General and Statistical Thermodynamics PDF
Preview General and Statistical Thermodynamics
Graduate Texts in Physics Raza Tahir-Kheli General and Statistical Thermodynamics Second Edition Graduate Texts in Physics SeriesEditors KurtH.Becker,NYUPolytechnicSchoolofEngineering,Brooklyn,NY,USA Jean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet, UniversitéParisDiderot,Paris,France SadriHassani,DepartmentofPhysics,IllinoisStateUniversity,Normal,IL,USA MortenHjorth-Jensen,DepartmentofPhysics,Blindern,UniversityofOslo,Oslo, Norway BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan RichardNeeds,CavendishLaboratory,UniversityofCambridge,Cambridge,UK William T. Rhodes, Department of Computer and Electrical Engineering and ComputerScience,FloridaAtlanticUniversity,BocaRaton,FL,USA SusanScott,AustralianNationalUniversity,Acton,Australia H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University,Boston,MA,USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching,Germany AndreasWipf,InstituteofTheoreticalPhysics,Friedrich-Schiller-UniversityJena, Jena,Germany Graduate Texts in Physics publishes core learning/teaching material for graduate- andadvanced-levelundergraduatecoursesontopicsofcurrentandemergingfields withinphysics,bothpureandapplied.ThesetextbooksservestudentsattheMS-or PhD-levelandtheirinstructorsascomprehensivesourcesofprinciples,definitions, derivations,experimentsandapplications(asrelevant)fortheirmasteryandteach- ing,respectively.Internationalinscopeandrelevance,thetextbookscorrespondto course syllabi sufficiently to serve as required reading. Their didactic style, com- prehensiveness and coverage of fundamental material also make them suitable as introductionsorreferencesforscientistsentering,orrequiringtimelyknowledgeof, aresearchfield. Moreinformationaboutthisseriesat http://www.springer.com/series/8431 Raza Tahir-Kheli General and Statistical Thermodynamics Second Edition RazaTahir-Kheli PhysicsDepartment TempleUniversity Philadelphia,PA,USA ISSN1868-4513 ISSN1868-4521(electronic) GraduateTextsinPhysics ISBN978-3-030-20699-4 ISBN978-3-030-20700-7(eBook) https://doi.org/10.1007/978-3-030-20700-7 ©SpringerNatureSwitzerlandAG2012,2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicatedtomychildren ShehraJ. BoldtandKazimR.Tahir-Kheli and(inmemoriam)tomyteacher MartinJ.Aitken,F.R.S. Preface to the Second Edition Except for a few minor corrections, Chaps. 1–11 are identical to those in the first edition.ThesecondeditionaddsChaps.12–16. Chapter12offersabriefintroductiontoLandau’stheoryofsecondorderphase transitions.Cooperpairs,whichplayavitalroleinsuperconductivity,arediscussed inChap.13.Bogolyubov’simportantworkisdescribedinChap.14.Londonbroth- ers’ theory, which is analyzed in Chap. 15, was the first to provide a viable theo- reticalexplanationfortheMeissnereffectinthesuperconductingstate.Chapter16 presentsatheoreticaldescriptionofsuperconductivity.Severaltopicsmentionedin thetextarediscussedintheAppendix. I am greatly indebted to miss Adelheid Duhm and Dr. Zachary Evenson of Springer Verlag for their superlative support in putting together this book. Special thanksareduetoVygintasVilimasformanyinsightfulsuggestions.Gratitudeisalso owedtoTempleUniversityDepartmentofPhysicsforhelpandencouragement. Philadelphia,Pennsylvania RazaTahir-Kheli March3,2020 vii Preface to the First Edition Thisbookisintendedforusebothasatextbookandasasourceforself-study. Asatextbook:MyexperienceteachingcoursesrelatedtoThermodynamicsand StatisticalMechanicsatTempleUniversityhasguideditswriting.TheThermody- namicscourseisofferedtoundergraduatesintheirjuniororsenioryear.Theseun- dergraduateshaveeitherasingleoradoublemajorinPhysics,Biology,Chemistry, Engineering,EarthSciences,orMathematics.ThecourseonStatisticalMechanics isattendedbygraduatestudentsintheirsecondyear.Feelingthatarobuststudyof statisticalthermodynamicsappropriatelybelongsinaseniorlevelgraduatecourse, onlyrelativelysimpleaspectsofStatisticalMechanicsareincludedhere. Asasourceforself-study:Yearsofteachinghavetaughtmeathingortwoabout what works for students and what does not. In particular, I have learned that the more attention a student pays to taking notes, the less he understands of the sub- ject matterof the lecture beingdelivered.Further,I have noticedthat when,a few daysinadvanceofthedeliveryofthelecture,astudentisprovidedwithdetailsof the algebra to be used, solutions to the problems to be discussed, and some brief information about the physics that is central to the lecture to be given, it obviates much of the need for note-taking during the delivery of the lecture. Another im- portant experience that has guided the writing of this textbook is the pedagogical benefitthataccruesfromanoccasional,quick,recapitulationoftherelevantresults that have already been presented in an earlier lecture. All this usually results in a bettercomprehensionofthesubjectmatter.Bothforpurposesofelucidationofthe conceptsintroducedinthetextandforprovidingpracticalproblemsolvingsupport, alargenumberofsolvedproblemshavebeenincluded.Manyofthesolutionspro- vided include greater detail than would be necessary for presentation in a lecture itself or needed by teachers or more advanced practitioners. They are there in the givenformtoofferencouragementandsupportbothforself-studyandindeedalso toallowforfullerunderstandingofthesubjectmatter.Therefore,itisasimportant toreadthroughandunderstandthesesolutionsasitistolearntherestofthetext. Part of the vocabulary of thermodynamics are such terms as a thermodynamic system, an adiabatic enclosure, an adiabatically isolated and adiabatically en- closed system, conducting and diathermal walls, isobaric, isochoric, isothermal, ix x PrefacetotheFirstEdition quasistatic, nonquasistatic, reversible, and irreversible processes. Also, there are the concepts of thermal and thermodynamic equilibria. These terms and concepts are commonly used in thermodynamic analyses. The study of thermodynamics is greatlyassistedbystandardmathematicaltechniques.Inparticular,exactdifferen- tials,andsomewell-knownidentitiesofdifferentialcalculus,arecentraltothethe- oreticaldescriptionofthesubject.Temperatureisnotonlyawordinnormaldaily use,italsositsatthecoreofthermodynamics.Thezerothlawbringstoforeoneof themanyreasonsforitsrelevancetothephysicsofthermalequilibria.Theforegoing arediscussedinChap.1. Arguably, the emphasis on constructing simple models with the hope that they may pertain to real—and necessarily more complicated—systems of interest is unique to the discipline of Physics. Usually, these models are uncomplicated and lacking the complexity of the real systems. But because they can often be exactly solved and understood, the hope always is that the predicted results for the model systemswillgiveinsightintothephysicsofrealsystems. A“perfectgas”representsanidealizedmodelsystem.Themodelwasoriginally motivatedbyexperimentalobservation,andconstructedtounderstandthebehavior of real gases. It has had a long and illustrious history and a great deal of success. WedescribeanddiscussitsimplicationsinChap.2. Caloric was thought to be a massless fluid whose increase warmed an object. Just like a fluid, it was thought to flow from a hot object—where there was more ofit—toacoldoneuntiltheamountsofcaloricinthetwoobjectsequalized.Also, like fluids, caloric could neither be created nor destroyed. Therefore, the reason a cannonmuzzlegothotwhenitwasbeingboredisthatthechipsthatwerecreated carried less caloric so more was left behindfor the muzzle.While supervising the boringofacannon,theCountRumfordofBavarianoticedthatthedullertheboring bits the hotter the muzzles. In other words, less chips, but more caloric! Rumford came to the obvious conclusion: Heat energy is not exchanged by the transfer of caloricbutratherbytheexpenditureofworkthathastobedoneforthedrillingof thecannon.Thusworkwasempiricallyobservedtoberelatedtoheatenergy. Conservation of energy is a concept as old as Aristotle.1 The first law of ther- modynamicsrecognizesthisconcept,aswellasCountRumford’sobservations,that heat energyand work are related. If some work is performed uponaddinga given amount of heat energy to a system, the portion of heat energy that is left over is calledthechangeintheinternalenergyofthesystem.Andwhileboththeamounts of heat energy added to a system and the work done by it depend on the details of how they were carried out2 the first law asserts that their difference, which is thechangeintheinternalenergy,iscompletelypathindependent.Theseissuesare discussedindetailinChap.3andsolutionstomanyhelpfulproblemsareprovided. Clearly,thehallmarkofthefirstlawofthermodynamicsisitsrecognitionofthe pathindependentfunction,the“internalenergy.”Arguably,thesecondlawhaseven moretoboastabout:namely,theidentificationofthestatefunction,the“entropy,” 1Approximately350BC. 2Meaningtheybothdependonthephysicalpathsthatweretakeninexecutingthetwopro- cesses. PrefacetotheFirstEdition xi anditsfamilialrelationshipwithCarnot’sideasaboutmaximumpossibleefficiency ofheatengines.Theseideasandtheirvariousaftereffectsarediscussedandtreated inChap.4. InChap.5,wemarrythefirstandsecondlawsofthermodynamicsinamanner thatachieves“aperfectunion”ofthetwo:aunionthatprovidesgreatinsightsinto theworkingsofthermodynamics. Many hold to the view that Johannes Diderik Van der Waals’ derivation of a possibleequationofstateforimperfectgaseswasthefirsteversignificantcontribu- tiontopredictivestatisticalthermodynamic.Henotedthatanyinteractionbetween microscopicconstituentsof abodymusthavetwodistinctfeatures.Becausethese constituentscongregatetoformmacroscopicentities,theoverallinterparticleinter- action must be attractive. Yet, because matter does condense to finite densities, at smallenoughdistancestheinteractionmustbecomestrongandrepulsive,meaning itmusthaveahardcore. He assumed both these interactions to be “short ranged.” It turns out, however, thathisequationofstateissomewhatmoremeaningfulforagasthathasbothahard core—much as he assumed—but, unlike his assumption, has attractive interaction whichislongranged.Theseideas,aswellasthemanyphysicalconsequencesofthe VanderWaalsequationofstate,areextensivelystudiedandanalyzedinChap.6. State variables such as the volume, pressure, and temperature are easy to mea- sure.Incontrast,thermodynamicstatefunctionssuchastheinternalenergyandthe enthalpy cannot be measured by straightforward procedures. Generally, therefore, oneneedstofollowsomewhatcircuitousroutesfortheirmeasurement. ThefamousGay–Lussac–Jouleexperimentattemptedtomeasuretheinternalen- ergy. The experiment was based on determining the amount of heat energy that is produced by free-expansion of a gas enclosed in vessels submerged in water. Un- fortunately,theheatcapacitiesofthewaterthatneededtobeusedplusthatofthe enclosing core were vastly greater than that of the gas being used. As a result no reliablemeasurementcouldbemade. Joule and Kelvin devised an experiment that overcame this difficulty. The ex- periment shifted the focus from the internal energy to the enthalpy. Additionally, ratherthandealingwithafixedamountofgasthatisstationary,itusedaprocedure thatinvolved“steady-flow.”Asaresult,reliablemeasurementscouldbemade.The relatedissuesandresultsarepresentedinChap.7. Chapter 8 deals with the Euler equation—namely the complete fundamental equation—anditsdependenceonthechemicalpotential,theGibbs–Duhemrelation inboththeenergyandtheentropyrepresentations,andthethreepossibleequations ofstatefortheidealgas,againinboththeentropyandenergyrepresentations. SeveralimportantissuesarebroachedinChap.9.Forinstance,werecallthatall spontaneous processes in isolated thermodynamic systems increase their total en- tropyandthisfactfollowsfromthesecondlaw.Ofcourse,spontaneousprocesses continue until the system ceases to change, that is, until it achieves thermal equi- librium. Accordingly, subject to the constraints under which the system has been maintained,andforthegivenvalueofitsvariousextensiveproperties,itstotalen- tropyinthermodynamicequilibriumisthemaximumpossible.Withthisknowledge
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