Statistical and thermal physics (1)

Developing microscopic statistical physics and macroscopic classical thermodynamic descriptions in tandem, Statistical and Thermal Physics: An Introduction provides insight into basi

Statistical and Thermal Physics

Concepts and relationships in thermal and statistical physics form the foundation for

describing systems consisting of macroscopically large numbers of particles Developing

microscopic statistical physics and macroscopic classical thermodynamic descriptions

in tandem, Statistical and Thermal Physics: An Introduction provides insight into

basic concepts at an advanced undergraduate level Highly detailed and profoundly

thorough, this comprehensive introduction includes exercises within the text as well as

end-of-chapter problems

The first section of the book covers the basics of equilibrium thermodynamics and

introduces the concepts of temperature, internal energy, and entropy using ideal gases

and ideal paramagnets as models The chemical potential is defined and the three

thermodynamic potentials are discussed with use of Legendre transforms The second

section presents a complementary microscopic approach to entropy and temperature,

with the general expression for entropy given in terms of the number of accessible

microstates in the fixed energy, microcanonical ensemble The third section emphasizes

the power of thermodynamics in the description of processes in gases and condensed

matter Phase transitions and critical phenomena are discussed phenomenologically.

In the second half of the text, the fourth section briefly introduces probability theory

and mean values and compares three statistical ensembles With a focus on quantum

statistics, the fifth section reviews the quantum distribution functions Ideal Fermi and

Bose gases are considered in separate chapters, followed by a discussion of the

“Planck” gas for photons and phonons The sixth section deals with ideal classical gases

and explores nonideal gases and spin systems using various approximations The final

section covers special topics, specifically the density matrix, chemical reactions, and

irreversible thermodynamics.

Thermal Physics

A TAY L O R & F R A N C I S B O O K

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Boca Raton London New York

Department of Physics, Florida State University Tallahassee, USA

and

School of Physics, University of the Witwatersrand Johannesburg, South Africa

Boca Raton, FL 33487-2742

© 2011 by Taylor & Francis Group, LLC

Taylor & Francis is an Informa business

No claim to original U.S Government works

Version Date: 20131107

International Standard Book Number-13: 978-1-4398-5054-1 (eBook - PDF)

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Contents

Acknowledgments xxiii PhysicalConstants xxv

Part I Classical Thermal Physics: The Microcanonical

Ensemble

SectIon Ia Introduction to Classical Thermal Physics

Concepts: The First and Second Laws of

SectIon IB Microstates and the Statistical Interpretation

SectIon Ic Applications of Thermodynamics to

Gases and Condensed Matter, Phase

Transitions, and Critical Phenomena

SectIon IIa The Canonical and Grand Canonical

Ensembles and Distributions

10.3 ENSEMBLES IN STATISTICAL PHYSICS 203

10.5 CALCULATION OF THERMODYNAMIC PROPERTIES FOR A SPIN SYSTEM USING THE CANONICAL

SectIon IIB Quantum Distribution Functions, Fermi–Dirac

and Bose–Einstein Statistics, Photons, and

SectIon IIc The Classical Ideal Gas, Maxwell–Boltzmann

Statistics, Nonideal Systems

SectIon IID The Density Matrix, Reactions and Related

Processes, and Introduction to Irreversible

Preface

mentalimportanceinthedescriptionofsystemsthatconsistofmacro-scopicallylargenumbersofparticles.Thisbookprovidesanintroductionto the subject at the advanced undergraduate level for students inter-estedincareersinbasicorappliedphysics.Thesubjectcanbedevelopedin different ways that take either macroscopic classical thermodynam-icsormicroscopicstatisticalphysicsastopicsforinitialdetailedstudy.Considerableinsightintothefundamentalconcepts,inparticulartem-peratureandentropy,canbegainedinacombinedapproachinwhichthemacroscopicandmicroscopicdescriptionsaredevelopedintandem.Thisistheapproachadoptedhere

Thermalandstatisticalphysicsconceptsandrelationshipsareoffunda-Thebookconsistsoftwomajorparts,withineachofwhichthereareseveralsections,asdetailedbelow.Aflowchartthatshowsthechaptersequenceandtheinterconnectionofmajortopicscoveredisgivenattheendofthisintroduction.PartIisdividedintothreesections,eachmadeupofthreechapters.ThebasicsofequilibriumthermodynamicsandthefirstandsecondlawsarecoveredinSectionIA.Thesethreechaptersintroducethereadertotheconceptsoftemperature,internalenergy,andentropy.Twosystems,idealgasesandidealnoninteractinglocalizedspins,areusedextensivelyasmodelsindevelopingthesubject.Useofidealequationsofstateforgasesandforparamagneticsystemsallowsillustrativeapplica-tions of the thermodynamic method. Magnetic systems and magneticworkaredealtwithinsomedetail.TheoperationofaCarnotrefrigeratorwithanidealparamagnetasworkingsubstanceispresentedalongwiththetraditionalidealgascase.ThechemicalpotentialisintroducedfromathermodynamicviewpointinChapter3andisdiscussedinsubsequentchaptersintermsofthemicroscopicstatisticalapproach

Chapters 4, 5, and 6 in Section IB provide a complementary scopic statistical approach to the macroscopic approach of Section IA.Considerableinsightintoboththeentropyandtemperatureconceptsisgained, and the general expression for the entropy is given in terms of

micro-the number of accessible microstates in micro-the fixed energy, calensembleapproach.Thisrelationshipisofcentralimportanceinthedevelopmentofthesubject.Explicitexpressionsfortheentropyofbothamonatomicidealgasandanidealspinsystemareobtained.Theentropyexpressionsleadtoresultsfortheothermacroscopicpropertiesforboththeidealgasandtheidealspinsystem.Itismadeclearthatforidealgasesinthehigh-temperature, low-density limit, quantum effects may be neglected.Theneedtoallowfortheindistinguishablenatureofidenticalparticlesinnonlocalizedsystemsisemphasized.Theexpressionsfortheentropyandthechemicalpotentialofanidealgasaregivenintermsoftheratioofthequantumvolume,whichisintroducedwithuseoftheHeisenberguncer-taintyprinciple,andtheatomicvolumeorvolumeperparticle.Theseformsfortheentropyandchemicalpotentialareeasilyrememberedandprovide

Thefinalsectioninthefirsthalfofthebook,SectionIC,emphasizesthepowerofthermodynamicsinthedescriptionofprocessesforbothgasesinChapter7andcondensedmatterinChapter8.TheMaxwellrelationsareobtainedandusedinanumberofsituationsthatinvolveadiabaticandisothermalprocesses.Chapter9concludesthissectionwithadiscussionofphasetransitionsandcriticalphenomena

ory,meanvalues,andthreestatisticalensemblesthatareusedinstatis-ticalphysics.Thepartitionfunctionisdefinedasasumoverstates,andtheideallocalizedspinsystemisusedtoillustratethecanonicalensembleapproach.Thegrandcanonicalensembleandthegrandsumarediscussed

Chapter10inSectionIIAgivesabriefintroductiontoprobabilitythe-inChapter11 Itisshownthatforsystemsoflargenumbersofparticles,for

whichfluctuationsinenergyandparticlenumberareextremelysmall,the

differentensemblesareequivalent.SectionIIB tumstatistics.Chapter12 reviewsthequantummechanicaldescription

isconcernedwithquan-ofsystemsofidenticalparticlesanddistinguishesfermionsandbosons.Chapters13and14dealwiththeidealFermigasandtheidealBosegas,

respectively.ExpressionsfortheheatcapacityandmagneticsusceptibilityareobtainedfortheFermigas,whereastheBose–EinsteincondensationatlowtemperaturesisdiscussedfortheBosegas.Thesechaptersareillus-tratedwithapplicationstoavarietyofsystems.Forexample,Fermi–Diracstatisticsisusedtotreatwhitedwarfstarsandneutronstars.TheradiationlawsandtheheatcapacityofsolidsarediscussedinChapter15,whichdeals with photons and phonons. The cosmic microwave backgroundradiationisconsideredasanillustrationofthePlanckdistribution.

InSectionIIC,Chapter16 returnstotheidealgastreatedintheclassical

tinguishablenatureofidenticalnonlocalizedparticles.Theinternalenergyofmoleculesisincludedinthepartitionfunctionfortheclassicalgas.Theequipartitionofenergytheoremforclassicalsystemsisdiscussedinsomedetail.NonidealsystemsaredealtwithinChapter17intermsoftheclus-termodelforgasesandthemeanfieldapproximationforspins.TheIsingmodelforinteractingspinsisintroducedandtheone-dimensionalsolutionoftheIsingmodelisgivenforthezeroappliedfieldcase.AnintroductiontoFermiliquidtheoryisfollowedbyadiscussionofthepropertiesofliquidhelium-3atlowtemperatures.Thechapterconcludeswithaphenomeno-logicaltreatmentofBoseliquidsandthepropertiesofliquidhelium-4.SectionIIDdealswithspecialtopicsthatincludethedensitymatrix,chemicalreactions,andanintroductiontoirreversiblethermodynamics.Chapter18introducesthedensitymatrixformulationwithapplicationstospinsystemsandmakesaconnectiontotheclassicalphasespaceapproach.TopicscoveredinChapter19arethelawofmassaction,adsorptiononsur-faces,andcarrierconcentrationsinsemiconductors.Chapter20dealswithirreversibleprocessesinsystemsnotfarfromequilibrium,suchasthermo-osmosisandthermoelectriceffects

limitofthequantumdistributions,whichautomaticallyallowsfortheindis-eredareSectionsIA,IB,IIA,andIIB.Ifstudentshavehadpriorexpo-suretoelementarythermodynamics,muchofSectionIAmaybetreatedasaself-studytopic.Problemsgivenattheendofeachchapterprovideopportunitiesforstudentstotestanddeveloptheirknowledgeofthesub-ject.Dependingonthenatureofthecourseandstudentinterest,materialsfromSectionsIC,IIC,andIIDcanbeadded

Foraone-semestercourse,theimportantsectionsthatshouldbecov-Adiagramthatillustratesthestructureandtheinterrelationshipsofthefirst16chaptersofthebookisgiveninthefollowingfigure

Ch 1 Introduction: Basic

Concepts

Ch 2 Energy: The First Law

Ch 3 Entropy: The Second Law

Ch 5 Entropy and Temperature: Microscopic Statistical Interpretation

Ch 4 Microstates for Large Systems

Ch 6 Zero Kelvin and the Third

Ch 9 Phase Transitions and

Critical Phenomena Ch 10 Ensembles and the

Canonical Distribution

Ch 11 The Grand Canonical Distribution

Ch 12 The Quantum Distribution Functions

Ch 13 Ideal Fermi Gas

Ch 14 Ideal Bose Gas

Ch 15 Photons and Phonons— The ‘‘Planck Gas’’

Statistical and Thermal Physics Topics Covered in Chapters 1 to 16

Thermodynamics Statistical Physics

Ch 16 The Classical Ideal Gas

Acknowledgments

My thanks go to numerous colleagues both in Johannesburg and inTallahasseeforhelpfuldiscussionsontheconceptsdescribedinthisbook.InteachingthematerialIhavelearntagreatdealfromtheinteractions

tionshaveoftenbeenenlightening

I havehadwithmanystudents.Theircommentsandresponsestoques-Finally,Iwishtothankmyfamilyfortheircontinuingsupportduringthisproject.Inparticular,IoweagreatdealtomywifeRenée,whoinadditiontopreparingmostofthefigures,providedthenecessaryencour-agementthathelpedmetocompletethebook

PART I

Classical Thermal Physics: The Microcanonical Ensemble

SECTION IA

Introduction to Classical Thermal Physics Concepts: The First and Second Laws of Thermodynamics

Introduction:

Basic Concepts

1.1 STATISTICAL AND THERMAL PHYSICS

tionofmacroscopicsystemsmadeupoflargenumbersofparticlesofthe

Thesubjectofstatisticalandthermalphysicsisconcernedwiththedescrip-orderofAvogadro’snumberNA=6.02×1023mol–1.Theparticlesmaybeatomsormoleculesingases,liquids,andsolidsorsystemsofsubatomicpar-ticlessuchaselectronsinmetalsandneutronsinneutronstars.Arichvari-etyofphenomenaareexhibitedbymany-particlesystemsofthissort.Theconceptsandrelationshipsthatareestablishedinthermalphysicsprovidethebasisfordiscussionofthepropertiesofthesesystemsandtheprocessesinwhichtheyareinvolved.Applicationscoverawiderangeofsituations,frombasicscience,inmanyimportantfieldsthatincludecondensedmatterphysics,astrophysics,andphysicalchemistrytopracticaldevicesinenergytechnology

Theoriginsofmodernthermalphysicsmaybetracedtotheanalysisofheatenginesinthenineteenthcentury.Followingthisearlyworkanum-berofresearcherscontributedtothedevelopmentofthesubjectofther-modynamicswithitsfamouslaws.Bytheendofthenineteenthcentury,thermodynamics,classicalmechanics,andelectrodynamicsprovidedthefoundationforallofclassicalphysics.Todaythermodynamicsisawell-developedsubject,withmodernresearchfocusedonspecialtopicssuchasnonequilibriumthermodynamics.Applicationofthemethodsofthermo-dynamicstocomplexsystemsfarfromequilibrium,whichincludelivingorganisms,presentsamajorchallenge

5

The microscopic classical statistical description of systems of largenumbersofparticlesbeganitsdevelopmentinthelatenineteenthcen-tury,particularlythroughtheworkofLudwigBoltzmann.Thisapproachwastransformedbythedevelopmentofquantummechanicsinthe1920s,whichthenledtoquantumstatisticsthatisoffundamentalimportanceinagreatdealofmodernresearchonbulkmatter.Statisticaltechniquesareusedtoobtainaveragevaluesforpropertiesexhibitedbymacroscopicsystems.Themicroscopicapproachonthebasisofclassicalorquantummechanics together with statistical results has given rise to the subjectknownasstatisticalmechanicsorstatisticalphysics.Bridgerelationshipsbetween statistical physics and thermodynamics have been establishedandprovideaunifiedsubject.

Underconditionsofhightemperatureandlowdensity,weshallfindthatitdoesnotmatterwhetherclassicalorquantummechanicaldescrip-tionsareusedforasystemofparticles.Athighdensitiesandlowtempera-turesthisisnolongertrue,becauseoftheoverlapoftheparticles’wavefunctionsandquantummechanicsmustbeused,givingrisetoquantumstatistics. Under high-density, low-temperature conditions, the prop-ertiesofasystemdependinacrucialwayonwhethertheparticlesthatmakeupthesystemarefermionsorbosons.Manyfascinatingphenom-enaoccurincondensedmatterasthetemperatureislowered.Examplesareferromagnetism,superconductivity,andsuperfluidity.Theseimpor-tantnewpropertiesappearfairlyabruptlyatphasetransitions.Progressin the microscopic understanding and description of the behavior ofthesesystemsinvolvesquantummechanicsandstatisticalphysicsideas.ApplicationsofquantumstatisticsarenotconfinedtoterrestrialsystemsandincludeastrophysicalphenomenasuchasthemicrowavebackgroundradiationfromtheBigBangandthemass–radiusrelationshipsforwhitedwarfstarsandneutronstars.Animportantconceptinthermalphysicsisthatofentropy,which,asweshallsee,increaseswithtimeassystemsbecomemoredisordered.Theincreaseoftheentropyoftheuniversewiththepassageoftimeprovideswhatistermedtime’sarrow.Aninterestingandunansweredquestionarisesastowhytheentropyoftheuniversewassolowatthebeginningoftime

Thereareanumberofwaysinwhichthesubjectofthermalphysicsmaybeapproached.Inthisbook,theelementsofclassicalthermody-namicsarepresentedinthefirstthreechaptersthatcompriseSectionIA.ThemicroscopicstatisticalapproachisintroducedinthethreechaptersofSectionIB,whichcomplementthethermodynamicsinSection IAand

provideadditionalinsightintofundamentalconcepts,specificallyentropyand temperature. Both classical and quantum mechanical descriptionsareintroducedindiscussingthemicrostatesoflargesystems,withthequantumstatedescriptionpreferredinthedevelopmentofthesubjectthatispresentedinthisbook.Thisapproachinvolvestheapplicationofthemicrocanonicalensemblemethodstotwomodelsystems,theidealgassystemandtheidealspinsystemforbothofwhichthequantumstatesare readily specified with use of elementary quantum mechanics. Thelaws of thermodynamics are stated in compact form, and their signifi-canceisheightenedbytheinterweavingofmacroscopicandmicroscopicapproaches. Expressions for the entropy and chemical potential of the

idealgasareexpressedintermsoftheratioofthequantumvolumeVQto

theatomicvolumeVA.VQistakentobethecubeofthethermaldeBroglie

wavelength,whereasVAisthemeanvolumeperparticleinthegas.Inthefinalthreechaptersinthefirsthalfofthebook(SectionIC),thethermo-dynamicapproachisappliedtothedescriptionofthepropertiesofgases,andcondensedmatter

ticalphysicsresultsandthecanonicalandgrandcanonicaldistributionspresentedinSectionIIA.FollowingtheintroductionofquantumstatisticsideasinSectionIIB,theFermi–DiracandBose–Einsteinquantumdistri-butionfunctionsarederived.ThesefunctionsareusedinthediscussionofthepropertiesofidealFermiandBosegases.Photonandphononsys-temsarethentreatedintermsofthePlanckdistribution.InSectionIIC,canonicalensembleresultsareapplied,first,totheidealgasintheclassicallimitofthequantumdistributionsandthentothenonidealgasesandspinsystems.Thefinalsection(SectionIID)ofthebookcontainsmoreadvancedtopicsandincludesanintroductiontothedensitymatrixandnonequilibriumthermodynamics

Inthesecondhalfofthebook,emphasisisplaced,initially,onstatis-Thematerialinthisbookisconcernedprimarilywiththedescriptionofsystemsatorclosetoequilibrium.Thisimpliesthat,apartfromfluc-tuations,whichformacroscopicsystemsaregenerallysmall,theproper-tiesdonotchangewithtimeorchangeonlyslightlywithsmallchangesin external conditions. Processes will often be considered to consistofalargenumberofinfinitesimalchanges,withthesystemofinterestalways close to equilibrium. As mentioned above, two special systemsareusedinthedevelopmentofthesubject:theidealgassystemandthe

idealspinsystem.Idealinthiscontextimpliesthatinteractionsbetween

theparticlesarenegligiblysmallandmaybeignored.Thesetwosystems

umeV.Becauseoftheirthermalmotion,theparticlesexertapressure,or forceperunitareaP,onthewallsofthecontainer.Theidealspinsystem

locatedinamagneticfieldB.

In the development of the subject, use is made of concepts such asvolume, pressure, work, and energy, which are familiar from classicalmechanics.Afurtherconceptoffundamentalimportanceisthatoftem-perature,andthisisdiscussedinthenextsection.Usefulrelationsthatinvolvetemperature,suchasequationsofstateandtheequipartitionofenergytheorem,aregiveninlatersectionsinthischapter.SIunitsareusedthroughoutthebookunlessotherwiseindicated

1.2 TEMPERATURE

Theconceptoftemperaturehasevolvedfromman’sexperienceofhotandcoldconditionswithtemperaturescalesdevisedonthebasisofchangesin the physical properties of substances that depend on temperature.Practicalexamplesofthermometersfortemperaturemeasurementincludethefollowing:

• Constant volume gas thermometers, which make use of the sureofafixedquantityofgasmaintainedatconstantvolumeasanindicator

pres-• Liquidinglassthermometers,whichusethevolumeofaliquid,suchasmercuryoralcohol,containedinareservoirattachedtoacapil-larytubewithacalibratedscale

• Electricalresistancethermometers,whichusethevariationoftheresistanceofametal,suchasplatinum,orofadopedsemiconductor,suchasGaAs,toobtaintemperature

• Vaporpressureandparamagnetthermometersforspecialpurposesparticularlyatlowtemperatures

Mostofthethermometerslistedabovearesecondarythermometersthatarecalibratedagainstagreedstandards.Theconstantvolumegastherm-ometerhasamorefundamentalsignificanceasexplainedinSection 1.3.

For everyday purposes, various empirical scales have been established.CommonlyusedscalesaretheFahrenheitandCelsiusscales.Forreasonsthatbecomeclearbelow,weconsidertheCelsiusscale,whichchoosestwofixedreferencetemperatures.Thelowerreferencepointisat0°C,whichcorrespondstothetriplepointofwater,thepointatwhichwater,ice,andwatervaporcoexist,andthehigherreferencepointat100°C,whichcor-respondstothesteampoint,wherewaterandsteamcoexistatapressureof1atm.DegreesCelsiusareobtainedbydividingtherangebetweenthetriplepointandthesteampointinto100°.Figure1.1showsaschematicdrawingofatriplepointcell.

Thermodynamicsshowsthatitispossibletoestablishanabsolute peraturescalecalledthekelvin scaleinhonorofLordKelvin,whointro-

tem-duceditandfirstappreciateditssignificance.Theabsolutescaledoesnotdependonthepropertiesofaparticularsubstance.Absolutezeroonthekelvinscale,designated0K,correspondsto–273.16°C.Forconvenience,

1 Kischosentocorrespondto1°C.ThisgivesT(K)=t(°C)+273.16.We

talimportancebyconsideringtheequationofstateforanidealgas

cangaininsightintowhytheabsolutezerooftemperatureisoffundamen-1.3 IDEAL GAS EQUATION OF STATE

An equation of state establishes a relationship among thermodynamic

umeV,andtheabsolutetemperatureT.Experimentscarriedoutonreal

variables.Foranidealgas,thevariableschosenarethepressureP,thevol-gases,suchashelium,underconditionsoflowdensityhaveshownthatthefollowingequationdescribesthebehaviorofmanygases:

Ice Water

Vapor Thermometer

FIGURE 1.1 Schematicdepictionofatriplepointcellinwhichwater,ice,andwatervaporcoexist.Thecellisusedtofix0°ContheCelsiusscale

wherenisthenumberofmolesofgasandRisaconstantcalledthegas

constantwithvalue8.314Jmol–1K–1.Asmentionedabove,theconstantvolumegasthermometerinvolvesthemeasurementofthepressureofaconstantvolumeofgasasafunctionoftemperature.Figure1.2givesa

sketchofaconstantvolumegasthermometerwitharepresentativeP susT plotshowninFigure1.3.

ver-Heat bath Gas Pressure gauge

FIGURE 1.2 Sketchofaconstantvolumegasthermometerinwhichthepressureofafixedvolumeofgas,heldatvariousfixedtemperaturebyuseofheatbaths,ismeasuredonapressuregauge

50 0 0 5 10 15 20 25

FromEquation1.1,itfollowsthatthetemperatureT=0Kcorresponds toapressureP=0Pa.Zeropressureimpliesthattheparticlesofthegas

havezerokineticenergyat0Kanddonotexchangemomentumwiththewallsofthecontainer.Wecanthereforeviewtheabsolutezerooftempera-tureasthetemperatureatwhichtheenergyofparticlesinthesystemiseffectivelyzero.Thisisoffundamentalsignificance

ifyattemperaturesmuchhigherthan0K.Thisisbecauserealgaseshaveinteractionsbetweenparticles,whichleadtodeparturesfromidealgasbehavior.Extrapolationfromthehigh-temperature,low-densityregion,wheregasesobeytheidealgasequationofstate,showswhatwouldhap-penatmuchlowertemperaturesifthegasweretoremainideal.Theidealgas equation of state, expressed in Equation 1.1, is extremely useful in

Asthetemperatureislowered,gasesnormallyliquefy,andmostsolid-consideringprocessesinwhichgasesareinvolved.P,V,andTarecalled

state variables,andbecausetheyarerelatedbytheequationofstate,the

specificationofanytwoofthevariablesimmediatelyfixesthevalueofthethirdvariable.Examplesofapplicationsoftheidealgaslaw,asitisalsocalled,aregiveninlaterchapters.Theidealgasequationprovidesafairlygooddescriptionofthebehaviorofmanygasesoverarangeofconditions.Underconditionsofhighdensity,however,thedescriptionmaynotbeadequate,andempiricalequationsofstatethatworkbetterundertheseconditionshavebeendeveloped.Twooftheseequationsarebrieflydis-cussedinthenextsection

1.4 EQUATIONS OF STATE FOR REAL GASES

An important empirical equation of state that provides a fairly gooddescriptionofthepropertiesofrealgasesathighdensitiesisthevanderWaalsequation,

surecorrectiontermallowsforinterparticleinteractions,andthevolumecorrection term allows for the finite volume occupied by the particlesthemselves

1.5 EQUATION OF STATE FOR A PARAMAGNET

AnidealparamagnetconsistsofNparticles,eachofwhichpossesses a

spin and an associated magnetic momentμ proportional to the spin,

with negligible interactions between spins. Real paramagnetic systemsapproximate ideal systems only under certain conditions, such as hightemperature,andinmagneticfieldsthatarenottoolarge.Amoredetaileddiscussionoftheseconditionsisgivenlaterinthisbook

usefulincalculationsrelatedtoprocessesthatinvolvechangesinthestate

variables.NotethatforT→0K,Equation1.4predictsthatMwilldiverge.

Thisunphysicalpredictionshowsthattheequationbreaksdownatlowtemperatures,wherethemagnetizationsaturatesafteritreachesamaxi-

mumvaluewithallspinsalignedparalleltoH.Inmanymagneticsystems,

thespinsinteracttosomeextentandorderbelowatemperaturecalledthe

1.6 KINETIC THEORY OF GASES AND THE

EQUIPARTITION OF ENERGY THEOREM

InChapter2,wedealwithworkandenergyforthermodynamicsystems,andthiswillleadtothefirstlawofthermodynamics.Itishelpfultohaveexpressionsforthetotalenergyofasystemintermsofthermodynamicvariables. For ideal gases, kinetic theory provides an important resultknownastheequipartitionofenergytheorem,whichwenowconsideralongwithotherkinetictheoryresults.Thekinetictheoryofgases,whichmakesuseofclassicalmechanics,isrelatedtocertaintopicsinstatisticalphysicsbutislessgeneralinscopeandapproach

Although thermodynamics concerns itself with the macroscopicdescriptionofsystemsoflargenumbersofparticles,kinetictheorycon-sidersthemicroscopicnatureoffluidsystems.Inparticular,forourpur-poses,kinetictheoryprovidesaclassicalmicroscopicdescriptionofthepropertiesofidealgasesintermsofthekineticenergyoftheparticlesinthesystem.Particlesinagaseoussystemareinconstantmotionandundergocollisionswitheachotherandwiththewallsofthecontainer.Foragasataparticulardensityandtemperature,thecollisionprocessesmay

collisions.Asweshallseelaterinthissection,

τistypicallyveryshortcom-paredwiththetimescaleonwhichmeasurementsofanypropertiesofthesystemaremade.Therapidexchangeofenergyandmomentumthattheparticlesundergomakesitmeaningfultoconsideraveragepropertiesoftheparticles,suchasthemeanenergy〈 ε 〉 definedbelow.IfthereareN

particlesinthegas,thetotalenergyissimplyE=N〈 ε 〉.