymechanicalones,arereversible.Whatdoesthismeanexactly?Itmeansthatthelawsofmechanicsdonotchangeifwechangethesignofthetime,thatis,ifwereplacetby−teverywhereintheformulaeofmechanics.Thiscanbeseenasfollows:Newton’ssecondlawstatesthattheforceonabodyisequaltoitsmasstimestheacceleration.Andwhatisacceleration?Theaverageaccelerationofabodyoveragiventimeintervalfromafixedtimettot+Δtisthechangeinvelocity8perunittime,overthattimeinterval:Δv.(1.4)ΔtSincevelocityisthechangeindisplacementperunittime,theaveragevelocityoveratimeintervalfromt1tot1+Δt1isΔsv=.Δt1IfweconsiderthisaverageforsmallerandsmallerΔt1,theninthelimitasΔt1goestozero,weobtain—accordingtothedifferentialcalculus—thevelocityofthebodyattheinstantt1,denotedbydsv=(1.5)dtt1toremindusofitsoriginsasthelimitingvalueoftheratioΔs.HencetheaverageΔt1accelerationovertheintervalfromttot+ΔtisΔ(ds/dt),ΔtwhereΔ(ds/dt)=Δvisthechangeinvelocitybetweentimestandt+Δt.InthelimitasΔtgoestozero,thisyieldstheaccelerationattheinstantt:d(ds/dt)d2sa==,(1.6)dtdt2inthenotationofthedifferentialcalculus.Observethatintheexpressionfortheacceleration—andhencealsotheforce—thetimechangeenterstothesecondpower.Thisisclearalsofromthedimensionsofacceleration,namelylengthperunittimesquared.Thusitisthatchangingthesignofthetimedoesnotchangetheoverallbehaviorofanidealmechanicalsystem.Mechanicalprocessesaretime-reversible.8Hereandbelowdisplacementandhencevelocityandaccelerationarevectorquantities.Or,forsimplicity,itmaybeassumedthatthemotiontakesplacealongastraightlineinthedirectionoftheforce.Trans. 8Chapter1.“Reflectionsonthemotivepoweroffire...”Butwhatdowemeanbythe“signofthetime”?Well,timeflowsfromthepast,throughthepresent,andintothefuture.Thereappearstobenogoingbackwardsintime.Atimemachinethattravelsinthenegativetimedirectionisimpossible.9Wenormallyordereventsintimesothatcausesprecedeeffects.Thisorderingwouldseemtobesecondarytotheunderlyingdirectionalityoftime.Butsuchmattersarepuzzling.Weshallnotattempttoprovideanswerstosuchcomplexquestions,lyingastheydoontheboundarybetweenphysicsandphilosophy.Herewedesireonlytostressthefundamentalnatureofthedifferencebe-tweenthereversiblephenomenaofclassicalmechanicsandirreversiblethermalphenomena.Agoodwayofdemonstratingthisdifferenceistoplayamovieback-wards.Theprocessesthendepictedinthefilmthatseemcompatiblewiththelawsofphysics,thatis,seemmoreorlesspossible,arejustthereversibleones,whilethosethatcontraveneoursenseofthepossiblearetheirreversibleones.Scenesofmechanicalprocessesinvolvingthemeredisplacementofanobjectretaintheirnaturalitywhenthefilmisrunbackwards.Thusifamanisseentojumpoffachair,theninthereversedirectionheisseentojumpupontothechairwithhisbacktoit.Althoughthesecondactionisactuallymoredifficult,itisnonethelessfeasible,anddoesnot,therefore,contraveneoursenseofwhatispossible.Ontheotherhand,processessuchastheburningofacigarette,thestirringofcreamintocoffee,andthehatchingofanegg,strikeusashighlyimprobablewhenviewedinreverse.Realityisirreversible.Henceamechanicsthatdoesnotincludetheconceptofirreversibilitycannotsufficeforunderstandingtheworld.Inthesequelweshalllinkthefactthattimehasaspecificdirectiontothisobservedirreversibilityofcertainprocesses.Apersonisborn,grows,reachesmaturity,growsold,anddies.Lifeisunde-niablyirreversible.Youmayobjectthatthesearebiologicalphenomenaunrelatedtophysics.Howeverthefactofthematteristhattheyarerelatedtophysics;wewilltakeupthisissuebelow.Forthemomentlet’sconsideraswingingpendulum.Soonerorlaterairresistanceandfrictionatthepointofsuspensionwillbringittoahalt,aprocessproducingheat,andthereforeirreversiblesincetheheaten-ergyproducedisnotreconvertedintotheoscillatoryenergyofthependulum.Toexpressthismathematically,onemustintroducetheforceduetofrictionintotheequationofmotionofthependulum.Experienceshowsthatingeneraltheforceoffrictionisproportionaltoacomponentofthevelocityofthebodyonwhichitacts,sothatthevelocityv=ds/dtentersintotheequationofmotiontothefirstpower,andtheinvarianceundersubstitutionof−tfortdisappears.Iftwobodiesofdifferenttemperaturesareincontact,thenheat(caloric?)flowsfromthewarmertothecoolerbodyuntiltheirtemperaturesbecomeequal.Thusthepastisrepresentedbyatemperaturedifferenceandthefuturebyequalityoftemperature.Theprocessofheatflowfromonebodytoanotherisirreversible.9Or,atleastseemstobeso?(SeeChapter4.)Trans. 9Inparticularitisnotpossibleforabodyatuniformtemperaturetospontaneouslydivideitselfintoahotpartandacoolpart.10Priortothediscoveryofheatenginestherewereothermachinesinuse,theearliestbeingtheleverandpulley.Thesemachinestransformedoneformofmotionintoanother,bytransferringenergyfromonesourcetoanother.Itseemedthatnothingwaslostinsuchprocesses11,whileontheotherhandcoalorwoodinthefireboxofasteamengineburnupirreversibly.Infactacorrectioniscalledforhere.Inourtalkof“mechanicalphenom-ena”andof“mechanicalengines”,weneglectedfriction,airresistance,andsoon.Howeverfrictionoccursineverymechanicaldevice,causingheattobeproduced,andirreversiblytransformingmechanicalenergyintoheat.Thedifferencebetweenideallymechanicalphenomenaandheatphenomenaturnedouttoberelatedtothedeepestquestionsofscience.Theelucidationofthenatureandmeaningofirreversibleprocessesbecameoneofthecentralproblemsof19th-centuryphysics—andnotonlyphysics.Sinceforanideal(frictionless)mechanicalsystemthesignofthetimeisim-material,suchasystemdoesnotoperateirreversibly.Howeverasysteminvolvingheatprocessesiscapableofirreversibledevelopment,thatis,ofevolving.Althoughaclearappreciationofthiscapacityemergedonly30yearsaftertheappearanceofCarnot’smemoir,thatworkhadpreparedtheground.Itisnoteworthythatthesesamedecadeswitnessedthediscoveryofthetheoryofevolutioninbiology,whichfounditsfullestexpressioninCharlesDarwin’sTheoriginofspecies,publishedin1859.Herethesubjectwasevolutioninnature,thatis,theirreversibledevelopmentoflivingthings.Theconnectionbetweenbiologyandphysics,whichseemedinitiallyoflittleconsequence,subsequentlyplayedanenormouspartinthedevelopmentofscience,and,surprisingly,inthisconnectionbiologyhadmoretoimparttophysicsthanviceversa.Weshalldiscussthisinthesequel.WhatdidCarnotdo?Carnotlaidthefoundationsof“thermodynamics”,nowoneofthemainareasofphysics.Thermodynamicsisconcernedwiththegenerallawsthatdeterminethemu-tualconnectionsbetweenthephysicalquantitiescharacterizingallthoseprocessesoccurringinnatureandtechnology,bymeansofwhichenergyistransferredfromonebodytoanotherandtransformedfromoneformtoanother.Thermalpro-cessesareincludedamongthese.Thermodynamicsisa“phenomenological”sci-ence,meaningthatitsconcernsareuniversalandnotdependentonthespecific10Orjusthighlyimprobable?(SeeChapter4.)Trans.11Forexample,thattheenergyusedtomoveoneendofaleverisfullyavailabletomovetheother.Trans. 10Chapter1.“Reflectionsonthemotivepoweroffire...”substancesinvolvedintherelevantprocessesofenergyexchange.Thusinthisre-spectthermodynamicscontrastswithmolecularandatomicphysics,whosetaskistoinvestigatetheconcretepropertiesandstructureofspecificmaterialbodies.Carnotusedthenotionofthecaloric.Inaccordancewiththeideasofthetime,heregardedthecaloricasweightlessandindestructible.Averystrangesubstanceindeed!Hewrote:“Themotivepowerinsteamenginesarisesnotfromanactuallossofcaloric,butfromitstransferencefromahotbodytoacoolone....Inorderformotivepowertoemerge,itisnotenoughtogenerateheat:onemustalsoprocurecoldness;withoutitheatwouldbeineffectual...”.Ofcourse,Carnot’sideaoftheindestructibilityofheat—whateveritsna-turemightbe—iswrong.Howevertherestofwhatwehavequotediscorrect:foramotiveforcetoemerge,capableofdoingwork,atemperaturedifferenceisindispensable.Carnotgoesontoconsiderareversible,cyclical,process:“IfwewishtoproducemotivepowerbymeansofthetransferenceofadefiniteamountofheatfromabodyAtoabodyB,wemayproceedasfollows:1.TakecaloricfrombodyAtogeneratesteam...weassumethatsteamisformedatthetemperatureofbodyA.2.Funnelthesteamintoanexpansiblevessel,forexampleacylinderwithapiston;thevolumeofthevesselwillthenincreasealongwiththevolumeofsteam.Thetemperatureofthesteamwilldecreaseasitexpands....WeassumethatthisrarefactioncontinuesuntilthetemperatureofthesteamreachesthatofthebodyB.3.CondensethesteambygettingitintocontactwithbodyBandsimulta-neouslyapplyingconstantpressuretillsuchtimeasitrevertstotheliquidstate....”ThusthebodyAistheheaterandbodyBthecooler.Carnotclaimsthatthesethreestepscanalsobecarriedoutinthereverseorder:“WecangeneratesteamusingthecaloricofbodyBatitstemperature,bycompressingthatbodyuntilitheatsuptotheoriginaltemperatureofbodyA,andthencondensingthesteambycontactwiththelatterbody.”12Howeverherethecycleisreversibleonlyinthesensethatthesystemcanbereturnedtoitsoriginalstateinthesameway.InfactthetransferofheatfromtheheaterAtothecoolerBisreallyirreversiblesinceexternalworkhastobedonetoreturnthesystemtoitsoriginalstate.Byappealingtotheimpossibilityofaperpetual-motionmachine(perpetuummobile),bythenalreadyestablishedasalawofnature,Carnotprovesthattheabove-describedprocessyieldsmaximalmotivepower,thatis,workperunittime.Hisreasoningisremarkable:“...Ifthereweremoreefficientmeansforutilizingheatthanthemethodwehaveused,thatis,ifitwerepossibletoobtainalargeramountofpower...then12Presumablyfollowingtheoriginalprocess,afterwhichbodyAhascooled.Trans. 11onecoulduseaportionofthispowertoreturnthecaloricbytheindicatedmethodfromthebodyBbacktothebodyA,...andtheoriginalstatewouldberestored;onecouldthenrepeattheoperationandgooninlikemannerindefinitely:thiswouldnotonlyconstituteperpetualmotion,butalsotheunlimitedgenerationofpowerwithouttheconsumptionofcaloricoranyotheragent.”Finally,hestateshismostimportantconclusion:“Thepowerobtainedfromheatisindependentoftheagentsusedforgeneratingit;ultimately,theamountofpowergeneratedisdeterminedexclusivelybythetemperaturesofthebodiesparticipatinginthetransferofthecaloric.”Inotherwords,the“coefficientofeffectiveaction”,or“efficiency”ofaheatengineisdeterminedbythetemperaturesofheaterandcooleralone,regardlessoftheparticulargasexpandinginsideit.Thegreaterthedifferencebetweenthesetemperatures,thegreatertheoutputofwork,or“motivepower”.HoweverCarnotwasunabletoobtainaquantitativeexpressionforthiscoefficient,beingpreventedfromdoingsobyhisassumptionthatcaloricisconserved.Carnot’sworkisremarkableforthebreadthanduniversalityoftheanalysis,thesystematicworkingoutofhisthoughtexperimentconcerningthecyclicityoftheprocess,andhisgeneralconclusions,validindependentlyofthenatureoftheparticularmaterialsinvolved.Carnot’spaperwentessentiallyunremarkedforanumberofyears;hiscon-temporariesfailedtounderstandorappreciateit.Thishasoccurredseveraltimesinthehistoryofsciencewheningeniousworksappearedaheadoftheirtime.Such,forinstance,wasthecasewithGregorMendel’sdiscoveryofthelawsofgenetics;hisworkremainedunappreciatedfor40years.Howevertoday,asaresultoftheeverincreasinginternationalizationofscienceandthewidespreadpropagationofinformationaboutscientists’work,suchsituationshavebecomeveryrare.TheCarnotcycleIn1834theFrenchphysicistandengineerBenoit-PierreClapeyronbecameinter-estedinCarnot’spaperandwasabletogiveitmathematicalform—theveryforminwhichthe“Carnotcycle”isexpoundedinmodernphysics.Suchareformulationwascrucial,sincenobetterwayhasbeenfoundofexplainingthefunctioningofaheatengineorintroducingthephysicalconceptofentropyintoscience.ThuswastheindestructiblecaloriceliminatedfromtheCarnotcycle.Butwhatreplacedcaloric?Theansweris:energy,whichwillbediscussedinthenextchapter.Lookingaheadalittle,weshallseethateverybody,inparticularaquan-tityofsteamorothergas,ischaracterizedundergivenconditionsaspossessingadefiniteamountofinternalenergy.Andwhatmightthephrase“undergivenconditions”meanforaquantityofgas?Answer:atprescribedtemperatureandvolume.Thusabody’sinternalenergyisafunctionofitsstate,whichcanchangeasaresultofitsinteractionwithotherbodies,morespecifically,throughbeing 12Chapter1.“Reflectionsonthemotivepoweroffire...”pbM0M1aVFigure1.1:Statesofagasandtransitionpathsbetweenthem.heatedorcooled,orworkbeingdone.ThusthechangeE1−E0intheinternalenergy13ofagivenquantityofgasisgivenbyΔE=E1−E0=Q−W,(1.7)whereQisthequantityofheattransferredtothegasfromtheheater,andWistheworkdonebythegasinexpanding.Inequation(1.7)(seealso(1.1))weseethattheheatenergyQandworkWhaveequalstatus.14Nowheatenergyismeasuredincaloriesorjoules15,whileworkismeasuredinkilogram-metres.Henceitisclearthatequation(1.7),whichexpressesthelawofconservationofenergywhentransformedfromoneformtoanother,makessenseonlyifthereisamechanicalequivalentofheatenergy,or,conversely,aheatequivalentofwork,thatis,aconversionfactorforconvertingunitsofheatintounitsofmechanicalenergy,andconversely.Whatwastheflawinthetheoryofthecaloric?Answer:Itwasamistaketoassumethatitispossibletodeterminetheamountofcaloric,thatis,“essenceofheat”,containedinabody.Weshallnowdeducefromequation(1.7)thatthestatement“theamountofheatcontainedinagivenquantityofgasisQ”ismeaningless.InFigure1.1thestatesofafixedquantityofgasarerepresentedbypointscoordinatizedbythepressurepandvolumeV.TheinternalenergyofthegasinthestateMisgreaterthanitsinternalenergyinthestateM.16Supposethatin01thesetwostatesthegascontainsdifferentamountsofheat.LetQ0betheamountofheatcontainedinthegasinstateM0,andQ1=Q0+Q01,(1.8)theamountwhenthegasisinstateM1;thusQ01isthechangeintheamountofheatinvolvedinthetransitionfromstateM0toM1,assumedindependentofthe13Seeanearlierfootnoteforthedefinition.14Thatis,measurethesamesortofentity.Trans.15AcalorieistheamountofheatenergyrequiredtoraisethetemperatureofonegramofwateratatmosphericpressurebyonedegreeCelsius.Onecalorie≈4.1868joules.Seeanearlierfootnoteforthedefinitionofajoule.Trans.16Thiscanbeseen,forexample,fromthedescriptionoftheCarnotcyclebelow.Trans. 13mannerinwhichthetransitioniseffected.Accordingto(1.7),Q01=E1−E0+W01,whereW01istheworkdonebythegasinthecourseofthetransitionfromstateM0toM1.ForaprescribeddifferenceE1−E0ininternalenergies,wecaneffectthetransitionatrelativelyhighpressures,alongapathsuchasb,inorderthatW01>E0−E1,whenceQ01>0,whichonemightinterpretasmeaningthat“thegascontainsmoreheatQ1instateM1thaninstateM0”.Howeverifweeffectthetransitionintheoppositedirection,viathepathaalongwhichthepressureislowenoughforthereverseinequalityW011.(Thesignificanceofγwillbeexplainedbelow.)Adiabaticexpansionisaccompaniedbyaloweringofthetemperature.Weallowthegastocontinueexpandingtothepoint3lyingontheisothermalcurvewherethevolumeisV3andthegasisatthecoolertemperatureT2(Figure1.3). 15ppT214TT2233V2V3VV4V3VFigure1.3:AdiabaticexpansionoftheFigure1.4:Isothermalcompressionofgas.thegas.c)AtthethirdstagewecompressthegasisothermallyatthetemperatureT2bybringingitintocontactwiththecooler.Thisrequiresanamountofworkequaltothecross-hatchedareainFigure1.4.WecontinuecompressingthegastillitreachesavolumeV4suchthatthecorrespondingpoint4ontheisothermalcurveinFigure1.4isalsoontheadiabaticcurvethroughtheinitialpoint1(Figures1.2,1.5).d)Finally,weagaininsulatethegasadiabaticallyandcompressitfurthertoitsoriginalvolumeV1alongtheadiabaticcurvethroughthepoint1.Thisrequiresworktobedone(seeFigure1.5).Thecycleisnowcomplete.Positiveworkhasbeendoneinanamountequaltothecross-hatchedareaenclosedbythecycle(Figure1.6),thatis,thedifferencebetweentheworkdonebytheexpandinggasinstagesa)andb)andtheworkrequiredtocompressitinstagesc)andd).p1T1p1T1T2244T23V1V4VVVVVV1423Figure1.5:AdiabaticcompressionofFigure1.6:TheCarnotcycle.thegas.Inthecourseofthisprocess,thegasabsorbedfromtheheateranamountQ1ofheatatthetemperatureT1,andgaveoutasmalleramountQ2ofheatatthetemperatureT2.HencetheportionoftheheattransformedintoworkisW=Q1−Q2,(1.10) 16Chapter1.“Reflectionsonthemotivepoweroffire...”whenceQ1=W+Q2>W.(1.10a)Theefficiencyηofthecycleisthentheratiooftheworkmadeavailabletotheheatinput:WQ1−Q2η==.(1.11)Q1Q1HowdoestheefficiencydependonthetemperaturesT1andT2oftheheaterandcooler?WeknowthatCarnotconsideredthisdependencedecisive.ThermodynamictemperatureButwhatsortoftemperaturearewetalkingabouthere?Thetemperatureinequation(1.2)isthetemperatureontheKelvinscale,measuredindegreesKelvin(◦K),whichareequaltodegreesCelsius,butstartfrom−273.15◦C;thatis0◦K≡−273.15◦C,so-called“absolutezero”.Howdoesthis“thermodynamic”temperaturearise?EveryidealgassatisfiesthestateequationpV=Φ(θ),(1.12)whereΦisauniversalfunctionofthetemperatureθmeasuredindegreesCel-sius.Experimentshowsthatthisfunctionincreaseslinearlywiththetemperature,whenceitfollowsthatthereareconstantsα,βsuchthat18θ=αpV+β.(1.13)TheCelsiusscaleisdefinedbytakingthetemperatureatwhichicemeltsatatmo-sphericpressuretobe0◦Candthatatwhichwaterboilstobe100◦C.Substitutingthesetwotemperaturesintheequation(1.13)foranidealgas,weobtain0=α(pV)0+β,100=α(pV)100+β.(1.14)Solvingforαandβfromthesetwoequations,weobtain100100(pV)0α=,β=−.(1.15)(pV)100−(pV)0(pV)100−(pV)0Substitutinginequation(1.13)from(1.15),weobtainpV−(pV)0◦θ=100C.(1.16)(pV)100−(pV)018SincepVisalinear(oraffine)functionofθ,θmustlikewisebesuchafunctionofpV.Trans. 17Experimentshowsthatforasufficientlyrarefied—thatis,ideal—gas(pV)100=1.366.(1.17)(pV)0Equations(1.16)and(1.17)nowgivepV=(pV)0(1+0.00366θ).(1.18)θ°C300200(pV)100100(pV)00pV–100–200–273Figure1.7:Dependenceofthe“idealgastemperature”θonpV.Figure1.7showsthegraphoftheidealgastemperatureθ(pV)asthefunctionofpVgivenby(1.18);pVtakesonthevaluezeroatθ=θ0,givenby◦θ0=−1/0.00366=−273.15(C).(1.19)Thusifwetake−273.15◦CasthezerotemperatureoftheKelvinscale,thenweobtain◦◦TK=θ−θ0=(θ+273.15)K(1.20)forthetemperaturemeasuredindegreesKelvin.AtT=0◦K,pVvanishes.Sub-stitutionfrom(1.20)in(1.18)yields(pV)0◦CpV=T.(1.21)273.15At0◦Candapressureof1atm(atmosphere),thepressuremultipliedbythevolumetakenupbyonemoleofanidealgasisknowntobe(pV)0◦C=22.414·atm/mole.(1.22) 18Chapter1.“Reflectionsonthemotivepoweroffire...”(Thismeansthatat0◦Cand1atm,aquantityofgasofmassxgramswherethemolecularweightofthegasisx,occupiesavolumeof22.414liters.)Dividingthisby273.15yields(pV)0◦C22.414·atm/moleR==273.15273.15◦K7◦◦=8.314·10ergs/(mole·K)=8.314joules/(mole·K).WenowexpresstheefficiencyoftheCarnotcycleintermsofT1andT2,thetemperaturesofheaterandcooler.TheamountofheatsuppliedbyaheaterattemperatureTtotheworkingsubstanceofsuchacycleisafunctionofthattemperature:Q=ϕ(T).(1.23)ThisisapositivefunctionofTwhichinfactturnsouttoincreaseproportionallytothetemperature:T=ξQ.(1.24)Itisimmediatefromthisandthesecondexpressionfortheefficiencyin(1.11),thatthelatterisgivenbyT1−T2η=.(1.25)T1Thus,inaccordancewithCarnot’sdeepideas,aheatenginewithanidealgasasworkingsubstanceischaracterizedashavingthemaximalcoefficientofusefulaction(efficiency)givenbytheformula(1.25).Observethattheefficiencyis1onlyforT=0◦K=−273.15◦C.2Finally,wederiveaformulafortheconstantξinequation(1.24),usingonceagaintheconventionthatthedifferenceintemperaturebetweenmeltingiceandboilingwaterbetakenas100degrees.Thus100100=ξ(Q100−Q0),whenceξ=>0.(1.26)Q100−Q0Henceequation(1.24)becomes100Q100−Q0T=QorQ=T.(1.27)Q100−Q0100 Chapter2Thelawsofthermodynamics“EverythingispossibleinthisworldExceptforprohibitions.”Sosingdullpoets,Strummingtheirlyres.Butnature’ssternlawsAremoreinspiringthansuchsongs.Theworld,trulywonderful,Obeysthoselawsunswervingly.LomonosovandtheconservationlawsThelawofconservation(undertransformation)ofenergyalsogoesunderthesomewhatold-fashionedandpompousname“thefirstlawofthermodynamics”.Thesecondandthirdlawswillsoonberevealed.Fornowweconsiderjustthefirst.Whodiscoveredthelaw?Doesitreallymatter?Whyispriorityindiscoveryofanyimportance?Weliveinarealworld,notautopia.Intherealworldthereexistsuchcon-ceptsasnationalconsciousness.Itisnaturalandreasonableforacountry,nation,orpeopletotakeprideintheachievementsofitscreativemembers—itswriters,artists,andscholars.1Henceasociety’sinterestinwhodidwhatfirstisnormalandappropriate—providedsuchpriorityisestablishedwiththestrictestaccuracy,andwithoutanyhintofthechauvinismthatseekstobelittletheachievementsofothernationsandpeoples.InordertoexaltRussianscienceithassometimesbeenclaimedthatM.V.Lomonosov2wasthediscovererofthelawofconservationofenergy.Thisis1Andtofeelshameatthemisdeedsofitsvillains?Trans.2MikhailVasilyevichLomonosov(1711–1765),Russianpolymath,scientist,andwriter. 20Chapter2.Thelawsofthermodynamicsfalse.Lomonosovdiscoveredthelawofconservationofmass3,foundedthekinetictheoryofheat,andmademanyotherimportantcontributionstoscienceandthehumanities.Pushkinwrotethat“Hewashimselfourfirstuniversity”.Butthefactremainsthathedidnotdiscoverthelawofconservationofenergy.ThegreatRussianscientist,poet,andartistdoesnotneedimaginarydiscoveriesfoistedonhim!Whatisthisrumour—whichhasappearedinprintmorethanonce—basedon?Answer:OnasinglesentenceinaletterLomonosovwrotetoLeonhardEuleronJuly5,1748.Havingdescribedhisdiscoveryofthelawofconservationofmassormatter,Lomonosovgoesonasfollows:“Thustheamountofmattergainedbyabodyislostbyanother...Sincethisisauniversallawofnature,itextendsalsototheprinciplesofmotion:abodythatimpingesonanotherlosesasmuchofitsownmotionasitimpartstotheoneitsetsinmotion”.Somuchfortheconservationofmotion.Butwhatdoeshemeanbymotionhere?Thekineticenergy(orvisvivaasitwascalled)mv2/2(m=mass,v=velocity)orthemomentummv(alsocalled“quantityofmotion”)?Aprecisenotionofenergydidnotexistinthe18thcentury,andindeedcouldnothavebeenformulateduntilthe19th,thecenturyofsteamandelectricity.Lomonosovwroteabouttheconservationofmotionasifitwereself-evident,orwellknown.Thisisnotsurprising,sinceahundredyearsearlier,theFrenchphilosopher,mathematician,andphysicistRen´eDescarteshadwritten:“Iclaimthatthereisaknownamountofmotioninallcreatedmatterwhichneverincreasesordecreases”.InhisPrinciplesofphilosophyof1644heformulatedthe“lawsofnature”,thethirdofwhichassertsthat“ifamovingbodyencountersanothermorepowerfulbody,itlosesnoneofitsmotion;ifitencountersaweakeronethatitcancausetomove,thenitlosestheamountitimpartstothelatter”.LomonosovknewtheworksofDescartesintimatelyandtheyimpressedhimmorethantherigorousassertionsandformulaeofNewton’sPrincipiamathemat-ica.Beallthisasitmay,whatistrueisthatLomonosovdiscoveredtheimportantlawofconservationofmass.Theideaofconservationlaws—theimpossibilityofgettingsomethingfromnothing—cametotheforeinthephysicsofthe18thcentury.(Incidentally,theideaofthe“indestructibility”ofthefictitioussubstancecalledthecaloricisrelatedtothisdevelopment.)In1775theFrenchAcademyofSciencesannounceditsrefusalhenceforthtoconsideranyandallprojectshavingtodowithperpetualmotionmachines.Suchlawsrepresentextremelyimportantgeneralprinciplespertainingtoallofphysics.Theyshowthattheuniverseisconstructedinadefiniteway,thatnatureisgovernedbycertainobjectivelaws.Thetaskofscienceistodiscovertheselaws,andnotatalltoseektorefutethem.4Sometimesoneseeslawsofnature3ThisdiscoveryisoftenattributedtotheFrenchchemistAntoineLavoisier(1743–1794).However,itisclearthatLomonosovhaspriority.Trans.4Somephilosophersofscienceclaimthatthescientificprocessconsistspreciselyinattemptstorefutesuchlaws,sothatourconfidenceinthecorrectnessofthelawsiscontinuallysustained 21interpretedasinterdictionsfromonhigh:Natureforbidsperpetualmotion,andthat’sthat!Othersunfamiliarwithscienceadoptadifferentstance.Whentoldthattheirassertionscontradictestablishedscientificlaws,theysay:“Oyes,yousay‘suchathingcanneverbe!’butthenwhatseemedimpossibleyesterdayisoftenrealizedtoday!”.Furtherdebateisfutile.Fromtimetotime,theBiophysicsInstituteoftheAcademyofSciencesoftheUSSRusedtoproducesplendidpopularfilmsonscientificsubjects.Oneofthebestofthese—ifnotthebest—showedthefirst-rateachievementsofmembersofthatinstitute.Itborethestrikingtitle“Neversay‘never’,”implyingthateverythingispossible,therearenoimpenetrablebarriers,nofortressesthatcannotbetakenbyscientists,andsoon.However,thisisnotso.Thelawsofthermodynamicsareforever,justastwicetwowillalwaysbefour.5ThelawofconservationofenergyHowwasthislawdiscovered?Wesaidearlierthat19thcenturybiologydidmoreforphysicsthanphysicsforbiology.Thishaspreciselytodowiththefirstlawofthermodynamics.In1840,ayoungGermandoctorbythenameofRobertMayerfoundhimselfinthetropics,ontheislandofJava.Henoticedthatthevenousbloodofpeoplelivingtherewascloseincolortothatofarterialblood—redratherthanbrown.Mayerknewthatthedifferenceincolourofvenousandarterialbloodisconnectedwiththeabsorptionofoxygen—oxygenatedarterialbloodisnormallyabrighterredthandeoxygenatedvenousblood.Bodyheatresultsfromoxidation,aprocessakintoburning.Mayerwrote:“Themaintenanceofthehumanbodyatasteadytemperaturerequiresthatitsproductionofheatbeinsomequantitativerelationtoitsheatloss,andhencealsotothetemperatureofthesurroundings;thereforetheproductionofheatandsothecolordifferenceinthetwokindsofbloodmust,onthewhole,belessintenseintropicallatitudesthanincoolercountries”.Thatishowitallbegan.Mayerarrivedatthefollowinggeneralconclusion:“Inallphysicalandchemicalprocessestheforcepresentremainsconstant”.WhatMayercalledforcewenowcallenergy.Hewrotefurther:“Thelocomotivepullingitstrainmaybecomparedtoadistillingapparatus:theheatfurnishedtotheboileristransformedintomotion,whichinturnleavesaresidueintheformofheatinthewheelaxles”.ButMayerwentbeyondgeneralities.Denotingtheheatcapacity6ofagivenspecifiedquantityofairatconstantpressurepbyCandpbythefactthattheysurvivesuchattempts!Trans.5SinceEinsteinweknowthatenergyisnotconserved,butcanbeconvertedtomassandviceversa.Moreoverphysicistshaveconsideredthepossibilitythatthelawsofnaturemightmutatewithtimeorbedifferentinthefarthestreachesoftheuniverse.That2×2=4wouldseemtobeatruthofadifferentsort—unlesseverybodywhohasdonethismultiplicationhasineverycaseerred,whicheventhasanon-zero,thoughsmall,probability.Trans.6DefinedinChapter1. 22Chapter2.ThelawsofthermodynamicsitsheatcapacityatconstantvolumeVbyCV,Mayerformulatedaquantitativedefinitionofthemechanicalequivalentofheat—theconversionfactorofheattowork.TothisendhesetthedifferenceoftheheatcapacitiesCp−CVequaltotheworkdonebytheairinexpandingatpressurep(1841).Hisfirstmeasurements(laterimproved)gaveakilocalorieasequivalentto365kilogram-meters.Thuswasthelawofconservationofenergydiscovered.In1843,unawareofMayer’swork,JamesJouledeterminedthemechanicalequivalentofheatbyadirectexperimentsubsequentlydescribedineverytextbook.Jouleheatedwaterinacalorimeter7bymeansoffriction—usingalittlepaddle-wheel—anddeterminedtheratiooftheworkdonetotheheatgenerated.Hefoundthatakilocaloriewasequivalentto460kilogram-meters.Theexactmodernvalueofthemechanicalequivalentofheatis427kilogram-meters,or4.18605·1010ergs=4186joules.In1847theGermanscientistHermannHelmholtzformulatedthelawofconservationofenergy(whentransformedfromoneformtoanother)ingeneralandrigorousmathematicalform.Inparticular,heprovedthatenergyisgivenbyanintegralofmotionoftheequationsofmechanics.ItisremarkablethatHelmholtz,likeMayertrainedtobeadoctor,arrivedatthelawviabiologicalphenomena.Hewrote:“AccordingtoStahl,theforcesoperatinginalivingorganismarephysicalandchemicalforcesarisingintheorgansandsubstances[ofwhichtheyaremade],butsomelifeforceorsoulinherentintheorganismcanarrestorreleasetheirfunctioning...IconcludedthatStahl’stheoryattributestoeverylivingcreaturethepropertiesofaso-calledperpetuummobile....Thissuggestedthefollowingquestiontome:Whatrelationsmustexistbetweenthedifferentforcesofnatureifoneassumesthattheperpetuummobileisimpossible...?”Thusbyabandoningvitalisminbiology,Helmholtzwasledtooneofthemostprofounddiscoveriesinphysics.Itwouldseem,therefore,thatthefirstlawofthermodynamics,orthelawofconservationofenergy,wasdiscoveredbyMayer,Joule,andHelmholtzintheperiod1841–1847.OnlymuchlaterdiditbecomeclearthatpriorityindiscoveryofthefirstlawbelongstoCarnot.Itwasonlyin1878inconnectionwithaneweditionofhisR´eflections,thathisnotes,hithertounpublished,appearedinprint.Therehehadwritten:“Heatisbutmotiveforce,or,morecorrectly,motionthathaschangeditsform;itisthemotionoftheparticlesofbodies;wherevermotiveforceisannihilatedtherecomesintobeingsimultaneouslyanamountofheatexactlyproportionaltotheamountofmotiveforcemissing.Conversely,wheneverheatdisappears,motiveforcearises.“Wemaythereforestatethefollowinggeneralprinciple:Theamountofmo-tiveforceinnatureisunchanging.Properlyspeaking,itisnevercreatedandnever7Acalorimeterisadeviceusedformeasuringtheheatcapacityofasubstance,aswellastheheatproducedinchemicalreactionsandphysicalchanges.Asimplecalorimeterconsistsofjustathermometerattachedtoaninsulatedcontainer.Fromthetemperaturechange,thechangeinheatiscalculatedasmass×specificheat×temperaturechange,wherethespecificheatofthesubstanceisitsheatcapacityperunitmass.Trans. 23destroyed;inrealityit[merely]changesform,thatis,assumesoneoranotherformofmotion,butnevervanishes.“FromcertainoftheideasIhaveformedconcerningthetheoryofheat,itfollowsthattheproductionofaunitofmotiveforcerequirestheuseof2.70unitsofheat.”ThefigureforthemechanicalequivalentofheatfoundbyCarnot(bymeansunknown)isequivalentto370kilogram-meters,whichisveryclosetoMayer’sestimate.ThusCarnothadmuchearlierabandonedthecaloricandgivenapreciseaccountofthefirstlaw.Unfortunatelythisremarkableworklongremainedun-publishedandunknown.InourdescriptionofacycleconcludingChapter1,wetacitlyassumedonemathematicalversionofthefirstlawinusingtheformula(1.7):ΔE=Q−W.TherewedescribedtheCarnotcycleinthemodernformduetoClausius,8startingwiththeformula(1.7)andendingwiththeexpression(1.25)fortheefficiencyofthecycle.ThesecondlawCarnot’spublishedarticlecontainedthesecondlawofthermodynamics.WhatwassaidonthisthemeinChapter1maybecompressedintothefollowingassertion:Thereisnoreversiblecyclicalprocessinvolvingtheconversionofheatintoworkthatisnotaccompaniedbythetransferofacertainamountofheatfromahottertoacoolerbody.Westressoncemorethattheword“reversible”isusedhereinadifferentsensefromtheoneusedinmechanics.Earlierwespokeofthereversibilityofidealmechanicalprocessesintime,inthesensethatthelawsofmechanicscontinuetoholdifsuchaprocessisrunbackwardsinthedirectionofthepastinsteadofthefuture.Inidealmechanicsonecanrunthefilmbackwards.Here,ontheotherhand,wearetalkingonlyofreturningathermodynamicalsystemtoitsinitialstatebymeanssimilartothoseusedinthedirectprocess.WesawinChapter1thattheCarnotcycleisreversibleinthissense,andfurthermorethatithasmaximalefficiencycomparedwithanyothercycleinvolvingatemperaturedropofT1−T2.Thereversibility—thatis,theclosednessofthecycle—comesdowntotheconditionthattheisothermallyexpandinggasisforthewholeofthatstageofthecycleinthermodynamicequilibriumwiththeheater,maintainingitstemperatureatT1,whileduringthestageofisothermalcompressionitisinthermodynamicequilibriumwiththecooler,andhasitstemperaturekeptatthevalueT2.To8RudolfJuliusEmmanuelClausius(1822–1888),Germanphysicistandmathematician.Trans. 24Chapter2.Thelawsofthermodynamicsensurethis,itisassumedthatthisexpansionandcompressiontakeplaceveryslowly—slowlyenoughfortheequilibriumbetweengasandheatsourcetoremainundisturbed.Suchprocessesarecalled“quasi-static”sincetimedoesnotenterintotherelationscharacterizingthem.Thereisapuzzlehere.Irreversibilitywithrespecttotimeisbuiltintothephysicsofheatphenomenafromthestart—heatdoesnotflowonitsownaccordfromacoolertoahotterbody—yettimedoesnotenterintoCarnot’sfundamentallawsofthermodynamics.Movingon,weobservethatthelawformulatedatthebeginningofthissectionimpliestheimpossibilityofaperpetualmotionmachine“ofthesecondkind”.Whatdoesthismean?Clearlythelawofconservationofenergywouldnotbecontradictedbythetransferofheatfromacoolertoahotterbody,orbyasituationwhereworkwasdoneasaresultofcoolingasingleheatreservoir,thatis,byutilizingheatfromaheatsourceintheabsenceofacooler.Itdoesnotexcludetheunlikelyeventthatasealedcontainerofwatersubmergedinabucketofwatercouldcometoaboilandthewaterinthebucketfreeze!Ineachcasetheamountofenergy,whetherexpressedincaloriesorjoules,remainsunaltered.Nordoconsiderationsofenergyruleoutthepossibilityofextractingvirtuallyunlimitedamountsofusefulworkfromthecoolingoftheworld’soceans.Sincetheoceans’temperaturesarehigherthan0◦C,or273.15◦K,theamountofheatenergytheycontainishuge.Itissuchhypotheticalproceduresforobtainingworkthatarecalledperpetualmotionmachinesofthesecondkind.Thefirstlawdoesnotprohibitthem,butnonethelessweknowthattheyalsoareimpossible—thisamountstoarestatementofthesecondlawofthermodynamics:Aprocesswhoseonlyoutcomeistheconversionintoworkofheatextractedfromsomesourceisimpossible.ThisisequivalenttothelawasCarnotformulatedit;werepeathisversion:AheatenginethatabsorbsanamountofheatQ1attemperatureT1andyieldsanamountofheatQ2attemperatureT2,cannotdomoreworkthanareversibleheatengine,whosework-yieldisT1−T2W=Q1−Q2=Q1.(2.1)T1This,thesecondlaw,isalawofnature.Althoughasstrictanduniversalasthefirst,itisofaradicallydifferentcharacter.ThepressureoflightThesecondlawprovidesthekeytothesolutionofagreatmanyproblemsofphysics,chemistry,technology,and,asweshallsee,ofbiology.Bywayofexampleweconsiderapurelythermodynamicalproofthatlightexertspressure.ThisproofisduetotheItalianphysicistAdolfoBartoli,whoproposeditin1876. 25ABTT12Figure2.1:ContainerinBartoli’sproofoflightpressure.Wehaveacontainer,twoofwhoseoppositewallsareattemperaturesT1andT2withT10(sincetheintegrand1/zisunboundedonintervalsincluding0)andf(x)<0for00for11.Wedigresstoderivethisequation. 31Ifnoheatissuppliedtoagasthen,bythelawofconservationofenergy,ΔQ=ΔE+W=0.(2.13)Inotherwords,theworkisdoneattheexpenseofsomeoftheinternalenergyofthegas:W=pΔV=−ΔE.Howeveronp.5wesawthatΔE=CVΔT,anddeducedfromthestateequationofanidealgasthatRT(Cp−CV)Tp==.VVSubstitutingtheseexpressionsforΔEandpin(2.13),weobtaintherelationΔT(Cp−CV)ΔV+=0,TCVVor,indifferentialform,dTCpdV+−1.(2.14)TCVVThisequationiseasytointegrate.Intheprecedingsectionwesawthattheanti-derivativedx/x=lnx+const.Henceintegration(thatis,antidifferentiation)ofbothsidesoftheidentity(2.14)yieldsCplnT+−1lnV=const,CVorCp(C−1)lnTVV=const,whenceCpC−1TVV=const.(2.16)SubstitutingforTfromT=pV/R,weobtainfinallythedesiredequation(1.9):γCppV=const,whereγ=>1.(2.17)CVWenowreturntotheCarnotcycle.Fortheend-pointsoftheadiabaticexpansionfromV2toV3wehavefrom(2.16)thatγ−1γ−1T1V2=T2V3, 32Chapter2.ThelawsofthermodynamicsandfortheadiabaticcompressionfromV3toV4,γ−1γ−1T1V1=T2V4.Dividingtheformerequationbythelatter,weobtainV2V3=.(2.18)V1V4Analogouslyto(2.12),oneshowsthatforstage3,theisothermalcompressionofthegasfromvolumeV3toV4,thereisadecreaseinentropybytheamountV4ΔS=Rln.(2.19)V3Henceinviewof(2.18),thedecreaseinentropyoverthepath3→4isexactlycompensatedbyitsincreaseoverthepath1→2.Thecycleclosesandthestatefunctionentropyremainsunchanged.Thereisoneobviousbutimportantpropertyofentropy,namelythattheentropyofahomogeneoussysteminthermalequilibriumincreasesinproportiontothemassofthesystem.Thisissobecauseduringthetransitionfromsomeinitialstatetothestateinquestion,theheatabsorbedateachstageoftheprocessisproportionaltothemassofthesystem.Thismeansthattheentropyofasystemisthesumoftheentropiesofitshomogeneoussubsystems.Forexample,ifwehaveanisolatedsystemconsistingofavesselcontainingwaterandwatervaporinmutualequilibrium,thentheentropyofthesystemisequaltothesumoftheentropyofthewaterandtheentropyofthevapor(andalso,ofcourse,theentropyofthematerialofthevessel).Thusentropyisadditive.Inthisargumentwehavenottakenintoconsiderationtheentropyoftheinterfacebetweenwaterandvapor,norofthatbetweenthewaterandthewallsofthecontainer.Theproportionofmoleculesinvolvedattheseinterfacesisrelativelysmallandmaybeneglected.Abovewecalculatedthechangeofentropywhenthetemperatureisheldconstant.Whatifthetemperaturevaries?WehavedQdEpdVdTdVdS==+=CV+R.(2.20)TTTTVIntegratingonceagain,weobtainS=RlnV+CVlnT+a,(2.21)whereaistheconstantofintegration;itisonlyuptoanadditiveconstantthattotalentropyisdetermined,sincewehaveonlychangesinentropy,not“absolute”entropy.From(2.20)weinferthattransitionfromastateV1,T1toastateV2,T2resultsinthechangeinentropygivenbyV2T2ΔS=Rln+CVln.(2.22)V1T1 33Henceariseintemperatureisaccompaniedbyariseinentropyprovidedthatthechangeisnotadiabatic.Aswehavealreadyseen,alonganadiabaticcurveentropydoesnotchange—inthecaseofadiabaticexpansionofagas,theincreaseinentropyduetotheincreaseinvolumeisexactlycompensatedbyitsdecreaseduetotheresultingcoolingofthegas.Theabovecalculationsaresubjecttocertainprovisos.ForinstancewetacitlyassumedthatCpandCVdonotvarywiththetemperature.Thisisinfactnotcompletelytrue;infacttheheatcapacityofabodyincreaseswithdecreasingtemperature,andthisincreaseisespeciallymarkedatlowtemperatures.Thustherigoroustheoryismorecomplicatedthantheonejustpresented.Wenextcomputethechangeinentropyresultingfromheatconductionequal-izingthetemperaturesofgases,andfromdiffusion,thatis,themixingofgases.ConsideranadiabaticallyinsulatedsystemconsistingoftwoidenticalvesselseachofvolumeVandeachcontainingamoleofanidealgasattemperaturesT1andT2.Thevesselsarebroughtintocontactandasaresultofheat-conductionintheirwallsthegasesreachastateofthermalequilibriumwithoutchangeinvolume.Accordingto(2.21)thetotalentropypriortothetimeofcontactisS=2RlnV+CplnT1+CplnT2+2a.AftercontacttheequilibriumtemperatureofbothgasesisT1+T2T=,2andtheentropyisT1+T2S=2RlnV+2Cpln+2a.2Thechangeinentropyistherefore2T1+T2ΔS=S−S=Cpln−ln(T1T2).2Itiseasytoseethattheentropyhasincreased:Thisisimmediatefromthefactthatthearithmeticmeanoftwopositivequantitiesisalwaysgreaterthanorequaltotheirgeometricmean:T1+T2≥T1T2.2Hereisaproof.Weneedtoshowthat(squaringbothsidesandmultiplyingby4)(T+T)2≥4TT,1212or,equivalently,that22T1+2T1T2+T2≥4T1T2, 34Chapter2.Thelawsofthermodynamicsthatis,T2−2TT+T2≥0.1122Butthisistruesincetheleft-handsideis(T−T)2,asquareandsonevernegative.12Thustheentropyhasincreasedbytheamount(T+T)212ΔS=Cpln.(2.23)4T1T2Wenowturntothemixingoftwogases.SupposethefirstgasoccupiesavolumeV1andcontainsn1moles,andthesecondcontainsn2molesatvolumeV2.WeassumethetwogasesareatthesametemperatureTandpressurep,andareseparatedbyapartition.Thepartitionisremovedandthegasesmingle.Howdoestheentropychangefromthesituationwherethegasesareseparatedtothatwheretheyaremixed?SincethevolumeofeachgasinthemixtureisV1+V2,theindividualchangesinentropyofthegasesareV1+V2V1+V2ΔS1=n1RlnandΔS2=n2Rln.V1V2Thecombinedchangeinentropyduetothemixingofthegases,theso-called“mixingentropy”,isthenthesum:ΔS=ΔS1+ΔS2.SinceRTRTV1=n1andV2=n2,ppthisbecomesn1+n2n1+n2ΔS=Rn1ln+n2ln.(2.24)n1n2ThusthemixingentropyΔSispositive.Weseefromthisthatentropyincreasesinprocessesoccurringspontaneously,suchasthoseinvolvingheatconductionanddiffusion.Ifsuchprocessescouldbereversedthenentropywoulddecrease,butsuchprocesses—producingadifferenceintemperatureorinconcentrationsofgasesinamixture—wouldrequireexternalworktobedone.MeasuringentropyexperimentallyHowcanentropybemeasuredinpractice?Bydefinition,theentropychangeinasubstancebetween0◦KandtemperatureTis1T1dQΔS==ST1−S0.0T 35Nowthereisathirdlawofthermodynamics,Nernst’s10“heattheorem”,postu-latingthattheentropySvanishesatabsolutezero.11(WeshalldiscussNernst’s0theoremfurtherbelow.)HencewemaywriteT1dQS=.(2.25)0TWhatistheincreasedQinheat?Ifheatingtakesplaceatconstantpressure,thendQ=CpdT,(2.26)whenceT1dTS=Cp.(2.27)0THowever,inhavingitstemperatureraisedfromabsolutezerotoT1,thesubstanceinquestionmaybesubjecttovariouseffectsanditsheatcapacitymaywellchange.Consider,forinstance,carbontetrachlorideCC4.Atroomtemperature,298.1◦K,thiscompoundisapartiallyvaporizedliquid.Nowatlowtemperaturestheheatcapacityofpurecrystallinesubstancesisknowntobeproportionaltothecubeofthetemperature:C=bT3.(2.28)pInthecaseofCCtheconstantbisapproximately0.75·10−3.At225.4◦Ka4changetakesplaceinthecrystallatticeofsolidfrozenCC4requiring1080.8caloriespermoletoeffect.Atthemeltingpoint250.2◦KofCC,thelatentheat4ofmelting,thatis,theamountofheatneededtochangethecrystallinesolidintoaliquid,is577.2calories/mole.Atitsboilingpointof298.1◦K,thelatentheatofvaporizationis7742.7calories/mole.Henceinunitsofcalories/(mole·◦K)wehave10◦K225.4◦K−32dT1080.8S298.1=0.75·10TdT+Cp+010◦KT225.4250.2◦K298.1◦KdT577.2dT7742.7+Cp++Cp+.225.4◦KT250.2250.2◦KT298.1At298.1◦Kthepressureofcarbontetrachloridevaporis114.5mmofthemercurycolumn,equivalentto0.15atm.Tobringthevaportoatmosphericpres-surewemustcompressit,andthisentailsadecreaseinentropyofV760ΔS298.1=Rln.V114.5UsingvaluesofCpfoundbyexperiment,theintegralswithintegrandCpdT/TcanbeexpressedexplicitlyasfunctionsofthetemperatureT,andthenintegrated10WaltherHermannNernst(1864-1941),Prussianphysicist,Nobellaureateinchemistry1920.Trans.11Andthusgivingmeaningto“absolute”entropySafterall.Trans. 36Chapter2.ThelawsofthermodynamicsTable1:Theentropyofcarbontetrachlorideat298◦Kand1atmosphere.Changeinentropyincalories/(mole·◦K)Scal./(mole·◦K)10S−S=0.75·10−3T2dT0.251000S225.4−S10(integratedgraphically)36.29ΔS225.4=1080.8/225.4(phasetransition)4.79S250.2−S225.4(integratedgraphically)3.08ΔS250.2=577.2/250.2(melting)2.31S298.1−S250.2(integratedgraphically)5.45ΔS298.1=7742.7/298.1(vaporization)25.94Rln(114.5/760)(compression)-3.76Total:S=74.35cal/(mole·◦K)=311.22joules/(mole·◦K)Table2:ValuesofCpandCp/Tforsilverandgraphiteatvarioustemperatures.Cincal/(mole·◦K)C/Tincal/(mole·◦K2)ppT◦Ksilvergraphitesilvergraphite502.690.130.05310.00261004.820.410.04820.00411505.540.790.03790.00532005.841.220.02920.00602505.971.650.02590.0066273.16.021.860.02210.0068298.16.042.080.02030.0069graphically,thatis,bydrawingthegraphofCp/TasafunctionofT,andestimat-ingtheareaunderthegraphbetweentherespectivelimitsofintegration(compareFigure2.5below).TheindividualtermsmakinguptheentropySofCCat298.1◦Kand298.14atmosphericpressure,togetherwiththeirsum(thevalueofS298.1)aregiveninTable1above.Inthiswaytheentropyofvarioussubstancesmaybeestimatedexperimen-tally.Weshallseethatoftenitsvaluecanbecalculatedtheoretically.Theairinaroom,asheetofpaper,anyobjectwhatsoever,containsadefiniteamountofentropy—justasitcontainsadefiniteamountofinternalenergy.Wenowgivetwofurtherexampleswheretheentropyiscalculatedbymeansofgraphicalintegration.Table2givesvaluesofCpandCp/Tforsilverandgraphiteatvarioustemperaturesandacertainfixedpressurep.TherespectivegraphsofCp/TasafunctionofTaresketchedinFigure2.5. 37AgGraphiteGraphite0,060,006Gp/T0,040,004Ag0,020,002T°K50100150200250300Figure2.5:Graphicalcalculationoftheentropyofsilverandgraphite.OnefirstcalculatestheapproximateaveragevaluesofCp/Tovervariousoftheindicatedintervals.Forexample,overtheintervalfrom50◦Kto100◦Ktheav-eragevalueC¯/T,incalories/(mole·◦K)forsilveris(0.0531+0.0482)/2=0.0506.pOnethenmultiplieseachoftheseaveragesbythelengthofthecorrespondingtemperatureinterval,andsumstheseproductstoobtainanapproximationofthedesiredintegral:nC¯TnCdTp,ipΔTi≈.TiT1Ti=1Forsilverthissumis50(0.0506+0.0425+0.0330+0.265)+0.0230·23.1+0.0212·25=8.69cal/(mole·◦K)or36.38joules/(mole·◦K).Forgraphitethesumis50(0.0034+0.0047+0.0056+0.0063)+0.0061·23.1+0.0068·25=1.31cal/(mole·◦K)or5.48joules/(mole·◦K).Thesefiguresrepresentthechangesinentropyofsilverandgraphitewhenheatedfrom50to298.1◦K.IrreversibleprocessesWestateoncemorethelawsofthermodynamics—ofwhichtherearenowthree,ratherthantwo. 38Chapter2.ThelawsofthermodynamicsThefirstlaw(Mayer,Joule,Helmholtz,Carnot).Anincreaseininternalenergyofasystemisthesumoftheheatabsorbedbythesystemandtheworkdoneonthesystem:dE=dQ+dW.Thesecondlaw(Carnot,Clausius).Aprocesswhoseexclusiveoutcomeistheextractionofheatfromaheatsource(anditsconversionintowork)isimpossible.Inotherwords,itisimpossibletoconstructanenginethatworkscyclicallyanddoesworkbydrawingheatfromasingleheatreservoir,withoutcausinganyotherchangesinthesystem(aperpetualmotionmachineofthesecondkind).WesawinChapter1thatthemaximumefficiencyareversibleheatenginecanhaveisWT1−T2η==.Q1T1Alternativeversionofthesecondlaw(Clausius,Thomson12).IfinareversibleprocessasystemabsorbsanamountΔQofheatattemperatureT,thentheentropyofthesystemincreasesbytheamountΔQΔS=.TEntropyisafunctionofthestateofthesystem.Thethirdlaw(Nernst,1906).AtT=0◦K,theentropyS=0.(Whythisissowillbeexplainedbelow.)Sofarinourstudyofentropywehaveencounterednothingespeciallyinter-esting,letalonemysterious.Howeverentropydoespossessonesurprisingpecu-liarity:While—likeenergy—itispreservedinreversibleprocesses,unlikeenergyitincreasesinirreversibleones.Supposethatwebringtwobodiesintocontact,havingtemperaturesT1andT2withT1>T2—forinstance,thatwedropanicecubeattemperatureT2inaglassofwaterattemperatureT1.ThewaterwillthentransmitanamountofheatΔQtotheice,causingthewater’sentropytodecreasebyΔQ/T1,whiletheheatΔQabsorbedbytheicecubewillincreaseitsentropybyΔQ/T.13Theoverall2changeinentropyispositive:ΔQΔQΔS=−>0.(2.29)T2T1Wehavealreadyseenthatentropyincreasesinspontaneousprocessessuchasdiffusionandheatconduction.Suchprocessescanbereversedonlyatthenetexpenseofwork.Andonemightadduceagreatmanyotherexamplesattesting12JosephJohnThomson(1856–1940),Britishphysicist,discovereroftheelectron.Trans.13Thisassumesthatthetemperaturesoficecubeandwaterarenotsubstantiallychanged.Trans. 39totheinvariableincreaseofentropyinirreversibleprocesses.Equalizationofthetemperaturesoftwobodiescanoccurreversibly(asinthefirsttwostagesofaCarnotcycle;see(2.12)),andirreversibly.Inthelattercasetheincreaseinentropywillbegreater.Atwhatpointdoestheincreaseinentropyofasystemstop?Answer:Itincreasesuntilthesystemreachesastateofequilibrium.Herewecometoquestionsofafundamentallynewsort.Inourdiscussionsofreversibleprocessesweavoidedconsideringtheflowofsuchprocessesintime,assumingthattheyproceedinfinitelyslowlyandthatthereisequilibriumateachstage.AlongtheisothermalsofaCarnotcycle,itwasassumedthatthetemperatureofthegaswaskeptthroughoutthesameasthatoftheheaterorcoolerrespectively.Strictlyspeaking,wehavebeendealingwiththermostaticsratherthanthermodynamics,sinceweignoredtime-relatedfeaturessuchastherateofprogressoftheprocesses.Bycontrast,anirreversibleprocessinvolvesinanessentialwayaprogressionintimetowardsequilibrium.Ittakestimeforthetemperaturesofahotandacoldbodybroughtintocontacttobecomeequal.Entropydoesnotattainitsmaximuminstantaneously.Sofar(sofar!)timehasnotappearedexplicitlyinourdiscussions.Yetwewerenonethelessconcernedwiththedynamicsofheatprocesses.Toremedythislack,wesupplementourearlierformulation(s)ofthesecondlawwithaprovisointhecaseofirreversibleprocesses:Addendumtothesecondlaw(Clausius,1865).Inirreversibleprocessesthetotalentropyofthesystemalwaysincreases,thatis,14dQdS>,(2.30)TorT1dQS>.(2.31).0TInarrivingatthisconclusionClausiusandThomsonbroughttolightafun-damentalpropertyoftheuniverseasawhole.Inactualitytherearenoreversibleprocesses.Atinyportion,atleast,ofthemechanicalenergyofeverymotionistransformedintofrictionalheat.Soonerorlaterallmovingbodiescometoahalt,andanequilibriumcorrespondingtomaximumentropyisreached.Thomsonconcludedthattheworldisultimatelysubjectto“heatdeath”—whileitsenergyremainsunchanged.Thereisafurtherverygeneralconclusiontobedrawn.Wesawearlyonthatinpuremechanicsthereareinprinciplenoirreversibleprocesses,whileinther-modynamicstherearesuchprocesses.Wehavetacitlyassumedthat,forisolatedirreversiblesystems,timeincreasesinthedirectionofincreaseofentropy.Weshall14Theequationsthatfollowareperhapsintendedtoindicatethatifasystemundergoesanirreversibleprocess,thenentropyisgeneratedinternally,inadditiontothatresultingfromheatinput.(Seealsoequation(6.5)below.)Trans. 40Chapter2.Thelawsofthermodynamicsseeinthesequelhowsuchmattersstandforopensystemsinteractingwiththesurroundingworldwithitsmatterandenergy.Thustheconceptofentropyhasledusfrommeretechnology(thesteamengine)tocosmologicalconsiderations(thedirectionoftimeandthefateoftheuniverse).Thusentropyturnsouttohaveremarkableproperties.Weshallinthese-quelsearchforthereasonbehindthis.Howeverfirstweturnagaintoreversibleprocessesandthermostatics,inordertoderivesomeinterestingandimportantconsequencesofthelawsofthermodynamics. Chapter3EntropyandfreeenergyEnergyisthemistressoftheworld,ButablackshadowFollowsherinexorably,Makingnightanddayone,Emptyingeverythingofvalue,Transformingalltosmoke-filledgloom....Atleastthat’showentropyHasbeeninvariablyrepresented.ButnowweknowThatthereisnosuchshadowAndneverwasnorwillbe,ThatoverthesuccessivegenerationsofstarsThereisonlyentropy—lifeandlight.ObtainingusefulworkAswehaveseen,theequationgivingasystem’schangeininternalenergywhenundergoingareversibleprocessisdE=dQ−dW,(3.1)wheredQistheheatabsorbedbythesystemanddWtheworkdonebythesystem.Wealsohave(see(2.4))dQ=TdS.(3.2)HencedE=TdS−dW,(3.3).ordW=−(dE−TdS)=−dF,(3.4) 42Chapter3.EntropyandfreeenergywhereF=E−TS(3.5)iscalledtheHelmholtzfreeenergyofthesystem.Thisformoftheequationineffectcombinesthefirsttwolawsofthermodynamics,assertingthattheworkdonebythesystemisnotmerelyattheexpenseofsomeofitsinternalenergybutattheexpenseoftheinternalenergylesstheheat.Thusthegreatestusefulworkthatasystemcandoisequalto(thelossof)itsfreeenergy.Wewishalsotoconsiderprocesses—reversibleornot—proceedingatconstantpressurep.1Insuchaprocessworkisdoneagainsttheconstantpressure,regardlessofreversibility.Aswehaveseen,thisworkisgivenbypdV.TheenergyremainingfordoingotherusefulworkisthendW=−dF−pdV.(3.6)Thusatconstantpressure,theamountofenergyavailablefordoingusefulwork2isequaltothequantityG=F+pV=E+pV−TS.(3.7)ThequantityGiscalledtheGibbs3freeenergy,orthermodynamicpotential,ofthesystem.Likeinternalenergyandentropy,bothkindsoffreeenergiesarefunctionsofthestateofthesystem,thatis,theirvaluesdependonlyonthestateofthesystem,4andnotonthetransitionpathtothatstatefromsomeother.Itfollowsfrom(3.6)and(3.7)thatthemaximalamountofuseful,non-expansiveworkobtainablefromthesystematconstantpressureisequaltothelossofthermodynamicpotential:5dW=−dG.(3.8)Ifthevolumeisalsoconstant,thenequation(3.6)reducestoequation(3.4),per-tainingtothecaseofanarbitraryreversibleprocess:themaximalamountofenergyavailableforusefulworkisequaltothedecreaseinHelmholtzfreeenergy:dW=−dF.Sofarwehavetalkedonlyofmechanicalwork,thatis,theworkpdVdonebyanexpandinggas.Howeverequation(3.8)isvalidforarbitrarykindsofwork,such1Ofparticularinterestinconnectionwithchemicalreactionsoccurringatatmosphericpres-sure.Trans.2Ofanon-mechanicalsort.Trans.3JosiahWillardGibbs(1839–1903),Americanphysicist,chemist,andmathematician,oneoftheoriginatorsofvectoranalysis.Trans.4Meaningthattheyaredeterminedbyanytwoofthe“statevariables”p,V,T.Thefactthattheyarestatefunctionsisthusimmediatefromtheirdefinitions.Trans.5Theworkhereisthatnotinvolvedinexpansion,thatis,themaximalobtainableworklesstheworkdoneinexpandingagainstthefixedpressure.Trans. 43asthatdonebyanelectricalcurrent,orastheresultofachemicalreaction,andsoon.Thevariouskindsofworktransformintooneanother.6Forinstance,thechemicalreactioninabatteryproduceselectriccurrent,whichinturncanbeusedtodomechanicalwork—suchaswhenthecurrentfromacarbatteryactivatesthestarter.Thesameresultcanbeobtainedbyusingacrank,inwhichcasemuscularenergyisbroughttobearinstead.Butwheredoesthatenergycomefrom?Amuscleisamechano-chemicalsysteminwhichfreechemicalenergyisuseddirectlyfordoingmechanicalwork.Aheatengineusesheatobtainedfromthechemicalreactionofburningfuel.InaccordancewithCarnot’slaw(2.1)some(butnotall)oftheheatischangedintomechanicalwork.Howeveralivingorganismexistsatconstanttem-peratureandpressure.Thismeansthatthemuscularworkperformedbypeopleandotheranimals—suchasthatcarriedoutinbuildingtheEgyptianandMex-icanpyramids—isnottobeexplainedintermsofatransferofheatfromheatertocooler.Theworkdonebymusclestransformschemicalfreeenergyproducedinspecificchemicalreactionstakingplaceinmusclesatconstanttemperature.Inwhatdirectiondothesechemicalreactionsproceed?Weshallnowanswerthisquestion.EquilibriumconditionsIfasystemisisolated(closedandinsulated),thatis,doesnotexchangematterorenergywiththesurroundingworld,thenspontaneousprocessestakingplacewithinthesystemwilltendtowardsequilibrium.7Apurelymechanicalprocessevolveswithoutgivingofforabsorbingheat;itisanadiabaticprocessinvolvingnochangeinentropy.Heretheroleoffreeenergyisplayedbythetotalmechanicalenergy.Asiswellknown,mechanicalequilibriumisreachedataminimumofpotentialenergy.Astonetossedupwardsfallstotheground.ThustheconditionformechanicalequilibriumundertheassumptionΔS=0,isthatpotentialenergybeataminimum:U=Umin,(3.9)andthedirectionofsuchaspontaneousprocessisthatofdecreasingpotentialenergy:ΔU<0.(3.10)Sincethetotalmechanicalenergyofasystemundergoingafrictionlesspro-cessisconstant,anydecreaseinthepotentialenergyisexactlymadeupbyanincreaseinthekineticenergy.Thistransformationcanbeexploitedasasourceof6Orworktransformsonekindofenergyintoanother,whichisthenoncemoreavailablefordoingwork.Trans.7Althoughtherearesystems—suchasplanetaryones—thatdonotseemevertosettleintoanysteadystate.Trans. 44Chapter3.Entropyandfreeenergyusefulwork—as,forexample,inhydroelectricpowerstations,wherethepotentialenergyofthewateristurnedintokineticenergyasitfalls,andthenbymeansofturbinesintoelectricenergy.Asnotedearlier,inanisolatedadiabaticsystemequilibriumisreachedatthegreatestvalueofentropy:S=Smax.(3.11)Agoodexampleofanisolatedsystemisaspaceship,sinceinconstructingtheshipeveryeffortismadetoensurethatitsinteriorisinsulatedasfaraspossiblefromthesurroundingcosmos.Ofcourse,thespaceship,withitsfunctioningcrewofastronauts,isfarfrombeinginastateofequilibrium.Lateronweshallseehowthenon-equilibriumstateofalivingorganismissustained.Afterall,forsuchasystemequilibriummeansdeath.Thusthedirectionofspontaneousirreversiblechangeinanisolatedsystemisthatofincreasingentropy:ΔS>0.(3.12)Forthetimebeingweleaveasidethequestionoftherateofincreaseofentropy,thatis,thespeedatwhichequilibriumisattained.Weturnnowtoaclosedsystematconstanttemperatureandpressure.(Asystemissaidtobeclosedifitcanexchangeenergybutnotmatterwiththesurroundingmedium.)Anexamplewouldbeachemicalreactiontakingplaceinaflaskwhosetemperatureiscontrolledbyathermostat.TheequilibriumconditionforsuchasystemistheminimalityoftheGibbsfreeenergy:G=Gmin,(3.13)andconsequentlythedirectionofchangeofstateofsuchasystemisthatofdecreasingfreeenergy:ΔG<0.(3.14)Wehaveseenabove(see(3.7))thattheGibbsfreeenergyconsistsoftwoparts:G=(E+pV)−TS=H−TS.TheinternalenergypluspVisastatefunctioncalledtheenthalpy8ofthesystem:H=E+pV.(3.15)Wementioninpassingthattheterm“heatcontent”sometimesusedforenthalpyisunfortunate,since,aswehaveseen,itmakesnosensetospeakoftheamountofheatinabody.Thusinaclosedsystematconstanttemperatureandpressure,wemusthaveΔG=ΔH−TΔS≤0.(3.14a)8Enthalpyissimplyanotherconvenientstatefunction.Foranisobaric(constantpressure)process,wehaveΔH=ΔE+pΔV,sofromtheenergyequationΔE=ΔQ−ΔWweobtainΔH=ΔQ,theheatabsorbedatconstantpressure.Trans. 45WeseethatsuchadecreaseinGibbsfreeenergycancomeaboutintwodistinctways:throughadecreaseinenthalpyoranincreaseinentropy.Ofcoursethesemayoccurtogether,orindeedtheenthalpymayincreasebuthaveitsgrowthmorethancompensatedbyanincreaseinentropy:ΔHΔS>>0.TAnd,finally,thereversemayoccur:enthalpyandentropymightdecreasetogetherwiththelossofenthalpyexceedingthatofentropy.Briefly,foraprocessofthesortweareconsideringtoberealizable,theGibbsfreeenergymustdecrease;itisnotsufficienttoconsiderthechangesinenthalpyandentropyseparately.AchemicalreactionTheburningofanykindoffuel—anoxidationprocess—isaccompaniedbyare-ductioninitsfreeenergy,whichisgivenoffintheformofheatandlight.Thustheburningofhydrogen—itscombinationwithoxygen—resultingintheformationofwater,involvesalargedischargeoffreeenergy.Specifically:1H2(gasat1atm)+O2(gasat1atm)→H2O(liquid)+236760joules/mole.2Ontheright-handsideofthisequationweseetheamountoffreeenergyliberated.Sincethereactiontakesplaceatconstantpressure,thismustbetheGibbsfreeenergy,orthermodynamicpotential.Beforeitwasunderstoodthatwhatcountsisfreeenergyratherthaninternalenergyorenthalpy,itwasthoughtthatachemicalreactionispossibleonlyifitisexothermic,thatis,involvestheproductionofheat.However,intimeitwasrealizedthatthereareendothermalreactions,involvingtheabsorptionofheat.Thedirectionofchangeinenthalpyisnotinitselfthecrucialfactor.Liberationoffreeenergyisanecessaryconditionforachemicalreactiontotakeplace,butitisnotsufficient.Forexample,amixtureofhydrogenandoxygencanexistforanindefinitelylongtimewithoutareactionoccurring.Howeverifalightedmatchisbroughtuptothemixtureanexplosivereactionoccurs.Whatisgoingonhere?Itturnsoutthat,althoughitfurnishesuswiththecondition(3.13),thermodynamicsdoesnottelluswhetherornotareactionwillproceed—onlywhetheritcanorcannotoccur.Forexample,underordinarycon-ditionsoxygenandnitrogenwillnotreactbecauseN2+O2→2NO−174556joules/mole,thatis,thefreeenergyincreasesratherthandecreasesinthisreaction.Atthesametimethisindicatesthethermodynamicinstabilityofnitricoxide,sincethereversereactionresultsintheliberationoffreeenergy:2NO→N2+O2+174556joules/mole. 46Chapter3.EntropyandfreeenergyFigure3.1:Modelofachemicalreactioninanisolatedsystem.Sowhyisitthatthermodynamicstellsonlythatareactioncaninprincipleoccurandnotwhetheritwillactuallytakeplace?Thefactofthematteristhataprocessmaybethermodynamicallypossible,yetactuallyimpossible.9Itwasnotedabovethat,althoughitisthermodynami-callypossibleforhydrogenandoxygentoreactwhenmixed—sincethereactionliberatesfreeenergy—,suchamixturemaynonethelessremaininertforanar-bitrarilylongtime.Thermodynamicshasnothingtosayaboutusignitingthemixturewithamatch—itmerelyfindshowthebalanceoffreeenergylies.Everyprocesstakestime.Ifthetimerequiredisinfinite,itwillnotproceed.However,timedoesnotenterintothermodynamics;asmentionedearlier,strictlyspeakingweshouldcallitthermostaticsratherthanthermodynamics.Imagineasystemconsistingofavesselcontainingaliquidwithatapfordrainingtheliquidintoasecondvesselplacedunderthetap(Figure3.1).Ther-modynamicstellsusthatsoonerorlatertheliquidfromtheuppervesselwillallflowintothelower,andofcoursewecancalculatethefinallevelofliquidinthelowervessel.Thermodynamicstellsusthattheliquidmustflowdownwards,notupwards.Butwillthisoccursoonerratherthanlater?Thermodynamicsissilentonthisquestion.Therateofflowdependsontheextenttowhichthetapisturnedon.Ifitisturnedoff,thentheliquidintheuppervesselisinastateofthermo-dynamicaldisequilibriumsinceitsenergyisabovetheequilibriumvalueforthesystem.Neverthelessthisdisequilibriuncanlastindefinitely.Thereisaclearsimilaritybetweenignitingamixtureofhydrogenandoxygenwithamatchandturningonthetap:openingthetapallowstheprocesstoproceed.Figure3.2showsthegraphofthefreeenergyagainsttimeoverthecourseofacertainchemicalreaction.Theinitialstate1ofthereagentshasgreaterfreeenergythanthefinalstate2oftheproductsofthereaction:G1−G2=ΔG>0.9Inthesensethatsomeexternalinputisnecessarytosetitoff?Trans. 47aGG11ΔGG22Figure3.2:Variationinfreeenergyduringachemicalreaction.Freeenergyhasultimatelybeenliberated.Thusthereactionispossible.Butwillitoccur?Thisisaquestionofkineticsratherthanthermodynamics.Inthereactioninquestiontheinitialstateisseparatedfromthefinalstatebyaso-called“activationalbarrier”,akindofridgewhichmustbesurmountedbythereagents.ThefreeenergyatthetopoftheridgeexceedstheinitialfreeenergyG1bytheamountGa.Thismeansthatforthereactiontoproceedthereactingmoleculesmustpossessasurplusoffreeenergy.Thehigherthebarrier,theslowerthepaceofthereaction.Ifthetemperatureisraised,thentheproportionofmoleculeswithsurplusenergyincreasesandthereactionspeedsup.(WeshallseelaterexactlyhowthespeedofachemicalreactiondependsonthefreeactivationenergyandthetemperatureT.)Thereaderdoubtlessknowsthatmanyreactionsarecarriedoutwiththeaidofcatalysts,substancesthatstimulatethereactionswhilethemselvesremainingunchanged.Underconditionsofchemicalequilibrium,theroleofthecatalystre-ducestotheloweringoftheactivationalbarrierandconsequentspeedingupofthereaction.ConsideragainthevesselsandliquidofFigure3.1.Hereacatalystmightbeanagentwho“opensthetap”more,therebyacceleratingtheflowofliquidfromtheuppertothelowervessel.Howeverthefinalresultisindependentofthiscatalyzingagent:soonerorlatertheliquidwillallflowintothelowervesselandreachapredeterminedlevelinit.(OfcoursethesystemdepictedinFigure3.1isnotanalogoustoachemicalreactionwithanactivationalbarrier—theflowofliquidishamperedonlybyfriction.However,ithelpstoclarifythedistinctionbetweenthermodynamicsandkinetics.)Theactivationfreeenergyisequaltothedifferencebetweentheactivationenthalpyandtheactivationentropytimesthetemperature(seetheequationfol-lowing(3.14)):Ga=Ha−TSa.(3.16)ThusareductioninGacanbebroughtaboutbyareductionintheactivationen-thalpyHa,oranincreaseintheactivationentropySa,orsomeothercombinationofchanges.Itisappropriatetomentionherethat,withoutexception,allthechemicalreactionstakingplaceinlivingorganisms—andwhichinfactconstitutelife— 48Chapter3.Entropyandfreeenergyinvolvecatalysts.Thesearethealbumins10andenzymes.MeltingofcrystalsandevaporationofliquidsInaclosedsystemunderequilibriumconditions,onbeingheatedcrystalsmeltintoaliquid,andonfurtherheatingtheliquidturnsintoavapor,thatis,agas.Themeltingofacrystalandtheevaporationofaliquidrequiretheinputofheat,thelatentheatoffusionandthelatentheatofevaporationrespectively.Ifoneplacesiceinavesselandheatsit,whenthetemperaturereaches0◦C(≡273.15◦K)itstayssteadyuntilalloftheicehasmelted.Onecanthencontinueheatingthewaterto100◦C(≡373.15◦K),whereuponthetemperatureagainceasesincreasinguntilsuchtimeasallthewaterhasturnedintosteam.Thelatentheatofmeltingoficeis5982joules/mole,andthelatentheatofevaporationofwaterat100◦Cis40613joules/mole.Thelatentheatofevaporationofasubstanceisalwayssignificantlygreaterthanitslatentheatofmelting.Crystal,liquid,andgasaredifferentphasesofasubstance.Theydifferintheirstateandexistenceconditions,andareseparatedbyboundarieswhentheycoexist.Amoretechnicalnamecoveringbothmeltingandevaporationisphasetransition.Weshallnowfindthermodynamicalconditionsforphasetransition.Whenacrystalisheated,itsinternalenergyincreases,andhencealsoitsenthalpyH.Sotoodoesitsentropy—itwasshowninChapter2(seeequation2.22)thatentropyincreaseswithincreasingtemperatureprovidedthechangeisnotadiabatic.Nev-ertheless,asarulethe(Gibbs)freeenergyG=H−TSalsoincreases.Asthetemperaturecontinuestorise,thefreeenergyfinallybecomesequaltothefreeenergyofthesamequantityofliquidatthemeltingpoint:Gcrystal=Gliquid,(3.17)orHcrystal−TmeltingScrystal=Hliquid−TmeltingSliquid.(3.18)Whenthefreeenergiesofthetwophasescoincide,aphasetransitionoccurs—melting,inthepresentcase.WecansolveforthemeltingpointTmeltingfrom(3.18):Hliquid−HcrystalΔHTmelting==.(3.19)Sliquid−ScrystalΔSAsnotedabove,bothΔHandΔSarepositive.ThegreaterΔH—whichisinfactthelatentheatoffusion—andthesmallerthechangeinentropyΔS,thehigherthemeltingpointTmelting.Iftheentropiesofcrystalandliquidshouldcoincide,sothatΔS=0,thenmeltingcouldnotoccur:Tmelting→∞.Thusanentropychangeatthemeltingpointiscrucialformeltingtobepossible.10Certainwater-solubleproteinsresponsibleforthefunctioningofcells.Trans. 49Let’scalculatethefusionentropyΔSmeltingforwater.ThelatentheatoffusionforiceisknowntobeΔH=5982joules/mole,andofcourseTmelting=273.15◦K.Hence5982joules/mole◦ΔSmelting==21.89joules/(mole·K).273.15◦K(WedidtheanalogouscalculationforcarbontetrachlorideinChapter2—seeTable1there.)Vaporizationentropyiscalculatedsimilarly.Forwateritis40613joules/mole◦ΔSboiling==108.84(joules/mole·K).373.15◦KWesee,therefore,howimportantaroleentropyplays.Withoutchangesinentropytherewouldbenophasetransitions,andtheworldweinhabitcouldnotexist.Inparticular,ifatleasttherewerewater,thenitcouldexistonlyasice—andtherewouldbenolife.Uptillnowwehavebeenconsideringthesephysicalprocessesmerelyphe-nomenologically,orformally.Weconsideredthechangesofenthalpyandentropyinvolvedinmeltingandevaporation,butdidnotask“Why?”,whichforphysicsisthefundamentalquestion.Whydoenthalpyandentropydecreasewhenasub-stanceiscooledandincreasewhenitisheated?Whyarethelatentheatandentropyofevaporationsomuchgreaterthanthelatentheatandentropyoffu-sion?Onceagain,thermodynamicsissilentonthesequestions;theygobeyondtheboundsofitscompetence.Thethreelawsbythemselvescanneverleadtoanswers.Ofcoursephysicscananswerthesequestions,butbymeansoftheoriesdevel-opedinotherareasofphysics,namelystatisticalmechanicsandthekinetictheoryofmatter.Weshalllookattheseinthenextchapter.Whydoesalcoholdissolveinwateryetgasolinenotdoso?Nowthatweknowaboutchemicalreactionsandphasetransitions,wemay,afterreflectingalittle,alsomasterthethermodynamicaltheoryunderlyingthedissolv-ingofonesubstanceinanother.Themostimportantanduniversalsolventisofcoursewater.Watersolutionsareeverywhereinourlives.Actually,tapwaterisasolution.Itcontainsvariousdissolvedsubstances,mainlyhardeningsalts—carbonates,silicates,phosphates,predominantlyofcalcium.Thesesaltsarenotverysoluble,andtheythereforegraduallysettleoutonthewallsofvesselsorwater-pipesasascale.Thisscaleisanuisance.Thoughnotmuchofaprobleminateapot,itcancauseagreatdealoftroubleinasteamboiler. 50Chapter3.EntropyandfreeenergySothesesaltshavelowsolubilityinwater.Forwhichsubstancesisithigh?Alcoholicdrinkscomeinmanydifferentstrengths,whichshowsthatethylalcoholdissolvesinwateroverawiderangeofproportionsofalcoholtowater.Weoftenneedtodissolvesugarandtablesaltinwater,andeveryoneknowsthattheirsolubilityincreaseswiththetemperature.Thesearejustthemostfamiliarexamplesofthegreatmanywater-solublesubstances—salts,acids,bases—whosesolubility,asarule,increaseswiththetemperature.However,therearemanysubstancesthatdonotdissolveinwater.Ithardlyneedsmentioningthatmercuryisonesuchsubstance—theideaofasolutionofmercuryinwaterstrikesoneasunnatural.(Yetsilverdoesdissolvetoaverysmallextentinwater,yieldingasolutionusefulasabactericide.)Agreatmanyorganiccompounds,aboveallhydrocarbons,arepracticallyinsolubleinwater.Thusgasolineandparaffin,whicharemixturesofhydrocarbons,formlayersinwater,asisshown,forexample,bytheiridescentfilmformedbygasolineonthesurfaceofpuddles.Asaresultofthehighsurfacetensionofwater,thelayerofgasolineisstretchedsoastoformathinfilmwhosecolorshavethesameoriginasthoseofsoapbubbles.Thecolorsvisibleinsuchthinfilmsresultfromthephenomenonofinterference,andsoprovideabeautifulproofofthewavenatureoflight.11Butwhatisthethermodynamicalsignificanceofgreaterorlessersolubility?Clearly,asolutionwillformifitsformationisaccompaniedbyareductioninfreeenergy,muchasinachemicalreaction:solublesubstance+solvent→solution.Sofreeenergymustbereleased—andthiscanonlycomeaboutthroughanin-creaseinentropy.12Itfollowsfromequation(2.22)thatatconstanttemperatureentropyincreaseswithincreasingvolume,andalsothatmixingentropyisposi-tive(equation(2.24)).Whenwemixaliterofalcoholandaliterofwaterwedonotrestrictthemixturevolume-wise,sotheentropyincreases.Inaddition,theenthalpydecreasesasaresultoftheinteractionofthemoleculesofthesoluteandthesolvent.Hencethefreeenergymustdecrease:ΔH<0andΔS>0,whenceΔG=ΔH−TΔS<0.Thatentropyincreaseswhenmanysubstancesaredissolvedisprovedbythein-creaseintheirsolubilitywithtemperature(asinthecaseofsugarorsaltdissolvedinwater).SincethecontributionΔHofthechangeinenthalpyturnsouttoberelativelysmall,theincreaseinsolubilityresultingfromatemperatureincrease11Actuallythefullexplanationcomesfromquantumelectrodynamics,whichconsiderslightasmadeupofphotons.SeeR.P.Feynman’sQED,PrincetonUniversityPress,Princeton,1985.Trans.12SincetheenthalpyHdecreases(seebelow)andthetemperatureTisunchangedwhenasolutionforms.Trans. 51mustbeduetotheterm−TΔShavingdecreased(thatis,becomelargerneg-atively).Hencethehigherthetemperaturethemorefreeenergyavailabletobeliberated—providedΔSispositive.Thusalcohol,sugar,andsaltdissolveinwaterbecausetheirdissolutionisaccompaniedbyadecreaseinfreeenergy.Butwhydoesgasolinenotdissolveinwater?Theobviousansweristhatthiswouldrequireanincreaseinfreeenergy.Andwhyisthat?Well,freeenergycanincreaseintwodistinctways—eitherthroughanincreaseinenthalpyoradecreaseinentropy.Whichoftheseismorepertinenttothecaseinquestion?Hydrocarbonsdodissolveinwater,butonlytoaminimalextent.Carefulinvestigationhasshownthatthisprocessisaccompaniedbyadecreaseinenthalpy:ΔH<0.Butwhatisveryunusualisthatthesolubilityofhydrocarbonsgoesdownratherthanupasaresultofheating.Itfollowsthattheentropymustalsodecreaseduringdissolution—moreoverbyanamountsufficienttocompensateforthereductioninenthalpy:ΔH<0andΔS<0,butΔG=ΔH−TΔS>0,thatis,thepositivequantity−TΔSexceedsthelossΔHofenthalpy:−TΔS>|ΔH|.Weconcludethatthelayeringofgasolineandwaterisduetoentropy!Theentropychangecausesthehydrocarbonmoleculestobeexpelledfromthewateryenvironment.Theentropychangeplaystheroleofanactingforce!HydrophobicforcesandthealbuminglobuleThis“entropicforce”isusuallycalledahydrophobicforce,andsubstancesthatthisforceexpelsfromwaterarealsotermedhydrophobic;theyare“inimical”towater,unlikehydrophilicsubstanceswhich“like”water.Manyimportantphenomenacanbeexplainedbythehydrophobicforce.Forexample,howdoessoapclean?Soapsareusuallymadefromsodiumandpotassiumsaltsoffattyacids,withchemicalformulaesimilartothefollowingone(forsodiumpalmitate,asodiumsaltofpalmiticacid):H3C—CH2—CH2—CH2—CH2—CH2—CH2—CH2—CH2——CH2—CH2—CH2—CH2—CH2—CH2—COONa.ThelonghydrocarbongroupH3C(CH2)14ishydrophobic,whiletheradicalCOONaishydrophilic.Inwater,soapsformcolloidalsolutions,13andthesus-pendedmoleculesformmycelia,14thatis,moleculeslikeHC(CH)COONaar-3214rangedinaspecificmanner,withthehydrophobichydrocarbongroupsoriented13Veryfinesuspensionsofparticlesinaliquid.Trans.14Amyceliumissomethinglikeamassoffibers.Trans. 52Chapter3.EntropyandfreeenergyFigure3.3:Modelofasoapmycelium,showingthehydrophilic“heads”andhy-drophobic“tails”ofmolecules.towardstheinteriorofthemycelium—beingrepelledbythewater—whilethehydrophilicgroupsremainonthesurfaceofthemycelium.SeeFigure3.3foraschematicpictureofasoapmycelium.Thecleansingactionofsoapisduetothemycelialstructureofitssolution.Thesurfacesofthemyceliaarehighlyactive,andreadilyadsorb15manysub-stances.Thepresenceofthehydrophilicgroupsenablessoaptowethydrophobicsurfacesandtoemulsifyfats,oils,andsoon.ThisiscoupledwiththealkalinereactionoftheCOONaradical.Howeverthemostimportanteffectsofhydrophobicentropicforcesarenotrainbowfilmsonwaterorthecleansingactionofsoap,butthesynthesisofalbu-mins,substancesdeterminingthefunctioningofalllifeprocesses.Analbuminmoleculeisachainofaminoacidresidua.Theyareallbuiltoutofthe20differentaminoacids,accordingtothefollowinggeneralscheme:H|H2N—C—COOH|RwhereRstandsforafunctionalgroupofatomsdistinguishingoneaminoacidfromanother.Whenaminoacidscombinetoformanalbuminchain,watermoleculesseparateoffandpeptidelinks—CO—NH—areformed.Hereisafragmentofanalbuminchain(tripeptide):HHH|||—NH—C—CO—NH—C—CO—NH—C—CO—|||R1R2R3whereR1,R2,andR3standfordifferentoridenticalradicals.Anentirechain,whichmaybeverylong—containingahundredormoreaminoacidresidua—isa15Thatis,adheretoandsurroundinathinlayerofparticles.Trans. 5321Figure3.4:Schematicsketchofanalbuminglobule.Region1consistsofhydropho-bicradicals,andregion2ofhydrophilicones.kindoftextwrittenusinga20-letteralphabet.Weshallhavemoretosayaboutthesetextsattheendofthebook.Albuminsfunctioninwatersolutionsasfermentingagents,catalystsofbio-chemicalreactions.TheypossessacertaindefiniteflexibilitysincerotationsarepossibleabouttheunitlinksC—NandC—C.Initsnaturally-occurringbiologi-callyfunctionalstate,analbuminchainisrolledupintoadenseglobularstructure,characteristicofthatparticularalbumin.Whatdeterminesthisstructure?Amongtheradicals,orfunctionalgroupsR1,R2,...,R20,therearehydrophobicones,con-taininghydrocarbongroups,andhydrophilicones,containinginparticularacidicandbasicgroups.Whathappenstothesedifferentsortsofradicalswhenanalbu-minchainisimmersedinwater?Tosomeextentthestructureofanalbuminglobuleresemblesthatofamycelium,inthesensethatthealbuminchainisrolledupsothatthehydrophobicgroups,whichspontaneouslywithdrawthemselvesfromthesurroundingwater,arelocatedintheinterioroftheglobule,whilethehydrophilicradicalsarelocatedonitssurface.Figure3.4givesaschematicpictureofsuchaglobule,andFigure3.5givesamoredetailedrepresentationofthestructureofaglobuleofthealbuminmyoglobin,establishedbymeansofX-rayanalysis.Manyalbuminsfunctionnaturallyinglobularform.Thisnaturally-occurringstatecanbeundonebymeansofacids,alkalis,ortheapplicationofheat.Thealbuminthenbecomesdenaturedandceasestofunction.Youcan’tgetachickenoutofaboiledegg.Wehavearrivedatahighlynon-trivialconclusion:thefunctioningofseveralofthealbuminscrucialtolifedependsontheirglobularstructure,andthisisde-terminedbyhydrophobic,thatis,entropic,forcesactinginawateryenvironment.Whatdorubberandanidealgashaveincommon?Thiswouldseemtobeasillyquestion,orpoorlyframedatbest.Whatcouldasolidsubstance—whichrubbercertainlyis—haveincommonwithagas,moreover 54Chapter3.Entropyandfreeenergy–CO2+NH3Figure3.5:Structureofaglobuleofmyoglobin.Thedotsrepresentaminoacidradicals.Inmyoglobinthealbuminchainhasaroughlyspiralstructure,asshownhere.aperfect,thatis,highlyrarefied,gas?Itmustbe,surely,thattheposerofthequestionhadinmindsomespecificpropertyofrubberinwhichitdiffersfromothersolidsbutresemblesanidealgas.However,itispreciselysuchseeminglyparadoxicalquestionsthatoftenlendimpetustoscientificprogress.L.D.Landau16usedtosaythatthetaskoftheoret-icalphysicsistoestablishnewconnectionsbetweenphenomenathatatfirstsighthavenothingincommon.Thefindingofsuchconnectionsinvariablyprovestobeapotentsourceofnewinsights.(AstrikingexampleisMaxwell’sdiscoveryoftheconnectionbetweenthewavetheoryoflightandtheoryofelectromagnetism.)Inthepresentcase,ourinterestlieswithelasticity:elasticforceanddefor-mation.Whenasteelspringisstretchedtherearisesanelasticforce,whichincreaseswiththeamountofstretching.AccordingtoHooke’slaw,thetensiondevelopedinaflexiblesolidisproportionaltothedeformation:fL−L0σ==,(3.20)sL0whereσdenotesthetension,thatis,theforcefperunitareaofcross-sectionofthedeformedbody,17Listhelengthofthebodyinitsstretchedstate,L016LevDavidovichLandau(1908–1968),Sovietphysicist,Nobellaureate1962.Trans.17Thatis,cross-sectionperpendiculartothelineofstretching.Trans. 55itslengthunstretched,andisYoung’smodulusofelasticityofthematerialofthebody.18Forsteelthemodulusisverylarge,around200gigapascals.19Thusasmalldeformation(orstrain)resultsinalargeelastictensileforce,or,conversely,alargeappliedforce(stress)isneededtoproduceanappreciabledeformationofthespring.Whencecomestheelasticforce?Whenwedeformastripofmetalelasticallyweraiseitsinternalenergy,byincreasingthepotentialenergyofitsatoms,heldtogetherinthecrystallatticeofthemetalbychemicalbonds.Sowhatchangesisjusttheinternalenergy.Nowweseefromequation(3.4)thattheworkdoneindeformingthemetalisequaltotheincreaseinits(Helmholtz)freeenergy:ΔW=ΔF=ΔE−TΔS.Butworkisforcetimesdistance,sowealsohaveΔW=fΔL=f(L−L0).(3.21)HencetheelasticforceisgivenbyΔFΔEΔSf=σs==−T.(3.22)ΔLΔLΔLAsalreadynoted,whenasteelspringisdeformed,itsinternalenergychangesbutnotitsentropy.20HenceΔS=0,andtheelasticforceisgivenbyf=ΔE/ΔL,sothatitisofapurelyenergeticcharacter.Mostsolidsbehavethiswayunderelasticdeformation—butnotrubber.Theelasticmodulusofrubberismanyordersofmagnitudelessthanthatofsteel;dependingonthedegreeofvulcanization,itrangesfrom200to8000kilopascals—whichmeanssimplythatapieceofrubber,arubberbandforinstance,caneasilybestretchedelasticallytoseveraltimesitslength.WenextcalculateYoung’smodulusforanidealgas.Agasmayalsobeconsideredanelasticbody,butonethatresistscompressionratherthanstretching.Supposeouridealgasiscontainedinacylinderwithapiston,asinFigure3.6.WhenthegasiscompressedisothermallytherearisesanelasticforceΔFf=ps=.ΔLSincehereL=V/s,wherenowsistheareaofcross-sectionofthecylinder,itfollowsthatΔFΔEΔSf=ps=s=s−T.(3.23)ΔVΔVΔV18Determinedbythisequation.Trans.19Since1pascal=1newton/meter2,agigapascalis109newtons/m2(≈30×106pounds/squareinch).Trans.20SinceΔQ=0?OristheauthormaintainingthatΔS=0followsfrom(3.22)?Thisisunclearintheoriginal.Trans. 56Chapter3.Entropyandfreeenergy0LLFigure3.6:Compressionofagasinacylinderwithapiston.Nowthebasicassumptionconcerninganidealgasisthattherebenointeractionbetweenitsmoleculesbeyondelasticcollisions.Hencetheinternalenergyofaquantityofidealgasisindependentoftheaveragedistancebetweenthemolecules,andhenceofthevolume.ThusΔE=0,and(3.23)yieldsΔSp=−T.(3.24)ΔVForanidealgaswehavetheequationofstatepV=RT.Sincethecompressionisisothermal,Tisfixed.HencepV=const,whenced(pV)=Vdp+pdV=0,yieldingdVdp=−p.(3.25)VThisequationisanalogoustoHooke’slaw,ifweinterpretdpasthe“elastictension”ofthegasanddV/Vasitsrelativedeformation;thenegativeofpressure,−p,isthentheanalogueoftheelasticmodulus.21Notethatthe“elasticmodulus”−pofthegasataparticularvolumeVisproportionaltothetemperatureT,sincebythestateequation,RTp=.V21Exceptthattheelasticmodulusformetals,forexample,isclosetobeingconstant,whereashere−pisvariable.Trans. 57Thismayalsobeseenfromequation(3.24).22Equation(3.25)showsthattheelasticityofanidealgasisnotenergy-basedlikethatofasteelspring,butentropic:agasresistscompressionnotbecausecompressionincreasesitsenergybutbecauseitdecreasesitsentropy.Atmosphericpressure,orthe“elasticmodulus”ofanidealgas,isaround100kilopascals,whichisofthesameorderofmagnitudeastheelasticmodulusofrubber.Itturnsoutthattheelasticityofrubberisalsoentropic.ExperimentshowsthattheelasticforcefofrubberisproportionaltotheabsolutetemperatureT,andfurthermoreisclosetozeroatabsolutezero.Hencein(3.22)onlytheterm−T(ΔS/ΔL)issignificant,thatis,ΔSf≈−T.(3.26)ΔLThusthehighelasticityofrubberisexplainedbythefactthatitsentropydecreasesverymarkedlyunderstretching.Inviewofthefactthattheelasticityofanidealgasandarubberbandarebothessentiallyentropic,onewouldexpectsimilarheatphenomenatobeobservableforthesetwosubstances.Andindeed,anyonewhohaspumpedupabicycletireorstretchedarubberbandheldagainsthisorherlipswillattesttothefactthatineachcaseheatisgivenoff.Quickcompressionofagasisanadiabaticprocessbecausethereisnotimeforheattobeabsorbedbythesurroundingmedium,andthesameistruewithregardtotherapidstretchingofarubberband.Thattheelasticityofrubberisessentiallyentropicisveryimportant:themainuseofrubber,namelyinautomobileandairplanetyres,dependsonthisproperty.Weconcludethissectionbyjuxtaposingintabularformtheelasticpropertiesofanidealgasandofrubber(Table3below).Howeverwehavestillnotpenetratedtotheunderlyingreasonforthissim-ilarityofsuchradicallydifferentmaterialbodies.Weshallconsiderthisinthesequel.Whydoweheataroom?Thisapparentlyverysimplequestionwillallowustobetterunderstandhowenergyandentropyareinterrelated.TheoutstandingtheoreticalphysicistsArnoldSommerfeldandRyogoKubobothincludedintheirmonographsonthermodynamicsanotewrittenbytheSwissgeophysicistRobertEmden23entitled“Whydowehavewinterheating?”,22Young’smodulusalsodecreaseswithtemperatureformetals,thoughrelativelyslightly.Forexample,forcarbonsteelitdecreasesbyabout6%between0◦Cand100◦C.Trans.23JacobRobertEmden(1862–1940),Swissastrophysicistandmeteorologist.Trans. 58Chapter3.EntropyandfreeenergyTable3:Propertiesofanidealgascomparedwiththoseofrubber.IdealgasRubberElasticmodulusisproportionaltoab-Elasticmodulusisproportionaltoab-solutetemperature,andequals100solutetemperature,andliesbetweenkilopascalsatoneatmosphere.200and8000kilopascals.VolumecanbechangedbyalargeLengthcanbechangedbyalargefactor.factor.Heatsupunderadiabaticcompres-Heatsupunderadiabaticstretching.sion.Internalenergyindependentofvol-Internalenergypracticallyindepen-ume.dentoflength.UndercompressionanentropicelasticUnderstretchinganentropicelasticforcearises.forcearises.publishedintheBritishjournalNaturein1938.WefollowtheexampleofthesescientistsandquoteEmden’snoteinitsentirety.24“Thelaymanwillanswer[tothequestionastowhywehavewinterheating]:‘Tomaketheroomwarmer.’Thestudentofthermodynamicswillperhapssoexpressit:‘Toimportthelacking(inner,thermal)energy.’Ifso,thenthelayman’sanswerisright,thescientist’swrong.“Wesuppose,tocorrespondtotheactualstateofaffairs,thatthepressureoftheairinaroomalwaysequalsthatoftheexternalair.Intheusualnotation,the(inner,thermal)energyis,perunitmass,E=CVT.(Anadditiveconstantmaybeneglected.)Thentheenergycontentis,perunitofvolume,E1=CVρT,[ρ=density]or,takingintoaccounttheequationofstate,p=RT,ρwehaveCVpE1=.R“Forairatatmosphericpressure,−3−3E1=0.0604cal·cm=60.4cal·m24Whatfollowsisreproducedverbatimfromtheoriginalarticle(writteninEnglish)inSupple-menttoNATURE,May21,1938,pp.908–909,exceptthatthesymbolismischangedtoconformwiththatusedinthisbook.Trans. 59[=2.528·105joules/m5].Theenergycontentoftheroomisthusindependentofthetemperature,solelydeterminedbythestateofthebarometer.Thewholeoftheenergyimportedbytheheatingescapesthroughtheporesofthewallsoftheroomtotheoutsideair.“Ifetchabottleofclaretfromthecoldcellarandputittobetemperedinthewarmroom.Itbecomeswarmer,buttheincreasedenergycontentisnotborrowedfromtheairoftheroombutisbroughtinfromoutside.“Thenwhydowehaveheating?Forthesamereasonthatlifeonearthneedstheradiationofthesun.Butthisdoesnotexistontheincidentenergy,forthelatterapartfromanegligibleamountisre-radiated,justasaman,inspiteofcontinualabsorptionofnourishment,maintainsaconstantbody-weight.Ourconditionsofexistencerequireadeterminatedegreeoftemperature,andforthemaintenanceofthisthereisneedednotadditionofenergybutadditionofentropy.“AsastudentIreadwithadvantageasmallbookbyF.Wald25entitled‘TheMistressoftheWorldandherShadow’.Thesemeantenergyandentropy.Inthecourseofouradvancingknowledgethetwoseemtometohavechangedplaces.Inthehugemanufactoryofnaturalprocesses,theprincipleofentropyoccupiesthepositionofmanager,foritdictatesthemannerandmethodofthewholebusiness,whilsttheprincipleofenergymerelydoesthebook-keeping,balancingcreditsanddebits.R.Emden.Kempterstrasse5,Z¨urich.”SommerfeldrefinedEmden’sargument,concludingthattheenergydensityofaroomdoesnotinfactremainconstant,butactuallydecreaseswithheating—whichgoestoshowallthemorethevalidityoftheconclusionthatentropyplaystheleadingroleoverthatofenergy.“Themistressoftheworldandhershadow”ThebookbyWaldthatEmdenreferstoisnottheonlybookwiththistitle.AsachildIreadabookwiththesametitlebyBertholdAuerbach.26Thankstopopularizerssuchmetaphorshaveretainedtheircurrency.ButEmden’sreversalofthemetaphoriscorrect.Energyiscalled“mistressoftheworld”becauseeverythingthathappensintheworlddoessoviachangesofoneformofenergyintoanother.EinsteinshowedthataquantityofmatterofmassmisequivalenttoanamountofenergygivenbyE=mc2,wherecisthespeedoflight.Energyisinanessentialwaycontainedinthema-terialoftheworld—matterandfields.Mostofthisenergyisliberatedandusedonlyincertainofthetransformationprocessesofatomicnuclei—whenceatomic25FrantiˇsekWald(1861–1930),Czechchemist.Trans.26BertholdAuerbach(1812–1882),German-Jewishpoetandauthor.Trans. 60Chapter3.Entropyandfreeenergyenergy.Theotherformsofenergy—potentialandkinetic,thermalandchemical,electricalandmagnetic—arethedirectsourcesoftheworkcarriedoutinnatureandtechnology.Workisdonewhenoneoftheseformsofenergyistransformedintoanother.Entropyiscalled“theshadow”ofthemistressoftheworldbecauseitcanbeusedasameasureofthedepreciationofenergy,ifweunderstandthevalueofenergytolieinitsavailabilityfortransformationintousefulwork.Aswesawearlier,themaximumamountofusefulworkisequaltothedecreaseinHelmholtzfreeenergy(see(3.4)).ButtheHelmholtzfreeenergyisthechangeininternalenergylesstheheatabsorbed(see(3.5)):F=E−TS.Thusthegreatertheheatabsorbed,or,equivalently,thegreatertheincreaseinentropy,thelessenergyavailablefordoingusefulwork,thatis,thelessvaluabletheinternalenergyE.Asmentionedearlier,inpurelymechanicalprocessesalloftheenergyisavailablefordoingwork,butinprocessesinvolvingheatexchange—suchasmechanicalprocesseswherefrictionoccurs—someoftheenergyistransformedintoheat,thatis,intoentropytimesthetemperature.WhywasEmdenright?Answer:Becausethedirectionofflowofallrealprocessesisdeterminedbythedi-rectionofchangeinentropy.Aswesaidearlier,allrealprocessesareirreversible,sothatinanisolatedsystemtheywillproceedinthedirectionofincreasingentropy.However,thisdoesnotmeanthatentropycannotdecrease.Thelawofincreasingentropyholdsonlyforisolatedsystems.Inopensystems,thatis,insystemsinwhichmatterandenergyareexchangedwiththesurroundingmedium,thesitua-tionisverydifferent.Allphenomenaofthebiosphere,thatis,occurringinlivingnature,involvechangesinentropy.Undernormalconditions,themassandsupplyofenergyof,forinstance,ahumanorganismremainconstant;theyareconstantlymaintainedthroughbreathingandeating.However,thisreplenishmentinvolvesadecreaseinentropyratherthananincreaseinenergy.AnimportantconsequenceofthesecondlawisthepositionsetoutinEmden’sarticle,alsoformulatedinEr-winSchr¨odinger’sfamousbook27Whatislife(fromthepointofviewofphysics)?:Alivingorganismfeedsonnegativeentropy.InChapter6weexplainwhatthismeansexactly.ThusEmdenthinksthatweshouldinterchangetheplacesofentropyandenergy,shadowandmistress—justasinHansChristianAndersen’sfairytale“Theshadow”,turnedintoamarvelousplaybyEvgeni˘ıSchwartz.28Infactthemetaphorofmistressandshadowisofnogreatsignificance.Itwouldperhapsbebettertoabandonitandrefertoenergyandentropyneitheras27PublishedasWhatislife?in1944.Trans.28Evgeni˘ıSchwartz(1896–1958),Russianplaywright.Trans. 61mistressnorshadow.Whatwedoretainisthediscoverythatentropyisjustasimportantasenergy,andespeciallysoincosmologyandbiology—whichiswhytimeflowsthewayitdoes,fromthepastintothefuture. Chapter4EntropyandprobabilityIfyouaspiretoconceiveapotentideaInthenameoftheadvancementofknowledge,YoumustfirstcarryoutAstatisticalsummation.Thiswillhelpyouineverything,Itwillrevealalightedpathinthegloom.But,tryasyoumight,withoutitTheessenceofphenomenawillremaininaccessible.Boltzmann’sformulaThusfarwehavestudiedonlyphenomenologicalphysics,thatis,thermodynamics,whereoursystemsaredescribedintermsofstatefunctionssuchasenergy,en-thalpy,entropy,andfreeenergy.Inparticular,wediscoveredthatentropyincreasesinspontaneouslyevolvingprocesses.Butwhyshouldthisbeso?TheanswertothisquestioniscontainedinBoltzmann’sformulaS=klnP,(4.1)wherePdenotestheso-called“statisticalweight”1ofthecurrentstateofthesysteminquestion,andkisaconstantcalledBoltzmann’sconstant.ItistheratioofthegasconstantR=8.31joules/(mole·◦K)totheAvogadronumber(thenumberofmoleculesinamoleofgas)N=6.06·1023permole:AR−23◦k==1.38·10joules/K.(4.2)NAAlthoughitisnotpossibletogiveafullyrigorousderivationofthiscelebratedformulainsuchapopularaccountasthis,weshallnonethelessattempttoshow1Definedbelow.Trans. 64Chapter4.Entropyandprobabilityhowtheentropyofagasinagivenstateandtheprobabilityofthatstatemustberelatedbysuchaformula.FirstwemustunderstandwhatismeantbyP,the“statisticalweight”ofthestateofthesystem.Clearly,thiscannotbetheusualprobabilityofoneoutcomeoutofseveralpossibleoutcomes,suchaswhenadieiscastandtheprobabilityoftheoutcome3isq=1/6.2Notethattheprobabilityislessthan1,asitmustbesincethesumoftheprobabilitiesofeachoftheoutcomes1,2,3,4,5,6mustbe1.3Forsimilarreasons,probabilitiesaregenerallylessthan1,andsincethelogarithmofa(positive)numberlessthan1isnegative,the“statisticalweight”Pappearinginformula(4.1)cannotbeordinaryprobability.Thestatisticalweightofastateofasystemisdefinedtobethenumberofwaysthatthestatecanberealized.Sincethestateofathrowncubicaldieisthesameregardlessofthenumeralsonitsfaces,thestatisticalweightofacastdieis6,P=6.Ifwethrowtwodice,thenthenumberofoutcomesis6·6=36.Theproba-bilityof,forinstance,obtaininga3ononedieanda4ontheother4isequaltotheproductofthetwoseparateprobabilities,sincethesetwooutcomesareinde-pendent.5ItfollowsthatherethenumberP,againinviewoftheindependenceofthetwoevents,isequaltotheproductofthenumbersP1andP2:P=P1·P2.(4.3)Boltzmann’sformulaisthenmadeplausiblebythefactthatityieldstheadditivityofentropiesdirectly:S=klnP=kln(P1P2)=klnP1+klnP2=S1+S2.(4.4)Supposenowthatwehavefourmoleculesdistributedamongtwoboxes,asinFigure4.1.Howmanydifferentstatescantherebe?Ifweassumethemoleculesidentical,thenclearlythenumberofstatesisfive:4|0,3|1,2|2,1|3,0|4.Thenumberofwaysinwhichthesestatescanberealized,thatis,theirstatisticalweights,aredifferent.Ifweassumethemoleculesdistinguishablebybeingdifferentlynumberedorcolored,thenitcanbeseenfromFigure4.1thatthestatisticalweightsarerespectively1,4,6,4,1.Themostlikelydistributionistheuniformone2|2,withtwomoleculesineachbox.2GivenanexperimentwithafinitenumberNofmutuallyexclusiveandequallylikelyout-comes,theprobabilityofaneventA,thatis,asubsetconsistingofnoutcomes,isdefinedtoben/N.Iftheprobabilityspaceofallpossibleoutcomesisinfinite,thenanappropriate“probabil-itymeasure”hastobedefinedonitsothatthemeasureofthewholespaceis1,andthentheprobabilityofanevent,thatis,ameasurablesubset,isjustitsmeasure.Trans.3Thatis,theprobabilitythatsomeoutcomeoccursmustbe1.Trans.4Herethediceareassumeddistinguishedfromoneanother.Trans.5TwoeventsAandBaresaidtobeindependentifprob(A∩B)=probA·probB,thatis,iftheprobabilityofbotheventsoccurringisequaltotheproductoftheirseparateprobabilities.Thiscapturesinpreciseformtheideathattheoccurrenceofeithereventshouldnotchangetheprobabilityoftheother.Trans. 65Box1Box2Figure4.1:Thepossiblestatesoffourparticlesdistributedovertwoboxes.Wheredothenumbers1,4,6,4,1comefrom?Theyarethebinomialcoeffi-cients4!4!4!4!4!=1,=4,=6,=4,=1.4!0!3!1!2!2!1!3!0!4!WeremindthereaderthatforanypositiveintegerN,thesymbolN!,read“N-factorial”,denotestheproductofallnumbersfrom1toN:N!=1·2·3···N,and0!=1.Thus1!=1,2!=2,3!=6,4!=24,andsoon.InthegeneralcaseofNmoleculesdistributedovertwoboxes,thenumberofdifferentwaysofobtainingthedistributionN1|N2,whereN1+N2=N,isN!N!P==.(4.5)N1!N2!N1!(N−N1)!Thisisnotdifficulttosee:thenumeratorN!isthetotalnumberofpermuta-tions(orderedlineararrangements)oftheNmolecules.IfweimaginethefirstN1moleculesineacharrangementasbeinginthefirstbox,andtherestinthesecond,thensincetheorderwithineachboxisimmaterial,andthereareN1!N2! 66Chapter4.EntropyandprobabilitypermutationsfixingeachpartitionoftheNmoleculesintotwosubsetsofsizesN1andN2respectively,weneedtodividebyN1!N2!.WenowuseBoltzmann’sformulatocomputethechangeinentropyduetothemixingoftwogasesconsistingofN1andN2moleculesrespectively.Beforebeingmixed,thegasesareassumedtobeseparatedbyapartition.Wedistinguishtheirstatesonlybylocation:theN1moleculesofthefirstgasareallinthelefthalfofthecontainer,andtheN2moleculesofthesecondintherighthalf.TherespectivestatisticalweightsarethenP1=N1!,P2=N2!,and,sinceentropiesareadditivefortheunmixedsystem,itsentropyisgivenbyS=S1+S2=k(lnN1!+lnN2!).Whenthepartitionisremoved,thegasesmix.TheentropyofthemixtureisS=kln(N+N)!=klnN!.12ThemixingentropyisthereforeN!ΔS=S−S=kln.(4.6)N1!(N−N1)!Thusthemixingentropyisgivenbyktimesthelogarithmoftheexpression(4.5),soinordertoestimateitweneedagoodapproximationtothefactorialsoflargenumbers,whereby“large”wemeanofanordermuchlargerthan1.Stirling’sformulaFromN!=1·2·3···N,weinferthatNlnN!=ln1+ln2+ln3+···+lnN=lni.(4.7)i=1ThefunctionlnNincreasesmoreandmoreslowlywithincreasingN,sincethedifferenceln(N+1)−lnN=ln(1+1/N)decreaseswithincreasingN.HenceforlargeNtheareaunderthegraphofy=lnxaffordsagoodapproximationoflnN!:NlnN!≈lnxdx.(4.8)1Thisintegralcanbeevaluatedusing“integrationbyparts”.Werecallhowthisisdone.Forfunctionsuandvofx,wehaved(uv)=udv+vdu. 67Hencebbbbudv=d(uv)−vdu=uv|b−vdu.aaaaaInourcase,wetakeu=lnxandv=x,obtainingNNNlnxdx=xlnx|1−xd(lnx)11Ndx=NlnN−x=NlnN−(N−1).1xSinceNandNlnNarelargecomparedwith1,wecanneglecttheterm−1.ThusweendupwiththeapproximationNlnN!≈NlnN−N=Nln,(4.9)eyieldinginturntheroughapproximationNNN!≈.(4.10)eAmoreaccurateapproximationisgivenbyStirling’sformulaN1/2NN!≈(2πN).(4.11)eTakinglogarithms,weobtain11lnN!≈NlnN−N+lnN+ln2π,(4.12)22whichisnotsomuchbetterthan(4.9),sinceforlargeNin(4.12)wecanneglecttheterms1lnNand1ln2πincomparisonwithNlnNandN.Thuswemayuse22(4.9)asanapproximationoflnN!.Havingsettledonanapproximationofthelogarithmoffactorials,wereturntotheformula(4.6).Thatformulayields,viatheapproximation(4.9),ΔS≈k(NlnN−N−N1lnN1−N2lnN2+N1+N2)=k[(N1+N2)ln(N1+N2)−N1lnN1−N2lnN2],orN1+N2N1+N2ΔS≈kN1ln+N2ln.(4.13)N1N2Ifweexpressthenumbersofmoleculesintermsofthenumbersn1,n2ofmoles:N1=NAn1,N2=NAn2, 68Chapter4.EntropyandprobabilitywhereNAisAvogadro’snumber,(4.13)becomesn1+n2n1+n2ΔS=kNAn1ln+n2ln.(4.14)n1n2SincekNA=R,wehavearrivedviaBoltzmann’sformulaattheformerexpression(2.24)forthemixingentropy,derivedbymeansofacompletelydifferentargument.ThemeaningofBoltzmann’sformulaThuswehaveprovidedconsiderableevidencefor,thoughofcoursenotproved,Boltzmann’sformula,expressingentropyasaconstanttimesthelogarithmofthestatisticalweightofthesystem.Agreatmanythingsfollowfromthis.Underequilibriumconditions,entropyisnotinanywayremarkable.Itisafunctionofthestateofthesystemwhichcanbemeasuredexperimentally(aswellascomputedtheoreticallyusingBoltzmann’sformula,asweshallseebelow).However,assoonasanisolatedsystemdeviatesfromequilibrium,aremarkablepropertyofentropyemerges,namely,itspropensitytoalwaysincreasetoamaxi-mum.ThispropertycanbedemonstratedusingtherelationbetweenSandthestatisticalweightP.WemayrewriteBoltzmann’sformulaintheexponentialformP=eS/k,(4.15)whichshowsthattheprobabilityofaparticularstateincreasesexponentiallywithitsentropy.(NotethattheentropyScanbedeterminedexperimentally,whilePcanbecalculatedfromitsdefinition.)Thustheincreaseinentropyinanirre-versibleprocessentailsachangetoamoreprobablestate.Thisisconfirmedbythefactthatadisorderedstateismorelikelythananorderedone.Supposewehaveinitially,asearlier,twobodiesatdifferenttemperatures.Thisimpliesacertainorder.Ifthetemperaturesofthebodiesareequalizedbymeansofheatconduction—theflowofheatfromthehottertothecolderbody—,thenthisorderisdestroyed.Thesamesortofthingoccurswhengasesorliquidsaremixed.Again,theunrestrictedexpansionofagassuchashydrogenorcarbondioxide,whenreleasedfromaballooncontainingit,representsincreasingdisorder.Whilethegaswastrappedintheballoon,itoccupiedlittlespace.Onbeingreleaseditexpandsfreely,forthesimplereasonthattheprobabilityisgreaterthatitshouldoccupyalargervolumethanasmaller,thatis,thatthestatisticalweightofthestateofthegasoccupyingalargervolumeisgreaterthanthatofitsstatewhenoccupyingasmallervolume.Foragivensystem,alessorderedstatehaslargerstatisticalweightsinceitcanberealizedinmorewaysthanamoreorderedstate.Ifnoconsciouseffortismadetoarrangethebooksandpapersonadeskneatly,theywillendupinastateofdisorder,asaresulttheirbeingrandomlymovedabout—randomly,sinceunsystematically. 69Intheseexamplesorderiscreatedartificially,whiledisorderarisessponta-neously,beingassociatedwithgreaterprobability,greaterentropy.Thusonemightsaythatentropyisameasureoftheamountofdisorderofastateofasystem.Theaimofpeople’sandanimal’srationalactivityistheovercomingofdis-order.Forexample,membersofaproductivefarmhavetostruggletothwartspontaneousprocessessuchassoilerosion,deteriorationofcrops,andsoon.Heretooisentropymoresignificantthanenergy.Fromallthisweseethatthesecondlawofthermodynamicsisofaquitedifferentsortfromthefirst.Thefirstlawholdsinallcases:energyisconservedininteractionsofelementaryparticlesjustasitisinmacroscopicsystems.Itisworthmentioningthatinthe1930stheideaarosethatenergymightbeconservedonlyonaverage,andnotnecessarilyincertainindividualprocessesinvolvingelementaryparticles.Asmightbeexpected,thisideaowedmuchtothenatureofthesecondlaw.TheAmericanphysicistShankland6,inhisinvestigationsoftheinteractionsofphotonsandelectrons,thoughtthathehadfoundevidenceforthenon-conservationofenergyinsuchbasicprocesses.EventhegreatNielsBohrthenallowedthepossibilitythatthelawofconservationofenergymightnotapplytoindividualeventsinthemicrocosm,butholdonlyonaverage.How-everShankland’sexperimentsweresoonshowntobeflawed,andsincethentherigorouslydeterministicnatureofthefirstlawhasremainedunquestioned.Incidentally,thislaw,togetherwiththelawofconservationofmomentum,servedasthebasisofagreatdiscovery,thatoftheneutrino.Thetheoryoftheα-decay7ofradiumwascreatedbyG.A.Gamow8in1928.Herenocontraventionofthelawofconservationofenergyarose.However,β-decay,involvingtheradioactiveemissionofanelectronorpositronfromanatomicnucleus,representedatthattimeaseeminglyinsolublepuzzle:theconservationlawsappearednottohold!Gamow,whowassomewhatofajokester,madearubberstampofaskullovertwocrossedβs(Figure4.2),whichheusedtostamponhisoffprints.Thissymbolizedthedifficultieswiththetheoryofβ-decay.ThisGordianknotwascutin1933byWolfgangPauli,whointroducedanewparticle,theneutrino,inordertosavetheconservationlaws.Theneutrinowaslaterdetectedexperimentally.Unlikethefirstlaw,thelawassertingtheimpossibilityofaperpetualmotionmachineofthesecondkind—thelawofincreaseofentropy—isnotdeterministic,butstatistical,probabilistic,sincetheimpossibilityofaperpetualmotionma-chineofthesecondkindisaconsequenceofitsimprobability.Letusestimate,forinstance,theprobabilitythatallmoleculesinacontainerofvolumeVspon-taneouslygathertogetherinonehalfofthecontainer.Theprobabilityofasinglemolecule’sbeingfoundin,say,therighthalfofthecontaineris1/2.IfthereareN6RobertSherwoodShankland(1908–1982),Americanphysicistandhistorianofscience.Trans.7Anα-particleisidenticaltoaheliumnucleus,consistingoftwoprotonsandtwoneutronsboundtogether.Trans.8GeorgeGamow(1904–1968),Russianphysicist,intheU.S.from1934.Trans. 70Chapter4.EntropyandprobabilityFigure4.2:Symbolofthetheoreticaldifficultiesassociatedwithβ-decay.moleculesinall,thentheprobabilitythatallNwillgathertogetherinthathalfofthecontaineristheproductoftheprobabilitiesfortheindividualmolecules,sincetheseareindependentevents.SupposethevolumeVisacubiccentimeter.Atordinarytemperatureandpressure,acubiccentimeterofagascontainsapproxi-mately2.7·1019molecules(Loschmidt’snumber).Hencethedesiredprobabilityisabout2.7·10191,2whichisvanishinglysmall.Violationofthesecondlawwouldrequireahighlyimprobableeventsuchasthistotakeplace—compressionofagaswithoutdoinganyworkonit.ThediscoveriesofBoltzmannandGibbs,thecreatorsofstatisticalphysics,heraldedascientificrevolution,abreakthroughintoanentirelynewfield.Ofcourse,asisalwaysthecaseinscience,thisrevolutiondidnotariseoutofnowhere.ThegroundwaspreparedbyGibbs’andBoltzmann’spredecessors:Carnot,Clau-sius,Thomson,andMaxwell.Themaininnovationconsistedintheprobabilistic,ratherthandeterministic,natureofthenewstatisticallaws.Wesaidearlierthatitisimpossibleforthewaterinakettlesubmergedinabucketofwatertocometoaboilatthesametimeasthewaterinthebucketfreeze.Thesenseoftheword“impossible”here,isthatofvanishinglysmallprobability,notthattheeventinquestioniscategoricallyruledout—passingstrangethoughitwouldbeifitoccurred.Afteranextremelylargenumberoftrials,itmaycometopass.9Itisjustthataspontaneousdrop,ratherthanrise,inentropyisarare—extremelyrare!—occurrence.Weshalllookfurtherintothisissueinthesequel.Fornow,sufficeittosaythat,astheabovecalculationshows,inamacroscopicsystemexceptionstotheprobabilisticsecondlawareveryfewandfarbetween.Nevertheless,suchexceptionsoccur,andwewitnessthemeveryday—forexample,inthestatisticsbehindthebluenessofthesky.Weshallexplainthisinthenextchapter.9Infact,itfollowsfromPoincar´e’srecurrencetheoremandatheoremofLiouville,thatitisalmostcertainthattheunlikelyeventinquestionwilleventuallyoccur.See,forexample,thepopularbookEasyasπ?byO.A.Ivanov,Springer,NewYork,etc.,1998,pp.105–110.Trans. 71IntheclassicalaccountofstatisticalmechanicswrittenbyJosephMayerandhiswifeMariaGoeppert-Mayer10adiscussionoftheseissuesisprecededbythefollowingepigraph:“Never?”“No,never.”“Absolutelynever?”“Well,hardlyever.”ThefusionofacrystalandtheevaporationofaliquidWediscussedthefusionofcrystalsandevaporationofliquidsintheprecedingchapter.Wearenowinapositiontounderstandthephysicalmeaningofsuchphasetransitions.WesawinChapter3thatphasetransitionsinvolvechangesΔHinenthalpyandΔSinentropy.Suchtransitionsaresaidtobe“ofthefirstkind.”Whenacrystalmelts,itsdisorderingasitassumesliquidformmightbecalled“entropi-callyadvantageous.”Thedisorderedstateoftheconstituentatomsormoleculesismoreprobablethantheirorderedstate,sinceitcanberealizedbyagreaternum-berofarrangementsoftheseparticlesthantheorderedcrystallinestate.Hencetheentropyoftheliquidstateofthesubstanceisgreaterthanitsentropyinthecrystallinestate.Ontheotherhand,theenthalpyoftheliquidislessthantheenthalpyofthecrystal:inordertofusethecrystalonemustbreaksomeoftheinteratomicorintermolecularbonds.Thusthecrystallinestateisadvantageousenergy-andenthalpy-wise,butdisadvantageousentropy-wise.Fusionbeginspre-ciselywhentheentropiccontributiontothedifferenceinfreeenergiesofcrystalandliquidexactlycompensatestheenthalpiccontribution:TmeltingΔS=ΔH.AthighertemperaturesT>Tmelting,wehaveTΔS>ΔH,sothattheentropiccontributionexceedsΔH,andthemoreprobableliquidstateisthemorestable.AtlowertemperaturesTiWeshallnowsimplifytheright-handsideusingelementarymethods.ObservethatΔ2=ΔiisinceΔiiseither0or1.HenceΔ2=Δ¯i=q,iwhenceforthefirstsumontheright-handsideabove,wehaveNNNΔ2=Δ¯i=q=Nq.ii=1i=1i=1Nowweturntothedoublesum.WhatisΔiΔj?WehaveΔΔ=1·1·q2+2·1·0·q(1−q)+0·0·(1−q)2=q2.ijHence2N(N−1)2n=Nq+2q,2sincethereareN(N−1)/2pairs(i,j)withi0,dS>0;deS<0,but|deS|0;deS<0,and|deS|>diS,dS<0;deS<0,and|deS|=diS,dS=0.Thelasttwoofthesefourcasesarethemostinteresting.Theyrepresentrespec-tivelythesituationswhereanopensystem’sentropydecreasesbecausetheeffluxofentropyexceedsitscreationinternally,andwheretheeffluxofentropyandinternallycreatedentropyexactlybalanceout.Inthesituationwherethereisnochangeintheinternalentropy,andonlyheatexchangewiththeenvironmentoccurs,thendQdS=deS=.(6.3)TThisisjusttheformula(2.4).Herethemeaningoftheexpression“flowofentropy”isespeciallysimple:itisessentiallyjusttheflowofheat.Ontheotherhand,ifentropyisproducedinternally,thentheformulabecomesdQdS=diS+deS=diS+,(6.4)Twhichwealsoencounteredearlier(see(2.30))asaninequalityholdingforirre-versibleprocesses:dQdS≥,(6.5)T 115Figure6.1:Modelofquasi-equilibrialexpansionofagas.withequalitypreciselyifdiS=0.Aswesawinearlierchapters,entropyisalwaysgeneratedinsystemswithinwhichphysical,chemical,orbiologicalprocessestakeplace.Thisproductionofentropyproceedsatadefiniterate,aratethatisnevernegative,andbecomeszero,thatis,satisfiesdiS=0,(6.6)dtonlyunderconditionsofinternalequilibrium.Denotingbyσtheamountofentropyproducedinternallyperunittimeperunitvolumeofanopensystem,wehavediS=σdV≥0.(6.7)dtThequantityσiscalledthe(specific)dissipationfunctionofthesystem.Theformulae(6.6)and(6.7)differfundamentallyfromthoseofthermo-dynamicsandstatisticalmechanicswhichwederivedandworkedwithearlier:formulae(6.6)and(6.7)involvethetime!Wearenowconcernedwiththerateofgenerationofentropywithrespecttotime.Thissignifiesatransitionfromthermostatics,thatis,classicalthermodynam-ics,todynamicsandkinetics.Thermostaticsconcernsprocessesinequilibrium,orratherprocessesthattakeplacesufficientlyslowlyforequilibriumtobeattainedateachstage.Forexample,wemaymodeltheequilibrialexpansionofagasinthefollowingway:apistonmovinginacylindersupportsaloadconsistingofapileofsand(Figure6.1),fromwhichweremoveonegrainatatime.Formulae(6.6)and(6.7)relatetonon-equilibrialprocesses,andinthistheyrepresentabetterapproximationofreality.Afterall,innaturetherearenoequi-librialprocesses;suchprocessesaremereidealizations.Thetaskofphysicsistodeterminethefactorsonwhichtherateofgenerationofentropyandtheassociatedspecificdissipationfunctiondepend,andthenatureofthisdependence. 116Chapter6.OpensystemsThedissipationfunctionAsalreadynoted,entropyisgeneratedinallphysical,chemical,andbiologicalprocesses.Supposeasystem—forthemomentitisimmaterialwhetheropenorclosed—containstwobodiesatdifferenttemperaturesincontactwithoneanother.Itisclearthatinthissystemheatwillflowfromthehottertothecoolerbodyimmediatelyuponcontact,andthattheflowwillcontinueuntilthetemperaturesareequalized.Inthissituationentropyisgeneratedasaresultofheatconduction.Thus,torepeat,thedrivingforceintheproductionofentropyistemperaturedifference.Inthisconnectionitisappropriatetomentionthefollowingverygeneralphysicalprinciple:Onlydifferencesgiverisetoeffects,andtheseinturnyieldnewdifferences.Returningtoourtheme,weobservethattemperaturedifferencesgiverisetoanonzeroflowofthermalenergy.Inprocessesinvolvingthegenerationofentropy,somesortof“flow”isalwayspresent—inthesenseofachangeinsomephysicalquantity—togetherwitha“force”causingtheflow.Inthepresentcase,theflowisthatofenergy,namelydE/dt(whereEisthethermalenergyperunitvolume),andthe“force”isthetemperaturedifference.Howeversincethedependenceofentropyontemperatureisoneofinverseproportionality,itisratherthedifferenceofthereciprocalsofthetemperaturesthatisofsignificanceforentropy:11T1−T2−=,T1>T2.T2T1T1T2Thusinthecaseofheatconduction,itisnotdifficulttoseethatthedissipationfunctionhastheformdE11σ=−.(6.8)dtT2T1NotethatsinceσisthequantityofentropyproducedperunitvolumeperunittimeandEistheenergyperunitvolume,thedimensionsofthetwosidesofequation(6.8)coincide.Observealsothatonealwayshasσ>0.For,theflowofthermalenergydE/dtbetweenthebodiesattemperaturesT1andT2ispositiveifT20ifX>0,whenceL>0,andsimilarlyforL.Itiseasytoshow11111122(andwellknown)thatthequadraticformin(6.15)ispositivedefiniteifandonlyifL20butJ2X2<0,thentheconditionσ>0issatisfiedonlyifJ1X1>|J2X2|.Thishasthefollowingimportantconse-quence.ThenegativityofJ2X2meansthat,takeninisolation,thecorrespondingprocess(No.2)isimpossible,sinceitwouldinvolveadecreaseinentropy.Yetthefactthattheotherprocess(No.1),forwhichJ1X1>0,istakingplaceatthesametimeandproducingsurplusentropysufficienttomorethanmakeupforthatdecrease,allowsprocessNo.2toproceedafterallintheopensystem.Bywayofexample,weconsiderthesurprisingphenomenonofthermodiffu-sion.Supposewehaveavesselfilledwithahomogeneousmixtureoftwogases.Ifthetemperatureofthevesselisuniform,thenthemixturewillbeinequilib-rium,withitsentropyatamaximum.However,ifoppositewallsofthevesselareatdifferenttemperatures,thentherewilloccurapartialseparationofthegases:onewilltendtoaccumulatenearthewarmerwallandtheothernearthecooler.Thelossofentropyoccasionedbythisseparationismorethancompensatedbythegaininentropyduetotheheatflow.Theflow(inoppositedirections)ofthegaseousmatterandtheflowofthethermalenergyareinterconnected.Inthissectionwehavediscussedcertainaspectsoftheso-called“linearther-modynamicsofopensystems”,validclosetoequilibrium.Wehavebecomeac-quaintedwithspecialfeaturesofsuchsystems,inparticular,withthepossibilityofprocessesinvolvingalossinentropyactuallyoccurringasaconsequenceoftheirbeinginterconnectedwithother“entropicallyadvantageous”processes.Inthenextsection,weshalldiscussother,equallyimportantandinteresting,peculiaritiesofopensystems.Theareaofnon-equilibrialthermodynamicswasdevelopedintheworkofL.Osager,T.deDonder,3andI.Prigogine.4However,thebasicideasofnon-equilibriallinearthermodynamicshadbeendescribedearlierintheworkofL.I.MandelshtamandM.A.Leontovichentitled“Towardsatheoryoftheab-sorptionofsoundinliquids”,publishedin1937.AnastronautlivesonnegativeentropyThisisallveryinteresting,important,anduseful,buthowcanoneapplythebasiclawsofthermodynamicstoopensystems?Afterall,thoselawsholdonlyforisolatedsystems.Theanswerisrathersimple:Alongwiththeopensystemoneisstudying,onemustconsideritsimmediateenvironment—includingallsourcesofmatterandenergyimpingingonthesystem—,andimaginetheresultingenlargedsystemtobeseparatedfromthesurroundingworldbymeansofanadiabatic,impermeable3Th´eophiledeDonder(1872–1957),Belgianmathematicianandphysicist.4IlyaPrigogine(1917–2003),Russian-born,Belgianchemist.1977Nobellaureateinchemistryforhisworkinnon-equilibrialthermodynamics. 120Chapter6.Opensystemsshell.Thenwecanapplythelawsofthermodynamicstothisenlargedsysteminordertoderiveimportantresults.Agoodmodelofthisisaffordedbyanastronautinthecabinofaspaceship.Here,indeed,noefforthasbeensparedtoisolatetheinteriorofthespaceshipfromthesurroundingspace.Theastronautissecureinthecabin,withasupplyoffood,water,andair.Let’scalculatethebalanceofentropy.Takenalone,theastronautrepresentsanopensystem.AnyinfinitesimalchangedS(a)inhisorherentropysatisfies(a)(a)dS=diS+deS,wheredeSistheentropiccontributionfromthesurroundingmedium(theinteriorofthespaceship)viaanyinterchangeofthermalenergyormatter.HencethechangeintheentropyofthemediumsurroundingtheastronautisdS(m)=−dS.eThetotalchangeinentropyoftheastronauttogetherwiththeinteriorofthespaceshipis(a)(m)(a)dS=dS+dS=diS>0,(6.17)where,onceagain,theinequalityobtainsbyvirtueofthesecondlaw.Thustheentropyofthecombinedsystemofastronautandspaceshipincreasesbyanamountequaltotheentropygeneratedbytheastronaut’sorganism.Nowanastronautmustbeahealthyyoungperson,sothathisorherstatewillremainthesameforthewholeoftheflight.Howeverthoughstationary,thisstateisnotoneofequilibrium.Thismeansthattheentropy(aswellasthemassandenergy)oftheastronautremainsunchanged,whencedS(a)=dS(a)+dS=0.(6.18)ieSinceby(6.17),dS(a)>0,itfollowsthatdSmustbenegative.Weconcludeiethatinastationaryopensystemtheentropygeneratedwithinthesystemmustbeexactlybalancedbytheamountthatflowsoutofthesystem.Infact,itcanbeshownthattheentropyofthesubstancesexcretedfromalivingorganismactuallyexceedstheentropyofthesubstancesitconsumes.In1944,E.Schr¨odinger,oneofthefoundingfathersofquantummechanics,wroteasmallbookentitledWhatislife?(Wementionedthisbookearlier,attheendofChapter3).Theappearanceofthisbookwasaneventofconsiderableimportance,sincetheideasSchr¨odingerexpressedinitplayedasignificantroleinthedevelopmentofmodernbiology.Inparticular,thebasicaspectsofthethermodynamicsofliving(andthereforeopen)systemsarediscussedinthebook.Whatdoesanorganismliveon?Weareusedtotalkingofthenumberofcaloriesweconsumewithfood.Doesthismeanthatwefeedoncalories,con-stantlyaddingenergytothatalreadypresentintheorganism?Ofcoursenot!If 121anorganismisinastationarystate,thentheamountofenergypresentinitisconstant.ToquoteSchr¨odinger:“Whatthenisthatprecioussomethingcontainedinourfoodthatkeepsusfromdeath?Thatiseasilyanswered.Everyprocess,event,happening—callitwhatyouwill;inaword,everythingthatisgoingoninNaturemeansanincreaseoftheentropyofthepartoftheworldwhereitisgoingon.Thusalivingorganismcontinuallyincreasesitsentropy—or,asyoumaysay,producespositiveentropy—andthustendstoapproachthedangerousstateofmaximumentropy,whichisdeath.Itcanonlykeepalooffromit,thatis,alive,bycontinuallydrawingfromitsenvironmentnegativeentropy—whichissomethingverypositiveasweshallimmediatelysee.Whatanorganismfeedsuponisnegativeentropy.Or,toputitlessparadoxically,theessentialthinginmetabolismisthattheorganismsucceedsinfreeingitselffromalltheentropyitcannothelpproducingwhilealive.”Consumptionofnegativeentropymeansexcretionofmoreentropythanen-terstheorganism,signifyinginturnthemaintenanceofastationarystatethroughaneffluxofentropy.NotethattheviewsofEmdenconcerningheatingaroom(seethefinaltwosectionsofChapter3)arefullyanalogoustothoseofSchr¨odingerquotedabove.Astationarystateispossibleonlyinanopensystem;suchastatemightbetermeda“flowingequilibrium”.Anopensysteminastationarystatehasanumberofspecialfeatures.Earlier(inthethirdsectionofChapter2)wemodeledachemicalreactionbytheflowofaliquidfromonecontainertoanother.Weshallnowrepresentanopenchemicalsystembyamodelofasimilarkind(Figure6.2).Thismodelisflowing(or“steadystate”)sincetheliquidiscontinuouslyreplenishedfrombelowintheuppervesselandflowscontinuouslydownwardsandoutofthelowervessel.5Asbefore,theroleofthecatalystisplayedbythepipeconnectingthevessels,regulatedbymeansofatap.Ifthesystemwere,asearlier,isolated,6thenthefinalstateofthesystem,representedbythelevelofliquidinthelowervessel,wouldbeindependentoftheextenttowhichthetapisopen—thecatalystaffectsonlytherateofthereaction,notthefinaloutcome.Bycontrast,inthepresentopensystemtheextenttowhichthetapsareopendeterminesnotonlytheratebutalsothelevelofliquidinthetwocontainers,thatis,thesteadystateoftheprocess.ItwasshownbyPrigoginethatifanopensystemisinastationarystateclosetoanequilibriumstate,thenthedissipationfunctionisatitsminimumvalue.Inotherwords,theamountofentropyproducedinanear-equilibrialstationarystateislessthanthatproducedinotherstatesofthesystem.Therefore,asthesystemapproachesastationarystate,itsdissipationfunctiondecreases:dσ<0,dt5Andisthenpumpeduptothehighervessel.Trans.6Withtheliquidintheuppervesselnotbeingreplenished,andliquidnotflowingoutofthelowervessel.Trans. 122Chapter6.OpensystemsFigure6.2:Modelofachemicalreactioninanopensystem.σtFigure6.3:Dependenceofthedissipationfunctionontime:approachtoastation-arystate. 123and,finally,takesonaminimumvalue,wheredσ=0,dtasshowninFigure6.3.Oncereached,thestationarystateisstable;alinearsystemwillnotleavesuchastatespontaneously.Thesteady-statesystemshowninFigure6.2behavesinthisway:Forgivendegreesofopennessofthetaps,thelevelsofliquidinthetwovesselswillachievestability,providedtheliquidcontinuestoflowforlongenough,andthenforaslongasitcontinuestoflow.Thestateofthebiosphereasawholecanberegardedasstationary.Thedestabilizingeffectofhumanactivityhasnotyetsignificantlycausedanychangeinthetemperatureorcompositionoftheatmosphere.7Therealizationofastationarystateinanopensystemrequirestwotimescales,thatis,thepresenceofafastprocessandaslowone.Weexplainwhatwemeanwiththefollowingsimpleexample:Supposewehavetwobodiesattemper-aturesT1andT2,thatareconnectedbyafinecopperwire,agoodconductorofheat.Heatwillflowalongthewireuntilthetemperaturesofthebodiesbecomeequalized.Thefasttimescaleisrepresentedbythecopperwire,whichquicklyentersastationarystateduringwhichtherateofflowofheatthroughacross-sectionofthewireisconstant.Thesecond,slow,timescaleisrepresentedbytheprocessofequalizationofthetemperatures.Ofcourse,ifwemaintainthetemperaturesofthetwobodiesatT1andT2(bymeansofaheaterandcooler,say),thenequalizationofthetemperatureswillnotoccur.Thestationarystateofthebiosphereprovidesanothersuchexample.Thisstatearoseandpersistsasaresult,ultimately,ofthestreamofradiationfromthesun.Thisstatewasestablishedrelativelyrapidly8asaconsequenceoftheappear-anceofgreenplantgrowth;photosynthesisintheseorganismsbroughtabouttheoxygenationoftheatmosphere.9Ontheotherhand,the“slow”timescaleishererepresentedbythemuchslowercosmicprocessoftheburningoutofthesun.Weshallnowimaginethecrewofourspaceshiptoconsistofababyandanoldman.Neitherofthesetwoorganismsisinastationarystate.Asthebabygrows,sodoitsmassandenergy,andeventhedegreeofitsorganicorder.Thus7Ifthiscouldbesaidin1986(thedateofpublicationoftheoriginalofthisbook)—whichisdoubtful—,itcertainlycannotbeseriouslymaintainednow.Globalwarmingduetothe“green-houseeffect”ofvastaccumulationsofcarbondioxideintheatmosphere,andthecountervailingdimmingofthesunlightfallingonmostplacesonearthduetoatmosphericpollution,arecon-cernsoneverybody’smindin2008.Trans.8Thatis,overaperiodofmorethanabillionyears.Trans.9Theprocessofoxygenationoftheatmosphereisnowthoughttobemuchmorecomplicated,involvingso-calledmonocellular“blue-greenalgae”datingfromabout2.5billionyearsago—whentheatmosphericoxygenhadreached10%ofitspresentlevel—andotherevenoldersingle-celledorganisms,toabout600millionyearsagowhenthefirstmulticellularorganismsappeared,andwhentheatmosphericoxygenhadreachedabout90%ofitspresentlevel.SeeEncyclopediaBritannica.Trans. 124Chapter6.Opensystemsthebaby’sentropyisdecreasing:(baby)(baby)dS=diS+deS<0,(6.19)thatis,owingtoamorepowerfulmetabolism,theeffluxofentropyexceedstheamountproduced,andthebabyabsorbsmore“negativeentropy”.Ontheotherhand,agingisaccompaniedbyanincreaseinentropythatisnotbalancedbyitseffluxintothesurroundings:dS(oldman)=dS(oldman)+dS>0.(6.20)ieAsanelderlyphysicistoncewrote:Entropy’sconsumingmeBitbybitandaltogether.MylastdullyearsaremeasuredoffGrambygramandmeterbymeter.Inthecaseofalivingorganism,entropyattainsitsmaximumintheequilib-riumstate—otherwiseknownasdeath.Whydocellsdivide?Elementaryconsiderationsofthethermodynamicsofopensystemsallowustounderstandwhyalivingcelldivides.Acellisanopensystem,sothebalanceofanychangeinentropyisgivenby(6.1),thatis:ΔS=ΔiS+ΔeS.Forthesakeofsimplicity,weshallassumethatacellisasphereofradiusr.TheamountofentropyΔiSproducedperunittimeinsidesuchasphereisproportionaltoitsvolume4πr3,andtheeffluxΔSofentropyisproportionalto3eitssurfacearea4πr2.Hence432ΔS=A·πr−B·4πr,3whereAandBareconstantsofappropriatedimensions.Asthecellgrows,rin-creases,andwhenr=3B/A,thecellentersastationarystate:ΔS=0.ForsmallervaluesofrwehaveΔS<0,thatis,theeffluxofentropyexceedstheamountgeneratedwithinthecell,andthecellcangrow.However,forr>3B/A,wehaveΔS>0,whichmeansthatsubstanceswithredundantentropyareaccu-mulatinginthecellandcausingittooverheat.Hencewhenrreachesthevalue3B/A,thecellmusteitherdivideorperish.Ondivision,whilethetotalvolumeremainsunchanged,thecombinedsurfaceareaofthetwodaughtercellsisgreaterthanthatofthemothercell.Thiscanbeseenasfollows.Denotingbyrtheradius 125ofeachdaughtercell,theequalityofvolumesbeforeandafterthedivisionyieldsr3=2r3,whencerr=√.32Thenewentropychangeperunittimeis832ΔS=A·πr−2B·4πr,3√3andwhenr=3B/A,wehaver=3B/(A2),whenceB3√3ΔS=36π(1−2)<0.A2Asaresultofdivision,theeffluxofentropyexceedstheamountgeneratedwithinthecell,bythefactor|ΔeS|√3=2≈1.26.ΔiSAlthoughthermodynamicsexplainswhyacellmustdivide,ittellsusnothingabouttheactualmechanismofthisextremelycomplicatedprocess.Thermody-namicsisaphenomenologicalscience.Thegrowthofalivingorganism,whichalwaysconsistsofcells,isfundamen-tallydifferentfromtheformationofacrystalfromtheliquidstateorasolution.Thecell-divisionandresultinggrowthofanorganismaredirectlyrelatedtotheeffluxofentropyintothesurroundingmedium.Suchprocessesarenon-equilibrial.Ontheotherhand,thegrowthofacrystalisanequilibrialprocessthattakesplacewhenthefreeenergiesofcrystalandliquidhavebecomeequal,andthusamountstoanequilibrialphasetransition.Biologicaldevelopment,whileitresemblesphasetransition,isfundamentallynon-equilibrial.FarfromequilibriumThedevelopmentofanembryoand,asnotedabove,thesubsequentpost-partumgrowthofthebaby,involveanincreaseinorder.Theembryostartsasasinglefer-tilizedcell—anovum—anditsfurtherdevelopment—ontogenesis—isaccompaniedbymorphogenesis,thatis,theformationofvariousspecificstructures—tissuesandorgans.Anotherexampleofstructure-formationisaffordedbytheriseofgalaxiesandstarsinthecosmos.Suchprocessesnecessarilyinvolvedecreasesintheentropyoftherelevantopensystems,thatis,theexportofentropyintothesurroundingmediuminaccordancewiththeinequality(6.19):deS<0,|deS|>diS>0.(6.21)Theseconditionscanholdonlyinstatesfarfromequilibrium—sincethetermdiSdominatesnearequilibrium;theextremecaseherewouldberepresentedbya 126Chapter6.Opensystemsstationarystate.Forexample,ourastronaut(seeabove)wasyoungandhealthy,andveryfarfromequilibrium.Inorderforstructure-formationtotakeplaceinanopensystem,thatis,foraradicalincreaseinordertooccur,theexportofentropymustexceedacertaincriticalvalue.Fortheexportofentropytoexceeditsinternalproduction,asortof“en-tropypump”isneeded,topumpentropyoutoftheopensystem.Sucha“pump”canworkeitherexternallyorinternally.Weshallnowfindthethermodynamicconditionsfortheeffectiveworkingofsuchapump.Aninfinitesimalchangeinthe(Helmholtz)freeenergyofanopensystematfixedtemperatureandvolumeisgivenby(seeequation(6.1)andthediscussionprecedingandfollowingit)dF=diF+deF=dE−TdS=diE+deE−TdiS−TdeS.NowdiE=0sincetheenergyofthesystemcanchangeonlyviaaninterchangeofenergywiththesurroundingmediumandnotinternally.HencedeF=deE−TdeS=dE−TdeS,andthenconditions(6.21)implythatdeF>dE+TdiS.(6.22)Thusinorderforanexportofentropytooccur,thatis,aneffluxfromthesystem,anamountoffreeenergyisneededgreaterthanthechangeininternalenergy10plusthecontributionresultingfromtheproductionofentropywithinthesystem.Ifweareconsideringinsteadaprocesstakingplaceatconstantpressureratherthanconstantvolume,thenitisappropriatetousetheGibbsfreeenergyinplaceoftheHelmholtzfreeenergy.TherelevantinequalityisthendeG>dH+TdiS.(6.23)Ifthesystemisinastationarystate,thenitsinternalenergydoesnotchange(andthereforeneitherdoesitsenthalpy):dE=dH=0,andtheaboveinequalitiesbecomedeF=deG>TdiS>0,(6.24)showingthatoneneedstosupplyfreeenergytothesysteminordertomaintainthestationarystate.10Thatis,theinternalenergyofthesystemtogetherwithitsactiveenvironment.Trans. 127T2T1Figure6.4:HowB´enardconvectionarises.“B´enardconvection”11affordsabeautifulexampleofstructure-formationunderrelativelysimpleconditions.Wehaveashallowvesselcontainingaviscousfluidsuchassiliconeoil.Weheatthevesselstronglyfrombelow,therebycausingatemperaturedifferenceΔT=T1−T2>0,betweentheloweranduppersurfacesoftheliquid.AslongasΔTissmall,theliquidremainsuntroubled,andheatistransferredfromthebottomtothetopbyheatconduction.Then,atacertaincriticaltemperaturedifferenceΔTcr,thereisasuddenchangeinthebehavioroftheliquid:convectionsetsin(Figure6.4)andtheliquidseparatesitselfintohexagonalcells(Figure6.5).Theresultisveryattractiveandtrulyremarkable:asaresultofheatingtheliquidtherearisesadynamic,organizedstructure,resemblingacrystallineone.Itiscreatedbythesimultaneouscooperativemotionsofthemoleculesoftheliquid.Figure6.6showsthedependenceoftherateofheattransferdQ/dtonthetemperaturedifferenceΔT.AtthecriticaltemperaturedifferenceΔTcrthereoc-cursasuddenchangeinthedependence,signallingtheformationofacellularstructure.Sincethissystemabsorbsheatfromitssurroundings,therateofflowofentropythroughtheexposedsurfaceoftheliquidisgivenbytheformula(see(6.3))deSdQ11dQT2−T1=−=<0.(6.25)dtdtT1T2dtT1T2Henceundertheseconditionsthesystemexportsentropy.Understationarycon-ditionsthisexportoreffluxofentropyexactlybalancestheamountofentropygeneratedwithintheliquidthroughinternalfrictionandheatconduction.AmoredetailedanalysisshowsthatthesurfacetensionoftheliquidplaysanessentialroleinB´enard’seffect.Prigoginecalledopensystemsthatarestructure-forming,self-organizing,andfarfromequilibriumdissipative.Suchsystemsformspatially(seebelow),aswellastemporally,stablestructuresasacertainparameterpassesthroughacriticalvalue.InthecaseofB´enard’seffect,theparameteristhetemperaturedifference.Anexampleofacompletelydifferentsortisaffordedbylasers.Forthesakeofconcreteness,weshallconsideronlysolidstatelasers,forexamplerubylasers.Arubylaserconsistsofacylindricalrubyrodwhoseendsaresilvered.Apulsatingxenonlampisusedfor“opticalpumping”ofthelaser:lightfromthexenonlamp11HenriB´enard(1874–1939),Frenchphysicist. 128Chapter6.OpensystemsFigure6.5:B´enardconvection.dQdtBénardcellsstableliquidΔTcrΔTFigure6.6:Thedependenceoftherateofheattransferonthetemperaturediffer-ence. 129KLKradiationRSIPFigure6.7:Diagramofarubylaser:Risasyntheticrubyrod,Kthecover,KLaxenonflashlampforopticalpumping,Sacapacitor,andIPasourceofconstantvoltage.LaserenergyLampEmittedradiationThresholdExcitationenergyFigure6.8:Thedependenceoftheemittedradiantenergyontheexcitationenergyofthelaser.isabsorbedbytherubyrodmainlyatwavelengthsaround410and560nanome-tersintwoabsorptionbands.Theopticalpumpingcausespulsatingradiationofwavelength694.3nmtobeemittedthroughtheendsoftherod(Figure6.7).ThexenonlampopticallyexcitestheionsCr3+oftrivalentchromiumrespon-siblefortheabsorptionspectrumofrubies,thatis,fortheircolor,andthesethenemitpulsesoflightafewmetersinlength.Eachpulselastsabout10−8seconds.Themirrors,thatis,thesilveredendsoftherubyrod,emittheradiationparalleltotherod’saxis.Atsmallamountsofopticalpumping,thelaseractslikealamp,sincetheemissionsfromtheindividualexcitedionsdonotcohere.However,atacertaincriticalvalueoftheenergyofdischargeofthexenonlampandacritical(threshold)valueofthefrequencyofpulsation,thepowerofthelaseremissionin-creasesabruptly.Theirradiatedionsnowgiveofflightcoherently—incooperation,asitwere—,emittingwavesinphasewithoneanother.Thelengthsofthepulsesgrowsto108or109meters,andtherubylasergoesoverfromaregimeofordinaryradiation,asfromalamp,tooneoflaserradiation.Ofcourse,thisprocessisveryfarfromequilibrium.TheschematicgraphofFigure6.8showsthetransitionfromordinarylamplikeemissiontolaseremission.Observethesimilaritytothegraph 130Chapter6.OpensystemsinFigure6.6relatingtoB´enard’seffect.Onecouldsaythatthemostimportantthingsintheuniversearoseintheformofdissipativeorderedstructuresfarfromequilibriumasaresultoftheexportofentropy.Inparticular,thegalaxiesandstarsoriginatedinthisway:gravitationalenergywastransformedintothermalenergy,leadingtoalocaleffluxofentropy.Thebiosphereasawhole,andeachlivingorganisminparticular,arehighlyor-deredandfarfromequilibrium;thuslifeexistsonearthasaconsequenceoftheexportofentropy.Inthemosthighlyself-organizingsystems,inthecourseoftheirevolutionovertime(relativetothesamevalueofthemeanenergy)theirentropydecreases—theamountofentropyexportedcomestoexceedtheamountproduced.TheSovietphysicistYu.L.Klimontovich12calledthisassertionthe“S-theorem”;thuswhileBoltzmann’s“H-theorem”(seethesectionentitled“TheageofDarwin”inChapter5)relatestoequilibriumsystems,theS-theoremhastodowithdissipativeones.Thebasicfeaturespeculiartodissipativestructuresareasfollows.First,theyoccurinopensystemsfarfromequilibriumandariseasaresultofanin-creaseinfluctuations—thatis,smalldeviationsfromthemostlikelystate—risingtoamacroscopiclevel.Inthiswayorderiscreatedoutofdisorder,outofchaos.Thiskindoforderisfundamentallydifferentfromtheordinarycrystallineorderthatarisesunderequilibriumconditions.Thedifferenceconsistspreciselyinthedisequilibriumofadissipativesystem,maintainedbyaforcedexportofentropy.Allthesame,theappearanceofspatialortemporalorderinadissipativesystemisanalogoustoaphasetransition.Phasetransitionsunderconditionsofequilib-rium,suchas,forinstance,crystallization,areduesolelytotheinteractionsofamultitudeofparticles,wherebytheparticleseffectthechangeofstatecoherently,asitwerecooperatively.Ontheotherhand,thetransitionofadissipativesystemtoanorderedstatefromapriorunstabledisorderedstatetakesplacewhensomeparameterreachesacriticalvalue.Itispreciselyinsuchsituationsthatsmallfluctuationsgrowtoamacroscopiclevel.Phasetransitionsconstituteanimportantandfarfromsimpleareaofphysics.Theyare“cooperative”phenomena.Thefailuretorealizethishasledinthepasttoveryseriouserrors.Forexample,itwasonceclaimedthatthedependenceontemperatureoftheratioofthenumberofmoleculesinthesolid(crystalline)statetothenumberstillintheliquidstatecanbederivedfromtheBoltzmanndistribution(4.20).LetNcrysdenotethenumberofcrystallizedmoleculesinagivenquantityofthesubstanceinquestionattemperatureT,andNliqthenumberintheliquidstate.IfweblindlyapplytheBoltzmanndistribution(4.20),thenweobtainNcrysexp(−Ecrys/kT)=,Nexp(−Ecrys/kT)+exp(−Eliq/kT)12YuriLvovichKlimontovich(1924(?)–2004(?)),Soviet/Russianphysicist. 131whereNisthetotalnumberofmolecules.HenceNliqN−NcrysEliq−Ecrys==exp−.NcrysNcryskTHowever,thismakesnosense,sinceaccordingtothisformulathesubstancewillfreezecompletelyonlyinthelimitasT→0◦K,whileasthetemperatureap-proaches∞thetwosortsofmoleculesbecomeequalinnumber:Nliq=Ncrys.Themistakeherearisesfromfailingtotakeintoaccounttheinteractionofthemoleculesandtheircoordinatedbehavior.Thestatisticalsumhastobemodifiedtoaccomodatethisfeature.Thetransitiontoadissipativestructureinanopensystemisanon-equilibrialphasetransition.Haken13calledtheareaofphysicsconcernedwithcoordinatedphasetransitions—bothequilibrialandnon-equilibrial,butprincipallythelatter—“synergetics”.TheBelousov–Zhabotinski˘ıreactionItcanbeshownthatifasystemdeviatesjustalittlefromequilibrium,thenitsre-turntotheequilibriumstatewillproceedsmoothlywithoutoscillationsaccordingtothefollowingexponentiallaw:IfΔameasuresthedeviationofsomephysicalparameterfromitsequilibriumvalueae,then−t/τΔa=a(t)−ae=(a(0)−ae)e,whereτisaconstantcalledtherelaxationtimeofthesystem.Thusast→∞,Δa→0,thatis,theperturbedvaluea(t)oftheparametertendstotheequilibriumvalue.Ifthesystemisinastationarystateclosetoequilibrium,andthesystemde-viatesslightlyfromthatstationarystate,itwillreturntothatstateinaccordancewiththesamelaw.Therelaxationtimeτdeterminestherateatwhichthesystemreturnstoequilibriumortothestationarystate.Att=τwehavea(0)−aea(τ)−ae=,ethatis,theinitialdeviationhasbytimeτdecreasede(≈2.78)times.Ontheotherhand,incertainsystemsfarfromequilibrium,where,aswehaveseen,dissipativespatialandtemporalstructures—inequilibrialorder—canarise,thisordermayconsistofoscillationsorwaves.Thisisespeciallystrikingincertaindissipativechemicalsystems.13HermannHaken(born1927),Germantheoreticalphysicist. 132Chapter6.OpensystemsImaginealecturerdemonstratingachemicalexperiment.Hefillsabeakerwithablueliquid,andusingapipetteaddsafewdropsofacolorlessliquid.Thesolutioninthebeakerturnspink.Well,there’snothingsoremarkableaboutthat.Inchemistrytherearemanymuchmoresurprisingtransformations!Butwait!What’shappening?Afteraboutaminutetheliquidinthebeakerturnsblueagain,thenpinkagain,thenblueagain,andsoon.Theliquid’scolorchangesperiodically,likeasortofchemicalclock.Thisremarkablephenomenon—aperiodicchemicalreactioninahomoge-neoussolution—wasdiscoveredbyB.P.Belousov14in1951.In1910Lotka15showedinimportanttheoreticalworkthatinachemicalsystemfarfromequilibrium,os-cillationsinthedegreesofconcentrationofthereagentsarepossible.Thenin1921Bray16observedforthefirsttimeaperiodicchemicalreactioninasolutionofhy-drogenperoxideH2O2,iodicacidHIO3,andsulphuricacidH2SO4.Heobservedaperiodicoscillationintheconcentrationofiodineinthesolutionassuccessiveoxidationsofiodinetoiodateandthenreductionsbacktoiodinetookplace:5H2O2+2HIO3→5O2+I2+6H2O,5H2O2+I2→2HIO3+4H2O.Thisreactionwasrathercomplex,andforsometimeitcouldnotberuledoutthatitmightbeheterogenous,17takingplaceonbubblesoftheiodatecatalyst.Belousovdiscoveredhisreaction,subsequentlytomakehimfamous,sometimeinthe1950s,andindependentlyoftheworkofthesescientists,butatthetimewasunabletopublishitexceptinanobscurejournal.Itwasintensivelyinvestigatedfurther,startingin1961,byA.M.Zhabotinski˘ı,18whowasabletosimplifythereaction,whenceitbecameknownasthe“Belousov-Zhabotinski˘ıreaction”.Inthisreactionthecolorchange(betweenyellowandclear)iscausedbyachangeinthechargeonametallicion.WegiveasimplifieddescriptionofBelousov’sreaction,reducingittojusttwostages.Atthefirststage,trivalentceriumisoxidizedbybromicacid:HBrO33+4+Ce−→Ce,andthesecondconsistsinthereductionofthefour-valentceriumbytheorganiccompoundmalonicacid:malonicacid4+3+Ce−→Ce.14BorisPavlovichBelousov(1893–1970),Sovietchemistandbiophysicist.15AlfredJamesLotka(Lviv1880–1949),Americanmathematician,statistician,andphysicalchemist.16WilliamCrowellBray(1979–1946),Canadianphysicalchemist.17Thatis,causedbyreagents’beingintwoormoredifferentphases.Trans.18AnatolM.ZhabotinskycurrentlyholdsapositionatauniversityintheU.S. 1334+[Ce]tFigure6.9:VariationoftheconcentrationofCe4+intheBelousov-Zhabotinski˘ıreaction.Thisperiodicprocesscomestoanendafteralargenumberofoscillationsasa−resultofirreversibleexhaustionofthesupplyofthebromateanionBrO3.ThefinalproductsofthereactionareCO2,H2O,andbromiumderivativesofmalonicacid.Noperpetuummobileisachieved!Figure6.9showstheoscillationsintheconcentrationofthefour-valentceriumion.Zhabotinski˘ıandA.N.Zaikinlaterdiscoveredandinvestigatedotherreac-tionsofthistype.Butthatisnotall:Byhavingperiodicreactionstakeplaceinnarrowtubes(one-dimensionalsystems)andinthinlayersofsolution(two-dimensionalsystems)intheabsenceofconvection,Zhabotinski˘ıandZaikinwerealsoabletoproducewavelikechemicalprocesses.Figure6.10showstheevolutionofsuchawaveintwodimensions.First,agerminatingorinitiatingcenter—agermoreye—appearsastheresultofalocalfluctuationintheconcentration,andfromthiscenterwavesofcoloremanateaccordingtovariationsinconcentration.Suchphenomenaofspatio-temporalorderingrepresentauto-oscillatoryandauto-waveproducingprocesses.Suchprocessesoccurinopennonlinearsystemsthatarefarfromequilibrium,asaresultofforcesdependingonthestateofmotionofthesystemitself,andfurthermoretheamplitudeoftheoscillationsisdeterminedbypropertiesofthesystem,andnotbyanyinitialconditions.Auto-oscillatoryandauto-waveprocessesinchemistry(andalsoinbiology—seeChapter8)aresustainedbyaneffluxofentropyfromtherelevantsystem.Ifabreakoccursinthewavefrontofachemicalwave,thenaspiralwavecalleda“reverberator”mayresult.Reverberatorsform,inparticular,whentwo-dimensionalwavespropagatenearanopening.Figure6.11isaphotographofchemicalreverberators.Figure6.12isaphotographofacertainspeciesoflichen—bywayofcomparison.OrganismsasdissipativesystemsOfcourse,thesurprisingsimilarityoftheabovephotographsdoesnotmeanthatthegrowthofalichenisinallrespectslikethepropagationofaspiralchemical 134Chapter6.OpensystemsFigure6.10:Initiatingcenters,andsubsequentstages.Figure6.11:Chemicalreverberators. 135Figure6.12:Thelichenparmeliacentrifuga. 136Chapter6.Opensystemswave.Nevertheless,attheheartofallbiologicalphenomenawefindthephysicsofopensystemsfarfromequilibrium.Althoughweshallbediscussingthistopicindetailinthefinalchapterofthebook,itisappropriateatourpresentjuncturetoconsidercertainspecialstructuralanddynamicalfeaturesoflivingorganisms.Wehaveseenthatinopensystemsfarfromequilibriumtherecanariseaspecificstructuringofthesystemasaresultofintensificationoffluctuationsuptothemacroscopiclevel,resultingfromtheeffluxofentropyfromthesystem.In1952,Turing19showedthattheconjunctionofanautocatalyticreactionwithdiffusioncancausespatialandtemporalordertoarise.Areactioniscalledautocatalyticifatleastoneoftheproductsofthereactionisalsoareagent,sothatthequantityofthatreagentincreaseswithtime.ProbablythebestknownexampleisthereplicationofDNA(deoxyribonucleicacid)macromoleculesthattakesplaceincell-division.TheinitialdoublehelixofDNAcatalyzes,viaaprocesscalled“matrixsynthesis”,theformationofacopyofitself.Turing’sreaction-diffusionmodelisbasedonareactionofthefollowingtype:A→X,2X+Y→3X,B+X→D+Y,X→E.HereAandBaretheinitialreagents,XandYtheintermediateones,andDandEthefinalproducts.Thesecondstageofthereactionistheautocatalyticone:AsaresultoftheactionofthesubstanceX,thesubstanceYistransformedintoX,whichthuscatalyzesitsownproduction.Itfollowseasilythattheoverallreactionis,insum,A+B→D+E.Prigogineandhiscollaboratorscalledsuchchemicalsystems“brusselators”forthesimplereasonthattheirearlytheoreticalinvestigationwasundertakeninBrussels.Theequationsdescribingthekineticbehaviorofbrusselatorsandthediffu-sionofthereagentsXandY(brusselatorsinadistributedsystem)arenonlinear,andsuchasystemisfarfromequilibrium.TheconcentrationsofthesubstancesXandYundergoperiodicoscillations,thusformingwavesofvaryingconcentration.Atacertainthresholdvalueoftheconcentration20aninitialfluctuationfromthestationarystateisreinforced,ultimatelybringingthesystemintoanewstation-arystatecorrespondingtothenewinhomogeneousdistributionofthesubstancesXandY.Figure6.13showsoneofthesolutionsofthekineticequationsofabrusselator:thedistributionoftheconcentration[X]withrespecttoaspatialcoordinate.Turing’soriginalarticlehadthearrestingtitle“Thechemicalbasisofmor-phogenesis”.21Theterm“morphogenesis”referstotheinitiationanddevelopmentofanorganism’scomplexstructureinthecourseofitsembryonicgrowth,thatis,19AlanMathisonTuring(1912–1954),Englishmathematician,logician,cryptographer,and“thefatherofmoderncomputerscience”.20OfsubstanceX?Trans.21Phil.TransactionsoftheRoyalSoc.London.SeriesB,Biologicalsciences,Vol.237(1952),pp.37–72. 137[X]rFigure6.13:Alocalizedstationarydissipativestructure.thedifferentiationofcellsintotissuesandorgans.Turingwasthefirsttoestablishthepossibilitythatmorphogenesishasachemicalbasis.Ofcourse,bothTur-ing’stheoryandthatofbrusselatorswerebasedonchemicalmodels,forallthatveryconvincing.Nowweknowforafactthatmorphogenesisinnaturereallyisdeterminedbymolecularinteractions,andthatcertainsubstances,namely“mor-phogenes”,functioningatspecifictimesandatspecificplaceswithintheorganism,areresponsiblefortheformationofthevariousorganicstructures.Theseinterestingnaturalphenomenahavemuchincommonwithauto-oscil-latoryandauto-waveprocesses,studiedinespeciallygreatdetailinthecaseoftheBelousov-Zhabotinski˘ıreaction.Occasionallyonehearsofclaimstotheeffectthatchemicalauto-oscillatoryprocessesandstandingwaves(suchastheso-called“Liesegangrings”22observedincolloidalsuspensions)canbeexplainedonlyonthebasisofquantummechanics,andeventhatthePlanckconstanthcanbedeterminedfromtheperiodsofsuchoscillations.Howeverthisisacrudemisapprehension:onemightattemptwithequalsuccesstoestimatePlanck’sconstantfromthezebra’sortiger’sstripes.Aswehaveseen,thereisnotraceofquantummechanicsinthetheoryunderlyingtheseperiodicphenomena;theyaremacroscopicphysicalphenomena,notatomicorsubatomic.Manyofthetissuesoflivingorganismsareexcitable,meaningthatexcitation22Anyofaseriesofusuallyconcentricbandsofaprecipitate(aninsolublesubstanceformedfromasolution)appearingingels(coagulatedcolloidsolutions).Thebandsstrikinglyresemblethoseoccurringinmanyminerals,suchasagate,andarebelievedtoexplainsuchmineralforma-tions.Theringsarenamedfortheirdiscoverer,theGermanchemistRaphaelEduardLiesegang(1869–1947).EncyclopediaBritannica. 138Chapter6.Opensystems—chemicalorelectrochemical—istransmittedacrossthemfrompointtopoint,propagatinglikeawave.Muscularandnervoustissueshavethisproperty:thepropagationofastimulusalonganervefiberandthesynchronousoscillationsofthewholeheartmusclearephenomenawithachemicalbasis.TheSovietphysicistV.I.Krinski˘ı23hasinvestigatedoneofthemostdan-gerousofhumanpathologies,heartfibrillation.24Insuchfibrillationthehearthasdepartedfromitsnormalregimeofregularcontractionsandenteredoneofchaoticoscillations,andwithoutemergencymedicalaidtheconditionresultsindeath.Itturnsoutthatthistypeof“cardiacarrhythmia”resultsfromthemultiplicationofspiralwavesofexcitation,thatis,ofreverberators.Inhisworks,notonlydoesKrinski˘ıconstructatheoreticalmodelofthemechanismoffibrillationsandreportonhisinvestigationsofthephenomenonbymeansofdelicateexperimentsontheheartsofsuitableanimals,butalsogivespracticaladviceastowhatoneshoulddotobringthefibrillationstoanendintheeventofanattack.ThethreestagesofthermodynamicsWehavebynowbecomethoroughlyacquaintedwiththedevelopmentofthermo-dynamicsfromitsoriginsalmosttwohundredyearsago.Weendthischapterwithanoverviewoftheconceptualcontentofthisdevelopment.Priortothebirthofthermodynamics,sciencewasdominatedbyNewtonianmechanics—amechanicswheretimewasreversibleandtheworlddidnotevolve.AtsomepointtheAlmightysettheuniverse’smechanismgoing,andthenceforthithasworkedunchanginglylikeawound-upclock.Livingnaturealsoappearedtobeunchangingandunchangeable,remainingasitwasatitsinitialcreation.Thefounderofscientificbiology,andtheauthoroftheclassificationoflifeforms,CarlLinnaeus,25consideredthatthebiologicallifeformsallabouthimwereimmutable,andcreatedsimultaneouslyatsometimeinthepast.Thenaturalsciencesastheywerethendidnotinvolvetime.Timewastheprerogativeofthehumanities,aboveallhistory.Thereitwasclearthattimechangeseverything.M.V.LomonosovwasanopponentofNewtonianphysics,espousingDescartes’theoryinstead.PerhapshisrejectionofNewtonianphysicshadsomethingtodowiththebreadthofhisinterests,thatis,withthefactthathewasnotonlyaphysi-cistandchemist,butalsoapoetandhistorian;inpoetryandhistorytimeflowedirreversibly.Attheturnofthe18thcenturythefirstscientific-technologicalrevolu-tionoccurred—orrathertechnological-scientificrevolution,sincethesteamenginewasinventedindependentlyofphysics,anditallstartedfromthat.SadiCarnot’sthoughtsonthesteamenginerepresentedthefoundingofthermodynamics.The23ValentinI.KrinskyisnowworkingataninstituteinNice,France.24Thelaytermis“heartpalpitations”.Trans.25OrCarlvonLinn´e(1707–1778),Swedishbotanist,physician,andzoologist.Thefatherofmoderntaxonomy. 139firstandsecondlawsofthermodynamicswerediscovered,andthensomewhatlaterathird—Nernst’stheorem.Andentropyappearedonthescene—initiallyasthegrayshadowofenergy,theruleroftheuniverse.Timefirstputinappearanceinthesecondlawofthermodynamics,formu-latedastheirreversiblegrowthofentropyinspontaneousprocesses.Apartfromthis,however,thermodynamicsremainedthermostatics,thescienceofequilibriumandequilibrialprocesses.ThenThomsonpredictedtheheatdeathoftheuniverse.Itfollowedthattheworldcannotbestandingstill,itmustbeevolving;itmovesinexorablytowardsitsdemisejustlikeeverylivingorganism.Inthiswaytheemphasismovedfromtechnologytocosmology,representingashiftinfocusfromwhatexiststowhatdevelops—“frombeingtobecoming”,inthewordsofPrigogine.ItwasthedawnoftheageofDarwin:ideasfrombiology(andthehumanities)ofthedevelopmentandgrowthtowardsthemostprobablestateofaphysicalsystemwereincorporatedintophysics.ThefirststageofthermodynamicsculminatedinthecreationofstatisticalphysicsintheworksofBoltzmannandGibbs.Atthisstage,entropyceasedtorepresentmerelyameasureofthedepreciationofenergy,andassumeditstrueroleasameasureofthedegreeofdisorderofasystem,anobjectivecharacterizationoftheunavailabilityinprincipleofinformationaboutasystem.Theimportanceofentropyasoneofthechiefcharacteristicsofarbitrarysystemsbegantogrowrapidly.Duringthesecondstageinthedevelopmentofthermodynamics,scientiststurnedtothestudyofopennon-equilibrialsystemsclosetoequilibrium.ThislinearthermodynamicsofopensystemswascreatedbyOnsager,Prigogine,andothersoftheircontemporaries.Inthisscience,thedependenceontimehadbecomequantitative:aswesaw,thisnon-equilibrialthermodynamicsdoesnotlimititselftothemereassertionthatentropyincreasesinirreversibleprocesses,butactuallyinvolvesexplicitcomputationoftherateofthisincrease,thatis,ofthederivativeoftheentropycontentwithrespecttotime—thedissipationfunction.Therearetwofundamentalfeaturesoflinearthermodynamics(nowdefinitelythermodynamicsandnotthermostatics)thatarenontrivialandessential.First,therenowarisesthepossibilityofanopensystem’sexistinginastationary,butnon-equilibrium,state,inwhichtheproductionofentropyisbalancedbyitseffluxfromthesystem.Second,thisthermodynamicsallowsfortheconjunctionofdy-namicalprocessesinasingleopensystem,wherebyaprocessthatcouldnottakeplacebyitself(inasmuchasitinvolvesadecreaseinentropy),isrealizedthroughthefreeenergymadeavailablebyother,entropicallyadvantageous,processes.Finally,thelast20or30years26havewitnessedathirdstageintheevolutionofthermodynamics,representedbythephysicsofnon-equilibrialdissipativepro-cesses.Wehaveseeninthepresentchapterthatopensystemsfarfromequilibriumpossessremarkableproperties:theyarecapableofcreatingorderfromchaosbyexportingentropy,thatis,throughitseffluxoutofthesystem.Alivingorganism26Thatis,startinginthe1960s. 140Chapter6.Opensystemsfeedsonnegativeentropyandnotonpositiveenergy.Thusatthisstage,entropyhasbeenpromotedfromthemereshadowofanomnipotentsovereigntoapower-fulentitydeterminingtheveryexistenceoflifeonearth,andtheevolutionoftheuniverse.Forthefirsttimeweareinapositiontounderstandhowitisthatorder,anal-ogoustothecrystallinevariety,ispossibleinopensystemsfarfromequilibrium—thatordercanemergeoutofchaosmuchlikeaphasechange.Anewbranchofsciencehasbeenborn,thephysicsofdissipativesystems(asPrigoginecallsit)orsynergetics(asHakencallsit).Thisrelativelynewareaofphysicsholdsoutgreatpromise.Thecreationofthisbranchofsciencebegantheprocessofintegrationofthesciencescharacteristicofourtime,supersedingtheformerrigidspecialization.InvestigationsoftheB´enardeffect,thelaser,ofperiodicchemicalprocesses,andheartfibrillation,areallpursuedtodayfromaunifiedscientificpointofview.Fromthesameunifiedstance,expertsinsynergeticsinvestigatestructure-formationinplasma27“ontheearth,intheheavens,andonthesea”.Yes,yes,fromthepe-riodicitysometimesevidentincloudformationstothenorthernlights(auroraborealis)—allsuchphenomenaresultfromasinglelawofnature,namelytheemer-genceoforderfromchaosontheanalogyofaphasechange.Inthelastanalysis,cosmologyitselfismerelyapartofthephysicsofdissipativesystems.Atlast,acenturyaftertheappearanceofDarwin’sTheoriginofspecies,physicshasbecomeunitedwithbiologyinthetaskofcomprehendingtheessenceofirreversibleprocesses.ThiswillbediscussedfurtherinChapter8.ItmightjustifiablybeclaimedthatthepublicationofTheoriginofspeciesheraldedtheappearanceofsynergeticsinscience.Darwinshowedhowtheorderlyprocessofevolution—culminating,forus,inthepresentbiosphere—arisesoutofthechaotic,disorderlymutabilityofnaturallivingorganisms.ItisinthissensethatonemightcallDarwinthefoundingfatherofsynergetics.Simultaneouslywiththeriseofthisnonlinearthermodynamics,atheoryofinformationwasbeingformulated,closelyalliedtothermodynamics.Wehavemorethanoncespokenofentropyasameasureoftheunavailabilityofinformation.Butwhatexactlyis“information”?27Inthegeneralsenseofagasorsuspension.Trans. Chapter7InformationWeunderstand,eventhoughhalf-heard,Ofquestions,answerseveryword.Butifthosehalf-heardwordsarenew,They’reirredundantthroughandthrough.InformationandprobabilityWhatisinformation?Theeverydaymeaningofthewordisclear:Itiswhatiscommunicated.Weobtainitusingeverysenseorgan.Wetransmitittoothers.Ourpersonallifeandthefunctioningofsocietyarebasedoncommunication,onthereceiptandtransmissionofinformation—andthisappliesnotonlytohumanbeings,buttoalldenizensofthebiosphere.Inthe1940sanewscienceappeared:“cybernetics”.1Oneofitsinventors,NorbertWiener,2entitledhisclassicalbookonthesubjectCyberneticsorcon-trolandcommunicationintheanimalandthemachine.Therehewrote:“Ifthe17thandearly18thcenturiesaretheageofclocks,andthelater18thand19thcenturiesconstitutetheageofsteamengines,thepresenttimeistheageofcom-municationandcontrol.”Herecommunication—withoutwhichtherecanbenocontrol—meanstransmissionofinformation.Oneoftheessentialtasksof19thcenturysciencewastheformulationofatheoryofheatengines,atheoryofheat.Inthe20thcentury,itwasatheoryofcommunication—orinformation—thatbeggedtobecreated(amongothers).And,surprisingly,itturnedoutthatthermodynamicsandinformationtheorywereconnected.1“Cyberneticsistheinterdisciplinarystudyofcomplexsystems,especiallycommunicationprocesses,controlmechanisms,andfeedbackprinciples.”Trans.2NorbertWiener(1894–1964),Americantheoreticalandappliedmathematician. 142Chapter7.InformationThechiefaimofinformationtheory3consistsfirstinclarifyingtheconceptofinformationanditsmeansofcommunication,andthendiscoveringconditionsforoptimalcommunication,thatis,optimaltransmissionofinformation.Thuswemustfirstformulateaprecisedefinitionof“information”thatreflectsitseverydaysense,whilehavingaprecisequantitativecharacter.Webeginwithsomeelementaryexamples.Ifwetossacoin,thentheresult—headsortails—representsthecommunicationofadefiniteamountofinformationaboutthetoss.Iftherollofadieresultsinathree,thenthisalsoconstitutesinformation.Thecrucialquestiontoaskhereis:Inwhichofthesetwosituationsdoweobtainmoreinformation—intossingacoinorrollingadie?Theobviousansweris:inrollingadie.For,inacoin-tosswehaveanex-perimentwithjusttwoequallylikelypossibilities,whileinrollingadietherearesixequiprobablepossibleoutcomes.Theprobabilityofobtainingheadsis1/2,whilethatofadiecomingupthreeis1/6.4Therealizationofalesslikelyeventrepresentsgreaterinformation.Or,equivalently,themoreuncertainaneventispriortoreceivinginformationaboutit,thegreatertheweightor“quantity”ofthatinformationwhenreceived.WehavethusarrivedattheconclusionthataquantitativemeasureofinformationshouldsomehowdependonthenumberP0of(equallylikely)possibilities.Inthecaseofacoin-tossP0=2,andinthatoftherollofadieP0=6.Itisalsointuitivelyclearthatifwerolladietwice(orrollapairofdice),theresultrepresentstwiceasmuchinformationastheresultofrollingitonlyonce.Weconcludethatinformationobtainedfromasequenceofindependenttrialsisadditive.Ifonthefirstrollthethree-spotcameup,andonthesecondthefive-spot,thenintotalthisrepresentstwiceasmuchinformationastheresultofthefirstrollbyitself.5Similarconclusionsapplyifwerolltwodice.Thusameasureofinformationshouldbeadditiveoverasetofindependentevents.Ontheotherhandthenumberofwaysasetofindependenteventscanoccurismultiplicative:Ifwerolladietwice(orrollapairofdice),thenP0=6·6=36.Ingeneral,ifwehavetwoindependenteventswhichcanoccurinP01andP02waysrespectively,thenthenumberofwaysbotheventscanoccurisP0=P01P02,(7.1)whilean—asyethypothetical—quantitativemeasureI(P0)ofinformationshouldsatisfyI(P0)=I(P01P02)=I(P01)+I(P02).(7.2)Itfollowsthatthedependenceofthequantityofinformationyieldedbyaneventonthenumberofwaystheeventcanoccur—thedependenceofIonP0—mustbe3InformationtheorywascreatedbytheAmericanmathematiciansClaudeElwoodShannon(1916–2001)andWarrenWeaver(1894–1978).Trans.4SeethefirstsectionofChapter4,includingthefootnotes,forthedefinitionsoftheproba-bilistictermsthatfollow.Trans.5Thatis,asthefirstrollbyitselfyieldsaboutthetworolls.Trans. 143Table5:Thebinaryformsofthenumbersfrom0to32.0=01=19=100117=1000125=110012=1010=101018=1001026=110103=1111=101119=1001127=110114=10012=110020=1010028=111005=10113=110121=1010129=111016=11014=111022=1011030=111107=11115=111123=1011131=111118=100016=1000024=1100032=100000logarithmic:I=KlogP0.(7.3)ThebaseofthelogarithmandtheconstantKarenotdeterminedby(7.1)and(7.2),sotheymaybechosenarbitrarily.TheestablishedconventionininformationtheoryhasK=1andthelogarithmicbaseequalto2.HenceI=log2P0.(7.4)Thenthebasicunitofinformation,calledabit,isthatobtainedfromacoin-toss,whereP0=2:log22=1bit.Thusinformationiscalculatedinbits,thatis,inbinarydigits.Thebinarysystemiswidelyusedinthetechnologyofcybernetics,inparticularindigitalcomputers,sincebreakingacomputationdownintoasequenceofoperationseachinvolvingjusttwopossibilities—inclusionorexclusion,say—greatlysimplifiesthecomputa-tion.Everynumbercanberepresentedinthebinarysystembyasequenceof0sand1s.Atableofthewholenumbersfrom0to32inbothdecimalandbinarynotationsisgivenabove.Inbinarynotationthemultiplicationtableisespeciallysimple:0·0=0,1·0=0·1=0,1·1=1.Theadditiontableis:0+0=0,1+0=0+1=1,1+1=10.Howmanybitsdoesanarbitrarythree-digitnumberhave?Sincethereare900suchnumbers—from100to999—wehaveI=log2900≈9.82bits. 144Chapter7.InformationAnotherwayoflookingatthiscalculationistoobservethatthefirstdigitinsuchanumber(indecimalnotation)canbeanyoftheninedigits1,...,9,whilethesecondandthirdeachtakeanyofthetenvaluesfrom0to9,sothatI=log2900=log29+2log210≈9.82bits.Notethatsincelog210≈3.32,thedecimalbaseisequivalentto3.32bits,sothatbinarynotationusesonaverage3.32timesasmanydigitsasdecimalnotation.Computinginformationintermsofbitsamountstoencodingananswertoaquestionasasequenceof“yes”sor“no”s.InhisbookAmathematicaltrilogy,A.R´enyi6describesthe“Bar-Kokhba”game,7popularinHungary,whichpur-portedlyoriginatedasfollows.In132A.D.theJewishleaderSimonbarKokhbaledarevoltoftheJewsagainsttheirRomanoverlords.8AscoutsenttospyontheRomancampwascaptured,andhadhistonguetornoutandhandscutoff.Returnedtohisowncamp,hewasthusunabletocommunicateeitherorallyorinwriting.However,SimonbarKokhbawasabletoextractimportantinformationfromhimbyaskingaseriesofyes-noquestions,whichthepoorscoutcouldanswerbynoddinghisheadappropriately.Usingthisapproach,weshallverifythefollowingassertionofHartley:9IfinagivensetcontainingNelements,aparticularelementxissingledout,aboutwhichitisknowninadvanceonlythatitbelongstotheset,thenthequantityofinformationrequiredtofindxislog2Nbits.ThusaccordingtoHartley,inordertodeterminewhichnumberbetween1and32yourpartnerinthegameisthinkingof,youneedtoaskonlyfivequestions,sincelog232=5.Andindeed,proceedingasintheBar-Kokhbagame,thefirstquestionshouldbe:“Isthenumbergreaterthan16,yesorno?”Thisreducesthesetofpossibilitiesbyahalf,to16.Youcontinuereducingbyhalfinthisway,reducingthenumberofpossibilitiesto8,then4,then2,andfinally1.Itisimportanttoobservethatbyproceedingsomewhatdifferentlywecanaskfivequestionsallatonce,thatis,withoutneedingtoknowtheanswertoanyquestioninadvance.Firstwritethenumbersfrom1to32inbinarynotation,usingfivedigitsforallofthem;thusthelistwillstartwith00001andendwith11111(seeTable5).Supposeyourpartneristhinkingof14,thatis,01110inbinarynotation.Youcanthenaskthefivequestionsalltogetherintheform:“Isittruethatinbinarynotationthefirstdigitofthenumberyouhaveinmindis1,andtheseconddigit,...,andthefifthdigit?”Theanswerwillbe:“no,yes,yes,yes,no”.6Alfr´edR´enyi(1921–1970),Hungarianmathematician.Workedmainlyinprobability.7Similarto“Twentyquestions”.Trans.8AsaresultofwhichanindependentJewishstatewasestablishedinIsrael,reconqueredbytheRomansin135A.D.BarKokhbawasthusthelastkingofIsrael.Trans.9RalphVintonLyonHartley(1888–1970),Americanresearcherinelectronics,andcontributortothefoundationsofinformationtheory. 145Theformula(7.4)providesthebasisforsolvingsuch“searchproblems”.Hereisanotherone.AgaintheapproachisessentiallythatofSimonbarKokhba.Supposethatwehave27coinsofwhichjustoneisfalse,weighinglessthantheothers.Whatistheleastnumberofweighingsonabalanceneededtofindthefalseone?Eachweighingwithanequalnumberofcoinsinthepansofthebalance,yieldsthequantityofinformationI=log23,sincethereareexactlythreepossibilities:thepansbalance,theleftpanislighter,ortherightislighter.Ontheotherhand,weknowfromHartley’sstatementthatfindingthefalsecoinrequireslog227bitsofinformation.Hencethefalsecoincanbefoundinzweighingsprovidedonlythatzlog23≥log227=3log23,thatis,providedz≥3.Thereforethreeweighingssuffice.Inthefirstweighing,oneplacesninecoinsineachpan,inthesecondthree,andinthethirdone.Ofcourse,inplayingtheBar-Kokhbagameefficiently,oneneedstoknowwhichquestionstoask!Insteadofnumbers,let’sconsiderletters.IntheRomanalphabetthereare26letters,andintheRussian33.Howmuchinformationdoesasingleletterofsometextyieldinthesetwocases?Thenaturalansweris:AletterofapassageinEnglishyieldslog226≈4.70bitsofinformation,whilealetterfromaRussiantextyieldslog233≈5.05bits.However,theseanswersmakesenseonlyundertherathercrudeassumptionthateveryletterappearswiththesameprobability.InformationalentropyLettersdonotoccurwithoverallequalprobability;inanylanguagewrittenusinganalphabet,someletterswilloccuronaveragemorefrequentlythanothers.Thisaveragefrequency—whichisthesameastheprobabilityofoccurrenceofaletter—reflectsthestructureofthelanguageinquestion.ThereadermayknowConanDoyle’sstoryTheadventureofthedancingmen,inwhichSherlockHolmesusestheknownaveragefrequencyofoccurrenceoflettersinEnglishtextstodecipheracodedmessage.(Actually,thisideaappearedinliteratureconsiderablyearlier,inEdgarAllanPoe’sThegoldbug,writtenbeforeConanDoylewasborn.)Inderivingformula(7.4)itwasassumedthattheindividualoutcomesoftheexperimentorsituationunderconsiderationwereequiprobable.Whatiftheyhavevariousprobabilities?SupposewehaveamessageconsistingofNsuccessivecells—atextmadeupofNletters.SupposefurtherthateachcellcancontainanyofMdifferentletters.(IfthetextisinEnglish,thenM=26.)SupposealsothatthemessagecontainsN1occurrencesofthelettera,N2oftheletterb,andsoon,uptoN26occurrencesofz.ClearlyMN=Ni.(7.5)i=1 146Chapter7.InformationIfthetextwearepresentedwithissufficientlylong,wemayassumethattheprobabilitypioftheithletteroccurringinanycellofanytextisapproximatelygivenbyNipi=,i=1,2,...,M.(7.6)NThenMpi=1,(7.7)i=1asshouldbethecase.ThenumberofN-lettersequencesaltogetherisN!P=.(7.8)N1!N2!···NM!(Weencounteredthisformulaearlier;see(4.5)and(4.16).)By(7.4)thequantityofinformationinasinglesuchmessageis10lnP1N!I=log2P==ln.(7.9)ln2ln2N1!N2!···NM!UsingtheapproximationofN!andtheNi!givenby(4.10)(andassumingNandtheNialllarge),weinferthatM1I≈NlnN−NilnNiln2i=11MM=−Npilnpi=−Npilog2pibits.(7.10)ln2i=1i=1HenceifN=1,thatis,inthecaseofasingleletter,wehave1MMI1=−pilnpi=−pilog2pi.(7.11)ln2i=1i=1Thequantity−ipilog2piwascalledbyShannon,oneofthefoundersofinfor-mationtheoryandcommunicationtheory,theentropyofanymessageinvolvingMsymbolsoccurringwithprobabilitiespi.Weshallseebelowthattheformula(7.11)doesindeedaffordameasureofthedegreeofuncertaintyassociatedwitharandomsuchtext,thusjustifyingthename“entropy”.WhatvaluedoestheentropytakeinthecaseofEnglish?Thefrequencies,thatis,theprobabilities,ofoccurrenceoftheletters(andalsoaspace)inEnglishtextsaregiveninTable6below.SubstitutionofthesevaluesinShannon’sformula(7.11)yieldsI1=−0.164log20.164−0.106log20.106−···−0.001log20.001≈4.0bits.10AssumingallsuchN-letterstringsequallyprobable.Trans. 147Table6:Probabilitiespiofoccurrenceofletters(andspace)inEnglishtexts.space.164s.055m.021v.008e.106h.053w.020k.007t.078r.052f.019j.001a.068d.036g.017x.001o.065l.035y.017q.001i.058c.024p.017z.001n.058u.024b.012ThisnumberisappreciablysmallerthanI0=log226≈4.7.Theamountofinfor-mationcommunicatedbythelettershasdecreasedsincewehaveincorporatedinthecalculationpreviouslyascertainedinformationaboutthefrequencyofoccur-renceoftheletters.However,inalanguagetherearealwayscorrelationsbetweenletters—definitefrequenciesofoccurrencenotjustofindividuallettersbutalsoofstringsoftwoletters(bigrams),threeletters(trigrams),fourletters(quadrigrams),andsoon.Alinguistictextrepresentsacomplicated“Markovchain”,sincetheprobabilityofagivenletteroccurringatagivenplaceinthetextdependsontheprecedingletter.ItisappropriatetosaysomethinghereaboutthetheoryofMarkovchains,oneofthegreatestachievementsofRussianscience.Thistheory,inwhichcon-nected,probabilisticallydependentchainsofeventsarestudied,wasfoundedbyA.A.Markov.11Thecalculationoftheprobabilitythatseveraleventswillalloccurisoftenmadeeasybythefactoftheeventsinquestionbeingindependent.12Forexample,ifwetossafaircoin,theprobabilityofgettingheadsis1/2regardlessoftheoutcomeoftheprevioustoss.Howeversuchindependenceisfarfromalwaysbeingthecase.Wechooseacardatrandomfromadeckofcards.Theprobabilityofthecardbeingofaspecifiedsuitis1/4.Ifwethenreturnthecardtothedeck,theprobabilitythatthenextcardchosenwillbeofthatsuitremains1/4.However,ifwedonotreturnourchosencardtothedeck,theprobabilitythatthenextcardchosenwillbeofthegivensuitwilldependonthesuitofthefirstcard.Forinstanceifthatsuitwasclubs,thenthereremaininthedeck13cardsofeachofthesuitshearts,diamonds,andspades,butonly12clubs.Hencetheprobabilitythatthesecondcardisalsoclubsisnow12/51,whichislessthan1/4,whileforeachoftheothersuitstheprobabilitythatthesecondcardisofthatsuitisnow13/51,whichisgreaterthan1/4.AMarkovchainconsistsofasequenceofevents,eachofwhoseprobabilitiesdependsontheoutcomeoftheprecedingevent,or,inthemorecomplexcase,on11Andre˘ıAndreevichMarkov(1856–1922),Russianmathematician.12Sothattheprobabilityofthecombinedeventsistheproductoftheirindividualprobabilities.Trans. 148Chapter7.Informationtheoutcomesofseveralofitspredecessors.Clearly,anylinguistictextconstitutesacomplexMarkovchain.A.A.MarkovhimselfappliedhistheorytoaprobabilisticanalysisofPushkin’sEugeneOneginandAksakov’sThechildhoodofBagrov’sgrandson.TodaythetheoryofMarkovchainsandMarkovprocessesisappliedverywidelyintheoreticalphysics,meteorology,statisticaleconomics,andsoon.Plansforatypewriterofmaximalefficiencywouldhavetotakeintoaccountthefrequencyofoccurrenceoftheindividualletters,bigrams,andeventrigramsofthelanguage.Thekeysofrarelyusedlettersshouldberelegatedtooneoranothersideofthekeyboard,andthekeysoflettersthatoftenoccurtogethershouldbeadjacent.Theimplementationofthesetwoobviousprinciplesisenoughtogreatlyspeedupthetypingprocess.Hereisanexperiment.13Writethe26lettersoftheEnglishalphabeton26slipsofpaper,andplacethese,togetherwithablankslip,inabag.Thentakeoutaslip,writedowntheletterontheslip(orleaveaspaceiftheslipistheblankone),returnthesliptothebag,givethebagagoodshake,anditeratethisprocedurealargishnumberoftimes.Youwillinallprobabilityobtainarandomtextexhibitingnocorrelations,suchas:QFEZRTGPIBWZSUYKVLCPFMTQAUHXPBDKRQOJWNIfyounowtakeaccountofthefrequencyofoccurrenceoftheletters(andspace)byplacing1000slipsofpaperinthebag,with164blank,106withtheletterE,78withtheletterT,andsoon,till,finally,justoneslipforeachofthelettersJ,X,Q,andZ(seeTable6),thentheresultofdrawingoutslips,withreplacement,andrecordingtheresults,ismorelikelytoresembleatextformally:ENHRIVTUXSMOEHDAKCOTESLTNEIfyounextinsomesimilarfashionalsoincludeinformationaboutthefrequenciesofbigrams,thenyourresultmightbesomethinglike:NTRETIERANALITRONDTIORQUCOSAALINESTHTakingaccountalsoofthefrequenciesofoccurrenceoftrigrams,mightyield:NDETHERMENSTHELHASITSTHEROFTNCEQUIEXTAnd,finally,includinginadditioninformationaboutthefrequenciesofquadri-grams,wemightobtain:THENSIONISTERNALLYATORENCEOPERTIFUL13Intheoriginal,therefollowsadescriptionofanactualexperimentusingRussiancarriedoutbythemathematicianP.L.Dobrushin,reproducedfromthebookProbabilityandInformationbyA.M.YaglomandI.M.Yaglom(Moscow:Nauka,1973).Themade-upEnglishversionpresentedhererepresentsanattempttoconveythesenseoftheoriginal.Trans. 149Themoreextensivethecorrelationstakenintoaccount,themorethe“texts”resembleEnglishformally—without,ofcourse,acquiringsense.TheLaputanaca-demicsinJonathanSwift’sGulliver’stravelsconfinedthemselvestogeneratingtextslikethefirstoneabove,wherenocorrelationswhateverhavebeentakenintoaccount,byturningaletter-wheelandcopyingthelettersinthehopeofobtainingatextofsubstance.InthecaseofRussian,thevaluesofinformationalentropyasthemoreex-tensivecorrelationsaresuccessivelytakenintoaccount,areasfollows:I0I1I2I35.004.353.523.01bitsForEnglish,Shannoncarriedtheestimatesmuchfurther:I0I1I2I3...I5...I84.764.033.323.10...2.1...1.9bitsHumanlanguagescharacteristicallyhaveaconsiderableamountofbuilt-inredun-dancyofinformation,forinstanceinthesensethatitispossibletoreadasensibletextevenifseverallettersarelacking.Thisredundancycanbequantifiedasfol-lows:ItwouldappearthatIntendstoalimitI∞asn→∞,thatis,asthescopeofthecorrelationsgrowswithoutbound.TheredundancyRofthelanguageinquestionisdefinedbyI∞R=1−.(7.12)I0Ratherthangoingallthewaytothelimit,wecanconsidertheredundancyrelativetocorrelationsuptolengthnofthelanguage,thatis,InRn=1−.(7.13)I0ForRussianthefirstfewvaluesofRnareasfollows:R0R1R2R300.130.300.40andforEnglish:R0R1R2R3...R5...R800.150.300.35...0.56...0.60ThusinEnglishtheredundancyiscertainlygreaterthan60%.ThismeansthatonecanunderstandanEnglishtextevenifonly40%ofthelettersarelegible—provided,ofcourse,thatthesearenotclusteredtogether.Toillustratethis,wedescribeanepisodefromChapter2ofJulesVerne’snovelCaptainGrant’schildren.14TheprotagonistsfindabottlefloatingontheseacontainingtextsinEnglish,German,andFrench.Thesearesomewhatsmudgedbysea-water,sothatonlythefollowingfragmentsofwordsarelegible:14LesenfantsduCapitaineGrant,Paris,1868. 150Chapter7.Information62BrigowsinkstraskippGrthatmonitoflongandssistancelost7JuniGlaszweiatrosengreusbringtihnentroisatstanniagonieaustralaborcontinprcruelindi◦jet´eongit37.11latThankstotheredundancyinthethreelanguages,theprotagonistsareabletoreconstituteallofthemessageexceptforthelongitude:“OnJune7,1862thethree-mastedvessel‘Britannia’,outofGlasgow,issinkingoffthecoastofPatagoniainthesouthernhemisphere.Makingfortheshore,twosailorsandCaptainGrantareabouttolandonthecontinent,wheretheywillbetakenprisonerbycruelindians.Theyhavethrownthisdocumentintotheseaatlongitude...andlatitude◦37.11.Bringassistance,ortheyarelost.”Andwiththistheiradventuresbegin.AmoreextremeexampleoftheredundancyoflanguageisaffordedbythescenefromL.N.Tolsto˘ı’sAnnaKareninaofaconversationbetweenLevinandKitty:15“Waitamoment”,hesaid,seatinghimselfatthetable,“ThereissomethingIhavebeenwantingtoaskyouaboutforalongtime”.Helookedstraightintohereyes,whichshowedaffectionbutalsoalarm.“Askaway,byallmeans”.“Lookhere”,hesaid,andwrotedownthefollowinginitialletters:w,y,r:t,c,b,d,t,m,j,t,o,n?Theselettersstoodfor:“Whenyoureplied:‘Thatcannotbe’,didthatmeanjustthen,ornever?”KittyunderstandswhatLevinwantedtoaskher,andtheircodedconver-sationthencontinues.HerepracticallyeverythingisredundantsinceLevinandKittyarefulltooverflowingwiththenecessaryinformationonthesubjectobsess-ingthemboth.ItisclaimedthatTolsto˘ıtookthisepisodeinthenovelfromhisownlife,thatis,fromhiscourtshipofSofiaAndreevnaBers.Ontheotherhand,redundancyisindispensableininformation,sinceintransmittingamessagebyanymeansofcommunication,therewillinevitablybe15AtthisjunctureinAnnaKarenina,Levin,agoodmanandefficienthusbandman,hasbeenencouragedtoapproachKittyonceagain,afteranearlierrebuff.Thispair,KittyandLevin,arecontrastedinthenovelwiththecentralpair,thebeautifulAnnaandthedashingVronski˘ı,whoseextramaritalaffairleadstoAnna’sostracismbypolitesociety,andherultimatedemise.Trans. 151“noise”,thatis,randomdisturbancesofonekindoranotherinterferingwiththetransmission.(Thisisthebasisofthechildren’sgameinvolvingadefectivetele-phone.)AshipindifficultytransmitsthemessageSOSoverandovertoincreasetheprobabilityofitsreception.InformationandentropyItisnotdifficulttoseethatShannon’sformula(7.11),MI1=−pilog2pi,i=1atleasthastherightformasanexpressionofentropy.RecallfromChapter4thatforanisolatedsystemconsistingofNmoleculeswithNimoleculesinstatei,i=1,2,...,M,theentropyisgivenbytheformula(4.19):MS=klnP=kNlnN−NilnNi.i=1Writingpi=Ni/N,theprobabilitythatanarbitrarymoleculeisintheithstate,wethenhaveMNMiS=−kNlnN−lnNi=−kNpilnpi,Ni=1i=1whencetheentropypermoleculeisMS1=−kpilnpi.(7.14)i=1ThisformalresemblancebetweentheformulaeforI1andS1isnotaccidental.Wehaveoftenspokenofentropyasbeingameasureofthedegreeofdisorderofasystem,orasameasureoftheextenttowhichinformationaboutthesystemisunavailable.Itisimpossibletoobtaininformationaboutanisolatedadiabaticsystem,sinceanyinstrument—athermometer,forinstance—broughtintocontactwiththesystemviolatesitsisolation.Similarly,inobtaininginformationaboutonepartofanopensystemoneinevitablyincreasestheentropyofsomeotherpartofthesystem.Considerthefreezingofwaterinavessel.Heretheentropyofthewaterdecreasesandinformationincreases:themoleculeswererandomlydistributedintheliquid,butarenow,intheice,fixedattheverticesoftheice’scrystallinelattice,sowehaveamuchbetterideaastowheretheyare.However,inorderto 152Chapter7.Informationfreezethewater,weneededarefrigerator,andthefreezingprocesswillcauseitstemperatureandentropytorise.16And,ofcourse,theriseintherefrigerator’sentropy17must—thesecondlawcannotbeviolated!—morethanbalancethedropintheentropyofthewaterduetofreezing.Thusforeverybitofinformationobtainedthereisacostinentropy.Nowtheunitsofinformation—bits—aredimensionless,whereasentropyismeasuredincalories/degreeK,orjoules/degreeK,orergs/degreeK.Toadjusttheunitssothatformulae(7.11)and(7.14)coincide,weneedtomultiplythequantityofinformationIin(7.11)bykln2≈10−23joules/◦K.Thisgivestheentropic1equivalentofabit.Weseethatinthermodynamicunitsabitisverycheap.Let’snowattempttoestimatethenumberofbitscontainedinthewholeofhumanculture.Weshallassumethatduringthewholeofitsexistence,thehumanracehasproduced108books(agreatlyexaggeratedfigure),andthateachbookconsistsof25pagesofauthorialoutput.Astandardsuchpagewouldcontainabout40,000symbols.Weshallignoreredundancyandattributetoeachsymbol5bits.Hence,altogetherweobtain108·25·4·104·5bits=5·1014bits,whichisequivalenttoareductioninentropyof5·10−9joules/◦K!No,humancultureshouldnotbemeasuredinthermodynamicunits.Butdoesthismeanthattheequivalenceofentropyandinformationmakesnosense?Tobeperturbedabouttherelativedifferenceinmagnitudesoftwoequivalentphysicalquantitiesisinappropriate.Forexample,Einstein’sequationgivingtheenergyequivalentofmass,Em=,c2wherecisthespeedoflight,isfundamentalinexplainingtheproductionofatomicenergy.However,theequivalencefactorhere,namelyc−2≈10−17sec2/m2isalsoextremelysmall:alittlemassisequivalenttoalotofenergy.Thesmallvalueofabitinthermodynamicunitsmaybeinterpretedassig-nifyingthataquantityofinformationisasmalldifferencebetweentwolargequantities,namelytheamountofentropybeforeandtheamountaftertheinfor-mationhasbeenobtained.Fromourdiscussionofinformationsofar,itisclearthatthisdifferencemustbenon-negative.Anincreaseininformationaboutasystemisequivalenttoadecreaseinitsentropy(oncetheunitsofmeasurementhavebeenadjustedtocoincide).Further-more,informationalwaysreferstothemicrostatesofastatisticalsystem;entropymeasurestheunavailabilityofsuchinformation.Anychangeintheindeterminacyofthemicrostatesofaphysicalsysteminagivenmacrostate,characterized,say,bythevolumeVandthetemperatureT,resultsinadecreaseofentropy,or,equivalently,toanincreaseofmicro-information.16Thecompressoratthebackoftherefrigeratorwillheatup.Trans.17Includingthatoftheairimmediatelysurroundingtherefrigerator.Trans. 153Weshallnowprovethisassertion.Ifthesystemgoesoverfromadistributionofprobabilitiesp1,p2,...,pMforitsmicrostates,toadistributionq1,q2,...,qM,howdoestheamountofinformationchange?Fromformula(7.4)itfollowsthatthechangeininformationaboutanevent(madeupofequiprobableoutcomes)resultingfromachangeintheprobabilityoftheeventfromptoqisln(q/p)ΔI=log2q−log2p=.ln2Theexpectedvalueoftheoverallchangeininformationduetothechangepi→qi,i=1,2,...,M,isthesumofthepartialchanges,eachmultipliedbythecorrespondingprobabilityqi.Hence1qiΔI=qiΔIi=qiln.ln2piiiThisquantityispositiveunlessqi=piforalli.Thisfollowsfromthefactthatforallx>0exceptx=1,1lnx>1−.xFor,providedqi=piforatleastonei,thisimpliesthatqiqiqiln>qi1−=qi−pi=0.pipiiiiiThusanontrivialchangeintheprobabilitiespialwaysresultsinanincreaseininformation.Thecomplementarityofinformationandentropyisillustratedbytheevapo-rationofaliquid:informationaboutthepositionsofthemoleculesislost,namelytheirlocationinacircumscribedportionofspace—thecontainingvessel—,whileentropyincreasesbyanequivalentamount.Theinformationthatthemoleculesofliquidwerelocatedinthevesselistransformedinto“negative”informationaboutthecorrelationsbetweentheirpositionsandvelocitiesinthevapor,affectedbytheircollisionswitheachother.Butthislackofinformationisjustentropy.Shannon’sformula(7.11)forinformationalentropy,andtheequivalentfor-mulaforthermodynamicentropy(7.14),sharecertainpeculiarfeatures.Entropyisdefinedintermsofacollectionofrandomquantities,namelythevaluesE(i)oftheenergyofthemicrosystemscomprisingthesysteminquestion.Wewishtoconsidertheextremecasesofsuchasystem.Astateofthesystemwillbeoneofzeroindeterminacy,thatis,willbefullydetermined,ifpj=1forsomej,andpi=0foralli=j.AccordingtoNernst’stheorem,thisisthesituationatabsolutezero:S(0◦K)=0(seethefinalsectionofChapter4).Thestateofthesystemwillhavemaximalindeterminacy,thatiswillsatisfyS=Smax,I=0,ifallpiareequal,thatis,ifthestatesareequallylikely.Thiscan 154Chapter7.Informationbeshownasfollows.Sincetheprobabilitiespisumto1,itfollowsthatpi=1/Mforalli.WeshowedabovethatΔI>0foranychangeintheprobabilitiespitonewprobabilitiesqi.Inthecasewherepi=1/Mfori=1,2,...,M,wehave(seeabove)1qi1ΔI=qiΔIi=ln2qilnM−1=ln2lnM+qilnqi.iiiSinceΔI>0andtheqiarearbitrary,itfollowsthatM11−ln=lnM>−pilnpi,MMi=1iforanyprobabilitydistributionpiotherthantheextremeonewithallpiequal,thatis,entropyislargestinthiscase.Incontrasttomicroscopicinformation,the“pricepaid”inentropyforinfor-mationaboutmacrostatesishigh,thatis,inanamountfarfromequivalenttothegainininformation:thegrowthinentropyismanytimesgreaterthanthequantityofinformationobtained.Forexample,whenwetossacoin,weobtainonebitofmacroscopicinformation,buttheproductionofentropyresultingfromtheworkofthemusclesusedintossingthecoinandfromitsimpactwiththefloor,ismanytimeslargerthan10−16ergs/◦K(thethermodynamicequivalentofabit),evenifthecoinisasmallone.Thelargerthecoin,thegreaterthediscrepancy.Thisindicateswhyitmakesnosensetoestimatetheinformationcontainedinbooksincaloriesperdegree.Maxwell’sdemonThestruggletosubvertthesecondlawofthermodynamics,topreventthegrowthinentropy,beganalongtimeago.Tothisendphysicistsrecruiteddemons.WemetwithLaplace’sdemoninChapter5.In1871,Maxwellintroducedhisdemontophysics,intheformof“abeingwhosefacultiesaresosharpenedthathecanfolloweverymoleculeinitscourse,suchabeing,whoseattributesareasessentiallyfiniteasourown,wouldbeabletodowhatisimpossibletous.Forwehaveseenthatmoleculesinavesselfullofairatuniformtemperaturearemovingwithvelocitiesbynomeansuniform,thoughthemeanvelocityofanynumberofthem,arbitrarilyselected,isalmostuniform.Nowletussupposethatsuchavesselisdividedintotwoportions,AandB,byadivisioninwhichthereisasmallhole,andthatabeingwhocanseetheindividualmolecules,opensandclosesthishole,soastoallowonlytheswiftermoleculestopassfromAtoB,andonlytheslowermoleculestopassfromBtoA.Hewillthus,withoutexpenditureofwork,raisethetemperatureofBandlowerthatofA,incontradictiontothesecondlawofthermodynamics”.1818JamesClerkMaxwell,Theoryofheat,London,1872. 155AnanalysisofthisactivityofMaxwell’sdemonwaspublishedbyL.Bril-louin19in1951,expandingonworkofLe´oSzil´ard20donein1929,showingthattheresolutiontotheparadoxliesininformationtheory.Brillouin’sanalysis21bringsoutthecloseconnectionbetweenentropyandinformationespeciallyclearly.Thefirstquestiontoaskis:Whatisrequiredforthedemontobeabletoseetheindividualmolecules?IfthesystemisisolatedandinequilibriumattheconstanttemperatureT0,thenthedemonwillhavetobeatthattemperaturealso.Undertheseconditionsanyradiationwillbeblack-bodyradiation,which,althoughhecanobserveit,willnotenablehimtoseeanindividualmoleculeordetermineitsvelocity.Forthat,themoleculewouldfirsthavetobeilluminated,sowewouldneedtosupplythedemonwithabattery-poweredflashlight,whosefilamentwouldthenneedtobeheatedtoatemperatureT1exceedingT0;wemayassume,infact,thatT1T0.Thisconditionisnecessarytoobtainvisiblelightthatcanbedistinguishedfromthebackgroundofblack-bodyradiationinthevesselattemperatureT0,andensuretheconditionthatonequantumhνofit22ismuchgreaterthankT,whichisof0theorderofthethermalenergyofamolecule.Neglectingthebattery,wehavethatifEistheenergyradiatedperunittimebythefilamentofthebulb,thenitlosesentropyattherateESfil=−.(7.15)T1Beforethedemonintervenes,theenergyEisabsorbedbythegasattemperatureT0,asaresultofwhichtheentropyofthegasincreasesbytheamountEEES=+Sfil=−>0.(7.16)T0T0T1Thedemonwillbeabletoseethemoleculeprovidedthatatleastonequantumhνoflightisscatteredbythemoleculetothedemon’seye.Theabsorptionofthisquantumofenergyraisesthedemon’sentropybytheamounthνΔSdemon=.(7.17)T0Thedemon’spurposeistousetheinformationheobtainstodecreasetheentropyofthesystem.NowtheinitialentropyofthesystemwasS0=klnP0.(7.18)19L´eonBrillouin(1889–1969),French/Americanphysicist.20Le´oSzil´ard(1898–1964),Hungarian/Americanphysicist.ItwasSzil´ardwho,in1939,pre-vailedonEinsteintocosignthefamouslettertoPresidentFranklinD.RooseveltpointingoutthefeasibilityofnuclearweaponsandencouragingtheinitiationofaprogramtodevelopthemaheadofHitler’sGermany—aninitiativeleadingeventuallytotheManhattanproject.21L.Brillouin,“Maxwell’sdemoncannotoperate:informationandentropy.”J.Appl.Phys.22(1951),pp.334–337.Laterworkonthistheme,forexamplebyCharlesBennett,maybefoundinthecollectionMaxwell’sdemon2:Classicalandquantuminformation,computing,H.S.LeffandA.F.Rex(Eds.),2002.Trans.22Accordingtoquantummechanics,lightoffrequencyνcomesinpacketsofenergyofminimumsizehν,wherehisPlanck’sconstant.Trans. 156Chapter7.InformationAfterreceiptoftheinformation,theindeterminacyofthesystemdecreases,sothatthestatisticalweightofthesystemisreducedfromP0toP1=P0−p,say.Thisentailsachangeinthesystem’sentropybytheamountΔSinfo=S−S0=−k(ln(P0−p)−lnP0).SinceforpP0(whichholdsinallpracticalcases),ppln(P0−p)=lnP0+ln1−≈lnP0−,P0P0wehave,approximately,kpΔSinfo=−.(7.19)P0HencethetotalchangeinentropyishνpΔS=ΔSdemon+ΔSinfo≈k−>0,(7.20)kT0P0sincehν>kT0andpP0.Weconcludethat,afterall,theentropyofthesystemincreasesjustasthesecondlawsaysitshould.Thedemonwasnotabletoviolatethatlaw.Brillouinsays:“Allthedemoncandoisrecuperateasmallpartoftheentropyandusetheinformationtodecreasethedegradationofenergy”.Tosummarize:Atthefirststageofthedemonicprocess,entropyincreasedbytheamountΔSdemon,atthesecondsomeofthisentropywastransformedintoinformation,and,finally,thiswasthenusedtopartiallydecreasetheentropy.Theefficiencyoftheworkofthedemonisgivenbytheratioofthedecreaseinentropyresultingfromtheinformationhereceived,tothetotalincreaseinentropy:|ΔSinfo|p/P0η==1.(7.21)ΔShν/kT0−p/P0Brillouin,andalsotheSovietphysicistR.P.Poplavski˘ı,wereabletoshowthatηdependsontherelativedecreaseintemperatureachievedbythedemon,thatis,onΔTTB−TA==θ,T0T0whereTB−TA=ΔTisthetemperaturedifferenceachievedbythedemonbetweenthetwohalvesofthevessel.Thusforinstanceifθ1,thenthecoefficientofusefulactioninthecorrespondingCarnotcycle(see(1.25))wouldbeTB−TA(T0+ΔT/2)−(T0−ΔT/2)ηC==TBT0+ΔT/2ΔTθ==≈θ.(7.22)T0+ΔT/21+θ/2 157Henceforθ1bothcoefficientsofusefulactionηandηKareessentiallyequal.Thecoefficientηmeasuresthedegreeofirreversibilityofaprocessinwhichorderiscreated,whiletheCarnotcoefficientηCmeasuresthedegreetowhichheatcanbereversiblytransformedintowork.However,asPoplavski˘ıstresses,toobtainworktwostagesarenecessary:aninformationalone,thatis,acontrolstage,andathermodynamicone.Maxwell’sdemon,likeLaplace’s,hasbeenthesubjectofpoetry.InAndre˘ıBely˘ı’sgreatestpoeticalwork“Firstencounter”,onefindsthefollowingverses:WithausterephysicsmymindWasoverfilledby:ProfessorBraine.Withneckbentback,andruffledmane,Hesangofcosmicgloom,AndofhowMaxwellannihilatedentropyWithhisparadoxes.Poetryisnotsofarfromscienceasthosepeoplewhosorigorouslyseparatephysicistsfromlyricalpoetsliketothink(seethelastsectionofChapter8).Somelineslaterinthepoem,Andre˘ıBely˘ıuttersaremarkablepoeticsurmise:IntheCuries’experimentstheworldburstforthWiththeexplodingatombombIntoelectronicstreamingsLikeadisembodiedhecatomb.Thispoemwaspublishedin1921.ObtainingandcreatinginformationWehavethusconvincedourselvesthatonemustpayforinformationwithanin-creaseinentropy.Inmacroscopicprocessesthiscostcanbeconsiderable.Theentropicequivalentkln2thatweobtained(seeabove)forasinglebitofinforma-tionisjustthelowerlimitofthiscost.Weshallnowcalculatethecostofinformationinunitsofenergy.SupposewehaveaquantityofanidealgasatpressurepandtemperatureT,consistingofNmolecules,andthatastheresultoffluctuationsthevolumehasdecreasedfromVtoV−ΔV.TheworkdoneinachievingthisisW=pΔV(seeChapter1).Theinformationobtainedtherebyiscalculatedasfollows:EachmoleculewasformerlycontainedintheregionofvolumeVwithprobability1,thatis,withcertainty.TheprobabilitythatitwasformerlyinthenowcontractedregionofvolumeV−ΔVisclearly1−ΔV/V.TheprobabilitythatallNmoleculeswereinthatsubregionoftheoriginalregionistherefore(1−ΔV/V)N.Asaresultofthecompressionofthegas,theNmoleculesoccupythesmallerregion,sothat 158Chapter7.Informationwenowhavegreaterinformationaboutthem,namelyintheamountN111ΔV1ΔVΔI=ln=−ln1−≈N.(7.23)ln2(1−ΔV/V)Nln2Vln2VHenceWpΔVpV≈=ln2=kTln2.(7.24)ΔI1NΔVNln2VThisrepresentstheworkdoneperbitofinformationobtained,thatis,thequantitykTln2istheminimalcostinenergyofonebitofinformation.AtT=300◦K(roomtemperature),wehavekTln2=2·10−21joules.Everyphysicalmeasurementyieldsinformation,andthereforeentailsalossinenergyaswellasagaininentropy.Measuringprocedures—thatis,processesforgatheringinformationaboutthemacrostateofasystemandcreatingorderinthesystem—areirreversibleinprinciple.Thestudyofthethermodynamicsofsuchinformationalprocesseshasbecomeofcrucialimportanceinphysicsinviewofthefactthatthecostsinenergyandentropyrisewiththeprecisionofphysicalmeasurements.However,physicsisnotlimitedtothetakingofmeasurements.Theresultsofmeasurementaretransmittedandused,thatis,theinformationobtainedthroughmeasurementisprocessed(inpartoncomputers).Andthistoomustbepaidfor.Thusweseethattheconcept“information”hasawell-definedthermody-namicsense.Thisbeingso,onecansafelyignoretheclaimsometimesfoundinprinttotheeffectthattheconceptofinformationisvagueandnon-physical.Inprevioussectionswespokeinpassingofobtainingorreceivinginformation,withoutmentioningthesignificanceofsuchprocesses.Wenowfillthebreach.Notefirstthattheconceptofinformationischaracterizedbythefollowingtwopostulates:1.Informationsignifiesachoiceofcertaineventsfromalargecollectionofevents(equiprobableorotherwise).2.Suchchoicesasin1areconsideredinformationonlyiftheycanbereceivedandremembered.InthedevelopmentofinformationtheoryintheworksofWiener,Shannon,andothers,wheretheaimwasthatofsolvingproblemsincommunications,moreattentionwaspaidtoprocessesoftransmissionofinformationthantothoseofreception.Inthestandardtheorythereceptorhasverylimitedcapabilities:allitcandoisdistinguishoneletterfromanother,onecodedsymbolfromanother.Thesituationisquitedifferentininformationalphysicsandinformationalbi-ology(tobediscussedinthenextchapter).Thereitbecomesespeciallyimportanttoinvestigatehowinformationisreceived.Wecanbeginourdiscussionofthereceptionofinformationbyconsideringoureverydayexperienceofit.Whatsortofprocessisit? 159First,itisclearthatreceptionofinformationisanirreversibleprocess(paidforbyanincreaseinentropy!).Informationmaybeforgottenorwastedbythereceptorinsomeotherway,butcannotberecouped.Second,receptionisanon-equilibrialprocess.Ifasourceofinformationanditsreceptorareinequilibrium,anexchangeofinformationistakingplacebetweenthem—bothdirectandinreverse—,andtheseflowsmustbalanceoneanother.Third,sincereceptionofinformationindicatesthecreationoforderinthereceptorsystem(Ireadapoemandrememberit!),thisisnotjustanon-equilibrialprocess,butonefarfromequilibrium.Areceptorsystemisadissipativesystem.Fourth,forthereceptionofinformationitisnecessarythatthereceptorpos-sessacertainlevelofcapability,ofcapacitytotakeintheinformation.However,althoughnecessary,thepresenceoftherequisitecapacityisnotsufficientforre-ceptiontooccur.BeforemethereisananthologyofpoetryinalanguageIamnotfamiliarwith.Inthiscase,Idonothavethenecessarylevelofreceptorcapacity(thatis,thepreliminarystoreofinformationthatmight,forinstance,resultfromthestudyofthelanguageinquestion)andsoamnotinastatetoreceivetheinformationcontainedintheanthology.Ontheotherhand,ifitisananthologyofRussianpoems,then,sinceRussianismynativetongue,Idopossesstherequi-sitelevelofreceptorcapacity(includingthelinguisticpreparationrequiredforanaestheticappreciation).However,Idonotwishtoreadpoetryjustnow,havingotherthingsinmind.Hencethefifthrequirementforthereceptionofinformation,namelythattherebeanelementofpurpose,anaim.Thepresenceofpurposeindicatesinsta-bility,sincetherealizationofanaimrepresentsatransitionfromalessstablestatetoamorestableone.Thusreceptionofinformationisanirreversibleprocessoftransitionofadissipativesystemfromalessstablestatetoamorestableone.Asalwaysinsuchsituations(seeChapter6)suchaprocessmustinvolvetheexportofentropyfromthereceptorsystem.Ofcourse,informationisconsideredashavingbeenreceivedonlyifitisrememberedforalongerorshorterperiod.Receptionandrememberingofinfor-mationgotogetherindissolubly.Informationcanbeaccumulatedandstored.Books,or,totakeaverydifferentexample,thegenomesofallspeciesonearth,representstoresofinformation.Wereceiveandrememberonlymacroscopicinformation.Thiskindofinfor-mationdemandspaymentnotinequivalentamountsofentropy,butinsignificantlylargeramounts.Thatiswhytheestimatewemadeearlieroftheequivalentinen-tropicunitsoftheinformationcontainedinallthebookseverwrittenisreallymeaningless.Howisinformationgenerated?AnanswertothisquestionhasbeengivenbythetheoreticianA.Koestler,23namely,thatinformationiscreatedthroughthe23ThismaybeAuthorKoestler(1905–1983),Jewish-Hungarian/Britishpolymathwriter.An-otherpossibilityis:G.Kastler,authorofTheoriginofbiologicalorganization,1967.Trans. 160Chapter7.Informationcommittingtomemoryofarandomchoice.Hegivesthefollowingillustration:SupposeIplacemysuitcaseinacloakroomlockerattherailwaystation,andencodethecombinationlockwithafour-digitcombination,knowledgeofwhichwillenablemetoopenthelocker.Bycommittingtomemory(orwritingdown)thatrandomlychosensequenceoffourdigitsIhavecreatedinformation,namelyintheamountlog29000=13.13bits.Thecreationofnewinformationalwaysproceedsinjumpsratherthangradu-ally.Althoughessentiallynon-equilibrial,itbearssimilaritiestophasetransition.However,ordinaryequilibrialphasetransitionssuchasthecrystallizationofaliquid,donotinvolvethegenerationofanynewinformation;thereisnochoiceinvolved,everythingproceedsinprescribedfashionaccordingtotheappropriatelawsofphysics.Ontheotherhand,thecreationofanactualcrystalwithpecu-liarflawsinitslattice—fissuresandimpurities,forinstance—maybeviewedasrepresentingarandomchoice,andthusasgivingrisetonewinformation.Wehaveintroducedthereadertotwodemons—Laplace’sandMaxwell’s—buttoonlyone“billiard”(seethesecond-lastsectionofChapter5).Hereisan-other,theso-called“Chinesebilliard”.24Thereisaboard,orfieldofplay,onwhichtherearedistributedfixedstudsorpinsandshallowpitsorwells.Nexttoeachwellthereisanumberindicatingthescoreforthatwell.Metalballsarefiredalongtheboard,oneatatime.Afterasuccessionofcollisionswiththepins—sothatitstrajectoryisusuallyquitechaotic—aballultimatelycomestorestinoneofthewells.Themotionofaballtillitcomestoahaltinawellprovidesamodelofentropy.Thecomingtorestinawellrepresentstheachievingofarelativelystablestateastheresultofarandomchoicefromamongamultitudeofpossibilities,andthereforemodelsthegenerationofinformation.Thecostinentropyforthisinformationishuge.For,ifthereare32,say,wellsontheboard,thentheamountofinformationgenerated(andreceived)is5bits,whiletheamountofheatperunittemperaturegeneratedbytheinitialpropulsionandsubsequentcollisionsoftheballuptothetimeitcomestorestexceeds5kln2=5·10−23joules/◦Kmanytimesover.Itshouldbeemphasizedthatatthepresenttimethecreationofagenuinephysicaltheoryofreception,remembering(orrecording),andgenerationofinfor-mation,thatis,ofthesespecificirreversibleprocessesinappropriatedissipativesystems,isencounteringsignificantdifficulties.ThevalueofinformationEverycommunicationofinformationhasadefinitecontentandmeaning,andisofsomevalueorothertotherecipient.Standardinformationtheory—whichiswhatwehavebeenconcernedwithuptillnow—completelyneglectsthequestionofthecontentofinformation.However,thisisactuallyavirtueratherthanadefectofthattheory;clearly,ifthemainconcernisthetransmissivecapabilityof24Thebasicformof“pinball”.Trans. 161acommunicationschannel—forexample,anelectrictelegraphsystem—thenitisnotatallappropriatetotakeintoconsiderationthecontentofthetransmittedtelegrams.Ontheotherhand,therearemanyscientificproblemshavingtododirectlywiththevalueofcertaininformation.Thesearemostlyproblemsarisinginbiologyandvariousaspectsofthehumanities,andwewillbediscussingthemtosomeextentinthenextchapter.ThequestionofthevalueofinformationhasbeeninvestigatedbyanumberofSovietscholars,includingM.M.Bongard,R.L.Stratonovich,andA.A.Kharkevich.ChapterVIIofMikhailMo˘ıseevichBongard’sveryinterestingbookTherecogni-tionproblem25(Moscow:Nauka,1967)isentitled“Usefulinformation”.Bongardlinksthedegreeofusefulnessofamessage,thatis,thevalueoftheinformationitcontains,withtheincreaseintheprobabilityofachievingsomeobjectiveasaresultofreceivingthemessage.ThusthevalueoftheinformationcontainedinthemessageisgivenbytheformulapV=log2,(7.25)pwherepandparerespectivelytheprobabilitiesofachievingtherelevantaimbeforeandafterreceiptoftheinformation.ThenotionsofvalueintroducedbyStratonovichandKharkevichintheirworksaresimilar.Itisclearthatthevalueofinformationcannotbedefinedindependentlyofitsreception,sincewecanjudgethevalueofamessageonlyfromtheconsequencesofitsapprehensionbythereceptor.Thus,incontrastwiththedefinitionsofquan-tityofinformationexpressedbyHartley’sformula(7.4)orShannon’s(7.11),itisimpossibletogiveauniversaldefinitionofthevalueofinformation.Thevalueisonlyrevealeduponreception,andisintimatelyconnectedwiththelevelofthatreception.Hereisasimpleexample:ConsiderVolume2ofV.I.Smirnov’sAcourseinhighermathematics.Thisbookcontainsawealthofinformation.Butwhatisitsvalue?Thenaturalresponseis:Forwhom?Forapreschoolertheansweriszero,sinceheorshelacksthenecessarypreparation,hencedoesnotpossessanade-quatelevelofreception,andisthereforenotinapositionorstatetoapprehendtheinformationcontainedinthebook.Foracompetentmathematicsprofessorthevalueisagainzerosinceheknowsthecontentsofthebookverywell.Theinforma-tioninquestionisofgreatestvalueforstudentstakingthecourseforwhichthistextbookwaswritten—sincethistextbookisindeedanexcellentone.Thegraphofthevalueoftheinformationagainstthelevelofpreparation—whichmightbecalledthe“thesaurus”fromtheLatinfortreasureorstore—herepassesthroughitsmaximum(Figure7.1).Clearly,thespecificvaluegivenbyformula(7.25)representsacharacteris-ticchieflyofthereceptionoftheinformation.Theincreaseintheprobabilityof25AnEnglishversionwaspublishedin1968.Trans. 162Chapter7.InformationValueThesaurusPrSPFigure7.1:Graphshowingthedependenceofthevalueoftheinformationonthe“thesaurus”:Prisforpreschooler,Sforstudent,andPforprofessor.achievementoftherelevantaimisdeterminedbythereceptorwhoseaimitis.Ofcourse,theconceptofaimorpurpose,thoughhighlysubjectivewhenappliedtoahumanreceptor,isentirelyobjectivewhenappliedtophenomenaofphysics,chemistry,andbiology.Asmentionedearlier,thepresenceofan“aim”incon-nectionwithsuchphenomenasignifiesmerelyadefiniteinstability.The“aim”ofgenesforstructureisthesynthesisofalbumins(seethethirdsectionofChapter8).Inthissenseoftheword,theconceptofaimorpurposedoesnotgobeyondtheboundsofphysicsorchemistry.Accordingtoformula(7.25),redundantorrepeatedinformationhaszerovaluesinceitdoesnotresultinanychangeintheprobabilityofachievingtheaim.Notealsothatthevalueoftheinformationreceivedcanbenegativeifitisactuallydisinformation,thatis,falseinformationmakingithardertorealizetheaim.Practicalapplicationofformula(7.25)presentsdifficulties.Letustrytosimplifytheproblemalittle.Startingfromthefactthatredundantinformationhaszerovalue(forexample,repetitionofthesamefactsconveysnonewinformation),weshallredefinethevalueofinformationasthedegreeofirredundancy,thatis,thedegreeofirreplaceabilityofthecommunicationasawholeorelseofasingleelementofit,forexample,asymbolofcode.SupposethemessagecontainsN0suchsymbolsorletters,andthetotalamountofinformationisN0I0.Atthenextlevelofreception,wherethefrequen-ciesofoccurrenceoflettersistakenintoconsideration(seethesecondsectionofthischapter),theamountofinformationconveyedreducestoN0I1whereI1