Thermodynamics by Callen

Asummary ofCallari" onThermoelectric Effects. 1.Myfirst step was togoback tohis chapter 2torelearn the two basic e representations fordoingthermo: (a)theenergy representation, wherein U=U(S,X))Xp-..) isthemainobject of interest, theX,arethevarious extensive parameters like volume, andthe first law of thermo has this form: Py=(dU/aX,) =generalized "force" intensive parameter P,=(dU/dS) =T ,thetemperature Pye (au/an; )=w+ thechemical potential, example ofintensite force a= Pydk =TdS +rest where Tds=aQ (b)theentropy representation ,wherein S=S(U,X,,X,....) imthemain object ofinterest, theX,=Unow, andtheothers arethesame asbefore. Then: Fy=(dS/ax,) =general force inthisrepresentation =-P,/T ifo Fy=(dS/aU) =1/T, theinverse temperature dS=Fydk=(1/f) dU+rest . 2.Next, weskip tochapter 16where welearn about affinities and fluxes for two coupled systems (discrete, not yet continuous). Assume there issome extensive paramter Xj; andrelated intensive force F,.Then you candefine: e fy=Fy-FL=thedifference betweentheforces(affinity ) Jj=(aX,/at) =thefluxrelated totheabove. . Think ofapartitioned box that isnot inTE.There isanetforce onthe sliding partition which causes ittomove. Thus, the volumes onthe two sides change, there isadV/dt. This would betheflux resulting from theaffinity. Thus weare making atheory todescribe things when weare close tobut not inThermal Equilibrium! Using the chain rule, you can show thats ds/at =£,Jy Te,asthings adjust (the partition moves) thetotal entropy must beincreasing and this gives the rate atwhich itdoes so, Here S=entory ofboth systems together. Note that entropy isnot converved inthe way Uis! 3.The next step istoextend these ideas tothe continuum case where your system isalittle differential piece ofsystem incontact with the rest. Callan switches tolower-case extensive paramters defined asper volume. The above equations become: a fy,=grad F,==theaffinity ordriving force. Gi=thefluxrelated todx,/dt bycontinuity, flows inresponse to .= thedrivingforceaffinity e ds/at =F, Jj,=Fate ofinerease ofentropy T onda ;MORplawom.bindahoawreManwoher. -~2- 4.Now wewould like toImow how the responding fluxes are related tothe affinity driving forces which are causing the flues. Ifyour system has e no"memory" (sayinthe€ormofcapacitance-like effects), itiscalled aMarkoff process and you may conclude that: Je=Je(tyr Fy) Ie, the fluxes are functions ofthe forces and their gradients ataninstant of time, there isnoneed tointegrate over the past somehow. Inmymind,thestateoftheparameters F,justsetsthestage, ie,sets the“operating pointx" oftheprocess, whereas thef,aretheactual driving forces. Clearly J,will depend more onf,than onanyother force, butwe should expect integactions. Note: J,isnotafunction ofU,X,,Xp+.. theextensive parameters. Should beable toprove this. 5.Now assume that all driving forces are very small insome sense, ie, you are nottoofaraway from equilibrium. Then youtreat theflux J,asafunction oftwo variables and expand inthe affinity like so: Yg= Lyfy+higher order inf Ifyou simply ignore higher order effects, you are now talking about alinear Markoff process and lmowledge ofwhat isgoing onisnow contained inthe matrix ofcoefficients #Ly,=Ly(Fy) Ie,these things areonlyfunctionsoftheoperatingpointintenilive forceparametersF.+ @ 6.Onsager's Theorem says that you don't really have tofigure out all ofthe entries inthis Lmatrix (called Kinttic Coefficients) because thefollowing istrue and can beproved from fluctiiatgon theory: LapFy)=Lyaltimereversed F,,) For example, anexternal magnetic field would bereversed insuch arelationship. Thus, you need tofind the diagonal entries ofLand one side only. 7,Simplest possible example ofthelinear Markoff application: LetX,=UandthenF,=(1/7) sothatf,=grad(1/T). Suppose there arenoother intensiveparamefers intheproblem: ThefluxJ,isofcourse a"total energy flux whichshowshowenergy flowsinresponse to°atenperature gradient. YourLmatrixhas only oneentry (for isotropic material, Iguess): 2 Jy=Ugg grad(1/t) =[-L,,/T°] grad(t). Thusyouwould identify L,,=Kappa xT°where kappa istheusual thermal conductivity that you look°tp intables for some material. Ohm's Law is another example. 8.The general philosophy ofusing these "Onsager Equations" istotry and identify alltheLj,coefficients withknown system parameters, thenyouwillgeta systemof’coupledlinearequations whichdescribe whathappens! Andall ecoefficients are known. -3- Notice that,asisalways thecaseinthermo, weneverhaveanymethod forcomputinganyoftheseconstants. Thermogivesuscorrelations between e@things, butdoes notexplain onthemicro-level what isactually happening. Ie, wemight like toknow exactly how the affinity creates the flux and so on. ThermoElectric Effects Wearenowgoing toapply alloftheabove formalism toasystem with basically only two intensive parameters: Xo=U=thetotal energy density FP,=(1/r) x,=ntheelectron density Fy=(-v/?) Here, muisthe electrochemical potential ofasingle electron. Recall that this issomehow theforce felt bythe electron. Weknow that such aforce has aérift part related toelectric fields andpotentials, andadiffusion part related todensity gradients. Both electric fields anddensity gradients can"push"anelectron around, sothey both contribute tothepotential mu. Ouropening equation (thefirst lawinentropy rep) says: ds=(1/7) du-(m/T) dn Obviously wehavevarious kinds offluxes: J,=flux ottotal energy (J,relatedtoJ,andJ,bytheabove)In=f1Uxofelectrons eJ,=flux oftotal entropy Ofcourse wecanalso write thefirst lawasdu=Tds+(ma) dn=da+(mu) dn 80wecanintroduce Jg=aheatcurrent related toJ,,andJ.,asshown, Actually, theelectric current ed,andtheheatcurrent J_arethe twothat aremostinteresting tousphysically, sowechoose themaour basic flux variables, Affinity associated with x.=4isF,=(ds/dq) =(1/7), sofy=grad(1/T) Affinity associated with x,=-n isFy=(m)/? so£;=grad(m/?) However, this point ofview isnot quite valid because heat isnot really animkenxextensive variable. ?WellmaybeitisOK.Whenwedograd(m/T) howeverwecancombinetheterm(m)grad(1/t) withthesecondtermandredefine theL's.Thus you get: “Jy=yy[(2/2)grad(m)] +—IypCgrad(1/t)] Sq=UyC(4/T)gred(m)] =+Igy[grad(2/t)} Here weseethat thermo-electric effects (effects involving heat ande electric currents) aremost simply modeled asatwo-dimensional linear Markov process! Thenext step istotryandidentify thevarious coefficientgs! -ke 1,Now, here ishowyouidentify L,,and'L,5'with known constants; . e (a)Consider enelectric current through awireatconstant temperature. The grad(1/T) terms intheMarkoff equations thus vanish. Ifn=n(T) ,then there will benodensity gradient, sothechemical mi, hasnogradient, andonly theelectrical m,hasagradient which isofcourse theelectric field. In this wayyoucanquickly relate L,,totheconductivity parameter sigma. (b) Now suppose you have atemperature gradient and aheat current through a wire, but noelectric current. You cannot inthis case set any terms equal tozero inthe Markov equations, but you can invert them asa2x2 matrix equation (in effect eliminating the unknown grad(mu) torelate thethermal conductivity kappatothethreeL's.Notethatkappa4£0)only. Indoing this matrix inversion, wealso get aspecial equation (17.24) which isofcourse valid only when there isnoelectric current. 2. There isstill one unknown parameter which wehave tofigure out how to measure. Let us define: epsilon =-1,,/(eTL,;) This you see isessentially the parameter that appears inthat equation (17.24) which applies when nocurrents. Ifyou integrate this equation around a standard thermocouple, you find: AA |e Vseebeck*»(ég-€,]at or dv/at =(€g-€,) ="the thermoelectric power ofthermocouple" | Note:Whilemeasuring thisthermocouple voltage, wemustmakesurenocurrent isdrawn, otherwise weinvalideate using (17.24). This voltage measurement with nocurrent iscalled the Seebeck Effect orSeebeck voltage. Note: although the absolute thermoelectric power € clearly exists and is calculable from first principles, the Seebeck Effect only allows you tomeasure relative epsilons. Atsome point you must know the epsilon for some metal (perhaps platimm), then you can measure the rest. Iwonder what experiment lets you measure absolute epsilon? 3.ThePeltier Effect. From (17-35), consider acurrent ruining through ajunction held atconstant temperature T.Since noTgradient, youhave J,=TepseJ,Butweknow that epsisdiscontinuous through thejunction, whefeas TandJ, are continuous; thus J,isdiscontinuous andheat isemmitted orabsorbed at thejunction. This isfhePeltier effect. Note that nomechanism isactually stated for this process. Weare only doing thermo. r Equation turnsouttober a~5q=Tea(eJy)whereWea=Tee-en ) Thus, the junction heat current generated inPeltier Effect isproportional to the electric current crammed through the junction (orgenerated byit!). The Peltier coefficient istrivially related tothe difference ofthe Seebeck Coefficients. e Jouleheatinghasbeenignoredhere. ~5- 1.Thomson Effect. Now consider ashort piece ofwire with atemperature difference aTbetween the ends. This might beapiece ofthe wire inathermocouple which e connects thetwotemperature reservoirs. Youcanshowthis: du/adt =rate orproduction ofenergy atsome point =tau. grad(T) +(eJ,) +Joule heating This says that there ishead created orabsorbed (depending onsign ofconstant tau) inaddition tousual I°R heating, andthis extra heat orlows iscalEd the Thomson Heat or Thomson Effect. You can see that the heat ofthis effect isproportional tothe current inthe wire and tothe temperature gradient the wire isin. The actual heating orcooling depends onthe relative directions ofcurreht and temp gradient. tau =-T(deps/d?) So this constant is the derivative of the Seebeck coefficient! 5.Numbers. Fortypical metals, (€,,) =20wV/degree which isvery omall andmakes practical reftigeration diffictft. Forcertain semiconductor materials however this number goes uptoperhaps 400wV/deg. Atypical Seebeck thermocouple voltage from metals isperhaps 6millivolts. Generally, thermocouple devices operate atvoltages like this, and relatively high currents like 40amperes. They are inherently low resistance devices. 6.Other effects r FourierConduction referssimplytothefactthat,whenyoutrytomakea Yefrigerator with Thomson orPeltier effects, even when you have noelectric currents, heat will leak through your material from the hot tothe cold, thus hurting your effieiency. Part ofthe game inthermo design istofind materials which have low Fourier conduction, but high Seebeck coefficient. Joule Heating. ThisisjusttheEI=I°Rheating inawireofjunction. References: Callan, HB. Chapters 2,16, 17 Zemansky, Chapters 9and 13. Reif has nothing except comments onOnsager symmetry. Angrist Direct Energy Conversion book has some stuff. Chanwoh 2Conddrina) 4,Sedetonnn. Eaease DWLAWS GN) Heet eaegy She. . BDyu.7 v Hong Uw. 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Y\Derrmumlare ontype, wonrand(00presNaefoegett onyaed @. @ .@ e an introduction tothe physical theories ofequilibrium thermostatics and irreversible thermodynamics JOHNWILEY&SONS,INC, HERBERT B.CALLEN [NewYork*London +Sydney Professor ofPhysics University ofPenntyivania ee x PREFACE sions ofthethermodynamic method have been made outside the domain ofequilibrium. Theassociated theory isgenerally referred toasirreversible thermodynamics orthetheory ofirreversible proc- esses, Itseems probable thatthisnewextension ofthetheory will Provide someofthemostsignificant rewards ofthethermodynamic >. method.Someresultsofthetheoryofirreversible thermodynamics arepresented inthefinalchapters ofthisbook. Contents Herverr B.Catten Philadelphia, Pennsyleonia : November 1959 : PART I : GENERAL PRINCIPLES OFCLASSICAL THERMODYNAMICS, 1, Basic Concepts and Postulates 3 1.1 The Nature ofThermodynamics 3 , 1.2. TheComposition ofThermodynamic Systems 7 1.3. TheInternal Energy 10 14 Thermodynamic Equilibrium 11 1.5 Walls arid Constraints 14 1.6 Measurabilty oftheEnergy 15 1.7 Quantitative DefinitionofHeat—Units 17 18 TheBasic Problem ofThermodynamics 22 19 The Entropy Maximum Postulates 24 2. ‘The Conditions ofEquilibrium 3t 21 Intensive Parameters 31 . 22 EquationsofState33, :2.3 Entropic Intensive Parameters 35, 24Thermal Equlibrium—Temperature 37 ’ 2.5 Agreement with Intuitive Concept ofTemperature 38 . 2.6 Temperature Units 40 2.7 Mechanical Equilibrium 43 2.8 Equilibrium with Respect toMatter Flow 45 :. 3. Some Formal Relationships 47 3. The Euler Equation 47 : a xii CONTENTS CONTENTS xilt 3.2TheGibbs-Duhem Relation48 9%First-andSecond-Order PhaseTransitions 14633°SummaryofFormalStructure 50 . 9.1First-Order PhaseTransitions inSingle-Component Systems3.4AnExample-—The IdealMonatomic Gas$1 14635. Specific Heats and Other Derivatives 54 9.2 The Discontinuity oftheVolume—The Lever Rule 154 .‘ 9.3TheDiscontinuity oftheEntcopy—Latent Heat 156 : 4.Processes andThermodynamic Engines 59 9.4PhaseLoci—The Clapeyron-Clausius Equation 1574.1Quasi-Static Processes 59 9.5Metastable StatesinPhaseTransitions 162+4.2Reversible andIrreversible Processes 63 : 9.6First-Order PhaseTransitions inMulticomponent Simple4.3,TheReversible WorkSourceandReservoirs 65 ‘Systems—Gibbs PhaseRule 163,44Maximum WorkProcesses 66 9.7PhaseDiagrams forBinarySystems 1674.5.Thermodynamic Engines 69 9.8Tisza’sTheoryofSecond-Order PhaseTransitions 1724.6AnIllustrative Problem 71 9.9Ehrenfest’s TheoryofSecond-Order PhaseTransitions 180 4.7 Refrigerators andHeat Pumps 74 4.8TheCarnotCycle 77 10.TheNernst Postulate 1824.9.Measurability oftheTemperature 79 10.1Qualitative Statistical Comments 1824.10TheT*MethodforLow‘Temperatures 81 10.2SpecificHeatsandOtherDerivatives atLowTemperature 184 5.Alternative Formulations andLegendre Transformations 85. 10.3.Thomsen andBerthelot's Principle 186S.A.TheEnergy Minimum Principle 85 104The“Unattainability"ofZeroTemperature 188 5.2 Legendre Transformations 90 . 53Thermodynamic Potentials 08 11,Summary ofPrinciples forGeneral Systems 191 54°Generalized Massieu Functions 101 111General Systems 191 11.2ThePostulates 192 6.TheExtremum Principle intheLegendre 11.3TheIntensive Parameters 192 Transformed Representations ‘ 103 iepesendeTransformswi? 5 Maxwell Relations 6.1 The Minimum Principles forthePotentials 103 11.6 Stability and Phase Transitions 194 6.2 The Helmholtz Potential 106 11.7. Properties atZero Temperature 195 63 The Enthalpy 110 6.4 ‘The Gibbs Function 115 6.5TheMaximum Principles fortheMassicu Functions 116 PART I 7. Maxwell Relations 7 REPRESENTATIVE APPLICATIONS TATheMaxwellRelations 117 12.ChemicalThermodynamics 199 2.2AThermodynamic Mnemonic Diagram 119 12.1Chemical Reactions 1997.3AProcedure fortheReduction ofDerivatives inSingle-Com- 42.2Chemical Equilibrium 201PonentSystems 121 123DegreeofReaction 202 7.4SomeSimpleApplications 124 124Simultaneous Reactions 203 7.3.Jacobian Transformations 128 12.5,HeatofReaction 204 126 Stability andtheLeChatelier Principle 205 8.Stability ofThermodynamic Systems ww 12.7GibbsPhaseRuleforChemical Systems 2068.1Intrinsic StabilityofSingle-Component Systems 131 12.8ChemicalReactionsinIdealGases207 82MutualStability ofSingle-Component Systems 137 129Temperature Dependence oftheEquilibrium Constant 209 83.TheLeChatelier-Braun Principle 139 12.10HeatofReactionofIdealGasReactions 210 84Intrinsic Stability ofGeneral Systems 141 12.11Additivity ofReactions 211 xiv contents .ConTENTS ” 174 The Peltier Effect 299 TaeBoldSyatereHasty as 17.5TheThomsonEffect300Elastic Strain ; 17.6TheThermomagnetic Dynamical Equations 302132 The Fundamental Equation 219 2 Theron tie baa 133TheStressComponents 220 171TheThermomagnetic Elects308 : 134 Maxwell Relations 224 13.5. TheElastic Coefficients 226 APPENDICES :13.6 Consequences ofPhysical Symmetry—Cubie and Isotropic : Systems 227 A.SomeRelations Involving PartialDerivatives +309 13.7Coeflcients ofThermal StrainandStress 232 Ad.PartialDerivatives 309 138SpecificHeats 233 ‘82Taylor's Expansion 310 139.OtherCoefficients 234 A3_Differentials of#310 : 13.10Hooke’s Equation ofState 236 ‘4Composite Functions 31% 14.Magnetic andElectricSystems 238 ASImplicitPanetlona 312 14.1 Magnetic Extensive andIntensive Parameters 238 B, Statistical Significance oftheEntropy 315142ElectricExtensiveandIntensiveParameters |243, :143 Magnetic Feee Energies andMaxwell Relations 24S €. Equilibrium with Internal Adiabatic Constraints 321 144 Specific Heats—Magnetie Susceptibility 247 14.5“Magnetic Equations ofState 249 D.Properties ofGases .94146 TheMagnetocalorie Effect 255 D.l TheSingle-Component General Ideal Gas 324 14.7 Superconductivity 258 : D.2_Compressibilicy and Expansion Coeficient ofGeneral Ideat Gases 326 D.3Gibbs-Duhem Relation forGeneral IdealGas327 PART Il Di The Monatomic Ideal Gas 328 D.SSpecific Heats ofIdealGases 329 FLUCTUATIONSANDIRREVERSIBLE THERMODYNAMICS D6TheMulticomponent Idea!Gas335 , D.7_ Nonideal Gases—The Virial Expansion 338 15.TheTheoryofFluctuations 267 D8ThevanderWaalsandBerthelot Equations ofState34113.4. ‘The Thermodynamic Distribution Function 267 15.2 Average orEquilibrium Values 271 E, Properties ofSimple Solids andLiquids 34315.3.MomentsandtheDistribution Function 272, E.LGeneralProperties -343,154Thermodynamic Fluctuation Moments 274 E2Liquids 346 15.5AnAlternate Form(ortheSecond Moments 279 E3Effect ofChanging Pressure 348156‘TheAssociated Gaussian Distribution 280 E4Simple Solids 360 16.Irreversible Thermodynamics 283 ESSpecieHeatsofSolids353 16.1 General Remarks 283 F, Several Common Cyclic Processes 387 16.2 Afinities andFluxes 284 F.lTheOttoCycle 357 16.3Markoffian Systems 288 F.2TheBrayton orJouleCycle 358164Linear Processes 289 F.3.TheAir-Standard Diesel Cycle 359165, ‘The Statistical Basis oftheOnsager Reciprocity 290 G.Matrices andtheStability Quadratic Form 361 17,Thermoclectric and Thermomagnatic Effects 293 17.1 TheDynamical Equations 293 Bibliography ofGeneral References 367 17.2 The Conductivities 296 17.3TheSeebeckEffectandtheThermoelectric Power297 Index ” I a 2 BASIC CONCEPTS AND POSTULATES ‘THE BASIC PROBLEM OFTHERMODYNAMICS 23 1.7-4, Asmall paddle wheel isinstalled inthesystem ofproblem 1.7-3. shallfindthatthemere statement oftheproblem suggests thepostulates ‘Theshaftofthepaddlewheeextendsthrough thewalsofthesystemandsan whichprovide itssolution. riven at240cpsbyanexternal motor. ‘The viscous torque onthe paddle i ined withi system while thevolumeiskeptconstantandthesystemisadiabatically enclosed, cylinder,separatedfromeachotherbyaninternalpiston.Assumethat tiepressureisfoundtoincreaseatarate thecylinderwallsandthepistonarerigid,impermeable tomatter,anddp 2 adiabatic andthattheposition ofthepiston isfirmly fixed. Each oftheaw systemsigclosed.Ifwenowfreethepiston,itwill,ingeneral,secksomeyst pi & s whereTistheviscoustorqueandaistheangularvelocityofthepaddlewheel. newposition. Similarly, iftheadiabatic coatingisstrippedfromthepistonUsingtheforegoingprocessandthepracessofadiabaticexpansiondescribed sothatheatcanflowbetweenthetwosystems,therewillbearedistribu- inproblem 1.7-3, findtheinternal energy ofanyequilibrium statewitharbitrary tionofenergy between thetwosystems. Again, ifholes arepunched inpressurePandvolumeV.Choosethestate4(P=32atm,V=Jliter)asthefiducial state.‘Whataretheheatfluxesineachseparatestepofprocessainproblem1.7-37 S Ineach stepofprocess c? ayyo)OD ‘Answer: Heat fiuxinthefirststep ofprocessa=560liter-atm. UOVON= UBVONP 1.7.5, Two moles ofaparticular single-component system arefound to stonhaveadependence ofinternal energy Uonpressure andvolume givenby =i U=APY* (forN=2) . a once - where A=10cm*, Note thatdoubling thesystem doubles thevolume, energy,andmolenumberbutleavesthepressureunaltered, andwritethecomplete : Figure1.3 dependence ofUonP,¥,andNforarbitrary mole number. . Answer: U=BPV*N, B=20/em?, thepiston, there willbe aredistribution ofmatter (andalsoofenergy)“ 1.7.6. Assuming thesystem ofproblems 1.7-3and1.7-4tohavebeena between thetwosystems. Thus theremoval ofaconstraintineachcase single-componentsystem,showthatthecomplete functional dependence of resultsintheonsetofsomespontaneous process, andwhenthesystems UonP,V,andNis,infact,independentofN. finallysettleintonewequilibriumstatestheydosowithnewvaluesof pdTeTeAparticular single-component systemofonemolehasadbats of theparameters U,VO,ND.+- andU®,V®,ND---. Thebasic theformPY" =constant. Itisfitted withastirrer, asin,problem 1.7-4. jynamics isthe i ‘libriWhenthesterdoceanamountofwork,iV,thepressure unereasedP(at problem ofthermodynamics isthécalculation oftheequilibrium values, constant volume) isgiven byssure ines ofthese parameters. : aw=v+VQaP Beforeformulating thepostulate thatprovides themeansofsolution of Find theinternal energy asafunctionofP,¥,andN. thisproblem,werephrasetheprobleminaslightlymoregeneralform17-8.Showthatitasingl i a withoutreferencetosuchspecialdevicesascylindersandpistons. in78, ShowthatiCasingle-component systemissuchthatPY*isconstant Giventwoormoresimplesystems, theymaybeconsidered asconsti- inanadiabatic process (kisapositive constant) theenergy is civen ‘ F i 1. tuting asingle composite system. Thecomposite system istermed closed UaeaPy +PVH ifitissurrounded byawallthatisrestrictive withrespecttothetotalherefisanarbitraryfuneti ‘energy,thetotalvolume,andthetotalmolenumbersofeachcomponent where/anarbitrary function.| ofthecompositesystem.‘Theindividualsimplesystemswithinaclosed Hint:FirstshowthatU~7+PVisconstantoneachadiabat. Lcomposite systemneednotthemselves beclosed.Thus,intheparticularexample referred to,thecomposite system isclosed even ifthe internal piston isfreetomove orhasholes init.Constraints thatprevent1.8TheBasicProblemofThermodynamics \ theflowofenergy,volume,ormatteramongthesimplesystemscon- “ stitutinig thecomposite system areknown asinternal constraints. Ifa Intermsofthedefinitions anddiscussions ofthepreceding sections we closed composite system isinequilibrium withrespect tocertain interrial arénowabletoformulate thebasicproblem ofthermodynamics. We constraints andifsomeoftheseconstraints arethenremoved, thesystem eS Te a BASIC CONCEPTS ANDPOSTULATES ‘THEENTROPY MAXIMUM POSTULATES 25 eventuallycomesintoanowequilibriumstate.Thatis,certainprocesses Itmustbestressedthatwepostulatetheexistenceoftheentropyonlywhichwerepreviously disallowed become allowed or,intheterminology forequilibrium states,andthatourpostulate makesnoreference what-Ofrechis, become virtualprocesses. Thebasicproblem ofthermo- soevertononequilibrium states.Intheabsence ofaconstraintthesystem dames thedetermination oftheequilibriwn statethateventually isfreetoselectanyoneof.number ofstates, eachofwhichmightalsobe resultsaftertheremovalofinternalconstraints inaclosedcomposite realizedinthepresence ofasuitableconstraint. Theentropyofeachof system.these constrained equilibrium states isdefinite, andtheentropy islargest i theconstraint itis1.9TheEnt insomeparticular stateoftheset.IntheabsenceofropyMaximumPostulates {thisstateofmaximumentropythatisselectedbythesystem.Theinductic xperi val snes 3 Jnthecaseoftwosystemsseparated byadiathermal wallwemajthat‘provides heeeepaaseattennoaeere |wishtopredictthemannerinwhichthetotalenergyUdistributesbeoween forward. Thelogical deviousness ofthehistorical method ismatched b thetwosystems. Wethenconsider thecomposite.system withtheinternal.thesubtletyoftheelegantmodernquantum statistical approach, Rathes diathermal wallreplaced byanadiabatic wallandwithparticular values ingtoi ra ‘-Rather acyis rictionthat shanatempigininducethesottionfomexprinetordevingit] Guyye0).Horeachsucheonstainedequibmette there ,istical theory,wethe F =0). postulate,dependinguponaposteriorrather|thaneehoelfeaibeaion: {entropyofthecompositesystem,andorsomeparticular valuesofU®andButevenbythismethod weshallseethatourpostulate isnotunmotivated; U®thisentropy ismaximum. These,then,arethevaluesofUDandUiiappearsathemostnaturalandplasileguesthatwemightmake. thatobrainintheBresenoeofthediatheemal wall,orintheabsenceof t , i t tic constraint. :ormheheportanceawenowadoptprovidethesimplestconceivable “allproblemsinthermodynamics ateessentialequivalenttothebasic ourbasicproblem.Onthisbasisalonetheproblem Pr yn y ‘ighthavebeensolved; thetentative postulation ofthesimplest formal Problem weformulated insection 1.8.Butthebasicproblém canbewrprmendae een“conventional andfrequentlysuccessfulmode sitigavenow23aivectionpeirergellaciayinteterin cd retical physics. . Whatthenisthesimplest criterion’ thateanreasonably beimagined relation thatgivestheentropy asafunction oftheextensive parametersforthedetermination ofthefinalequilibrium state?Fromourexperience 1isknownasafundamental relation. Ittherefore followsthatifthe withmany physical theories wemight expectthat themosteconomi I fundamental relation of@particular system isknown allconceivableformfortheequilibrium criterionisintermsofanextremunepincile \thermodynamic informationaboutthesystemisascertainable therefrom.Thatis,wemight hopethatthevalues oftheextensive paramet i ‘Theimportance oftheforegoing statement cannot beoveremphasized.ss of nsive tersinthe finalequilibrium statearesimplythosethatmaximise’ sonefuncinn ‘Theinformation contained inafundamental relation isall-inclusive—it And,straining ouroptimism tothelimit,wemight hopethatthis : isequivalent toallconceivable numerical data,toallcharts, andtoallhypothetical function hasseveral particular simple mathematical proper imaginable typesofdescriptions ofthermodynamic properties. Iftheties,designed toguarantec simplicity ofthederived theory. Wedevelo} fundamental relation ofasystem isknown, thereremains notasinglethisproposed solution inaseriesofpostulates. % p thermodynamic attribute thatisnotcompletely andprecisely determined. j Postulate If.There exists afunction (called theentropy S)ofthe | Postulate I.Theentropy ofacomposite system isadditive overtheextensiveparameters ofanycomposite system,definedforailequilibrium constituent subsystems. Theentropyiscontinuous anddifferentiable andstatesandhavingthéfollowing property. Thevaluesassumedbytheextensive isamonotonically increasingfunctionoftheenergy. Parameters inthe absenceofaninternalconstrai i . theentropyoverthemanoofconstrainedrainarethosethatmaximize Severalmathematical consequences followimmediately. Theadditivity mes. . + property states thattheentropy $ofthecomposite system ismerely the corm ; sumoftheentropies S"oftheconstituent subsystems:theSeminisinefonction,thisbeingpurelyamatterofconvention inthechoiceof ignoftheFunction,havin rucncewhateverinthelogical the savingnoconsequence whateverinthelogicalstructureofthe s=ds° (14) |2 | 26 BASICCONCEPTS ANDPOSTULATES ‘THEENTROPY MAXIMUM POSTULATES 2 Theentropy ofeach subsystem isafunction ofth i r i ofthatsubsystem aloney ionoftheextensive parameters ButU/Nistheenergy petmole, which wedenote byu. SO=SHUM, YO,NE...NE) (5) usUIN (2) ‘Theadditivity property whenapplied toconceptually distinct (rather Also,V[NVisthevolume permole,whichwedenotebye-thanactuallyphysicallydistinct)subsystems requiresthefollowing vaviN (1.13) property. Theentropyofasimplesystemisahomogeneous first-order ke ingJinctionoftheextensiveparameters. Thatis,alltheextensiveeinai ThvsSCN,VIN,1)=S(u,,1)istheentropyofasystemofasingle‘ofasystemaremultiplied byaconstant 4,theentropy ismultiplied by mole,tobedenoted bys(x,0). thissame constant. Or,omitting thesuperscript (a), s(4,0)=S(u,v,1) (14) thSU,AV,AN+++2N,)=AS(U,V,Nyo++N,) (1.6) Equation1.11nowbecomes 1monotonic property postulated implies that the partialderivative .(@S[AU)y.9,..-n, 18&positivequantity, een S(U,VN)=Nou,0) (1s) (3s)so PostulateIV.Theentropyofanysystemvanishesinthestateforwhich NGO) rare, an BUPS)y.y,.-.9, =0hatis,athezerooftemperature) Asthetheorydevelops inaubsoquent sections, weshallseethatthe Thispostulate isanextension, duetoPlanck,oftheso-called Nernstreciprocal ofthispartialderivative istakenasthedefnion ofthe postulateorthirdlawofthermodynamics. Historically, itwasthelatestofsapere Thusthetemperature ispostulated tobenonnegative.* thepostulatestobedeveloped, beinginconsistent withclassicalstatisticalacontinuity, ifferentiability, andmonotonic property implythat mechanics andrequiring thepriorestablishment ofquantum statistics in heentropyfunctioncanbeinvertedwithrespecttotheenergyandthat orderthatitcouldbeproperlyappreciated.Thebulkofthermodynamics 4ureyiasingle-oalued, continuous, anddifferentiable functionof doesnotrequirethispostulate,andwemakenofurtherreferencetoit My (,.Thefunction untilChapter 10,Nevertheless, wehavechosen topresent thepostulate S=SU,VN eM) (8) atthispoint toclose thepostulatory basis. canbesolved uniquely forUinthefor i ‘Theforegoing postulates arethelogical bases ofourdevelopment ofwey m thedynamics"InthelightoftheiNthen,itbewellte _ tee ermodynamics. Inthelightofthesepostulates, then,itmaybewellto .Y=US,¥,Myr-*Me) (9) ireiteratebrieflythemethodofsolutionofthestandardtypeofthermo- Equations1.8and1.9arealternative formsofthefundamental relation, dynamic problem, asformulated insection1.8.Wearegivenacomposite andeachcontainsal!thermodynamic information aboutthesystem. systemandweassumethefundamental equation ofeach oftheconstituent Wenotethattheextensivity oftheentropy permitsustoscalethe systemstobeknowninprinciple, Thesefundamental equations determineProperties ofasystemofNmolesfromtheproperties ofasystemof theindividual entropies ofthesubsystems whenthesesystemsareinImole. Thefundamental equation issubject totheidentity equilibrium. Ifthetotalcomposite system isinaconstrained equilibrium S(U,VNNy"N,)=NSCOIN,VIN,NINNIN)(1-10) sete,withparticularvaluesftheetesiveparameter,ofeachcon: inwhichweha 4 : _ stituentsystem,thetotalentropyisobtainedbyadditionoftheindividualSingle-component similestaninparton (0UNMENeFora entropies. Thistotalentropyisknownasafunctionofthevarious ipiesystem,inparticular, |extensiveparameters ofthesubsystems. Bystraightforward differentia~ SU,V,N)=NS(UIN, VIN,i) (Lip, { tion,wecompute theextremaofthetotalentropyfunction, andthen,on. =.. i i ivati lassify these extrema *Teponyofnegativevaleofthisderivativeati,oaegativetemperatures): -—=~=«t‘basisthesignofthesevondderivative,weclassify ; hasbeendzessodtyNEF.Kamsny,Pye:feo30070(1950.Socksateseaebyiasminima,maxima,orashorizontalinflections.Inanappropriate producedandimaintainedforshortimesincertainuniquesystems.Toaccommodate 'physicalterminologywefirstfindtheequilibriumstatesandwethen theiextnce requiressullerandmorabstract postales, andintheinterestof t ‘classify themonthebasisofstability. Itshould benotedthatinthesimplicity weexclude consideration oftheseveryspecialized states. adoption ofthisconventional terminology weaugment ourprevious 4 ee OO ee eee ee ee 28 BASICCONCEPTS ANDPOSTULATES ‘THEENTROPY MAXIMUM POSTULATES 29 definition ofequilibrium; thatwhichwaspreviously termedequilibrium isgivenanatomistic significance intermsofstatistical mechanical concepts. isnowtermed stable equilibrium, whereas unstable equilibrium states are ‘ Ourfurther considerations donotdepend inanywayonthematerial innewlydefinedintermsofextremaotherthanmaxima. }theAppendix. However,thereaderwhoisplaguedbythequestion“ButItisperhaps appropriate atthispoint toacknowledge thatalthough | whatistheentropy?” maywishtoallaythiscuriosity byreading allapplications ofthermodynamics areequivalent inprinciple tothe Appendix Batthispoint. procedure outlined thereareseveral alternative procedures thatfrequently 7. Prove moreconvenient. These alternate procedures aredeveloped in Problems—Section 1.9subsequent chapters, Thus weshall show thatunder appropriate con- " ; ditionstheenergyU(S,V,Ny-*-)maybeminimizedratherthanthe 19-1Thefollowingte,cquaonsarepurportedtobefundamental entropySU¥,Na)maximized, astthesetwoProceduresdetermine ‘wlthoneormoreofpostulates, IU,andIVandconsequently arenot physically, thesamefinalstateis analogous tothefactthatacirclemaybecharac- oceptabie, ndthefivethatarenotphysically pemaieibleandInsite theterized eitherastheclosed curveofminimum perimeter foragivenarea posthlate violated byeach. —_ orastheclosed curveofmaximum areaforagivenperimeter. Inlater ‘Thequantities vg,6,andRarepositive constants, andinallcasesinwhichchapters wealso introduce several new functions, theminimization ofLfractionalexponentsappearonlytherealpositiverootistobetaken. whichislogically equivalent totheminimization oftheenergy ortothe Rtmaximization oftheentropy. @s=(5)wrups.Theinversion ofthe fundamental equation andthealternative statement . ofthebasicextremum principle intermsofaminimum oftheenergy ©s=(2)*PNeye :ratherthanamaximum oftheentropysuggestsanotherviewpoint from | a)|>whichtheextremum postulateperhapsmayappearplausible. Inthe 1 “4pornstheoriesofelectricity andmechanics, ignoringthermaleffects,theenergy ©s-()[xe-al isafunction ofvarious mechanical parameters, and thecondition of 8 Pa! equilibrium isthattheenergy shallbea.minimum. ‘Thus aconeisstable 00lyingonitsside,ratherthanstandingonitspoint,becausethefirst @s=(3)WINGposition isoflower energy. Ifthermal effects aretobeincluded, theenergy ° ‘ceases tobeafunction simply ofthemechanical parameters. According _(B)* “ totheinverted fundamental equation, however, theenergy isafunction ©S=(ia)Weve . ofthemechanical parameters and ofoneadditional parameter (the entropy). Bytheintroduction ofthisadditional parameter theformof (DS =NRIn(UVIN?RO09)theenergy-minimum principleisextended tothedomainofthermal effects 4 . aswellastopuremechanicalphenomena. Inthismannerweobtain '@S=(7)INU}exp(—V4/2N*0g") a-sortofcorrespondence principle between thermodynamics and ' mechanics—insuring thatthethermodynamic equilibrium principle Wyose)woreop(-uy). reduces tothemechanical equilibrium principle when thermal effects can U NR00p, beneglected. ; _ ; oyS* Weshallseethatthemathematical condition thatamaximum of ( U=(FR)Sexpsinay SU.V,N+*)implies aminimum ofU(S,V,Ny*+)is thatthederivative |(@5/@U)y,v,... bepositive. Themotivation fortheintroduction ofthis 1 RO SsstatementinpostulateIITmaybeunderstood intermsofourdesireto'QUR(Z)er(+mR)exp(—SINR) ..insurethattheentropymaximum principle willgooverintoanenergy | oominimum principle oninversion ofthefundamental equation. | 1.9-2. Foreachofthefivephysically acceptable fundamental equations in ‘Theattention ofthereader iscalled toAppendix B,inwhich theentropy problem 1.9-1 findUas#function ofS,V,andN. A a eee 7 30 BASIC CONCEPTS At ;NDPOSTULATES CHAPTER 2 1.9-3.Thefundamental equation ofsystemAis ——_———- nwo 1[RY\4 . Sa(3)WVAUal i . |andthatofsystem2is TheConditions . Spo(55)nrava i ofEquilibrium SO What isthefundamental equation ofthecomposite system A+B? i . 1.9-4, Assume thattheinternal wallbetween subsystems 4andBinproblem 1.9-3 isrestrictive with respect toboth volume andmole number butnon-+restrictive withrespect toenergy; thats,itisrigid,impermeable, anddiathermal. iAssume thatsystem Ahasavolume of9cm* andamole number of3moles. i System Bhasavolume of4cm? andamole number of2moles. Thetotal nergyinthecomposite system's 20cal,Pottheentropy againsttheFractionUglUg. +Un)oftheencrey insubsystem A.When thesystem hascome to j‘equilibrium, whataretheinternalenergiesofeachofthe individual subsystems? 2.1 Intensive Parameters 4 Byvirtue ofourinterest inprocesses andintheassociated changes of theextensive parameters, weanticipate thatweshall beconcerned with } thedifferential form ofthefundamental equation. Writing thefunda- f ‘mental equation intheform r U=US, V,NyNoo N) 1) wecompute thefirstdifferential: i aU) aU) z(au1wea(2) dS+@) ave(2) an, : BS)vay, BV)sxyooMy BiBN)s.yomy| : (2.2) ‘Thevarious partial derivatives appearing intheforegoing equation recur sofrequently thatitisconvenient tointroduce special symbols forthem. They arecalled intensive parameters, andthefollowing notation is conventional: :. (2) =T,thetemperature (23) OS)yarn, \: } eu) .' —(28 =P, thepressure 4){ (iSM, -! au _ theelectrochemical potential of «: BIG)sye-caryMP thejthcomponent @5) m ee heetseomceetateRA anasoit AAI NEAA EDOS = eeeeeee 32 THECONDITIONS OFEQUILIBRIUM EQUATIONS OFSTATE 33 With this notation, equation 2.2bec3 comes 2.2EquationsofState dU=TdS —PdV +dN, +--* +2,dN, 26 F ; ‘Thetemperature, pressure, andtheelectrochemical potentials areTheformaldefinitionofthetemperature soonwillbeshowntoagree partialderivatives ofafunctionofS,¥,Ny"N,andconsequently arewithourintuitive qualitative concept, basedonthephysiological notion itofunctions of5,V,N,-+"N, Wethushaveasetoffunctionalof“hot”and“cold.”Wecertainlywouldbereluctanttoadoptadefini- relationships, :tionofthetemperature thatwouldcontradict suchstrongly entrenched T=1S,¥,M “ND Qu)although qualitative notions. Forthemoment, however, wemerely Pas. VN, «-N) @12) introduce theconcept oftemperature bytheformal definition (2.3). Mop Me ’Similarly, weshallsooncorroborate thatthepressure definedby y=aSVeNyNe) (2.13)‘equation 2.4agreesineveryrespectwiththepressure definedinmechanics. Suchrelationships, expressing intensive parameters intermsoftheWithrespecttotheseveralelectrochemical potentials, wehavenoprior independent extensive parameters, arecalledequations ofstate.definitioris orconcepts andwearefreetoadoptthedefinition (equation Knowledge ofasingleequation ofstatedoesnotconstitute complete |2.5)forthwith. knowledge ofthethermodynamic properties ofasystem. Weshallsec,Forbrevity,theelectrochemical potential isoftenreferred tosimplyas subsequently, thatknowledge ofaiftheequations ofstateofasystemisthechemical potential, andweshallusethesetwotermsinterchangeably. equivalent toknowledge ofthefundamental equation andconsequentlyHowever itshould benotedthatoccasionally, andparticularly inthe isthermodynamically complete.theoryofsolids,thechemical potential isdefined asjzminusthemolar ‘Thefactthatthefundamental equation ofasystemishomogeneouselectrostatic energy. first-order hasdirectimplications forthefurictional formoftheequations ‘Theterm—PdVinequation 2.6isidentified asthequasi-static work ofstate. Itfollows immediately thattheequations ofstatearehomogeneous AWsz,25givenbyequation1.1. zero-order. Thatis,multiplication ofeachoftheindependent extensiveInthespecial caseofconstant molenumbers equation 2.6canthenbe parameters byascalarAleavesthefunction unchanged.written as T(AS, AV,AN,» =»AN,)=T(S,VsNgo Ny) (2.14) TdS=dU—dWy ifdN,= dN,=dN,=0 Qn Ittherefore follows thatthetemperature ofacomposite system com- Recall definition ofetank |. posedoftwoidenticalsubsystems isequaltothetemperature ofeitherFeeTT eae a eecatcatebatten subsystem. Thisiscertainlyinagreement withourintuitiveconceptofquation1.2,wenowrecognize TdSasthequest-static heatflux, temperature. Thepressureandtheelectrochemical potentials alsohave . theproperty (2.14). 4Q=Tds 28) ‘Tosummarize theforegoing considerations, itisconvenient toadopt a Aquasi-static fluxofheatintoasystemisassociatedwithanincreaseof condensed notation.Wedenotetheextensiveparameters ¥,Na+°Neentropyofthatsystern, bythesymbolsX;,Xp°**X,sothatthefundamental relationtakestheTheremaining termsinequation2.6represent anincreaseofinternal form U=US,XpXeXY) (2.15)energy associated with theaddition ofmatter toasystem. This type of : «ensenergyflux,although intuitively meaningful, isnotfrequently discussed ‘Theintensive parameters aredenotedbyoutside thermodynamics anddoesnothaveafamiliar distinctive name. au ere aeWeshallcall5:dN,thequasi-static chemicalwork, FS)pongPTSMeMe1D (2.16) :au ;dW,=>uN, 29) (2)yeePHPASNoeMyX)f=hy2ee217) siI5 8Therefore whence ; dU=dQ+dWy,+dW, (2.10) dU=TdS+EPydX, (2.18)int Ferme hgeeAAS EES erenceneeeet aemeee ee eee 36 ‘THECONDITIONS OFEQUILIBRIUM THERMAL EQUILIDRIUM—-TEMPERATURE, 7 is,inonecasetheentropy isamember ofthesetofindependent para- aimemeters,andinthesecondcasetheenergyissuchamember.Inperforming 24ThermalEquilibrium —Temperatureformal manipulations inthermodynamics, itisextremely important to Wearenowinapositiontoillustrateseveralinteresting implications makeadefinite’commitment tooneortheotherofthesechoicesandto oftheextremum principlewhichwehavepostulated fortheentropy.adhere rigorously tothatchoice, Agreatdealofconfusion results from Consider aclosed composite system consisting oftwosimple systemsavacillation between thesetwoalfernatives withinasingleproblem, separated byawallthatisrigidandimpermeable tomatter butthatdoes.Iftheentropy isconsidered dependent andtheenergy,independent, allowtheflowofheat.Thevolumes andmolenumbers ofeachofthe asinS=S(U+-- X,*+-), weshallrefertotheanalysis asbeinginthe simplesystems arefixed,buttheenergies UandU®arefreetochange,entropy representation. Iftheenergy isdependent andtheentropy is subject totheconservation resteictionindependent, asinU=U(S-++ X,+*-), weshall refer totheanalysis as being intheenergy representation. Um+UM=constant (2.30) - Theformal development ofthermodynamics canbecarried outin imposed bytheclosure ofthecomposite system asawhole.. Assuming eithertheenergy orentropy representations alone, butforthesolution of thatthesystem hascometoequilibrium, wewishtofindthevalues of«particular problemeitheroneortheotherrepresentation mayproveto U®andU",Now,byourfundamental postulate, thevaluesofU®bebyfarthemoreconvenient. Accordingly, weshalldevelop the andU®are.suchastomaximize theentropy. Therefore, bytheusualtworepresentations inparallel, although adiscussion presented inone mathematical condition foranextremum, weconclude thatinthe representation generally requires onlyabriefoutline inthealternate equilibrium stateavirtual infinitesimal transfer ofenergy fromsystem 1representation. tosystem2willproducenochangeintheentropyofthewholesystem. ‘Therelation S=S(Xq-+X,+++)issaidtobetheentropicfundamental Thatis, ds<0 eanrelation, thesetofvariables Xq:-- X,~+iscalled theentropic extensive = ~ parameters, andthesetofvariables Fy-+ F,+* iscalled theentropic ‘Theadditivity oftheentropy forthetwosubsystems givestherelationintensiveparameters. Similarly,therelationU=U(S,X,-*-.X,++)issaid =SOKOYO...ND.)4SO™(YM,YO...NE--3,(2.32) tobetheenergetic fundamental relation, thesetofvariables S,X,***X,-*+ S=SOU,V'NYP)+SKU,HN+), (2.32) iscalled theenergetic extensive parameters, andthesetofvariables AsUCandU%arechanged bythevirtual energy transfer, theentropy PyP,+++ iscalled theenergetic intensive parameters, change is as(3) du+(3) du®2.33) Problems—Seetion2.3 BTR) yom... OF) yo... ye. 23-1. Findthethreeequations ofstateintheentropy representation fora or,employing ourdefinition ofthetemperature, systemwiththefundamental equation 1 1io as=Aya+hau (2.34), 7)on Bytheconservation condition(equation2.30),wehave 23-2. Show byadiagram (drawn toarbitrary scale) thedependence of dU =—ay@ (2.35)temperature onvolumeforfixedpressureforthesystemofproblem2.3-1. whence 14Drawtwosuch“isobars,"corresponding totwovaluesofthepressure,and ds=(Fs_=)aun 02.36) indicatewhichisobarcorresponds tothehigherpressure. : TH~Tei,23-3. |Findthethreeequations ofstateintheentropyrepresentation fora .Thecondition ofequilibrium (equation 2.31)demands thatdSvanishfor systemwiththefundamental equation ubitrary valuesOEEUWwhence " 0 : tia ue(are a 37) i _—— tr,_ ne 38 ‘THECONDITIONS OFEQUILIBRIUM AGREEMENT WITH INTUITIVE CONCEPT OFTEMPERATURE, 39 ‘Thisisthecondition ofequilibrium. Ifthefundamental equations of thetwosubsystems areinitially separated byanadiabatic wallandthat eachofthesubsystems wereknown, then1/7" would beaknown function thetemperatures ofthetwosubsystems arealmost, butnotquite, equal. ofU%(andalsoof9andNf «which, however, aremerely constants). Inparticular letusassume that Similarly, 1/Twould beaknown function ofU®,andtheequation THs 7m (2.39) 1/T® =1/T™ would beoneequation inU"andU. Theconservation condition U™+U®=constant provides asecond equation, andthese Thesystem isconsidered tobeinequilibrium withrespect totheinternal twoequations completely determine, inprinciple, thevalues ofUand adiabatic constraint. If,now, theinternal adiabatic constraint isremoved, ofU®, Toproceed further andactually toobtain thevalues ofU® ~thesystem isnolonger inequilibrium, heatflows across thewall, andthe andofU®would require knowledge oftheexplicit forms ofthefunda- entropy ofthecomposite system increases. Finally thesystem comes to | mentalequations ofthesystems. Inthermodynamic theory,however, we anewequilibrium state,determined bythecondition thatthefinalvaluesaccept theexistence ofthefundamental equations, butwedonotassume ofT®andT®areequal andwith themaximum possible value ofthe explicit forms forthem, andwetherefore donotobtain explicit answers. entropy that isconsistent with theremaining constraints. Letusnow Inpractical applications ofthermodynamics thefundamental equations compare theinitial andthefinal states. ifASdenotes theentropy may beknown, either byempirical observations (interms ofmeasure- difference between thefinal andinitial states, wehave ments tobedescribed later) oronthebasis ofstatistical mechanicalcalculations basedonsimplemodels. Inthiswayapplied thermodynamics As>0 2.40) isabletoleadtoexplicit numerical answers. But,asinequation 2.34,wefind Equation 2.37 could also bewritten asT=7, Wewrite itinthe : form1/7=1/T®)tostressthefactthatouranalysisiscouchedinthe As~(fs-zm)Aum (2.41) entropyrepresentation. Bywriting1/7,weindicateafunctionofU%, TH~Tavil..., whereas T” would imply afunction ofSY, V®--+. The nes, Gphysica!significance ofequation2.37however,remainstheequalityof whereTW)and7)aretheinitialvaluesofthetemperatures. "Bythethetemperatures ofthetwosubsystems.1 conditionthat7%)>7%,wenowfindthat Asecond phase oftheproblem istheinvestigation ofthestability of Au <0 (2.42) thepredicted final state. Inthesolution given wehave notexploited ; aor fullythebasicpostulate thattheentropy isamaximum inequilibrium, ‘Thismeans thatthespontaneous process thatoccurred wasoneihwhichbutwemerely haveinvestigated theconsequences ofthefactthatitis heatflowed fromsubsystem Itosubsystem 2,Weconclude thereforeanextremum, ‘Thecondition thatitbeamaximum requires, inaddition thatheattendstoflowfromasystemwithahighvalueofTtoasystemtothecondition dS=0,that withafowvalue ofT.Thisisagain inagreement withourintuitive notion ofthetemperature. Itshouldbenotedthattheseconclusions donot #S<0 (2.38) depend onourassumption thatT®isapproximately equal toT®; this ‘Theconsequences ofthiscondition leadtoconsiderations ofstability, to assumption wasmademerely forthepurpose ofobtaining mathematicalwhichweshallgiveexplicit attention inChapter 8. simplicity inequation 2.41,whichotherwise would require aformulationinterms ofintegrals. iy on Ifwenowtakestockofourintuitive notion oftemperature, basedon 2.5 Agreement with Intuitive ConceptofTemperature thephysiological sensations ofhotandcold,werealizethatitisbased Intheforegoing example wehave scen thatiftwosystemsareseparated upontwoessentialproperties. First,weexpecttemperature tobean byadiathermal wailheatwillflowuntileachofthesystems attains the intensive parameter, having thesamevalueinapartofasystem asithassame temperature. Thisprediction isinagreement withourintuitive intheentire system. Second, weexpect thatheatshould tendtoflownotionoftemperature, anditisthefirstofseveralobservations that fromregionsofhightemperature towardregionsoflowtemperature.corroborate theplausibility ofourformal definition ofthetemperature, Theseproperties implythatthermal equilibrium isassociated withequality Inquiring intoourexample inslightly moredetail, wesuppose that andhomogeneity ofthetemperature. Wehavedemonstrated thatour - oe 1 _ a a eaeacacia sccsaacsaaaaaaaassasaaaaaaill 40 ‘THE CONDITIONS OFEQUILIDRIUM TEMPERATURE UNITS 4 formal definition ofthetemperature possesses each ofthese properties, unit, denoted by°R,iscalled theabsolute Fahrenheit degree, orthedegree sothat wehave now corroborated that ourdefinition isintuitively Rankine, Absolute Fahrenheit temperatures aremerely times thesatisfactory. corresponding absoluteKelvintemperature.ThedefinitionscitedarethoseadoptedbytheTenthGeneralConference 2.6‘Temperature Units ofWeights andMeasures in1954, andbyuniversal acceptance they supplant theearlierdefinitions intermsoftwofixedpoints. : ‘Thephysicaldimensions oftemperature arethoseofenergydividedby Closelyrelatedtotheabsolute Kelvinscaleoftemperature isthe thoseofentropy. Butwehave notyetcommitted ourselves onthe international Kelvin scale, which isa“practical” scale, defined interms dimensions ofentropy; infactitsdimensions canbeselected quite oftheproperties ofparticular systems invarious temperature ranges andarbitrarily. Foriftheentropyismultiplied byanydimensional constant, contrived tocoincideascloselyaspossiblewiththeabsoluteKelvin ~weobtain anewfunction ofdifferent dimensions butwith exactly the scale. Thepractical advantage of.the international Kelvin scale isthatit~ same extremum properties andtherefore alsoquite acceptable asthe provides reproducible laboratory standards fortemperature measurement entropy. Theonly restriction wemust maintain isthattheproduct of throughout thetemperature range. However, from thethermodynamic temperature andentropy havethedimensions ofenergy. Wesummarily point ofviewitisnotatruetemperature scale atall,andtotheextent resolve thearbitrariness simply byadopting theconvention thatthe thatitdeviates from theabsolute scale itwillnotyield temperature entropy isdimensionless; from themore incisive viewpoint ofstatistical ratios thatareconsistent withthose demanded bythethermodynamic mechanics, this isaphysically reasonable choice. Consequently the formalism. dimensions oftemperature areidentical tothose ofenergy. However, Stillanother scale oftemperature isthethermodynamic Celsius scale. justastorque andwork have thesame dimensions, butareverydifferent, Theunitoftemperature isthedegree Celsius, denoted by°C,andthis types ofquantities andaremeasured indifferent units (theem-dyne and unitisidentical insizewith theabsolute degree Kelvin. Thethermo-theerg,respectively), sothetemperature andtheenergymustbecarefully dynamic Celsiustemperatures aredefined(bythe1954agreement) asdistinguished. Thedimensions ofbothenergyandtemperature are 273.15lessthanKelvintemperatures. Thetemperature ofice,water,and [mass «(length)*/(time)}]. Theunits ofenergy arejoules, ergs, calories, water-vapor inequilibrium is0.01°C. However, thetemperature ofa etc.Theunitsoftemperature remain tobediscussed. mixture oficeandwater at1atmpressure isverynearly 0°C,withthe Inourlaterdiscussion ofthermodynamic engines andoftheCarnot error appearing onlyinthethird decimal place, andthetemperature of cycle weshow thattheratio ofthetemperatures oftwogiven systems can boiling water at1atmpressure isapproximately 100°C. Consequently, ‘bemeasured directly andunambiguously. Themeasurability oftheratio theCelsius scale provides convenient numbers forgeneral use. oftemperatures determines thescale oftemperature, except foran Clearly theCelsius scale yields different temperature ratios thanthe arbitrary multiplicative constant. Thetemperature ofsome arbitrarily Kelvin scale, andconsequently theCelsius scaleisnotanacceptable scale chosen standard system maybeassigned atwill,andthetemperatures of forthermodynamic use. Celsius temperatures must beconverted intoallothersystemsarethenuniquely determined, withvaluesdirectly Kelvintemperatures (merelybytheadditionof273.15)beforesubstitutionproportional tothechosen temperature ofthefiducial system. inthermodynamic formulas. Various choices ofafiducial system, andvarious assignments ofits Prior tothe1954international agreement, theCelsius scale wasreferred temperature, leadtodifferent scales oftemperature. Theabsolute Kelvin toinmany countries, including theUnited States, astheCentigrade scale. scale oftemperature isobtained byassigning thenumber 273.16 tothe ‘Aninternational Celsius scale isdefined asbearingthesamerelationship temperatureofamixtureofpureice,water,andwatervaporinmutual totheinternational Kelvinscalea8thethermodynamic Celsiusscalebears equilibrium;astatewhichweshowinourlaterdiscussion of“triple totheabsoluteKelvinscale.Totheaccuracythatwillgenerally concern points”todetermine auniquetemperature. Thecorresponding unitof us,weneedmakenodistinction between the two Kelvin scales nor between temperature iscalled adegree Kelvin, designated bythenotation °K. thetwoCelsius scales, butthedifference inprinciple should bekeptin ‘The temperature oftheice-water vapor system then is273.16°K. mind, Theabsolute Fahrenheit scale isobtained byassigning thetemperature Finally, aFahrenheitscaleisdefinedintermsoftheabsoluteFahrenheit 9/5(273.16) =491.688°R totheice-water-vapor system referred to.The scale bysubtraction of459.67. This number isexactly 9/5(273.15) —32. a THECONDITIONS OFEQUILIBRIUM MECHANICAL EQUILIBRIUM “3 Itfollowsthatconversion ofaCelsiustemperature toaFahrenheit areseparatedby2diathermalwall,andthetotalenergyin.thecompositetemperature involvesmultiplication by$andaddition of32.Thetem- systemis6000cal.Whatistheinternalenergyofeachsystemineq!peratureoficeandwateratIatmpressureisabout32°F,thetemperature 2.6-4,Twosystemswththeequationsofsaegiveninproblem246.3are | ofboilingwaterat|atmpressureisabout212°F,androomtemperatures separacsby2xan) iuperainges aoeTa)250KandT!)=350°K, areinthevicinity of70°F.i , What arethevalues ofU"andUafter equilibrium hasbeenestablished? | Although wehavedefinedthetemperature formallyintermsofapartial Whatistheequilibrium temperature? . derivative ofthefundamental relation, wenote briefly, inconclusion, the conventional method ofintroduction ofthetemperature concept, as : , itibe+ fecha ib developed byKelvin andCaratheodory. TheheatfluxdQisfirstdefined 2.7"Mechanical Eqellibriom ; ) verymuch aswehaveintroduced itinconnection withtheenergy ‘Asecond application oftheextremum principle fortheentropy yields conservation principle. Fromtheconsideration ofcertain cyclic processes anevensimpler result andtherefore isuseful inmaking theprocedureitistheninferredthatthereexistsanintegrating factor(1/7)suchthatthe clear.Weconsideraclosedcomposite systemconsisting oftwosimple .Product ofthisintegrating factor withtheimperfect differential dQisa systems separated byamovable diathermal wallthatisimpervious tothe perfect differential (45). flow ofmatter. Thevalues ofthemole numbers aréfixed andconstant, ds=ha (2.43) butthevaluesofU®andU®maychange, subject onlytotheclosurer condition U®4UP)=constant - (2.44) ‘The temperature andtheentropy thereby areintroduced byanalysis of . theexistence ofintegrating factors inparticular typesofdifferential andthevalues ofV4and¥@maychange, subject onlytotheclosure i . condition equationscalledPfaffianforms. 004ve=constant aan Problems—Section 2.6' ‘Theextremum principle requires thatnochange inentropy result from 2.6-1. Thetemperature ofasystem composed ofice,water,andwater- infinitesimal virtualprocesses consisting oftransfer ofheatacrossthewall vapor inmutual equilibrium hasatemperature ofexactly 273.16°K, bydefinition, anddisplacement ofthewall. ‘Thetemperature ofatemoficeandvnterat‘atmofpressure isthen ‘Then measured as273.15°K, with thethird andlater decimal places uncertain, The ds=0 (2.46) temperature ofasystem ofwater andwater-vapor (i.boiling water) at1atmismeasured as373.15°K 40,01°K. Compute thetemperature ofwater-water- where vapor atIatm,withitsprobable error, ontheCelsius, absolute Fahrenheit, asm ‘au‘andFahrenheitscales, d=(3) au®4(5)cy7 .26-2. ‘The“gasconstant” Risaconstant having thevalueR=1.986 BU Aa acai ccalfmole*K. SincethesizeoftheCelsius degree isthesameastheKelvin degree, 0 [email protected]*C. ExpressRinunitsofjoules/mole’F. as: oy(So av™ (2.47)js +(Fa, dU+57)gs.yin 2.6-3,Twoparticular systemshavethefollowing equations ofstate: BUTyon OMee pe Pirie Bytheclosure conditionsFo 38ow du® =—dum (2.48) and ida.5pi an av®=ayn (2.49)=3RGHmee whence cypaar whereRisaconstanthavingthevalue1.986calfmole*K.Themolenumberof ds=(fs-7s)du4-)dv=0°(2.50) thefirstsystem isNO)=2,andthat ofthesecond isVN=3,Thetwosystems THTH, THTe 44, ‘THECONDITIONS OFEQUILIBRIUM EQUILIBRIUM WITHRESPECT TOMATTER FLOW 45 Ashsexpressionmstvasishforarbitraryandindependentvaluesof 28EquilibriumwithRespecttoMatterFlow 14 {Afinalexample employing theentropy maximum principle gives some Tu FH? 251) insight intothenature ofthechemical potential. Weconsider the and equilibrium state oftwosimple systems connected byarigid anddia- pa pa thermal wall, permeable toonetypeofmaterial (N,)andimpermeable Fo wm? (2.52) toallothers (N»,Ny*-*N,). Wethussecktheequilibrium values of U®andU®andofN{”andNf.Thevirtualchangeinentropyinthe Althoughthesetwoequations aretheequilibrium conditions intheproper appropriate virtualprocessis form,appropriate totheentropyrepresentation, wenotethattheyimply 1 a? 1 e thephysical conditions ofequality ofbothtemperature andpressure. ds=pau —Bian) +edu —Fane (258) TO=TO (2.53) andtheclosureconditions gemand pw=peo (254) . du =—qu© (2.56) ‘The equality ofthetemperatures isjust ourprevious result forequili- and - brium with adiathermal wall. Theequality ofthepressures isthenew dN® =—dn sn feature introduced bythefact that thewall ismovable. Ofcourse, the whence . . equality ofthe pressures isprecisely theresult thatwewould expect on to uPthebaclsofmeshanics, andthisresultcorroborates ouridentifeation of ds=(&-7)dum—(&-#)any? (2.58)thefunction Pasthemechanical pressure. Thereader mayaskwhywehaveconsidered theproblem ofamovable ‘AsdSmust vanish forarbiteary values ofbothdU anddN{?, wefinddiathermal wallratherthantheostensibly simplercaseofamovable astheconditions ofequilibrium adiabatic wall. Thelatter, unfortunately, isasubtle problem Jacking a 1 1 unique physical answer. Asthedifficulties ofthisproblem arequite rn (2.59)specialized, weshallnotconsider ithere, butadiscussion isgiven in ToT Appendix Cfortheinterestedreader. and wg?og? ef 2.60; Fispa (ovhence also44=wl) (2.60)Problems—Section 2.7 . iati ‘Thus,justasthetemperature canbelooked uponasasortof“potential” 2-7-1.Twoparticular systemshavethefollowing equations ofstate: forheatfluxandthepressure eanbelookeduponasasortof“potential” 1 3.Nm pm ym forvolume changes, sothechemical potential canbelooked upon asa Fa TRyw, FH Ryo sortof“potential” formatter flux. Adifference inchemical potential and provides a“generalized force” formatter flow. 1s Nm pm Nm Thedirection ofthematter flowcanbeanalyzed bythesame methodTH=jR owFARyw usedinsection2.5toanalyzethedirection oftheheatflow.Ifweassume thatthetemperatures TandT®areequal,equation 2.58becomes where R=1.986 cal/mole"K. The mole number ofthefirstsystem isN=0.5, ‘andthatofthesecond isN“=0.75, Thetwosystems arecontained ina pO—Closed cylinder, separated byamovable diathermal piston. ‘Theintial tempera- ds=ALBLayy 61 tures areT)=200°K andT!)=300°K, andthetotal-volume is20liters. T - +t tennant etmenatall NGtstattttinann tame et AA A A 46 THECONDITIONS OFEQUILIBRIUM CHAPTER 3 positive. Thusmatter tendstoflowfromregions ofhighchemical potential CO toregions oflowchemical potential.Inlaterchaptersweshallseethatthechemicalpotentialprovidesthe fi generalized forcenotonlyfortheflowofmatter frompointtopointbut Some Forma: alsoforitschanges ofphase andforchemical reactions. Thechemical . ,potential thusplaysadominant roleintheoretical chemistry. Relationships : Theunits ofelectrochemical potential arecalories permole, joules per mole, oranydesired energy unitpermole. . Problems—Section 2.8 2.8-1, Thefundamental equation ofaparticular type oftwo-component system is Piux My M S=NA+RING —NyRinFt—NyRInF . NeN+™ where R=1.986 cal/mole*K andwhere Aisanunspecified constant. Aclosedrigidcylinder oftotalvolume {0litesisdivided intotwochambers ofequal 3.1TheEalerEquationvolumebyadiathermal rigidmembrane, permeable ¢othefirstcomponent i atesleadtoasolution ofthe butimpermeable tothesecond.Inonechariberisplaced‘sampleofthesystem Havingseenhowthefundamental postulates leadtoewhatgreaterwith original parameters N{0 =0.5,N{0=0.75,V®=5liters,and70)= equilibrium problem,wenowpausetoexamineinsomewhat g 300°K.Inthesecondchamberisplacedasamplewithoriginalparameters detailthemathematical properties offundamental equations. . NO=1,NO=05,VO=5liters,andTH=250K, Afterequilibrium ‘Thehomogeneous first-order property ofthefundamental relation isestablished, whatarethevaluesofN{¥,N{®,7,P?andPm? permits thatequation tobewritten inaparticularly convenient form, called the Euler form. From thedefinition ofthehomogeneous first-order property wehave, q forany2, UCAS, AX, +AX) =AUS, Xo XD GD Differentiating with respect to4, BUG AK)AAS),WUEAKQOD9yes,xyesXD aa) at a) (SMa 80 (3.2) or BUC AX) uC--ax)eSEX $0=US,KHXD)33) was)oax) * ‘Thisequation istrueforany4andinpatticular for =1,inwhich case ittakes the form a au U = eee core U 3.4) :aot ay, 6a : U=aTS+ 3PX, G5) : rot a 48 SOME FORMAL RELATIONSHIPS ‘THEGIDBS-DUHEM RELATION 49 Forasimple system inparticular wehave f , jielding,inturn,¢+1equatiofstate USTS=PV+pyNytoo+Ny G6) yielding,inturn,¢+1equationsofstate ‘Therelation 3.5or3.6istheparticularization tothermodynamics of Py=PAS,XyXoXD (3.10) theEulertheorem onhomogencous first-order forms. Theforegoing Ifwechoosetheparameter Aofequation2.14as4=1/X; mGevelopment merelyreproduces thestandardmathematical derivation. pa equation24 1%wethenhaveWerefertoequation 3.5or3.6astheEulerrelation. Py=PAS|Xy XI/Xp0+* XalXy 1) GAD. 1c tiontheEulk ‘lationtakesthe Intheentropyrepresentation theEulerrelationtakes©form ‘Thuseachofthe(¢+1)intensiveparameters isafunctionofjusts=DRX, G7 variables, Elimination ofthese¢variables among the(t+1)equationssso yields thedesired relation among theintensive parameters. or : p . Tofindtheexplicitfunctional relationship thatexistsamongthesetofse(2)us(2)v-z(8)Ny (8) intensiveparameterswouldrequireknowledgeoftheexplicitfundamental T) TheaXT. equation ofthesystem. Thatis,theanalytic formoftherelationshipvaries from system tosystem. Given thefundamental relation, the Problems—Section 3.1 procedure isevident andfollows thesequence ofstepsindicated by ; ;; ‘equations 3.9-3.11. abiesWriteeachoefivephysicallyacceptablefundamental equations Adifferential formoftherelationamongtheintensiveparameters can ofproblem 1.9-1intheEulerform, beobtained directlyfromtheEulerrelationandisknownastheGibbs-Duhem relation. Taking theinfinitesimal variation of equation 3.5,3.2TheGibbs-Duhem Relation find “a ve ‘ ‘ InChapter 2wearrived atequilibrium criteria involving thetempera- dU=TdS +SdT+¥P,dX, +¥X,aP, @.12) ture,pressure, andthechemical potentials. Eachoftheintensive para~ |. mi sh metersentered thetheoryinasimilat way,andtheformalism is,infact, But,inaccordance withequation 2.6,wecertainly knowthatsymmetric intheseveralintensiveparameters. Inspiteofthissymmetry, |however,thereaderisapttofeelthathehasanintuitiveresponsetothe dU=TaS+3P,dX, concepts oftemperatures andpressure, which islacking, atleasttosomeS+BPax 6.13) degree, inthecaseofthechemical potential. Itisofsomeinterest, then, .tonotethattheintensiveparameters arenotallindependent, Thereisa whence,bysubtraction wefindtheGibbs-Duhem relation relation among theintensive parameters, andforasingle-component “ system pisafunction ofTandP. Sat+XX,dPs=0 (3.14) ‘Theexistence ofarelationship among thevarious intensive parameters | . |isaconsequence ofthehomogeneous first-order roperty ofthefunda- |Forasingle-component simplesystem,inparticular, wehave mental relation. Forasingle component system thisproperty permits the | SaT—~VaP+Ndp=0 G.15)fundamental relation tobewritten intheformu=«(s,0),asinequation |or7 2.19.Eachofthethreeintensive parameters isthenalsoafunction of 5 du=—sdT +v4P G19 and v.Elimination ofsandvfromamongthethreeequationsofstate ao. yields arelation amongT,P,and |Thevariationinchemicalpotentialisnotindependent ofthevariations ‘Theargument caneasilybeextended tothemoregeneral caseand|Oftemperature andpressure, butthevatiation ofanyonecanbecom-againconsists ofastraightforward counting ofvariables. Suppose we putedintermsofthevariations oftheothertwo. .haveafundamental equation in(¢+1)extensive variables __TheGibbs-Duhem relation presents therelationship among theU=US,XpXa--X) os) intensive parameters indifferential form.Integration ofthisequationyields therelation inexplicit form, andthisisaprocedure alternative to |Aetemeer amaynee aCmmm tliat Oe 50 SOME FORMAL RELATIONSHIPS AN EXAMPLE—THE IDEAL MONATOMIC GAS 51 that presented inequations 3.9-3.11. Inorder tointegrate theGibbs- equation and contains allthermodynamic information about asystem. Duhem relation, onemust know theequations ofstate which enable one Any single equation ofstate contains lessthermodynamic information towrite thejsinterms oftheP's, ofviceversa. ’ than thefundamental equation. Thenumberofintensive parameters capableofindependent variation F Iftwoequations ofstateareknown,theGibbs~Duhem relationmaybe iscalled thenumber ofthermodynamic degrees offreedom ofagiven integrated toobtain thethird. The equation ofstate soobtained will system. Asimple system ofrcomponents hasr-+Ithermodynamic contain anundetermined integration constant. Thus twoequations ofdegreesoffreedom. statesufficetodetermine thefundamental equation, exceptforanIntheentropy representation theGibbs-Duhem relation again states undetermined constant. thatthesumofproducts oftheextensive parameters andthedifferentials Analternative procedure forobtaining thefundamental equation when ofthecorresponding intensive parameters vanishes. only twoequations ofstate aregiven isbydirect integration ofthemolar relation $x,ar,=0 Gin du=Tds—Pdo 3.23) or iro Clearly, knowledge ofT=T(s,0)andP=P(s,0)yields adifferential 1 PY on fa) equation inthethreevariables u,s,andv,andintegration givesUd\— Vd|?)-aM)=0 (3.18) (e)+eG)~Snel Guy w=5,0) 62which isafundamental equation. Again, ofcourse, wehave an Problems—Section 3.2 undetermined constant ofintegration. -Itisalwayspossibletoexpresstheinternalenergyasafunctionof 32. Findtherelation among7,P,and1forthesystemwiththefunda- parameters otherthanS,V,andN.Thuswecouldeliminate Sfrom mentalequationoy U=US,¥,N)andT=T(S,V,N)toobtainanequation oftheform u=(@)ar U=U(T,V,N). However,wewishtostressthatsuchanequationisfi notafundamental relation anddoesnot.contain allpossible thermo- dynamic information aboutthesystem. Infact,recalling thedefinition of 3.3Summary ofFormalStructure Tas0U/@S, weseethatU=U(T,V,N)actually isapartialdifferential Letusnowsummarize thestructureofthethermodynamic formalism cauation,yenifthisequationwereintegrable,itwouldyieldafunda: intheenergyrepresentation. Forthesakeofclarity,andinorderto folationv-Ut5oi»‘lowsone.toCompute.therelationUe beexplici ijingle- imple |.Thefunda- oenon itone beexplicit, Wecomer «single-component simplesystem. Thefunda U(T,V,N),butknowledge ofU=UT,V,N)doesnotpermitone 9veus.ym oy) inversely tocompute U=U(S,V,N). Associated witheveryequationaa i there isboth atruth-value and aninformatiorial content. Each ofthe contains aifthermodynamic information about asystem. With the equations U=U(S,¥,N)andU=U(T,V,N)maybetrue,butonly definitions T=9U/@S, etc.,thefundamental equation implies three theformer hastheoptimum informational content, equations ofstate:T=1S,V,N)=1,0) 3.20) 3.4AnExample—The IdealMonatomic Gas P=P(S,VN)=Ps,0) G21) Anidealmonatomic gasischaracterized bythetwoequations 1=1S,¥,N)=ws) (3.22) PY=NRT 025 Ifallthreeequations ofstateareknown,theymaybesubstituted intothe and 49Eulerrelation, thereby recovering thefundamental equation, Thusthe U=NRT +6.26) totality ofallthree equations ofstate isequivalent tothefundamental inwhich X&isaconstant with thevalue 1.986 cal/mole °K, nea “THTHEORY OFFLUCTUATIONS CHAPTER 16 Now let$beafuvetion oftheform. “4=(GAN IM-+OR" @ snwhichthemarenonnegative integers. Usingtheresultprovedabove,show Irreversible that .Goty=-24»=-kmipiohy © Thermodynamics .Xe, ~ Considernowthequantity(#8,)and,usingequations (b)and(@),showthat Goky =—K>ShyGloXy : i inO%,if¢=6%;weobtain nally,showthatif¢=1weobtain(0%)=0,thatit6=3%in equation iavandthatotherchoicesof¢permitcalculation ofhighermoments i | 16.1.GeneralRemarks | Asusefulasthecharacterization ofequilibrium statesbythermostatic . theory hasproved tobe,itmust beconceded thatourprimary interest is frequently inprocesses rather than instates. Inbiology, particularly, it isthelifeprocess thatcaptures ourimagination rather than theeventual equiilbrium statetowhich eachorganism inevitably proceeds. Thetmo- 'statics does provide two methods that permit ustoinfer some limited information about processes, buteach ofthese methods isindirect and each yields only themost meager return. First, bystudying theinitial . and terminal equilibrium states, itissometimes possible tobracket a process and thence todetermine theeffect oftheprocess initstotality. Second, ifsome process occurs extremely slowly, wemay compare it with anidealized, nonphysical, quasi-static process. Butneither ofthese methods confronts thecentral problem ofrates ofrealphysical processes. The extension ofthermodynamics which hasreference totherates of : physical processes isthetheory ofirreversible thermodynamics. ‘Two basic postulates underlie equilibrium statistical mechanics, The N firstpostulate concerns theexistence ofanenormous number ofatomistic . states among which continual spontaneous transitions occur inthecourse ofamacroscopic observation. Thesecond istheassumption ofequalaprioriprobability ofeachoftheatomistic states.Fromtheseextremely -\ . general hypotheses follows theentire general theoryofstatisticalmechanics, culminating inthetheorems that, inturn, constitute thepostulates of thermostatics. Because ofthegeneral nature ofthepostulates, the predictions ofthermostatics aresimilarly general. The numerical values 23 <1:entail et SEEDS ALLe e- et IRREVERSIBLE THERMODYNAMICS AFFINITIES AND FLUXES 285 ofspecific heats,compressibilities, andthelikearenotpredicted, but theclosure condition requires that certain general relationships among thesequantities arepredicted. be‘Thetheoryofnonequilibrium statistical mechanics isbasedonthetwo A+%=X", aconstant (16.1)stulates oftheequilibrium theory, plustheadditional postulate ofrime IfX,andX;'areunconstrained, their equilibrium v redeteonmetryofphysicallaws.Thisadditionalpostulatestatesthatallthe bythevanishingofthe quantitycauilbsium values aredetermined lawsofphysics remain unchanged ifthetime1iseverywhere replaced by . .ttsnegative1andifsimultaneously themagneticfieldH,isreplacedby F,=()=A) <38Ln 63 itsnegative —H,,Melae Ox) OX, Oy FHA (182) Fromthegeneralpostulates ofnonequilibrium statistical mechanics - Thus,ifF,iszero,thesystemisinequilibrium, butifF,isnonzero antherefollowsanextensive theory,culminating inseveraltheorems that, irreversible processoccurs,takingthesystemtowardtheequilibrium state.inturn,constitute thepostulates ofirreversible thermodynamics. From Thequantity F,,whichisthedifference intheentropy-representation ” these wederive thermodynamic theorems ofageneralnature,expressing intensiveparameters,actsasa“generalized force”which“drives”the relationships among various dynamical quantities. process. Suchgeneralized forces arecalled affinities. ‘Thefirstsuch result wastheOnsager reciprocity theorem, which Fordefiniteness, consider twosystems separated byadiathermal wall,‘expresses acertainsymmetry intheresponse oftwosimultaneously andletX,betheenergyU.Thentheaffinityis °‘occurring processes. ©Another theorem isthefluctuation-dissipationtheorem ofH.CallenandT.Welton, whichexpresses arelation between F=1-1 63irreversible response andequilibrium fluctuations. Othertheorems { ror ,063) relating tothefluctuations during anirreversible process, andinvolving, Noheatflowsacross thediathermal wallifthedifference ininverseextensions tononlinear processes, havebeendeveloped byW.Bernard temperatures vanishes, Butanonzero difference ininversetemperature,andH,CallenandbyM.Lax. actingasageneralized force,drivesaflowofheatbetween thesubsystems, Despite theconsiderable number ofresults ofnonequilibrium thermo- Similarly, ifX,isthevolume, theaffinity F,is[P/T—(P'/T)],andifdynamics, byfarthemostpractically significant resultistheOnsager X,isamolenumber theassociated affinity is[j4/T”—(i4/T)).reciprocity theorem. Consequently, werestrictourattention tothis | Wecharacterize theresponse totheappliedforcebytherateofchangetheoremalone.Weindicateinsection16.5howthisthtoremisrelated oftheextensive parameter X,.ThefluxJ,isthendefinedby e totheunderlying postulate oftimesymmetry. aX,ale (16) 16.2 Affinities and Fluxes. Therefore, thefluxvanishes iftheaffinity vanishes, andanonzero affinit:Preparatory toourdiscussion ‘oftheOnsagertheorem,wedefine leadstoanonzeroflux.Itistherelationship betweenfluxesandainitiescertain quantities thatappropriately describe irreversible processes. thatcharacterizes theratesofirreversible processes.i Basically werequiretwotypesofparameters: onetodescribe the“force” Theidentification oftheaffinities inaparticular typeofsystem ithatdrivesaprocessandonetodescribetheresponsetothisforce. frequentlyrenderedmoreconvenientbyconsidering therateofoduct2“Theprocessesofmostgeneralinterestoccurincontinuoussystems,such ofentropy. Differentiating theentropy S(XyX,---)withrespecttotheastheflowofenergyinabarwithacontinuous temperature gradient. time,wehave oat pecttothe However, tosuggest the-proper waytochoose parameters insuch a _asax,continuous systems,wefirstconsidertherelatively simplecaseofa anaKa (16.5)discrete system. Atypicalprocessinadiscrete systemwouldbetheflow or te‘ofenergy fromonehomogeneous subsystem toanother through an - ooSafnitelythindiathermalpartition. S=3Fie (16.6)Consider @composite system composed oftwosubsystems. An Thus, therateofproduction ofentropy is 2extensive parameterhasvalves%,and.Xjinthetwosubsystems,and withitscsoeuiedayoFenreny1sthesumofproductofeachfux 5 IRREVERSIBLE THERMODYNAMICS. AFFINITIES AND FLUXES 287 |Theentropyproductionequationisparticularlyusefulinextendingthe ofthisconvention, incidentally, thatwecanspeakofthetemperature definition ofaffinities tocontinuous systems rather than todiseréte varying continuously inabar,despite thefactthatthermostatics implies systems. Ifheat flows from one homogeneous. subsystem toanother, theexistence oftemperature only inequilibrium systems. through aninfinitely thin diathermal partition, thegeneralized force is Equation 16.7 immediately suggests areasonable definition ofthe thedifference [I/7—(1/T")];butifheatflowsalongametalrod,in entropycurrentdensityJ. which thetemperature varies inacontinuous fashion, itisdifficult to Js=BEd, (6.9)applyourpreviousdefinition oftheaffinity.Nevertheless, wecancompute en ;therateofproduction ofentropy, andtherebywecanidentify theaffinity. inwhichJ,isthecurrent densityoftheextensiveparameterX,.The With theforegoing considerations toguide us,wenowturnour - ‘magnitude oftheentropy fluxJyistheentropy transported through unit attention tocontinuous systems, Weconsider athreedimensional system areaperunittime. ; ; inwhichenergy andmatter flow,driven byappropriate forces. Asfluxes, Therateoflocalproduction ofentropy isequaltotheentropy leaving‘wechoosethecomponents ofthevector current densities ofenergy and theregion, plustherateofincrease ofentropy within theregion. IC$ matter. Thus, associated withtheenergy U,wehavethethreeenergy denotes therateofproduction ofentropy perunitvolume ands/8¢ fluxes Jye,JyysJyyThese quantities arethex,y,andzcomponents of denotes theincrease inentropy perunitvolume, then thevector current density J,.Bydefinition themagnitude ofJ,isthe ap 8s amount ofenergy which flowsacross theunitareainunittime,andthe sagt Veds (16.10) direction ofJ,isthedirection ofthisenergy flow. Similarly, thecurrent density J,maydescribe theflowofaparticular chemical component per ‘Thevarious extensive parameters canbeneither produced nordestroyed, unitareaandperunittime; thecomponents JzasJey,andJ,arefluxes. sothattheequations-of continuity forthese parameters become Inorder toidentify theaffinities, wenowseek towrite therateof 7 production ofentropy inaform analogous toequation 16.6. On tvs (16.11) One problem that immediately arises isthat ofdefining entropy ina nonequilibrium system. This problem issolved inaformal manner as Wearenow prepared tocompute $explicitly andthence toidentify thefollows. affinitiesincontinuous systems,Toanyinfinitesimal region weassociate alocal entropy S(Xq Xi**)s ‘The firstterm inequation 16.10 iseasily computed from equation 16.8. where,bydefinition, thefunctional dependence ofSonthelocalextensive | ss ae, parameters X,,X,*~+istaken tobeidenticaltothedependence inequilibrivan. Peaaisa (16.12) Thatis,wemerelyadopttheequilibriumfundamentalequationtoassociate ME alocalentropy withthelocal parameters Xo,X,-+*. Then ‘Thesecond terminequation’ 16.10 iscomputed bytaking thedivergence is=SFeaX, (16:7) ofequation 16.9. * Vidg=V- (ZFsh)=EVA TAVSs(16.13) or,taking allquantities perunitvolume,* * = = : hi ia aeAids 68) Thusequation16.10beromeyd . :‘Thesummation inthisequation omitsthetermforvolume andcon- SeZAG ZV LTTAVA (16.14) sequently hasonelessterm than that inequation 16.7. i i ,SpainthelocalintensiveparameterF,istakentobethesamefunction cnetgene "Gitemeoberthatthersandthie‘erms ofthelocalextensive parameters asitwouldbeinequilibrium. Itisbecause F=DVA (16.15) — t toindentetaeeephatesperensomeatesenperme Althoughtheaffinityisdefinedasthedifferenceintheentropy-representation 288 IRREVERSIBLE THERMODYNAMICS LINEARPROCESSES 289 intensiveparameters fordiscretesystems,itisthegradientoftheentropy EachfluxJ,isknowntovanishastheaffinitiesvanish,sowecanrepresentation intensive parameters incontinuous systems. expand J,inpowers oftheaffinities withnoconstant term.IfJ,,denotes thezcomponent oftheenergycurrentdensity,theassociated 1affinity ¥,,iV,(I/T), thezcomponent ofthegradient oftheinverse SesShinFs+5LLlAto (16.17) temperature.AndifJ,denotesthekthmolenumbercurrentdensity hn 7 TF " (thenumber ofmolesofthekthcomponent flowing through unitarea where 2,Bpersecond),theaffinityassociatedwithJy,isFy,=-v,(H). Ln=(#), (16.18)- and : ay,16.3Markoffian Systems Lin=BF,aF,)y (16.19) Forcertain systems thefluxesatagiveninstant depend onlyonthe Thefunctions Ly,arecalledkinetic coefficients. Theyarefunctions ofvalues oftheaffinities atthatinstant. Wecallsuehsystems Markoffian, thelocalintensive parameters. borrowing theterminology fromthetheory ofrandom processes, andwe Ly=LglFosFy) (16.20) restrict ourattention tothistypeofsystem. . ; Foranon-Markoffian systemthefluxesmaydependuponthevalues ‘ThefunctionsLyarecalledsecond-orderkineticcoefficients,andtheyare oftheaffinities atprevious timesaswellasuponthevaluesatthepresent alsofunctions ofthelocal.intensive parameters. Third-order andhigher-time,Intheelectrical caseapureresistor isaMarkoffian system, orderkinetic coefficients aresimilarly defined.whereas acircuitwithcapacitance orinductance isnon-Markoffian. A Forthepurposes oftheOnsager theorem, whichweareabouttorhon-Markoffian system hasa“memory.” cenunciate, itisconvenient toadoptanotation thatexhibits thefunctional‘Although itmightappearthattherestriction toMarkoffian systems is dependence ofthekinetic coefficients onanexternally applied magneticaverysevererestriction indeed, itisfoundinpractice thatalmost all fieldH,,suppressing thedependence ontheotherintensive parameters. systems ofinterest, otherthanelectrical systems, areMarkoffian. The Ly=Lyf) (1621)extension ofthetheory tonon-Markoffian systems, which weshallnot TheOn:present here,hasbeenmoreimportant foritselucidation ofprinciples ieOnsagertheorem statesthat thanforitsapplication torealsystems. . «Enfll) =Le) (16.22) ForaMarkoffian system, bydefinition, eachlocalfluxdepends only Thati inet .upontheinstantaneous localaffinities anduponthelocalintensive Htis,thevalueofthekinetic coefficient Lymeasured inanexternalparameters. Thatis,droppingtheindicesdenotingvectorcomponents, necKeaidenticaltothevalueofLy,measuredinthereversed NWF Fy Fy FyFyBye) (16.16) TheOnsager theorem statesasymmetry between thelineareffectofthethaffinity onthekthfluxandthelineareffectofthekthaffinity on Thus,thelocalmolenumber current density ofthekthcomponent thejthfluxwhentheseeffectsaremeasured inopposite magnetic fields. depends onthegradient oftheinverse temperature, onthegradients of . 44,foreachcomponent, anduponthelocaltemperature, pressure, etc. 16.4 Linear Processes Itshould benoted that wedonotassume that each flux depends only-on ‘euatic a ene if Selisownaffinitybuteatherthateachfluxdepends onollafinites. 1s Asituation ofgreatpractical interestarisesiftheaffinities aresosmaliy pendsona thatallquadratic andhigher-order termsinequation16.17canbe truethateachfluxtends todepend moststrongly onitsownassociated accede Eeee erty dee nye.affinity,butthedependence ofafluxonotheraffinitiesaswellisthe approximate equationsiatcanbeadequately describedbythetruncatedsourceofsomeofthemostinteresting phenomena inthefieldofirreversi- + bility. BP heEhaFs (16.23) 290 IRREVERSIBLE THERMODYNAMICS THESTATISTICAL BASISOFTHEONSAGER RECIPROCITY 291 iscalled alinear Markoff process. Fortheanalysis ofsuchprocesses the which istheaverage product ofthedeviation 5%,andofthedeviation ‘Onsager theorem isaparticulsrly powerful tool. 6%,thelatterbeingobserved atime+raftertheformer. Assuming, for Itisperhaps surprising thatsomany physical processes ofinterest are simplicity, thatnomagnetic fieldispresent, theprinciple oftimesymmetry linear. Buttheaffinities thatwecommonly encounter inthelaboratory requires thatthecorrelation moment (16.26) beunchanged ifwereplace arequite small inthesense ofequation 16.17, andwetherefore recognize rby —7. thatwegenerally dealwith systems thatdeviate only slightly from (WR, IX0))=(6%, 5(—)) (16:27)equilibrium. . .‘Phenomenologically, itisfound thattheflowofenergy inathermally or,since onlytherelative times inthetwofactors aresignificant, conducting body isproportional tothegradient ofthetemperature. -Denoting theenergycurrent density byJ,,experiment yieldsthelinearlaw (OR,OXAr)=ORAr) OX) (16.28) S=—e VT (16.24) Htwenowsubtract (5.2,6X,)fromeachsideoftheequation anddivide 7,wefindinwhichxisthethermalconductivity ofthebody.Wecanrewritethis ¥eg¢¢inthemoreappropriateform (08,i) -Ci=0%,5)(16.29)Sop=KT?v(t) (16.25) _ . . T| Inthelimitas7»0wecanwritetheforegoing equation intermsofandsimilarly forxandycomponents, andweseethat(17)isthekinetic timederivatives. 5 :coefficient. Theabsence ofhigher-order terms, suchas[V(I/7)]* and (OR,OX)=(0%,0) (16.30) [V(I/T)P, inthephenomenological !awshows thatcommonly employed og temperature gradients aresmallinthesenseofequation16.17. wanneassumeiatthedecayofafluctuation3%,isgovernedbythe Ohm’s lawofelectrical conduction andFick's lawofdiffusion are ‘Yaamical'lawsasaremacroscopic processes, otherlinearphenomenological laws:whichdemonstrate thatforthe ok,=1,0F,‘common valuesoftheaffinitiesintheseprocesseshigher-ordertermsare =BludF (16.31) negligible.Ontheotherhand,boththelinearregionandthenonlinear Insertingtheseequations inequation16.30gives region canberealized easily inchemical systems, depending upon the . ideviations ofthemolarconcentrations fromtheirequilibrium values. YLldk, IF)=TLylSF,d%,) (16.32) Although theclass oflinear processes issufficiently common tomerit v 7 special attention, itisbynomeans all-inclusive, andcontrary totheusual However, weshall show below, bythemethods of‘Chapter 15,that statement theOnsager theorem isnotrestricted tothisspecial classof H kitsystems.%,)={— ifisj ysOL,8F)={0itinj (16.33) 165 The Statistical Basis oftheOnsager Reciprocity . inwhich kjsBoltzmann's constant. Thus equation 16.32 becomes ‘TheOnsager reciprocity hasbeenstated without proof inthepreceding Ly=Ly (16.34) sections. Applications aremadeinChapter17.Inthissectionweindicate - : therelationship oftheOnsager reciprocity totheunderlying principle of which istheOnsager theorem intheabsence ofamagneticfield. timesymmetry ofphysical laws. Tocomplete theproof wenowdemonstrate equation 16.33. From Weconsider asystem inequilibrium, andweaddress ourattention to equation 15.1wecanwrite thedistribution function forthespontaneous thespontaneous fluctuations, asinChapter 15.Consider, inparticular, fluctuations intheform acorrelation moment such as 1 ‘W=Qexp 7(08—5F,5%) OF,dRAr) (16.26) PZ z1OX) (16.35) e 292 IRREVERSIBLE THERMODYNAMICS CHAPTER 17 Combining thiswith equation 15.4indicates that “2 OW WF, =k—= (16.36). aor,Thermoelectric i wverage value inequation 16.33. . .Considernowtheaverag : andThermomagnetic .08,04)=fi32,FW dX, ddky >> (46.37) Effects ow|= sepdR,dbXy+> (16.38) afo%,052, 5 Ifij,theintegral over d5, vanishes atboth limits. Ifi=j, we integrate byparts, immediately obtaining theresult 16.33. Analternative, andeasier, derivation ofequation 16.33 isreadily obtained onthebasis oftheapproximate Gaussian distribution ofsection 15.6rather thanby therigorous distribution usedabove. Such analternative derivation is H essentially giveninproblem 15.6-1. | ; |Ourderivation oftheOnsager reciprocity isthatgivenbyomerExtension tothecaseinwhich amagnetic fieldispresent isquite simple. However, justification oftheassumption thatthemacroscopic dynamical ‘ 17.1TheDynamicat Equations Jawscanbeapplied tothedecayof@spontaneous Ructuation fe Asapplications oftheOnsager theorem weshallconsider thethermo-simple.“Justification ofthiaeaneeateeoviewedasa electricandthermomagnetic effects,whicharephenomena associated withticalmechanical analysis,sothatthissectionshouldpoet nical thesimultaneous flowofelectriccurrentandheatinasystem.These demonstrationofplausibility ratherthanasastatisticalme phenomena, andcertainrelationsamongthem,wereproposedin1854by derivation.LordKelvinonthebasisofempiricalobservations. KelvinalsoPresented { 4suggestive argument leading totherelations, carefully pointing out, t however, thattheargument wasnotonly unjustifiable butthatitcould i bemade toyield incorrect relations aswellascorrect ones. Itisacurious f factthatmany ‘modern thermodynamics textsstillpresent Kelvin's, argument asarigorous proof, completely ignoring Kelvin’s ownadmoni- t tiontothecontrary andignoring themethods ofmodern irreversibility ‘ theory. Asremarked above, thethermoelectric andthermomagnetic effects are Phenomena associated with thesimultaneous flow ofheat and electric current inasystem. Fordefiniteness ofexpression weshall think ofa solid inwhich thecharge carriers areelectrons. Then, ifsisthelocal ‘entropy density, wehave a! # () : ds=duol2Tdn, 7.1) inwhichujsthelocalenergydensity,jxistheelectrochemical potential(perparticle) oftheelectrons, nisthenumber ofelectrons perunit 293 = OOT—ov—_—ee ALRcael tnttinene atts teaaiaianette Rhine.sate atthd 294 THERMOELECTRIC AND THERMOMAGNETIC EFFECTS. ‘THE DYNAMICAL EQUATIONS 295 volume, andinwhich thesumrefers toother “components.” These Jyisacurrent density oftotalinternal energy, wegenerally prefer to other components refertothevarious types ofatomic nuclei thattogether discuss thecurrent density ofheat. Inanalogy withtherelation dQ=TdS withtheelectrons constitute thesolid. Itwillbenoted thatwehavetaken wetherefore define aheatcurrent density Sgbytherelation‘nasthenumberofelectrons ratherthanthenumberofmolesofelectrons, Jo=TB. any andjxisaccordingly theelectrochemical potentialperparticleratherthan or,byequation172 @s : permole.Inthisregardwedeviatefromthemoreusualparameters u 2, Je=3yai, ara: merelybymultiplication anddivisionbyAvogadro's rrumber, respectively. fo=Ju—45S z Justa§equation 16.7ledtoequation 16.9,equation 17.1nowleadsto ‘Inaveryrough intuitive waywecanlookonj«asthepotential energy .perparticleandonJyasacurrentdensityofpotentialenergy;sub- JgaASy —45y (17.2) traction ofthepotential energy current density fromthetotalenergy. rer current density yieldstheheatcurrent density as.asortofkinetic energy inwhich Jg,Jy,andJyarecurrent densities ofentropy, energy, and current density. Atanyrate,eliminating JyinfavorofJgfromequation number ofelectrons, respectively. Theothercomponents inequation 17.1 17.3gives areassumed immobile andconsequently donotcontribute fluxtermsto gees —Lup- dy ans)‘equation 17.2. T r Repeating thelogicleadingtoequation 16.15,wefind .Itfollows fromthisequation thatifthecomponents ofJgandof—Jy 1 x arechosen asfluxes theassociated affinities arethecorresponding $=Vidy—VGIw (17.3) components ofV(I/T)andof(I/T)Vy, respectively. Thedynamical equations canthenbewritten, intheone-dimensional case,as Thus,ifthecomponents ofJyand—Jyyaretakenasfluxes,theassociated j 'affinitiesarethecomponents ofveandVfoAssuming forsimplicity 2IwLaVE+TVFa (17.10) thatallflowsandforces areparallel tothex-direction, andomitting the =t.4y asubscript 2,thelineardynamical lawsbecome Jo=Fant La5 ann cpt andtheOnsagerrelationis JyLyVELiV5 a7) +Ly(H) =La(—H) (17.12) - 71 ‘Thereadershould verifythatthedynamical equations 17.10and17.11 Jo=LyvetlnIE (17.5) canalsobeobtainedbydirectsubstitution ofequation17.8intothe . .‘ previous pairofdynamical equations 17.4and17.5without recourse to andtheOnsager theorem givestherelation theentropy production equation 17.9.Lif)=La(-H) (17.6) Thesignificance oftheheatcurrentcanbeexhibited inanothermanner. Inwriting theforegoing dynamical equations wehaveassumed one- Weconsider, foramoment, asteady-state flow.ThenbothJyandJydimensional Now,suchasoccursinwiresorbars.Thisisthecaseof aredivergenceless andtakingthedivergence ofequation 17.8gives interest tousintheanalysis ofthethermoelectric effects.-Whenwe ViJo=—Vu-Sy —(inthesteadystate) (17.13) considerthethermomagnetic effectsinsection17.6,however, weshall whichstatesthatinthesteadystatetherateofincrease inheatcurrent havetotakeexplicit cognizance ofthefactthatthekineticcoefficients isequaltotherateofdecreaseinthepotentialenergycurrent.Further- forthex-directed currents maydifferfrom.the kineticcoefficients forthe ‘more,theinsertion ofthisequation intoequation 17.9givesy-directed currents. Beforedrawing physical conclusions fromequation 17.6,werecastthe gevi. Jotdyes (i714dynamical equations into anequivalent butinstructive form. Although T rT“? ’ nn OR Attl tt ttt tcliett tattthe tharAnta Niatheatta 296 THERMOELECTRIC ANDTHERMOMAGNETIC BFFECTS ‘THESEEDECK EFFECT ANDTHETHERMOELECTRIC POWER 297 which canbeinterpreted asstating thattheproduction ofentropy isdue isothy = ae totwo causes; thefirst termistheproductionofentropyductotheflow isothermalsystemVia,=0andVj=Vit,.Thus,bydefinition, ofheatfromhightolowtemperature, andthesecondtermistheincrease 1 inentropyduetotheappearance ofheatcurrent. os-ery/3vefor=Wr=0 (17.19)We now accept thedynamical equations 17.0 and 17.11 and the ; symmetry condition (equation 17.12) asthebasic equations withwhich whence equation 17.15 gives : tostudy theflowofheatandelectric current inasystem. o=eLy/T (17.20) . Similarly theheatconductivity xisdefinedastheheatcurrentdensityperunittemperature gradient forzeroelectric current. 17.2.TheConductivities K=S—JoVT forJy=0 EWe'consider asysteminwhichanelectriccurrentandaheatcurrent . en " (17.21)flowparallel tothez-axis inasteady state, withnoapplied magnetic Solving thetwokinetic equations simultaneously, wefind field. Then, omitting thesubscript x, D 1 1 “a (17.22) mJylnVtLan aay! Phywhere Ddenotes thedeterminant ofthekinetic coefficients 1 1 Jo=LipVe+LanV5 (17.16) DaLyla—Uh (17.23) where theOnsager theorem hasreduced tothesimple symmetry 17.3 TheSeebeck Effect andtheThermoelectric Power la=Ln any _TheSeebeck effect refers totheproduction ofanelectromotive forceinathermocouple underconditions ofzero electric current. - Thethreekineticcoefficientsappearinginthedynamicalequationscan Considerathermocouple withjunctionsattemperatures 7,and berelatedtomorefamiliar quantitics, suchasconductivities. Indeveloping T,(Ta>7),asindicated inFigure 17.1.Avoltmeter isinserted inonethisconnection wefirstcomment briefly onthenature oftheelectro j armofthethermocouple atapointatwhich thetemperature is7".This chemicalpotentialpoftheelectrons, Wecanconsiderjxascomposed ivoltmeter issuchthatitallowsnopassageofelectriccurrentbutoffers oftwoparts, achemical portion jz,andanelectrical portion 2, | noresistance totheflow ofheat. Wedesignate thetwomaterials * composing thethermocouple byAandB.With Jy=0,weobtain from hehe the (1718) thekinetic equations, foreither conductor, Ifthechargeonanelectronise,thenj,issimplyef,where¢isthe| Va=227 «724ordinary electrostatic potential. The chemical potential 1,isafunction Thy je ofthetemperature andoftheelectron concentration. Restating these Thus factsintermsofgradients,theelectrochemicat potentialperunitcharge n-ne fLhop (1725 is(I/e)y4; itsgradient (1/e)Vy isthesumoftheelectric field(Ife)Vi am hhTA 25) plusaneffective drivingforce(I/e)Vy, arisingfromaconcentration Bgradient. . - faaa 17.2 7 Theelectric conductivity ¢isdefined astheelectric. current density 1}TL (17.26) (eds) perunitpotential gradient (1/e)Vxinanisothermalsystem.Itis . “EB : easilyseenthat(1/e)Visactuallytheemf,forinahomogeneous Hh-fame 7.2 h TLE 298 ‘THERMOELECTRIC AND THERMOMAGNETIC EFFECTS “THE PELTIER EFFECT 299 Eliminating 44,andj2,from these equations, Aninteresting insight tothe’physical meaning oftheabsolute thermo- ( f(th ob electric power canbeobtained byeliminating (1/7) Vizbetween thetwoBa=f(&7a)T (17.28) foregoingdynamicalequationsandwritingJgintermsofJyandV(I/7): But,because thereisnotemperature difference across' thevoltmeter, the Ja=Teedy+TeV2 (1735) voltage issimply1 vere or,recalling thatJs=Jg/T, .alut— aps[(4-2) ar 17.29) 1 v=ou) f(aaa) (6725) Js=ely+TV5 (17.36) Thethermoelectric power ofthethermocouple, <4,isdefined asthe According tothisequation, eachelectron involved intheelectric currentchange involtage perunitchange intemperature difference. Thesignof carries withitanentropy ofee.Thisflowofentropy isinaddition to._ theentropy current TxV(I/T), which isindependent oftheelectronic A current. The thermoelectric power can belooked onastheentropytte hi Ww. sy HT ransported petcoulomb bytheelectron flo : | 17.4 ThePeltier Effect * i ‘ThePeltier effectreferstotheevolution ofheataccompanying theflow@) ofanelectriccurrentacrossanisothermal junctionoftwomaterials. Consider anisothermal junction oftwoconductors AandBandanelectric Figure17.1 current eJytoflowasindicated inFigure 17.2. Then thetotalenergy 4aischosen aspositive ifthevoltage increment issuch astodrive the : current from A(oBatthehotjunction. Then : A B av(4) (4) nnd Ide?cay&=(=2)-(= (17.30) ' oT,~Neri)~\eith, ' *pagure12 Defining theabsolute thermoelectric power ofasingle medium bythe \relation tcurrentwillbediscontinuous acrossthejunction, andtheenergydifference _lk i731 appears asPeltier heatatthejunction. WehaveJy=Jo+Wy,and“=TA (731) sincebothyzandJyarecontinuous acrossthejunction itfollows thatthe thethermoelectic powerofthethermocouple is discontinuity inJyisequaltothediscontinuity inJg.A gyBa Joh— Joh .fan=enfa (07.32) seathentee evom Ifweaccepttheelectricconductivity «,theheatconductivity x,and Becauseoftheisothermal condition thedynamical equations 17.33andc “ 11.34give,ineither conductor, theabsolute thermoelectric power €asthethree physically significant dynamicat properties ofamedium, wecaneliminate thethree kinetic Tq=TdeIy) (17.38)coefficients infavorofthesequantities andrewritethekineticequations in whence: aathefollowing form: Jo”~Jot=Men—eaeln) (17.39)To\tg,_(Tee)yh 1733 ThePeltiercoefficient74pisdefinedastheheatthatmustbesupplied ww=ale) Ye (17.33) tothejunction whenunitelectric current passes fromconductor 4to Toe\| t conductor B.Thus : Iga(F)hvm+(Pod+PV5,(17.34) nan=Vo"—Joely=Ten—€4) (17.40) 300 THERMOELECTRIC AND THERMOMAGNETIC EFFECTS ‘THE THOMSON EFFECT 301 Equation 17.40, which relates thePeltier coefficient totheabsolute However thetemperature distribution isthatwhich isdetermined bythe thermoelectric powers, isoneoftherelations presented onempirical steady state with noelectric current, andweknow that V+Jyvanishesevidence byKelvinin1854.ItiscalledthesecondKelvin,relation. inthatstate,ByputtingJy=0andV-Jy=0inequation 17.43,we Themethod bywhich wehave derived equation 17.40 istypical ofall conclude thatthetemperature distribution issuch astomake thesecond applications oftheOnsager relations, sothatitmaybeappropriate to term vanish, andconsequently review theprocedure. Wefirstwrite thelinear dynamical equations, 1 .reducing thenumber ofkineticcoefficients appearing therein byinvoking Vidy =TVe-(eSy) —2(edn) (17.44) theOnsager relations. Wethen proceed toanalyze various effects, . ; ‘expressing eachintermsofthekineticcoefficients. Whenwehaveanalyzed 4 Furthermore, notingthatthethermoelectric powerisafunction oftheasmanyeffectsastherearekinetic coefficients, werewrite thedynamical localtemperature, wewrite equations intermsofthoseeffects rather thaninterms ofthekinetic vente y,coefficients (asinequations 17.33and17.34). ‘Thereafter everyadditional iar ace (17.45) effect analyzed onthebasis ofthedynamical equations results inarelation and analogous toequation 17.40 andexpresses thisneweffect interms ofthe de 1coefficients inthedynamical equation. Vidy =TG.VT:(Iw)—5(eu (17.46) 175 ‘TheThomson Effect / Thesecond termistheJoule heat, produced bytheflowofelectric current, even intheabsence ofatemperature gradient. ‘Thefirstterm ‘TheThomson effect refers totheevolution ofheat asanelectric current represents theThomson heat, absorbed from theheat reservoirs when the traverses atemperature gradient inamaterial. current eJy traverses thetemperature gradient V7. The Thomson Consider aconductor carrying aheatcurrent butnoelectric current, coefficient +isdefined astheThomson heatabsorbed perunitelectric Atemperature distribution governed bythetemperature dependence of current andperunittemperature gradient. thekinetic coefficients willbesetup.Let.the conductor now beplaced incontact ateachpointwith2heatreservoir ofthesametemperature as pmThomson heat dethatpoint,sothatthereisnoheatinterchange between conductor and Wren) a (17.471) reservoirs. Now letanelectric current pass through theconductor. An |interchange ofheatwilltakeplacebetween conductor andreservoirs. | Although wehaveobtained theforegoing relation between 7and«byThisheatexchange consists oftwoparts—the JouleheatandtheThomson ‘ourstandard procedure, itisofinterest thatthisrelation canbederivedheat 4byanalternateprocedure, combining thesecondKelvinrelation(equation{AStheelectriccurrent passesalongtheconductor, anychangeintotal | 17.40)withconsiderations ofenergyconservation.energyflowmustbesupplied byanenergyinterchange withthereservoirs. Tocarryoutthisalternate derivation, weconsider athermocoupleThuswemustcomputeV°Jy. 'subjectedtoaverysmalltemperature differencedT.Intoonearmofthisthermocouple isinserted abatteryofjustsuchavoltageastonegatethe Vidy=V'Gotwy)=ViIg+ VarIy (17.41) Seebeckvoltage,causingno‘leciviecorrent10flowinthethermocouplewhich canbeexpressed interms ofJyandV(1/T) byusing equations circuit. 17.35 and17.33. Anaccordance with equations 17.30 and 17.32, therequired emfofthe 1 e 1 battery iseg—79‘Thethermocouple circuitisindicated inFigure 17.3.Vdy=V-(reedy41%vt)+(-©5y+Teevi):Jy(1742) _Wenowconsiderthevirtualtransferofaunitofchargearoundthe7,3 Tr circuit,andwelistthevariouseftergytransfersthattakeplaceinthe orga process. Asthechargeerostesthecoldjunction, fromBto4,aPele Ny <T Ver - (revt)2 ; \eatof-+7r isabsorbed fromthereservoir. Asthecharge traverses the Vly=TVe(ey)+9(%4)oe (1743) legAofthethermocouple, theThomson heat+4dT'isabsorbedfromthe 302 THERMOELECTRIC ANDTHERMOMAGNETIC EFFECTS : THETHERMOMAGNETIC DYNAMICAL EQUATIONS 303 reservoirs. However, weignore theJoule heatbecause weassume the componenis. Tosimplify thenotation wewriteN,forJyaQsforJoes charge tobetransported soslowly thattheJouleheat,which isquadratic andsimilarly forthey-components.* inthecurrents, isnegligible compared totheThomson heat, which is 1 1 1 1linearinthecurrents. Intraversing thehotjunctionfromAtoBthe | f=(v.*)Q.+(%.5)Q-(bv)Me-Gvt),(17.51)Peltierheat2.4,(T-+dT)isabsorbed. Intraversingtheleg#theThomson hereV. TT Tr. heat rpdTisabsorbed..Andintraversing thebatteryanamountof whereV,denotesthepartialderivative 8/@x.Weconcludethatif—Na, =Ny: Oyand Q,arefluxes theassociated affinities are(1/T)V.u, 4 . m Tr T+dT - ::¥ i B Ms : on%@ Figure 173. y workequal totheemfofthebattery, or¢,—€4,isperformed. Equating .7 thetotalenergyabsorbedtotheworkdoneonthebatterygives) . Figure17.4y (UT)V,1,V.Ci/T),andV(I/T), respectively.7 icalwlaf»a,ively.Thed;trti mpdT+TdT+al+AT)—19dTlen—Soap CTRaNall {U/T),respectively.Thedynamicalequations But 1 1 1 1dr -N, =Luz, nazVy i3aVex Ver py ayhT+aD)=ylT)+Sanar (17.49) fa=LingVettLiagVolt+LisVar+LVore a a 1 rolapg t pxawhence és ‘ <M,=LinVat+LiaVo+LesVeg+LuVe7.© Sbtranteam (17.50) y1. v1 11,:2752) Q.=LingVolt+LingVart+LaVernetLao Finally, inserting equation 17.40 forthePeltier coefficient gives the v1 v1 rol uelequivalent ofequation 17.47. Oy=Lar7Vat+LengVo+LaVernetLutes ‘Equation 17.50,whichdepends onenergy conservation aloneandwhich TheO: th thdoesnotrequiretheOnsagertheorem foritsdemonstration, wasgivenby ynsager theoremstatesthatKelvinandiscalledthefirstKelvinrelation. | Ly)=j-H) (17.53),Ifweassume isotropy inthe2y-plane, thesymmetry ofthe2and 17.6TheThermomagnetic Dynamical Equations ,y-axesrequires thatLj,=Ly,etc.,reducing thedynamical equations toal at sg ling! Havingdemonstrated thetechnique ofapplication oftheOnsager Ng=LipVe+LiaVote+LegtLulz theorem insome detail, wecannow treat thethermomagnetic effects . i 1- 1 H relatively concisely. Theseeffectsoccurwhenelectric andheatcurrents Ny=LapMate+Lage +LaVag +LisVyarepermitted toflowinaplaneperpendicular toanappliedmagnetic ' | , Trsa) field,asshowninFigure17.4. erypetlvetEVEL L:Thedynamicalequations17.10and17.11applyspecificallytoone- OQ,=LayVolt+LingVolt+LisVarn+LuV7 dimensional flow.‘Toobtain theiranalogues fortwo-dimensional flow, pl 1 rolgpe tlwwereturntoequation17.9,whichwerewriteexplicitlyinitsCartesian Oy=LeneVat+LanVell—LVeg+LaVeg 304 ‘THERMOELECTRIC ANDTHERMOMAGNETIC EFFECTS ‘THETHERMOMAGNETIC EFFECTS ws inwhich,furthermore, L,,Li,Lix,andLgareidentified asevenfunctions simplified ifwefirstinvertthedynamical equations insuchawaythat ofthemagnetic field, whereas Lis,Lig,Lyq,andLy,areidentified asodd theelectrical current andtemperature gradients, which aretheexperi= functions ofthemagnetic field mentally controlled variables, appear ontheright. From theOnsager relation (equation 17.53) wenowhave Incarrying outtheinversion ofthedynamical equations itisconvenient : , alsotointroduce explicitly theelectric current eV,andtheelectrochemicalLylH)=Ly(HD (17.55) potential perunitcharge(L/e)V_u andtowritetheequations interms-of. ‘However Lj,isanevenfunction ofthefield,sothat V.Tinstead ofV,(I/T). Thevarious resulting factors ofI/e,e,andLH) =Ln) (17.56) —I/T?areabsorbed intothenewkinetic coefficients. Wethusobtain 4 1 These twoequations imply that Vet =Lee +Laxey —Ly3VeT —LuV,T Ly)=LH (17.57) ly LyseN,+LeeN, yy—5Vee=—LyeNg+LyueNy+LuVeT—Ly3V,T Similarly, fromtheOnsagerrelation, e aufas *(17.62) LH)=Lul-H) (17.58) Q_=—ThayeN, ~Tye,—LaVeT—LuViT Butexamination ofequations 17.54showsthatwehavealready identified Q,=ThyeN, —ThyyeN, +LyVeT—Ly,V,T Li,asequal to—L5, bysymmetry considerations, sothat Thenewkinetic coefficients Ly,are,ofcourse, fairly complicated functions a A ftheoldkinetic coefficients Lj,Buttheprecise form ofthese relation-=-Lyf- 17.55 orne ws Pr relation- Li(Hl)=~Lgl) «78 “ships isofnoconsequence tous.Theonlysignificant factisthatthe ‘Again, Ligisanoddfunction ofthemagnetic field, implying finally that symmétry ofthecoefficients inequation 17.62 isimplied bythesymmetry 7=u 17.60) ofthecoefficients inequation 17.61,asthereadermayconvince himselfLH)=La) (17.60) withoutrecoursetodetailedalgebraicinversionoftheequations. ‘TheOnsager relation therefore enables ustocliminate twokinetic . coefficients fromthedynamical equations, which thereby become 17.7TheThermomagnetic Effects Ng=LighYap+LigVy+EgVen +Lays Wenowproceed withthedefinition ofanumber ofeffectsandoftheirT T T T associated descriptive coefficients. i 1 1+1 AM,=LinVell+DinVet—LuVeg+LisVo7 gingVelt+EaVolt~Var F ansy 1.TheAbsoluteThermoelectric Power 1 l rola pg Inthe presenceofamagneticfieldthedefinition oftheabsol ert a lopgl P igneticfieldthedefinitionoftheabsolute Qe=LispVee+LiaVol+LasVeg +Lawn7 thermoelectric powerisconveniently takenas a a roliagl 1 Oy=LagVert+LiszVat—Lae+Leo c=VuulV.T withN,=N,=V,T=0 (17.63) ‘Wearenowprepared toundertake ouranalysis ofthevarious thermo- whence magnetic effects inprinciple. There aresixkinetic coefficients inthe e=Ly (17.64) dynamical equations. Byintroducing sixeffects anddefining sixdescrip- , tive coefficients for thuse effects, we can climinate the kinetic 2.TheIsothermal Electric Conductivity coefficients infavor ofthese sixdescriptive coefficients. Anyfurther effects analyzed willresultinrelations analogous totheKelvin relations. a=—eN,/*Vinwith=-V,T=V,T=N, =0(17.65) However, Mazur andPrigogine haveshown thatthealgebra.is greatly ; e w= Ny 306 ‘THERMOELECTRIC ANDTHERMOMAGNETIC EFFECTS ‘THETHERMOMAGNETIC EFFECTS (307 whence . iabati 1 9.The Acoy=Ly (17.66) diabatic NernstEffect 1=--V, it =N,=0,=0 (17; 3,TheAdiabaticElectricConductivity eS—ZVoulH.VeT withNe=Ny=Q,=0(17.79)1 whenceo,=—eN,/'v.nwith|V;,T=Q,=N,=0 (17.67) Na=Las—LraksalLaa) He (17.80) whence n0,=(Lay—TLighLgs)* (17.68), 10.TheEttingshausen Effect E=V,T/HeN, with N,=0,=¥,T=0 (1781) 4,TheIsothermal HeatConductivity whence. . = —O1V.T with N,=N,=V,T=0 (17.69) EsThylHLay (17.82) whence Ky=Lag (17.70) 11.TheLeduc-Righi Effect 2=V,TH,V.T with N,=N,=Q,=0 (17.83) 5.TheAdiabatic Heat Conductivity whence x=-0,/V.T withN,=N,=Q,=0 (1771) CalaLas *7.84) whence ‘The fundamental setofdynamical equations cannow bewritten inthe Wg=(Es+E50Las 7.72) matrixform 1 : 6.TheIsothermal HallEffect , —Ver ot HR =e -An eN, 1 ‘ - RewVyulHeN, withVFV\T=N,=0 (17:73) _ivel=|aR ofHn Le eN, whence R=LH, (17.74) OQ. —Te-THy — —Hyw& VT Q, THTeAxe Ky vr leliabatic Hall Effect ‘1.TheAdi icHalfect (17.83) R=ty,wlHeN, with V,0=Q,=N,=0 (17.75) Wefurther findtherelations gVelHeeNy j KE=Ty (17.86) whence kyee=HyeLt 7.87) Ry=(Lin+Thaalyalladl He (17.76) aoesHane 4 ts=(Lia+ThyalralLoo genet ming 7s) 8,TheIsothermal Nernst Effect RaRemck (17.89) 1 . UAI=L (17.90)n=—-VwlHV.T with Ng=Ny=V,T=0 (17.77) Eachofthecoefficients isanevenfunction ofthemiagnetic field. Thus whencee theisothermal conductivity canbeexpanded in’powers ofH,togive 1=LlH, (17.78) oy=o+OPHE+OMHE++ (17.91), 308 ‘THERMOELECTRIC ANDTHERMOMAGNETIC EFFECTS APPENDIX A andthecoefficient ofthequadratic term iscalled amagneto-conductivity eros“a“$@: coefficient,: ; Anyexperimental situation canbedirectly analyzed byequation 17.85, . ;relating ittothethermomagnetic coefficients appearing therein. The Some Relations resultant equations, analogous tothethermomagnetic relations 17.86- ° :17.90,represent thefruitsofthetheoryofirreversible processes applied Involving Partial 7 toconductivity phenomena. oeDerivatives | AL Partial Derivatives Inthermodynamics weareinterested incontinuous functions ofthree (ormore) variables. i : v= vy 2) (AD Iftwoindependent variables, sayyand2,areheld constant, ybecomes afunction ofonly oneindependent variable 2,andthederivative ofy with respect to#may bedefined andcomputed inthestandard fashion. Thederivative soobtained iscalled thepartial derivative ofywithrespect to2andisdenoted bythesymbol (2y/22),,, orsimply by@y/@z. The derivative depends upon xandalsoupon thevalues atwhich yand2are held during thedifferentiation; thatisdy/2 isafunction of2,y,andz, ‘The derivatives @y/@y and y/@: aredefined inanidentical manner, The function y/@x, ifcontinuous, may itselfbedifferentiated toyield threederivatives which arecalled thesecond partial derivatives ofp: 2(2)2 Ga\ax) ~oa . : a28)_ae=k a2) .a(22)7 O2\dz) ™32Oz 09 ’ ied OR aaALR, 0SEmme —ireceeteenth iti ttn tinea nma ant 310 APPENDIX A SOME RELATIONS INVOLVING PARTIAL DERIVATIVES au Bypartial differentiation ofthefunctions 3/ay and@y/dz, weobtain whereothersecondpartialderivatives ofy: ay ay. ap dyat = padty tyty te Pe ayBY Ya aytat asGedy’ Oy”ey’ Beds’Dyae’ e ay ay ay a = e eItmaybeshownthatunderthecontinuity conditions whichwehave Pomas(b+ aUN+aa tae dedy assumed forpand itspartial derivatives theorder ofdifferentiation isimmaterial, othat 422%aed22ayde (AD)a a OxOz OyOz - ayoyye egy andgenerally eaydyae’BedsBeds’ Bye”Hedy aaaye . ‘There aretherefore justsixnonequivalent second partial derivatives ofa ay=(@utente 2)vy,2)(A.10) function ofthree independent variables (three forafunction oftwo . variables, and}n(n+1)forafunction ofnvariables). Thesequantities dy,d*y---,d"p-+- arecalledthefirst,second,andnthorder differentials ofy. A2. Taylor's Expansion. A.4 Composite Functions Therelationship between (2,y,2)andy(z+dz,y-+dy,2 +dz),where - 7dx,dy,anddzdenote arbitrary increments inx,y,and2,isgivenby Returning tothefirst-order differential, . ‘Taylor’s expansion: Qt a atay,Op..ap ae=(2)aes(58).av+(2)@(aap yle+de,y+dy,2+de)=le,4.2)+Beta Utes, we Yee Oe)ey . aninteresting cas¢ariseswhen’, y,andzarenotvaried independently but+ye(opeFray+e(a42ssdedy arethemselvesconetobefeteofsomevariableu.Thenaty ay] deme, dyeduanddzxdu de nam tee 4 at 42goedede+255dyde+ (aay venence ai du a Thisexpansion canbewritten inaconvenient symbolic form: y ay) de dh ay) a yle+de,y+dy,2+de)=ERA yx,4,2)(A.5) or va1)«de sad Expansionofthesymboliccrponentisorordngtotheusualseries dy()a,(2)“yy()aaryeatsetdege thee a6 au~\de),.du* \oy),.du*Be),de . 7 - Ifzand functi ft i chen reproduces theTaylor expnsion (quaion A4) anyeeunetion‘two(ormore)variables,sayuandv,thendz=&)dut(%)dv,ete. A3Differentials of ‘Bul a0} an ‘TheTaylorexpansion(equationA.4)canalsobewrittenintheform ip[(2)(2):(@)(2):)()]te ye+de,y+dy,2+de)—yx,y,2) Oz)y,.\Ou),* \Oy)\du) \O2),,,\du «. 1 1 By) (az), (ay) (au) EZmaentnec eee as (8),.8+8).8+ 88) oa nl > 22),\B0).*\ay),,Ad0) ,*Vee),,\@o)J?O19 312 APPENDIX A SOME RELATIONS INVOLVING PARTIAL DERIVATIVES. 313 or Bysuccessively putting dy=0anddz==0inequation A.20, wefindthe 2)(2) twosimilarrelationssl) du =} do (ALL a=(Z)a+(3), ay (2)=ere ans where ;Belog OVIDuy (A23) 3)-2)+(i).+@),,L@), . (3),~(), +GE).G0+Da), @9 a)_Craven \° : 20)pavl an andsimilarly for(3/30)... ve 1y/02)2.yItmayhappenthatwisidentical to«itself. Then . Returning toequation A.20 weagain putdz=0,butwenow divide )_)(®)):(*))an throughbydyratherthanbydz. Ba),\Ge)2°Nay).\22),*\@e)ye), ay)(22 ay)."’ . =a ES+a, (A.25) Otherspecialcasescanbetreatedsimilarly. whence wt\OH)ye\OY/eye ‘citFuneti )—@vlDes ASImplicitFunctions &etOVI),ie (A.26) Ifpisheldconstant, thevariations ofz,y,andzarenotindependent, . and,oncomparison withequation A.21,wefindthe.very reasonable result andtherelation that v(@,y,2)=constant (A.18) ax) 1stfanctii wsrelati (&)=a (A271) givesanimplicit functional relation among2,y,andz.Thisrelation may WW)2(Oy/PX)y,x ‘besolved foronevariable, say2,interms oftheother two. From equations A.22-A.24 wethenfind 22fz,y (A.19) Ox) ay (azim (8) GG) (420 ‘Thisfunction canthenbetreated bythetechniques described above to 1!2\O2)oe ve derive certain relations among thepartial derivatives. However, amore Finally wereturn toourbasicequation, which defines thedifferential direct method ofobtaining theappropriate relations among thepartial dy,andconsider thecaseinwhich 2,y,and2arethemselves functions of derivatives ismerely toputdy=0inequation A.8. | avariable u(asinequation A.12) ~(2), G+ (2) v=(5),+CH) a] Ifwenowputdz=0anddividethrough bydz,wefind Ifwistobeconstant, theremustbearelation among=,y,andz,hence a 4 aalsoamong dz/du, dy[du, anddz/du. Wefind v) ‘a ry) v=(8).8). am =)22)+(a) e Oy]2xNOXye O=tae),Aaa),+au)edi),*Fe)eady(A.30) inwhich thesymbol (@y/4z),,. appropriately indicates thattheimplied . stant ifunctional relationbetweenyandzisthatdetermined bytheconstancy Ifwefurtherrequirethat=shallbeaconstantindependent ofuwefind ofyandz.EquationA.2Icanbewrittenintheconvenient form o=(2)(2)+()2) (aanBx)y,\Ou)yg©\OY/e,2\0U/y,2 ” (2)aOH (a2) or “_— _— BeloOPI ules __OvleDys rer)‘Thisequation playsaveryprominent roleinthermodynamic calculations. . Gzfu),, Ovex " aus APERNDNE A APPENDIX B ‘Comparison with equation A.22 shows that OTT po )aORD (A.33) ; Gz}y,2 (Ox/0u),,4, Statistical Equations A.22, A.27, andA.33areamong themost useful formal wo.manipulations inthermodynamic calculations. Significance : Lo : oftheEntropy Thepurpose ofthisappendix istoprovide thereader withadescriptive, intuitive insight tothestatistical significance oftheentropy. Anapprecia. tionofthequalitative meaning oftheentropy attheatomic levelgives valuable perspective tomacroscopic thermodynamic theory., InChapter1itispointedoutthateachmacroscopic systempossesses anenormous number ofatomic coordinates. ‘Transitions among different atomic states occur extremely rapidly, whereas macroscopic observations arerelatively slow. Macroscopic observations consequently correspond | tostatistical averages over theatomic coordinates. Among themyriads ofatomic coordinates-there exist averyfewwith suchsymmetry andcoherence properties that,unlike thevastmajority, they arenot“averaged out” inamacroscopic measurement. These coordinates aretheenergy, volume, mole numbers, andother macro- scopic thermodynamic extensive parameters. The observed values of these parameters characterize athermodynamic state oramacrostate of asystem. Each such macrostate is,then, consistent with avery large number ofmicrostates, orunderlying atomic states. Asanexample, consider twosimple systems contained within aclosed cylinder andseparated byaninternal piston. The macrostate ofsuch a system iscompletely characterized bytheextensive parameters ofeach of thesimple systems; UY, VD,NQ--- NMandUM,Ve,NO-NOHowever, foranygivensetofvaluesofthese variables there isanenormous number ofways inwhich themolecules ofthesystem canbedistributed at anyinstant. ‘These different positions correspond todifferent microstates, us