Table Of ContentGraduate 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-
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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
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Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
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Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
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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
