Statistical Thermodynamics and Microscale Thermophysics Manyof theexcitingnewdevelopments inmicroscale engineeringarebased on the application of traditionalprinciples of statistical thermodynamics.This book offersamodemviewof thermodynamics,interweavingclassicalandstatistical thermodynamicprinciplesandapplyingthem to current engineeringsystems.It begins withcoverage of microscale energystoragemechanisms froma quantum mechanicsperspectiveandthendevelopsthefundamentalelements of classical and statistical thelmodynamics.Next,applications of equilibrium statistical ther-modynamics tosolid, liquid, and gas phase systems are discussed.The remainder of the book is devoted to nonequilibrium thermodynamics of transport phenomena andanintroduction tononequilibrium effects andnoncontinuum behavior at the micro scale. Althoughthetextemphasizesmathematicaldevelopment,itincludesmany examples and exercises that illustrate howthetheoretical concepts areapplied to systems of scientificandengineering interest.It offers a freshviewof statistical thermodynamicsforadvancedundergraduateandgraduatestudents,aswellas practitioners, in mechanical, chemical, and materials engineering. VanP.CareyisaProfessorintheMechanicalEngineelingDepartmentatthe University of California, Berkeley.The main focus of his research is development of advanced computationalmodelsof micro scalethermophysics andtransport in multi phase systems. Statistical Thermodynamics and MicroscaleThermophysics VANP.CAREY University of Califomia.Berkeley ..... .....CAMBRIDGE UNIVERSITY PRESS PUBLISHEDBYTHEPRESSSYNDICATEOFTHEUNIVERSITYOFCAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGEUNIVERSITYPRESS The Edinburgh Building, Cambridge CB2 2RU, UKhttp://www.cup.cam.ac.uk 40 West 20th Street, New York,NY10011-4211, USAhttp://www.cup.org 10 Stamford Road, Oakleigh, Melboume 3166, Australia Cambridge University Press1999 This book isin copyright.Subject to statutory exception and tothe provisions of relevant collective licensing agreements, no reproduction of any part maytakeplace without the written permission of Cambridge University Press. First published1999 Typset inTimes Roman10/12 pt.in ~ T E X 2 E[TBl A catalog record for this book is amilable from the British Library. Library of Congress Cataloging-ill-Publicatioll Data Carey,V.P.(VanP) Statistical thermodynamics and microscalethermophysics / VanP Carey. p.cm. ISBN 0-521-65277-4 (hb).- ISBN 0-521-65420-3(pb) 1.Statistical thermodynamics.2.Thermodynamics.I. Title. QC311.5.C361999 621.402/1- dc21 ISBN 0 52165277 4hardback ISBN 0 521654203 paperback Transferred to digital printing 2004 98-45449 CIP ToLee F.Carey,Bart Conta,and Dennis G.Shepherd, three of the best applied thermodynamicists I ever met Contents Nomenclature Preface 1Quantum Mechanics and Energy Storage in Particles 1.1Microscale Energy Storage 1.2A Review of Classical Mechanics 1.3Quantum Analysis Using the Schrodinger Equation 1.4Model Solutions of the Time-Independent Schrodinger Equation 1.5A Quantum Mechanics Model of theHydrogen Atom 1.6The Uncertainty Principle 1.7Quantum Energy Levels and Degeneracy 1.8Other Important Results of Quantum Theory 2Statistical Treatment of Multiparticle Systems 2.1Microstates and Macrostates 2.2The Microcanonical Ensemble and Boltzmann Statistics 2.3Entropy and Temperature 2.4The Role of Distinguishability 2.5More on Entropy and Equilibrium 2.6Maxwell Statistics and Thermodynamic Properties for a Monatomic Gas 3A Macroscopic Framework 3.1Necessary Conditions for Thermodynamic Equilibrium 3.2The Fundamental Equation and Equations of State 3.3The Euler Equation and theGibbs-Duhem Equation 3.4Legendre Transforms and Thermodynamic Functions 3.5Quasistatic and Reversible Processes 3.6Alternate Forms of theExtremum Principle 3.7Maxwell Relations 3.8Other Properties 4Other Ensemble Formulations 4.1Microstates and Energy Levels 4.2The Canonical Ensemble 4.3The Grand Canonical Ensemble 4.4Fluctuations 4.5Distinguishability and Evaluation of thePartition Function VB x xv 1 5 8 11 17 20 23 25 29 29 30 37 43 50 61 71 71 75 80 83 90 93 95 99 107 107 108 117 127 137 viiiContents 5Ideal Gases145 145 145 149 160 164 167 170 5.1Energy Storage and the Molecular Partition Function 5.2Ideal Monatomic Gases 5.3Ideal Diatomic Gases 5.4Polyatomic Gases 5.5Equipartition of Energy 5.6Ideal Gas Mixtures 5.7Chemical Equilibrium inGas Mixtures 6Dense Gases, Liquids, and Quantum Fluids 6.1Behavior of Gases in the Classical Limit 6.2Van der Waals Models of Dense Gases and Liquids 6.3Other Models of Dense Gases and Liquids 6.4Analysis of Fluids with Significant Quantum Effects 6.5Fermion andBoson Gases 7Solid Crystals 7.1Monatomic Crystals 7.2Einstein's Model 7.3Lattice Vibrations in Crystalline Solids 7.4The Debye Model 7.5Electron Gas Theory for Metals 7.6Entropy and the Third Law 179 179 182 196 205 212 225 225 228 229 233 238 241 8Phase Transitions and Phase Equilibrium245 8.1Fluctuations and Phase Stability245 8.2Phase Transitions and Saturation Conditions264 8.3Phase Equilibria inBinary Mixtures271 8,4Thermodynamic Similitude and the Principle of Corresponding States284 9Nonequilibrium Thermodynamics297 9.1Properties in Nonequilibrium Systems297 9.2Entropy Production, Affinities,and Fluxes298 9.3Analysis of Linear Systems301 9.4Fluctuations and Correlation Moments304 9.5Onsager Reciprocity of Kinetic Coefficients309 9.6Thermoelectric Effects311 10Nonequilibrium and Noncontinuum Elements of Microscale Systems325 10.1Basic Kinetic Theory325 10.2The Boltzmann Transport Equation337 10.3Thermodynamics of Interfaces346 10.4Molecular Transport at Interfaces351 10.5Phase Equilibria in Microscale Multiphase Systems357 ContentsIX 10,6Microscale Aspects of Electron Transport in Conducting Solids375 10.7The Breakdown of Classical and Continuum Theories at Small Length and Time Scales384 AppendixISome Mathematical Fundamentals391 AppendixIIPhysical Constants and Prefix Designations397 AppendixIIIThermodynamics Properties of Selected Materials399 AppendixIVTypical Force Constants for the Lennard-Jones 6-12 Potential409 Index411 Nomenclature c Cp Cv C ee E E, f f(E:) f(w, z,t) fep fes fpv f>c F g g(r) g(v) g*(E:) ge g gi G h 17 hi flv 11 H fI I i im x normalized bulk velocity,= wo/(2NAkB/M)I/2 Redlich-Kwong constant van der Waals constant interface area Redlich-Kwong constant van der Waals constant second virial coefficient particle speed molar specific heat at constant pressure molar specific heat at constant volume third virial coefficient electron charge energy electric fieldin the xdirection fugacity average occupancy of a microstate with energy E: fractionalparticle velocity distribution distribution function for an ensemble of particles ensemble number density particle velocity distribution fraction of molecules with speeds greater than c Helmholtz free energy degeneracy radial distribution function frequency distribution function distribution function for electron quantum states gravitational acceleration molar Gibbs function degeneracy of energy level i Gibbs function Planck's constant molar specific enthalpy molar specific enthalpy of saturated liquid molar specific enthalpy of saturated vapor = h/2rr enthalpy Hamiltonian moment of ineltia number fluxof molecules mass fluxof molecules Nomenclature la,x ls,x lv.x Ja Js Jv kB kt k< K Kc Kp L Lij Lms LVT Lz III M M I1j (11[') N Na N",i NA Na N ~ P Pi Px P P Pj P Pa Pc Pr Psat p q qa qe qi qint qnuci qrot flux of species ain thexdirection fluxof entropy in the xdirection flux of energy in the xdirection flux of species a flux of entropy flux of energy Boltzmann constant thermal conductivity forceconstant for linear restoringforce wavenumber, =2rr /A equilibrium constant equilibrium constant characteristic system dimension kinetic coefficients mean spacing between particles temperature gradient length scale, =T /VT Lorentz number,=kt!aeT mass per particle mass per molecule for polyatomic molecules molecular mass number of ensemble members in energy level E: j mean number of particles in microstate [' number of particles number of species aparticles number of species aparticles in energy leveli Avogadro's number number of species aparticles per unit volume instantaneous number of species aparticles momentum generalized momentum momentum in the xdirection = Pr - I fluctuation probability density function generalized momentum pressure partial pressure for species a critical pressure reduced pressure,=P / Pc equilibrium saturation pressure probability molecular partition function molecular partition function for particle species a partition function for electronic energy storage generalized coordinate partition function for internal energy storage partition function for nuclear energy storage partition function for rotational energy storage xi xii qrot. nuc1 qtr qvib qj Q re R RH S Sf Sg S S S' S'E' e t T Tc Tm Tm Tr Tsat u zl fif fig V [; V' (; V'E' e V V Vg D Dc Dr V V W Wo W X Xi y Z Z Nomenclature partition function for rotational and nuclear energy storage partition function for translational energy storage partition function for vibrational energy storage generalized coordinate canonical partition function equilibrium bubble or droplet radius universal gas constant,= NAkB Rydberg constant molar entropy molar internal energy for saturated liquid molar entropy for saturated vapor system entropy entropy per unit volume instantaneous entropy surface excess entropy = Tr- I temperature critical temperature equilibrium melting temperature temperature in phase III reduced temperature = T I Tc equilibrium saturation temperature velocity in the xdirection molar specific internal energy molar specific internal energy for saturated liquid molar specific internal energy for saturated vapor system internal energy internal energy per unit volume instantaneous internal energy potential function in quantum system surface excess internal energy velocity in theydirection mass specific volume group velocity molar specific volume critical molar specific volume reduced specific volume, =DI Dc = Dr- I system volume velocity in the z direction bulk velocity normal to interface number of microstates corresponding to a particular macrostate Cartesian spatial coordinate mole fraction of species i Cartesian spatial coordinate Cartesian spatial coordinate compressibility factor,=PDI RT Nomenclature Greek partition function for the microcanonical ensemble partition function for the microcanonical ensemble for species a configuration integral for single-species system configuration integral for a binarymixture f3=11 kBT f3Tcoefficient of thermal expansion yanalysis parameter yratio of specificheats, =cplc" y"activity coefficient for species a r abulk motion correction factor for a>0 r -abulk motion correction factor for a.:,y. z).(1.2) where0 (x,y. z)isthepotential energy of theparticle andKis thekineticenergy of the particle given by m222 K(u, v,w)=2[u+ v+ w].(1.3) In theaboverelation, m is the mass of theparticle.Noting,for the xdirection, that aLaK - =- =mu=px auau.. aLaO -=--=F, axax., (1.4) Newton's equation can bewritten as (1.5) If wegeneralize thenotation as (1.6) the equations of motion can then be written as :t ( : ~ )=: ~ , j=1,2.3.(1.7) Anextraordinary andveryusefulproperty of Lagrange's equations of motion isthat they havethesameforminanycoordinatesystem(usefulsinceit issometimeseasier mathe-matically todefine0inaparticular coordinatesystem).Aset of equationslike(1.7)can bewritten for each particle in a multiparticle system. A third formulation of classical mechanics is the Hamiltonian formulation. To construct theHamiltonianformulationof classicalmechanics,webeginbydefininggeneralized momenta as aL Pj=a4/ j= 1,2 .... ,3N(for a system of Nparticles).(1.8) Each generalized momentum P jis said to be conjugate to coordinate qj. Since each momen-tum is linearly proportional to the velocity 4j' it is a simple task to replace any velocity terms in the formulation with the corresponding generalized momentum P j. We do so throughout 61I Qualltum Mechanics and Energy StorageinParticles thedefinition of theLagrangian and definethe HamiltonianHfora system of Nparticles tobe 3N iI(Pt, P2., P3N.qt. q2.. q3N)=Lpilj - L(pt. P2.. P3N.qt, q2,. q3N). j=t (1.9) Ingeneral, for systems of particles we expect that the total kinetic energy of the pru1icles is given by 3N K=L aj(qt, q2.q3 . ... , q3N) q;, j=1 and by definition aLaK. Pj=-. =-. =2ajqj. aqjaqj 0.10) (1.11) The coefficient a jvaries depending on whether the generalized velocity is linear or angular. Substituting thisresult and the definition of Linto the relation definingiI yields (1.12) Note that the Hamiltonian equals the total system energy. If the Lagrangian is not an explicit function of time,wecan differentiate therelation definingiI to obtain A 3N3NaL3NaL dH=L (qjdpj+ pjdqj) - Laq dqj - L[kidqj jjjjj Using the following relations obtained above: aL -a' =Pj qj. therelation for d iI becomes 3N3N dil=L qjdpj - Lpjdqj. j (1.13) (1.14) (1.15) SinceiI = iI (PI.P2,... , P3N.qt. q2,... , q3N)it also can be mathematically stated that A 3N(ail)3N(ail) dH=Lapjdpj+ Laqjdqj. jj 0.16) Byequatingcoefficientsof thedpjanddqjtermsinEqs.(1.15)and(1.16).weobtain Hamilton's equations of motion (ail). apj= qj' j=1,2 ..... 3N.(1.17) These6Nfirst-orderequationscompletelydescribethedynamicsof asystemof N particles.Given theinitial conditions,wecould trytosolvetheseequationstopredict the 1.2I A Review of Classical Mechanics7 Figure 1.4 macroscopic behavior of a system of molecules.However,todoso isundesirable for two reasons.First,solving somany equations isvery difficult,andsecond,at themicroscopic levelquantumeffectsarenotincluded.Hamilton'sequationshavebeenpresentedhere because they demonstrate that the energy of the system is a function of appropriately chosen coordinates and conjugate momenta. This concept will becentral toour development of a statistical view of the thermodynamics of thesystem in subsequent chapters. Example1.2Themassinthemass-springsysteminFigure1.4oscillatesaboutthe position x=O.The spring exerts a restoring force Fthat is linearly dependent on x, obeying therelation F=-kx, wherek isa constant that characterizesthespring.Find theLagrangian andHamiltonian for thesystem and show that Eq.(1.5) reduces to Newton's equation of motion. SolutionAtanyinstant,thekineticenergyKis(l/2)mu2,whereu =dx/dt. The potential energy is 0=-!ofF dx =-fo'-kXdX =The Lagrangian and Hamiltonian are therefore given by and Substituting the relation for Linto Eq.(1.5),we obtain :t = d -(mu) =-kx =F. dt Thus, Eq. (1.5) reduces to the time derivative of momentum being equal to the applied force, which is essentially Newton's law of motion. 81 I Quantum Mechanics and Energy Storage inParticles 1.3Quantum Analysis Using theSchrodinger Equation As a result of the work of Planck and others, by 1920, the dual nature (wave/particle) of radiant energy was widely accepted. Radiant energy is carried in packets called photons, eachwithenergyhv,whichhaveparticleandwavelikequalities.Herevisfrequency andh=6.63X10-34 JsisPlanck's constant.In1924,deBrogliearguedthatif radiant energycouldhavebothwaveandparticleproperties,thenperhapsmatterparticlesalso havewavelikequalities.By analogy,hepostulated that if a particle has total energy E:and momentum p, then it hasa wavelength A and frequencyv associated with it such that E:=hv(l.18a) and p= hiA. (1.18b) For a particle travelingwith momentum mv this implies that A = hlmv. (1.19) Example1.3In1927DavissonandGermerdemonstratedthewavelikescatteringof anelectronbeamreflectedfromanickelsurface.Thewavelengthof theelectronbeam determined from the interference pattern and the known spacing of the nickel atoms agreed well with that predicted by de Broglie's formula. The mass of an electron is 9.109 x 10-31 kg. If they travel at O.OStimes thespeed of light, it followsthat 6.63x10-34 A==4.8SxlO-llm=0.48SA. 9.109x1O-31(.OSx3 x108) ForvisiblelightA =4,000-8,000A.Thusanelectronbeamcanprovidehigherimage resolutionsinceitswavelengthismuchsh0l1erthanvisiblelight.Thisisthemotivation behind the development of the electron microscope. The coordinates q I, Q2,... and momenta PI, P2,... that quantify the dynamics of an N-particlesystem definea multidimensional spacerefelTedtoas phase space.Thewavelike natureof particlesimpliesthattheyarereallysmearedinphasespacewithnodefinite positionandmomentum.It canbeargued,basedonthedeBroglierelations,thatfor anymeasurement,theuncertaintyinaparticle'sposition,llx,andinitscOlTesponding momentum,llpx, areconstrained by therelation where h11 Illxllpl>- =-.t- 4rr2' h n=-. 2rr (1.20a) (1.20b) Equation(1.20a)isa quantitativestatement of theHeisenberguncertainty principle.We will consider this issue fUl1herin Section1.6. 1.3I Quamum Analysis Usillgthe Schrodinger Equation9 Becausematter hassome wavecharacteristics,it isexpected that somewaveequation governing its behavior should exist.If weconsider the following mathematical representa-tion of a one-dimensional (I-D) wave: \If=Cexp{i[2rrxl)..)- 2rrvt]}, and differentiate, it is easy toshow that and a\Ifh -in- =-\If = p,.\If ax)... a\If in - = hv\lf= c\lf. at (1.21) (1.22) (1.23) Thissuggeststheidentification of -in(alax) withPxandi1i(alat) withc.Ina nonrela-tivistic,classical(3-D)system,thetotal Hamiltonian(total energy)fora panicleisgiven by where ry A A p- AH= K+ U= - + U(r), 2m 2222 P=p, + Py+ Po' r= xi + yj + zk. The above arguments imply that 2.a(.a)2a2 p.=-In- -111- = -n-, .\axaxax2 rya(.a)ry a2 P- = -In- -In- = -11--yayayay2' rya(.a)rya2 p: = -In- -ITI- = -n--. - azazaz2 (1.24) (1.25a) (1.25b) (l.26a) (1.26b) ( l.26c) Substituting Eqs. (1.26) and (1.25a) into (1.24), the right-hand side becomes the Hamiltonian operator A -112[ a2 a2 a2 ]AH=- -+-ry+-ry+U(r). 2max2 ay- az-( 1.27) The Hamiltonian fi is interpreted as equaling the total energy c, which, in tum, is identified with in (a lat). Replacing fi with in (a lat), and applying the operators on both sides to the wave function\If,yields a\If-112ryAin- =-V-\If + U(r)\If.(1.28) at2m With a little rearranging this can be written as ry 2mA (2m)a\lf V-\If - -U(r)\If =- --. n2 inat ( 1.29) This isthe Schrodillger equation for the system wave function\If. 101 I Quantum Mechanics and Energy StorageinParticles Thewave version of quantum mechanics theory isbuilt upon the following postulate: Thestate of aquantum-mechanical systemiscompletely specified by a function \If (r, t) that depends on the coordinates of the particles and on time. This function, called thewavefunctionor state function.has the property that\If*(r,t) \If(r, t) dx dy d zisthe probabilitythatthe particleliesinthevolumeelement dx dy dz located at locationrat time t.(Notethat \If*is the complex conjugate of \If.) The wave function\Ifis determined by solving the Schrodinger equation. For \Ifto havethe property indicated in thepostulate,\Ifmust satisfy i \If \If * dV=1. ( 1.30) If solution of the Schrodinger equation is postulated to beof the form \If(r,t) =f(t)1/I(r)(1.31) then substituting into Eq.(1.29) and rearranging leads to ( ili)df=~[0 (r)1/I_1'12\/21/1]. fdt1/12m (1.32) Note that the left side of the above equation is a function only of time whereas the right side is a function only of position. Hence, each side must be a constant, which we will designate asCo.Setting theleft side equal to Coyields f=foexp{-iCot/ti},(1.33) wherefois the initial value of fat t= O.Substituting this result into Eq.U .31) yields \If(r,t) =f01/l(r)exp{-iCot/h}. Differentiating gives a\If ifl- = Co\lf= .s\lf, at fromwhich we conclude that the separation constant Coisthe energy, Co=.s. The relation forfistherefore f=foexp{-ict/h}. (1.34) ( 1.35) (1.36) (1.37) Thus,thetimevariation isoscillatory withfixedamplitude.Boundary conditionsandpo-tentialenergyeffectsaremanifestedinthespatialvariation.Settingtherightsideof the separated Schrodinger equation equal to .syields A t12, U(r)1/I- 2m \/-1/1=.s1/l. (1.38) Theaboveequation isthetime-independent Sc/zrodinger equation.Thetime-independent SchrodingerequationwithappropriateboundaryconditionsgenerallyformsaSturm-Liouville system for which .sisan eigenvalue.For such a system it is known that solutions exist only for discretevalues of .s.Solution of the time-independent Schrodinger equation will specify the allowable quantum energy levels for thesystem. i.4 I Model Solutions of the Time-independent Schrodinger Equation11 1.4Model Solutions of the Time-Independent SchrOdinger Equation Several simple model solutions of the time-independent SchrOdinger equation can be used as models of energy storage in atoms and molecules. For this reason we will examine these simple model solutions in thissection. AFree Particle in a Box -J-D Case For the one-dimensional particle-in-a-box system, the potential function that con-finesthe particle to a finitexdomain is defined as o =0for 0 d2' _1_!!..- (sin edP)+ (211c_n ~ 2)P=O. sine dedetl- sin- e The 2rrperiodicity of thesolutions requires that >(O)=>(2rr), P(O)=P(2rr), Solutions to Eq.0.67) are of the form >=Asinm + Bcosm, (1.65) ( 1.66) (1.67) (1.68) (1.69) (1.70) 161 I Quantum Mechanics alld Ellergy Storage in Particles where m= 1, 2, 3, ... to satisfy theperiodic boundary conditions. Transforming the independent variable in the second Eq.(1. 68)to =cos e yields 0- _ (dP) + (21E:P=o. fl21 _(1.71) For integer values of m , this is Legendre's equatioll. For a given m , solutions can be obtained for an infinite number of E:values specified as h2 E:=-1(1 + 1), 21 I=integer:::1m I. Solutions arethe associated Legendre polynomials (1-dl+1Il pm(l:)___(1:2 _1)1 I,,- 211!de+m". ( 1.72) (1.73) If theseresultsareback-substitutedintotheseparationof variablessolutions,andthe normalization condition (J JJ 1/1* 1/1dV=I) is imposed, the wave functions take the form _1[(21+ 1)(1- m)! ]1/2IIIi Ill 1/1- (;CPI(cose) e. '\f2rr2(1+ m)! (1.74) Note that each I corresponds to one E:(energy level) and I is therefore the principal (rotational) quantumnumber.For each Itherecan beany mintherangespecifiedby I:::1m I.There are21+ 1 values of mpermissible for each I and hence for each E:.Thus, the degeneracy gl of energy level E:Iis gl=21+ l. (1. 75) Increased degeneracy corresponds to increased degrees of freedom for energy storage. Example 1.5The rigid-rotor solution is commonly used to model rotational energy storage in molecules in an ideal gas. Use the rigid-rotor solution to determine the separation between the first two energy levels for nitrogen molecules in an ideal gas. SolutionTheenergy differencebetweenthelowest twononzero energylevels specified by Eq.(1.72) isgiven by /).E:= [2 (2+ 1)- 1 (1+ 1)]=4 . Thenucleiof theatomsof nitrogenwithmass2.33x10-26 kgareseparated byamean distance of aboutl.6x10-10 m.The moment of intertia of thenitrogen molecule isgiven by 1 =!mz6=!(2.33x10-26)(1.6x10-10)2=3.0X10-46 kg m2. Substituting the result for 1 and h=6.63X1O-34/2rrIs in therelation for/).E:yields (6.63x1O-34/2rr)2)-23 /).E:=42(3.0x10-46)=7.42x101. Wewill showlater that themeanrotational energy of a nitrogenmolecule in anideal gas iskB T, where Tisthe absolute temperature and kB=1.38X10-23 11K.By comparison, at 1.5 I AQuantum Mechanics Model of theHydrogen Atom17 290 K (near room temperature) themean rotational energy of a nitrogen molecule is 4.0x 10-21 J,which is about 50 times larger than the separation between levels computed above. The simple model systems for which wehave developed quantum mechanical solutions to theSchrodinger equation will be used in later sections to model specific energy storage mechanismswithinatomsandmolecules.Beforeproceedingfurtherit isilluminatingto examine howan entire atomcan betreated quantum mechanically.The simplest treatment of this type isthe model of thehydrogen atom described in thenext section. 1.5A Quantum Mechanics Model of the Hydrogen Atom The Schrodinger equation described inprevious sections can be used todescribe thehydrogen atomasa whole.For thehydrogenatomthenucleusconsists of oneproton and one electron is in orbit around thenucleus (see Figure1.8). The attractive forcebetween electron and nucleus is given by Coulomb's law: F=Cce;/r2, where ee=1.602x10-19 coulombs is the charge of one electron.Wetherefore define the potential energy for thesystem as A 1r C1 U=- Fdr= --, r 00 (1. 76) The negative sign is applied because stored energy increases asr.....,.00. Note that the mass of anelectron=9.1x10-31 kgandthemassof aproton=1.67x10-27 kg.Sincethe proton mass mpis much larger than the electron mass me, we idealize the nucleus asbeing stationary,withtheelectronorbitingaroundit.TheSchrOdingerequationthentakesthe form (1.77) Postulating a solution of theform1/1= Y (e, )R (r), substituting, and executing theusual separation of variable technique results in the following equations for Yand R: ( 1)d(dY)(1)d2y(21) -- - sine- +-- --=- - .sY sinededesin2 ed2112 (1.78) (1)d (.zdR)(2me)[2CI 1(1+ 1)112]_ --::;- - 1- +-,- .s+ - - R - O. r ~drdr1 1 ~r21 (1.79) Figure 1.8 181 I Quantum Mechanics and Energy Storage inParticles where (1.80) The equation for Yis the rigid-rotor equation (with the same periodic boundary conditions) solved above.It follows directly that solutions forYare Y=_1_[(21+ 1)(1- m)!] 1/2pilicose) eifll. J2ir2(/+m)!1( Tosolve theradial Eq.(1.79),wedefine rr;z a=V-s;;;;i' r YJ=-, a (1.81) (1.82) For thissystem,theenergy zeropoint istakentocorrespond toinfiniteseparation of the electron andnucleus.Thedefinitionof aaboveincludestheminussigninsidethesquare root because theenergy levels for finiteseparation arenegative.Invoking these definitions, the equation forRbecomes d2LdL YJdYJ2+ (k + 1 - yJ)dll+ nLL=0, (1.83) where k=2/+1,(1.84) Thisequation,theassociated Laguerreequation,hassolutionsknownastheassociated Laguerre polynomials: IIL-k(-1))(IlL!)2YJ) (II)= - L (nL- k - j)!(k + j)!j! )=0 Rearranging therelation for nLand substituting for ayields -meC? E:=ry 2fz-(nL+ 1+ 1)2 Setting n= ilL+ I+ 1 yields ( 1.85) ( 1.86) (1.87) Thenegativesign on theright sideof therelation forE:isconsistent with therequirement that increasing n correspond to increasing energy.Solving for R(r) yields R(r)=[(n-1- I)!] 1/2(r/a)le-r/2aL2/+I(r/a). 2na[(n+ I)!P11+1 Notethat aisa function of nand 11-1-1(_1)+1[(11+/)!]2(I"/a) =L (11-1-1- j)!(21+ 1 + j)!j! )=0 ( 1.88) ( 1.89) 1.5 I AQuantum Meclzallics Model of the Hydrogell Atom19 For a given 11,Ican vary fromzero to n- 1 and givea valid radial solution.Asshown fortherigid-rotorsolution,foreachI,mcanhaveanyof 21+ 1 values.Hence,forthis solution: 11= principal quantum number (specifying energy level), I =angular quantum number, m =magnetic quantum number, and fora given n, thelimits on Iand mare: 0.:::/':::11-1, It followsdirectly that thetotal degeneracy for a given 11value isgiven by II-I gll=L(2/+1)=n2. 1=0 (1.90) Radiation emission or absorption results in a change of energy level for the hydrogen atom. For emission, thephoton energy Izvis given by 2rr2meC?[11 ] hv=1- 2=---,- ---,. h2 112III (1.91) This relation can be written in terms of thewavenumber k12as Vl2[11 ] k12=- = RH---,- ---,, CInz Ili (1.92a) where RHis the Rydberg constant given by (1.92b) Based on experimental detelminations of the(Balmer)series of emission spectral lines forhydrogen,JohannesRydberg,inastudydoneinthe1890s,concludedthatRH= 109,720 cm-I.The quantum-theory relation indicated above predicts RH= 109,680 cm-I. The agreement of thetheoretical prediction with the measurements isindeed impressive. Example1.6TheBalmer seriesof emissionspectrallinesforhydrogencorresponds to electrons fallingfromlevelsaboven=2 down tothen=2 level.Showthat theBalmer series corresponds to wavelengths between 3,640 and 6,563A. SolutionSince thewavenumber k12=l/A12, Eq.(1.92a) can bewritten =RH -For theBalmer series,n2= 2,and so AI2=_1 RH4nl 201 I Quantum Mechanics and Energy Storage ill Particles Thelargestwavelength in theseriescorresponds tonl = 3.SubstitutingandusingRH= 109,720 em-I, wefind A12= 6.562x10-5 em= 6,562A. The smallest wavelength in the series corresponds to Ill""'"00.Inthis case, therelation for Au yields A12=3.646x10-5 em=3,646A. Thus, the Balmer series spans thewavelength range from3,646Ato 6,562 A. 1.6The Uncertainty Principle To further explore the nature of the uncertainty principle, we now wish to reconsider the one-dimensional particle in a box discussed earlier.For that system, we foundthat {2.(rrn\.x) 1/1=VL sm-L- , Ilx=l,2,3, ... ,(1.93) (1.94) Because1/1*1/1dxisinterpretedasaprobability,it canbeusedtocalculateaveragesand variances.For theparticle inaI-D box,1/1hasnoimaginary part and is thereforeitsown complex conjugate: *Ii.(rrn0) for such systems. From the definition (2.47) it is clear that the absolute temperature could only be negative if the entropy of the system decreased as its internal energy increased. The discussion above impliesthat thiscannot occur ina system that hasaninfinitenumber of energylevels.In such a system, an increase of temperature produces increasing occupancy of higher energy levels. However, Eq. (2.49a) indicates that the ratio of the mean number of particles (of type a) in two adjacent energy levels iand i+ 1 isgiven by Na,l+!=g,+! e-(,+l-f ,ljkBT Na,igi (2.112) In a system with infinitely many energy levels, if the system energy is finite, the higher energy levels ofthe system must be less populated than lower energy levels. Since gi+! / giis slightly larger than one and Ci+!>Ci,Na.i+J/ Na,ican be less than one only if the temperature Tis positive.AsTapproaches infinity,Na.i+J/ N,',iapproaches one. This ratio could equal one only ifthe system had infinite energy. Equation (2.112) implies that negative Twould require thatNa.i+J/Na.i begreater than one,whichwouldrequiremoreenergythaninthelimit of infinite temperature. Wetherefore conclude that negative temperatures are impossible in a system that hasaninfinitenumber of energy levels.Since most common systems areof thistype,negativetemperaturesareregardedasanimpossibilityforvirtuallyallsystems encountered in nature and in technological applications. 2.5 I More all Entropy alld Equilibrium59 However,suppose wefounda system that had only a finitenumber of energy levels.It may then be possible to cause a population inversion in a system with finite energy, resulting in Na.i+I/ Na,i>1, which would imply thatthe system has a negative temperature. A system having only two possible energy states isanalyzed in Example 2.4. This typeof system is shown to have positive temperatures at low energy levels and negative temperatures at high system energylevelsin whichthemajorityof theparticlesareinthehigher energy state. Thus. in a system with only two energy states, the population inversion that exists at higher energies results in negativeabsolute temperatures. Example 2.4Particlesinasystemaredistinguishableandcan exist in oneof onlytwo possibleenergylevels,each havinga degeneracy of one.The energy of thelower ground state istaken to bezero.Theupper statehas energy CI.The system contains N aparticles. Detelmine how the entropy and temperature vary with energy for this system. SolutionTheinternal energy of thesystem iszero if all theparticlesarein the ground state. If N a.1is the mean number of particles in state1, then N a - N a.1must be the mean number in the ground state. It followsthat the internal energy of the system must be given by U= Na,ICI. With N a.1particles in state 1, the number of microsates Wis the number of ways of putting N"distinguishable particles into two bins such that thereareNa,lin one andNa- N".Iin the other. It followsfromEq.(2.4) that W= Na!(1) (_1 ). (Na- Na,I)!N",I! Taking thelog of both sides and using the fact that S= kBIn W yields SI kB=In[N,,!]-In[(N,,- Na,I)!]-In[Na,I!]. Using the Stirling approximation, this relation becomes SI kB= NaIn Na- Na- (N"- Na,l) In(N"- Na,l)- (N"- Na.l)- Na,llnNa,1- Na,l. SinceU =Na.lcl,wereplaceNa,1withU ICI.Doingsoandrearrangingsomewhat,we obtain _S_ _-(1 -~ )In(1-~ )- ( ~ )In( ~ ) NakB- N"cINaciN"cIN"cI' The variation of entropy with internal energy for this system predicted by the above relation is shown in Figure 2.5. It can be seen that at low energies, the entropy increases with increasing energy. At higher energies, the trend reverses, however. At zero energy, the system is perfectly ordered with all particles in theground state. At thehighest possible energy for thesystem, U= N aCI,the system isalso perfectly ordered with all particles in the higher energy level. The entropy is zero for both of these perfectly ordered states. The entropy is maximum when the pal1icles are evenly divided between the two accessible energy levels. The temperature is the inverse of theslope of thecurve inFigure2.5.At lowenergiestheslope ispositive and therefore thetemperature ispositive.TincreasesasUincreasesatlowenergies,approaching+00 602 I Statistical Treatment of Multiparticle Systems S 0.8 Nak8 0.6 0.4 0.2 0 00.5U N"E:1 Figure 2.5 asU.....,.(l/2)NaE:J.At high energies,theslope andhencethetemperaturearenegative.T increases as Uincreases. As Udecreases toward (l/2)NaE:J,Tapproaches-00. Thus, the system has positive temperatures at low energies and negative temperatures at high energies, and U= (1/2)NaE:Jisa singular point that corresponds toT= oo. Asnotedabove,mostcommonsystemscontainparticleswithanunlimitednumber of increasing energylevels,andabsolutetemperaturesforsuchsystemsarepositive.If a system with unbounded energy levels at a positive temperature is brought into contact with a system at a negative temperature, the equilibrium temperature reached when the systems are brought into contact must bepositivebecause thecomposite system will haveunbounded energy levels. For thesystem considered in Example 2.4,negative temperatures correspond to higher systemenergies thanpositivetemperatures.When asystem of thetypeconsidered inthe example at a positive temperature is brought into contact with a similar system at a negative temperature, energy will transfer from the system having a negative temperature to the sys-temhavingapositivetemperature.Thistransferincreasestheentropyof thecomposite system, which must be maximized at equilibrium. Thus, negative absolute temperatures are hotter than positive absolute temperatures. The scale of absolute temperature therefore runs from+0 K to+00 K/-oo K to-0 K. Thetwo-energy-statesystemconsideredinExample2.4canbeusedasamodelof thenuclearspinenergystorageinacrystalexposedtoamagneticfield.Inthepresence of amagneticfield,thelowest nuclear statemaybesplit intoafinitenumber of nuclear magnetic states. For this model to apply, the energy storage in the nuclear spin states must be decoupled from energy storage elsewhere in the crystal lattice. The nuclear spin storage must reachequilibriumquicklyandtransferof energy betweennuclear spinstorageandother storagemodes must beweak.When theseconditionsaremet,thenuclear spinsubsystem behaveslikeasa finite-statesystemlikethat considered inExample2.4andmay exhibit negative temperatures.Abragram and Proctor [1]conducted experiments with LiF crystals that meet these conditions. Their investigation indicated that the lithium and fluorine nuclei 2.6/ Ma:n1!ellStatistics and Thermodynamic Properties for a Monatomic Gas61 mayactastwoseparatesubsystemsthat exhibit differentnegativetemperatures.Further information on the thermodynamics of systems that exhibit negative absolute temperatures can be found in the references by Ramsey [2],Klein [3], Abragam and Proctor [1], Proctor [4], and Muschik [5]. 2.6Maxwell Statistics and Thermodynamic Properties for a Monatomic Gas Up to thispoint inthischapter wehavetalked ingeneral termsabout the energy storage in a system of particles. In this section we want to consider a specific system type. We will specifically consider a system containing two species of boson-type particles that store energyonlybytranslational kineticenergy.Wewilllimitthemodeltoconditionswhere thesystem exhibitsdiluteoccupancy.Thisisagoodmodel of amonatomicideal gasfor moderatetemperatureswhere electronic and nuclear energy storage effects arenegligible. Webegin with theBoltzmann distribution relations generated in Section 2.3: where gie-./kBT Za gje-j/kBT Zb 00 Za= L gie-,/kBT, i=O 00 Zb=L gje-j/kBT. j=O (2.49a) (2.49b) (2.50a) (2.50b) For translational storage only,the degeneracy isgiven by Eq.(1.12l) fromtheparticle-in-a-box quantum solution inSection1.7: dnIT(SmV2/3)3/2 g(c)= -oc = - cl/20c. dc4h2 ( 1.121) Theenergylevelsfortranslationalstoragebecomemorecloselyspacedastheenergy increases. For translation of atom-sized particles at room temperature or above, the spacing of thequantumenergylevelsisextremelyclose.For practical purposes,wecan consider energy to bea continuous variable.In this continuum limit, thesummations in therelation for thepartition functions become integrals: 00100(SV2/3)3/2 Za= L gie-,/kBT= L g(c)e-/kBT=~m ~ 2cl/2e-/kBT dc, i=O.110 (2.113a) (2.113b) 622 / Statistical Treatment of MultiparticleSystems Evaluating the integrals on the right side of theaboverelations yields (2rrmUkBT)3/2 Zu= V h2 (2.114a) (2.114b) Inthecontinuumlimit,Nu.i,thenumber of aparticles inenergylevel i, isreplacedwith dNa,thedifferentialnumberof particlesintheintervalfromE:toE:+ dE:,andDE:inthe degeneracyrelationsbecomesadifferentialdE:.Asimilartransformationappliestothe distribution for b particles. The discrete distributions (2.49a,b) are thereby transformed into the followingcontinuous distributions: dNa 3/' (rrv)ema)~E:l/2e-E/kBT dE: (2.115a) Na 4Zuh2 ' dNb (rr V)(Smb )3/2E:1/2e-E/ kBT dE:. (2.115b) Nb4Zb h2 Substituting the partition function relations(2. 114a,b) into (2.115a,b), weobtain (2.116a) (2.116b) Thus both species obey the same energy distribution at equilibrium. Note that the right side of the above equations are the probabilities that a particle has energy between E:and E:+ dE:. Wecan determine the mean energy of a particleas (E:)=roooE:(~ )_E:_l_/2--::-;-:-e_E/kBT dE:. ioyJ. _Na.Nh.} Na.Nh.}- ",",00*. L..j=OnNa.Nh.j (4.76) Using Eq.(4.74)to evaluate nNa.Nb.jin the above relation,we obtain - exp( -YaNa- YbNb- fJENa.Nb.j) P Na.Nh.j=",",00(I)() , L..j=O'8nEgNa.Nb.jexp-YuNa- YbNb- fJENa.Nh,j (4.77) which reduces to _gNNe-fJENa.Nb.J p. - ah.}(478) Na.Nb.}- ",",00-fJEN N.. L..j=O gNa.Nh.jeahJ But we haveshown in the previous section that for a canonical ensemble with specified Na and N b,the probability that a system is in energy leveljis given by (4.79) where 00 Q=L gje-EJ/kBT. (4.40) j=O Clearly, Eqs.(4.78) and (4.79)areequivalent only if fJ=1/ kBT.Since thismust betrue forall thecanonical ensemble subsets within thegrand canonical ensemble, wetherefore conclude that for the grand canonical ensemble asa whole 1 fJ=kBT'(4.80) To evaluate Yuand Yb, we return to the definition of the grand canonical partition function, Eq.(4.75).For convenience, we define Ya= YakBT, Yh=YhkBT. (4.81) (4.82) Substituting(4.80)-(4.82)into Eq.(4.75),thedefinitionof thegrand canonical ensemble becomes 000000 S=LLLgNa.Nb.jexp[-(YaNa+ YbNb+ ENa.Nb.j)/kBTj. (4.83) Na=ONh=Oj=O 1224I Other Ellsemble Formulatiolls Wenotethat forspecified NaN b.andj. theenergy levelsand degeneraciesareexpected to beat most functions of the volumeV. The other parameters that appear in the definition ofSareT.Ya.andYb.If weconsider thenaturallogof S. it toowouldbea functionof V.T.Ya.and Yb.Expanding InStofirst order in these variables.we obtain dOn S) =(aOn S)dT + (aOn S)dV aTV.YJ.YbavT.Yd.Yh + ( a ( l ~S)dYa+ ( a o ~S)dYb. aYaT.v.}>haYbT.v.}>" (4.84) We must bring to this analysis some additional information on properties. The ensemble average of any property is computed by mUltiplying thevalue of the property at each com-bination of N a.NI" and energy leveljby the probability that a system has that combination andsummingsuchtelmsforallpossiblecombinationsof Na Nh.andj. Designatingan arbitrary property asY.this definition of its ensemble average(Y)iswritten as 000000 (Y)=LLLY (ENd.Nh.). V. Na Nb)P(EN".Nh.}. Na Nb) N,,=ONb=O}=o 000000 =LLLY(EN".Nh.}. V.Na.N/ N,,=ONb=O}=o (4.85) The grand canonical ensemble analysis is based on the premise that the average property value computed using definition (4.85) is the value of that property that would be observed macroscopicallyforasystemat equilibriumatthe/-La./-Lb.V.andTvaluesspecifiedfor the ensemble. The ensemble averagevalues of Na Nh.internal energy U. and pressureP are therefore given by (Na)=fffNagN".Nb.}exp[-(YaNa +ihN/> + EN".Nh.i)/kBT]. N,,=ONh=O}=o~ (4.86) 000000[(ANANE) /kT] (Nb)=LLLNb gN".Nb.jexp- Yaa +!hb +Na.Nh.}B ~ (4.87) 000000 (4.89) We will now use Eq. (4.83) together with Eqs. (4.86)-(4.89) to evaluate the partial deriva-tivesin Eq.(4.84).The derivatives with respect to T. Ya.andYbarefairlystraightforward 4.3 I The Grand Canonical Ensemble123 to evaluate. 1000000 = kBT2LLL(YaNa+ YhNb+ ENd.Nb.i) Nd=ONb=O}=o (4.90) (4.91) (aons)1 (as) ~VoToY"=SaYhV.ToYd 000000(Nb) =LLL- kBT Na=ONh=Oj=O (4.92) The derivative with respect to Vis a bit trickier to evaluate: 1244 I Other Ensemble Formulations At this point we note that Eq. (4.49) from our analysis of the canonical ensemble implies that the Vderivative of Qin the above equation is just Q / kB Ttimes the pressure for a sys-tem at specified Nu, Nb,V,and T. Wecan therefore write the relation for theVderivative of InSas (a(ln S)=P(Na, Nb,V, T)Q(Na, Nb.V.T) e-(YaNa+YhNhl/kBT. avT"SkBT .Ya.YhNa=ONh=O (4.94) The factor P(Na, Nb, V,T) in the above relation is equal to the ensemble average pressure foracanonical ensembleat thespecifiedNa, Nh,V, and T.It followsfromthe definition of the canonical ensemble average properties (4.26) that 00gje-ENa.Nh.dkBT P(Na, Nb,V.T) = LP(ENa.Nh.j,V. Na, Nh)Q j=O (4.95) Substituting this result into Eq.(4.94) yields (a(lns)000000[ -----av.. =LLLP(ENa.Nh.j.V. Na. Nb) T.Ya.YbNa=ONb=Oj=O gNa.Nb.iexp[-(YaNa + YhNh+ ENa.Nb.j)/kBT]] X. SkBT (4.96) Comparing the right side of the above equation with Eq.(4.89), we see that the triplesum-mationinEq.(4.96)isjusttheensembleaveragepressure(P)forthegrandcanonical ensemble.Replacing the triplesummation with(P)simplifies Eq.(4.96) to (a(lns) avT.Ya,YbkBT (P) (4.97) SubstitutingEqs.(4.90)-(4.92)and(4.97)forthepartialderivativesinEq.(4.84).we obtain thefollowingrelation: d(ln S)=(Ya(Na)+++ (P)dV _(Na) dYa_(Nb) dYh. kBT2kBT- kBT2kBTkBTkBT (4.98) Weadd d( (U) / kBT), d(yu (Nu) / kBT), and d(Yb(Nb) / kBT) to both sides and rearrange to get d (In S+ (U)+ Ya(Na) + Yb(Nb)) =d(U)+ (P)dV +Yad(Nu) +Ybd(Nb)' kBTkBTkBTkBTkBTkBTkBT (4.99) As afinalstep,wedrop theensemble average brackets, acknowledging that the ensemble average properties are expected to equal the real physical properties for the system of interest: ( UYaNaYbNb)dUPYaYb d =-+-dV +-dNa + -dNb. kBTkBTkBTkBTkBTkBTkBT (4.100) 4.3 I TheGrand Callonical Ensemble125 We now compare Eq. (4.100) with the following equation, developed in Chapter 3, which applies to a binary mixture of two species: TdS= dU+ PdV - /-LadNa- /-LbdNb.(4.101) Equation (4.101) can be written intheform ( S)dUPdV/-La/-Lb dkB=kBT+kBT- kBTdNa - kBTdNb. (4.102) Equations (4.1 00) and (4.102) both must be valid for any choices of the differential terms. If we set dNaand d N b both to zero, the right sides of both equations are equal and it follows directly that d- = dInS+ _+ Yaa+ Ybb. (S)(UANAN) ~~ T~ T~ T (4.103) Since thedifferentials on theleft side of Eqs.(4.100) and (4.102) must be equal, theright sides must also be equal for any choices of the differentials. For this to be true, the coefficients of corresponding differentials on the right sides must be equal.Equality of the coefficients for the dNaand d N b terms requires that Ya=-/-La, Yb=-/-Lb (4.104) (4.105) Equations (4.104) and (4.105) can then be used with Eqs.(4.81) and (4.82) to evaluate the Lagrange multipliersYaand Yh: Yo=-/-La/ kBT, Yb=-/-Lb/ kBT. (4.106) (4.107) Substituting Eqs.(4.80),(4.106),and(4.107) into Eqs.(4.74) and (4.75), we obtain the following relations for the occupancy distribution and the grand canonical partition function: (4.108) whereSisthe grand canonical partition function defined as 000000 S=LLL gN".Nh.}exp[(/-LaNa+ /-LbNb- ENa.Nh.})/kBTj. (4.109) Na=ONh=O}=o The distribution (4.108) can also be interpreted asthe probabilityj5 Na.Nh.}that a system at equilibrium with specified T. V, /-La,and /-Lbwill,at anarbitrarily chosen time, contain N a species aparticles, N bspecies b particles and be in energy levelj: j5._gNa.Nho}exp [ (/-La N a + /-LbN h - E Na.Nh.}) / kB T j Na.Nh.}- ~. ~ (4.110) Havingevaluatedallof theLagrangemultipliers,wenowwilltumtotheproblemof obtaining relations for thermodynamic properties. Using Eqs. (4.104) and (4.105) to evaluate Yaand Yb,the differential relation (4.103) can be integrated to obtain ~U/-LaNa/-LbNb S=kBIn~+ - - -- - -- + So. TTT 1264 I Other Ensemble Formulations In theaboverelationSoisa constant of integration.Wenote,however,that in thelimit of N aand N bboth going to zero, thesystem has only one accessible microstate: that with zero internal energy and zero particles. In that limit, the partition function defined by Eq. (4.109) goes to one and InS iszero.With the exception of the Soterm. the other telms on the right sideof theabove equation also vanish asNoand Nbapproach zero, implying that Sois the limitingvalueof entropyasthetotal number of particlesapproacheszero.Ingeneral our definitionof entropy hasbeenbased onthenotion that it isan indicator of thenumber of accessible microstates for thesystem. This notion suggests that a perfectly ordered system with only one microstate should havezero entropy. We therefore set the integration constant Soequal to zero.Our relation for entropy then becomes ~U/-LaNu/-LbNb S=kBIn1:1+ - - -- - --.(4.111) TTT Replacingtheensembleaveragepropertieswiththeactualphysicalpropertiesandusing Eqs.(4.104) and (4.105) to evaluateYaandYt"Eq.(4.90) becomes (ann S) kBT--aT1'./1" ./1h U TTT (4.112) Theright sideof Eq.(4.112) isidentical to thelast three terms on theright of Eq.(4.111). Wereplace those three telms by the left side of Eq.(4.112) to obtain ~(annS) S=kBIn1:1+ kB T----aT,. \' .I'a./1h (4.113) Equation(4.113)explicitlyrelatesthesystementropytothegrandcanonicalpartition function. InEqs.(4.91),(4.92).and(4.97).wesimilarlyreplaceYawith-/-La.replaceYhwith -/-Lb,and replace each ensemble average property with the corresponding system physical property. Rearranging the resulting equations a bit yields the following additional property relations: Na- kBT--- _.(ann S) a/-LaI',T,/1h (4.114) (4.115) (4.116) Solving forU, Eq.(4.111) becomes U=TS- kBT InS+ /-LaNa+ /-LbNb. (4.117) InChapter 3,wederived theEuler equation.which takes thefollowingformfora system containing a binary mixture of components aand b: (4.118) Comparing Eqs.(4.117) and(4.118), wesee that both relations can bevalid only if PV=kBT InS.(4.119) 4.4 I Fluctuations127 Solving Eq.(4.112)forUandusing(4.114)and(4.115)to evaluateNaandNh,wealso obtain Equations (4.113 )-(4.116), (4.119),and (4.120) provide thelink between the grand canon-ical ensemble statistics and macroscopic thelmodynamic properties. Example 4.2Determine the value of the grand canonical partition functionSfor a binary mixture of monatomic ideal gases containing one kmol of each species. SolutionSince the mixture must obey Eq.(4.119), PV= kBT InS, and the ideal gas equation of state, wecan combine these relations to obtain InS=Nu+ Nh. This relation can besolved forS, yielding Substituting for NuandNh,weobtain Clearly the numerical value of Sis extraordinarily large, reflecting the fact that the number of accessiblesystem energy microstates with energies comparable to kB Tis enormous. 4.4Fluctuations Inthissectionwearegoingtoexaminemicro scalefluctuationsinasystemat macroscopic equiliblium.Wewill consider fluctuationsina system that contains a binary mixture of species aand species b particles. The degree to which a system fluctuates about themean value of a theml0dynamic propelty Ycan be quantified in terms of itsvariance af;(where ayis thestandard deviation): 22 ay =(Y- (Y)).(4.121) Herewewillspecificallyconsider fluctuationsof Nainagrand canonical ensemble(for which V,T,/-La,and /-Lbareheld fixed).Applying Eq.(4.121) to fluctuations inNuyields a ~ a=(Nu- (Nu))2=(N};- 2Nu(Nu) + (N,,)2)= (N;)- 2(Na)(Nu) + (Nu)2 =(N;)- (N,,)2.(4.122) 1284 I Other Ensemble Formulations The ensemble average of a property Yfor thegrand canonical ensemble for a two-species system is 000000 (Y)=LLLY(Na Nh.ENa.Nh.,)PNu.Nh.i Na=ONh=Oi=O 000000 where thepartition function for the two-species system is 000000 s= LLLgN".Nb.iexp[(ILaNa+ILbNh-ENu.Nh.i)/kBT]. (4.123) Na=ONb=Oi=O Treating N; as the property and applying this relation. wehave a x-a- (NagNu.Nh,) exp[ (ILaNa+ ILhNb- ENu.Nh.i)1 kBT]). ILa Wesimplify this relation by manipulating the right sideas follows: 000000 x[_a_ (NagNU,Nb.ieXP[(ILaNa+;:"Nh - ENu.Nh.i)/kBT]) aILa~ +as(NagNu,Nb,ieXP[(ILaNa:ILhNh - ENu,Nh,i)/kBT])] aILa~ 2 a000000 =kBT-LLLNa a ILaNa=ONh=Oi=O XgNa,Nb,ieXp[(ILaNa+ ~ h N b- ENa,Nh.i)/kBT]+ k ~ Tas ~~aILa xfff NagNa,Nb,iexp[(ILaNa+;:hNh- ENa,Nh,i)/kBT] ~ (4.124) 4.4 I Fluctuations129 UsingEq.(4.114)fromtheprevioussectiontosimplifythesecondtermontheright of Eq.(4.124), we obtain (N;)= kBT(a(Na))+ (Na)2. a/-LaV.T.llh Substituting the right side of theabove equation for(N;)in Eq.(4.122) yields which reduces to 2_/.T(a(Na)) aN-I\B. aa/-LaV.T.l1h (4.125) It should be obvious that a similar relation applies for the variance for species b, which can beobtainedbyswitchingtheroles of aandbinEq.(4.125).Equation(4.125)linksthe mean magnitude of the fluctuations in Nato the derivative (aNa/a/-La)V.T.l1h'This provides information about the fluctuationsin the number of particles for one species in thesystem. Wemayalsobeinterestedinfluctuationsinthetotalnumberof particlesinthesystem N = Na+ Nh.Withsomeadditional effort,thelineof analysisabovecan beextended to show that the variance in the total number of particles in the binary system is given by With a similar (but longer) line of analysis, it can beshown that the variance in thesystem internal energy for the binary system is where (4.127b) Wewill now explore theinterpretation of Eq.(4.125) in terms of other thermodynamic properties.From basic calculus, for a function z =z(x, y) we know that (see Appendix I) (az)(az)(ax) ayx=- ax.I"ay:' Holding T and/-Lbfixed,we can therefore write (4.128) 1304I Other Ensemble Formulations At constant (Nu) and T, we expand (av /allu)(Nu),T,llhusing the chain rule (::J (Nu),T,llh=( : ~ )(Nu),T,llh( : ~\Nu),T.!': (4.129) Substituting theright side of Eq.(4.129) for(av /a Ilu )(Nu),T.!'hin Eq.(4.128) gives The firstand third partial derivatives on the right side of the above relation can be evaluated using the following two Maxwell relations: (4.131) (4.132) ReplacingthefirstandthirdderivativesinEq.(4.130)withthederivativesindicatedin Eqs.(4.131)and (4.132),weobtain (4.133) Solving Eq. (4.119) for InSand substituting the resulting expression for InSin Eq.(4.114) yields (a(Pv / kBT) N"= kBT allu V,T,llh =v (::)F,T,Il: (4.134) It isclearfromthisresultthatthefirstpartialderivativeinEq.(4.133)isjust N a /V . Using thisresult to replace this derivative in Eq.(4.133)leads to (a(Nu))Nu(av)(a(Na)) a;::1',T,llh= --yap(Nu),T,llh----avP,T,I''" (4.135) To evaluate the rightmost derivative in the above equation, weproceed as follows.First, we rewrite Eq.(4.134) in the form (4.136) Differentiating both sides of Eq.(4.136)with respect toV, we obtain (4.137) 4.4I Fluctuations131 Switching the order of differentiation yields (aNa)(a[(a(PV)]) avP.T./-Lh=afla----av- P.T./-LhI'.T./-Lb (a)(ap) =-[P]=- . aflaI'.T./-Lhafla1'.T,!lb (4.138) Since Eq.(4.134) implies that (a P/ afla kT./-Lh= Na/ V, the above equation can be written (aNa) avP.T./-Lb Nu V (4.139) Substitutingtheright sideof Eq.(4.l39)fortherightmost derivativeon therightsideof (4.135),weobtain (4.l40) Replacing the derivative in Eq.(4.125)with the right side of Eq.(4.140)and replacing the ensemble average properties withsystem physical properties, therelation forthevariance of N abecomes (4.141) Dividingboth sideby N,;and takingthesquareroot.weobtain thefollowingrelation for the fractionalstandard deviation of Nafromits mean value: aN[kBT] 1/2 N:=-tap /aV)Nd.T./-Lh V2 (4.142) Therelationsobtained fora binary systemapply tosingle-speciessystem if weset N h tozeroanddropflhasaparameter.For asingle-speciessystem,herewewilldropthea subscript and designate the number of paIticles and chemical potential simply asNaIldfl. respectively. Conversion of relations derived earlier thus requires that we set N hto zero and replaceNaandflawithNand flrespectively.It followsthat forasingle-speciessystem. therelation for the fractional standard deviation in thenumber of particles Nisgiven by (4.143) Usinga similar strategy to simplify Eq.(4.127). it followsthat for a single-species system thevariance of the internal energy about its mean value isgiven by 22(aU)2(aU)2 au=kB T- + aN- , aTj1NaNTI' .. (4.144) where a ~is the variance of N , which can be computed using Eq.(4.143). With some minor manipulation of Eq.(4.144), thefollowingrelation for the fractionalstandard deviation of 1324/ Other Ensemble Formulations Ucan be obtained: au=[ k B ~ 2(au)+ ak (aU)2]1/2 uuaTV.Nu- aNT.V ( 4.145) Atthispoint,acommentonthesignificanceof theseresultsiswarranted.Notethat theanalysisof fluctuationspresentedinthissectionwasbasedonresultsforthegrand canonical ensemble formulation.The grand canonical ensemble fOlmulationappliesto an opensysteminthermaland massexchangecommunicationwithahost of other systems subject to the same constraints.Within ina largebody of fluid,such astheair in a typical room, if we define a subsystem with a specified volume that is small compared to the overall sizeof thebody of fluid,that subsystem issubject totheconstraintsimposed on systems in agrand canonical ensemble. For that reason,theresults of thestatistical analysis of the grand canonical ensemble provide useful tools for thermodynamic analysis of fluid elements within a larger body of fluid. Our previous analysis of the grand canonical ensemble led to relations for the number of ensemblemembersforeachparticle-number-energy levelcombination.Thedistribution we found for a single-component system has a sharp maximum at a paIticular combination of Nand U, which we interpret asbeing the equilibrium values for a system at the volume, temperature, and chemical potential values for the ensemble. The shape of the distribution is shown schematically in Figure 4.4. The relations for aN / Nand au / U derived in this section directly indicate the sharpness of the peak in the distribution.If the standard deviation from themean property value is very small,thepeak is verysharp,and theprobability that the system will beobserved with a value of Uor Nother than themean value isvery remote. Figure 4.4 4.4 I Fluctuations133 However,if thefractionalstandarddeviationissignificantlygreater thanzero,thereisa nonnegligible chancethat thesystem may be observed towander away fromthemean U or Nvalue. The sharpness of the distribution for Ncan be easily assessed for an ideal gas.Since the equation of state for anideal gas isP V= N kB T, it follows that NkBT -(ap /aV)N.T=-----y2' (4.146) Substituting the right side of Eq.(4.146) for -(ap /aV)N.Tin Eq.(4.142), weobtain = Nlj2 . (4.147) Thus for an ideal gas, as the number of paI1icles increases, the fractional level of fluctuations decreases. For a system containing Avogadro's number of particles, the fractional deviation from the mean number of particles is of the order of 10-13.It is therefore not surprising that any attempts to measure the number of particles in a system of macroscopic size will always givearesult in closeagreement with themeanvalueof N, and themeasuredvalue for an equilibrium system will not measurably change with time. Furthermore, to detect deviations from the mean in macroscopic systems, we must have instrumentation that can resolve such measurements to better than one part in1013. This level of precision is generally far beyond common instrumentation.These results imply, however,that if thesystem is small so that Nis not large, the fluctuations may bea significant fraction of the mean value. Example4.3Estimatethelevelof densityfluctuationsinacubiccentimeterof airat normal atmospheric pressure and room temperature. SolutionFor normal atmospheric pressureand room temperaturewetakeP= 101kPa and T= 293 K.Using the ideal gas equation of state we find PV(101,000)(10-2)3 N=-- =23= 2.50X1019 molecules, kBT1.38x10- (293) I v'ii - - - - r : : = : = = = = = = ~= 2.00x10 -10 . .J2.50X1019 This implies that the standard deviation in the density in the1 cm3 volume is only two parts in1010,anamount so small it isvirtually impossible to measure. Example 4.4Estimate the level of density fluctuations in a spherical control volume with a diameter of 200 A at conditions typical of the earth's atmosphere at 3,000 meters altitude. SolutionFor thespecified altitude, typical temperature and pressure conditions areP= 70 kPa and T= 268K.Thevolume of the control volume isgiven by 4rrr3 4rr(100x10-10)3 V=-- == 4.2X10-24 m3. 33 1344 I Other Ensemble Formulations U sing the ideal gas equation of statewefind PV(70,000)(4.2x10-24) N=kBT=1.38x10-23(268)= 79molecules, aN11 Ii =IN =,J79= 0.11. Thus,densityshiftsof11 %overregionsof thissizearecommonintheatmosphereat thisaltitude.Thesefluctuationsaresignificantbecausethevariationindensityresultsin avariationinthedielectricconstant,andthevariationof thedielectricconstantcauses theregiontoscatterlightpassingthroughtheatmosphere.Visiblelightcorrespondsto valuesof wavelengthA between 4,000and7,000A.For scatteringobjects that aresmall compared tothewavelength of theradiation,thefractionof theincident radiationthatis scattered is strongly wavelength dependent, being proportional to A-4. This regime is known as Rayleigh scattering.Rayleigh scattering does occur fromregions of anomalous density intheatmospherethatresultfromdensityfluctuations.Becauseof itsstrongwavelength dependence,shorterwavelengthbluelight isscatteredtoagreater degreethantheother colorsinthevisiblerange,makingtheskyappearblue.Theblueskyisaresultof the combined effects of density fluctuationsand Rayleigh scattering. Equation(4.142)alsoimpliesthatfluctuationsmaybesignificant if -(ap /aV)N.Tis small.For purefluids,-(ap /aV)N.Tbecomessmallinthevicinityof thecriticalpoint and formetastable superheated liquid or supersaturated vapor.Wewill discuss metastable fluidsin more detail in a Chapter 8.However, it isworth noting here that thelarge density fluctuations that occur when -(ap /aV)N.Tbecomes small may raise the density of a vapor nearlytothat of aliquidphase or may lower thedensity of aliquidnear that fora vapor phase. Large density fluctuations that may initiate a change of phase are sometimes referred to asheterophase fluctuations. Example 4.5A system contains saturated argon vapor at atmospheric pressure and 87.3 K. Treatingthevaporasanidealgas,estimatetheprobability that a heterophasefluctuation may occur within a spherical region of thesystem having a diameter of 5 nanometers. SolutionWeconsidersubsystemsof theoverallsystemthatpelmitmassand energy to cross their boundaries with each subsystem held at the temperature and chemical potential of the overall system. The subsystems are modeled as members of a grand canonical ensemble. The probability that a subsystem has N"argon atoms is obtained by setting N b=0 in Eq.(4.110) and summing over all energy levelsj. Doing so yields where S, the grand canonical partition function defined by Eq. (4.109) with N bset to zero, is 4.4 / Fluctuations135 Since the summation in therelation for PNuis overj, it can be rewritten as Thesummationintheaboverelationisequaltothecanonicalpartitionfunctionforan ensemble of systems having volume V, N aparticles, and temperature T. The above relation is therefore equivalent to PN =elluNa/kBTQ(T, V, Na). uS(T, V,fla) In this equation, the dependencies of thepru1ition functions have been indicated explicitly. Taking thenatural logof both sides converts therelation to - flaNa In PN =-- + InQ- InS. akBT In thismodel analysis,wewill treat thevapor asif it isan ideal gas.Wewill therefore userelations derivedpreviously foramonatomic ideal gasto evaluatefla,InQ,andInS in theaboverelations.Equation (4.119) isused to evaluate InS: InS= P V I kB T. For InQ, weuse therelation derived in Example 4.1with Nhset to zero, and the relation obtained in Example 3.3 for a monatomic gas is used to evaluate the chemical potential, II_- - kBT In-. _(3){2rrmakBT(V)2/3} ,.-"2h2 Na In this analysis, the subsystem is a control volume with volume V. The values of flaand T for the subsystem are taken to be the same as those in the overall bulk system. In calculating flawe therefore use Tand VI N"values associated with the bulk system. The overall system isassumed to obey theideal gas equation of stateP V= N akB T, and toget theflavalue forthebulk,wecanreplaceV IN"intheaboverelationwithkBT I PsinceTandPare specified for thebulk system: fl- - - kBT In-. _ (3){2rrmakBT(kBT)2/3} a2h2P Substitutingthisrelationforthechemical potentialandtheequationsaboveforInSand InQinto the equation forPNa'weget,after some simplification, InPNa = N"In( ~ )+Na _(PV). NakBTkBT In the subsystem, the chemical potential and the temperature are taken to be fixed at the bulk systemvalues,but weexpect that thenumber of atomsin thevolumemay fluctuatewith 1364 I Other Ensemble Formulations timeabout a mean,whichistheequilibriumvalueof NuSince thegasin thesubsystem volume obeys theideal gaslaw,and themean pressurein thevolumeisexpected toequal thebulksystem pressureP, weinterpretP V / kB Tasbeing equal to themean number of atoms in thesubsystem(Nu).Wecan therefore rewrite theaboverelation as - ((Nu)) lnPNa=NalnNu+Na-(Nu). Takingtheexponentialfunction(inversenaturallog)of eachsideof theaboveequation, we get the following relation for the probability that thesubsystem volume will contain N a atoms: Inasphericalvolumewithadiameterof 5nm,themeannumberof argonatomsin saturated vapor at atmospheric pressure and 87.3K is computed as (Na)=PV / kBT= (101, 000)(rr/6)(5x1O-9)3/(l.38X10-23 x87.3) = 5.50 molecules. The density of saturated argon liquid at these conditions is1,394 kg/m3 (see Appendix III). Dividing this mass density by the molecular mass of39.9 kg/kmol for argon and multiplying by Avogadro's number yields a density of 2.10x1028 molecules/m3. To fillthesubsystem with molecules at theliquid density would require a fluctuation such that Na= (2.10x1028)(rr/6)(5x10-9)3= 1,380 molecules. Puttingthecalculatedvaluesof (Na)andN aintotherelationforPNa'wefindthatthe probability that such a fluctuation would occur is _(5.50)1.380, P= el.380-5.50-- ~1 O - ~ . 7 1 4 . Na1 380 , Thus,theprobabilitythatrandomfluctuationsinthevaporwillproducenearliquid density in a region nanometers in dimension is virtually zero. The accuracy of this prediction is limited because we treat the vapor asan ideal gas, and its behavior will deviate fromthat for an ideal gas asit approaches saturation. The result nevertheless suggests that fonnation of a liquid phaseina tinysubregion of thesystemby randomfluctuationsin thevapor is very unlikely, even at saturation conditions. This observation is qualitatively correct for most fluids.Wewill retUl11to consider this issue inmore detail in Chapter 8.In that chapter, we will showthat if thesystem ispushed intoa supersaturated state, heterophase fluctuations may havehigh enough probability to belikely initiators of a phase transition. The ensemble fonnulations discussed so far in this chapter have considered system energy levels.Toapply the results of these fonnulationsto real systems of atoms, molecules, and other particles,weneedtoestablish thelinkbetweenmolecular orparticleenergylevels and system energy levels.Wewill examine this issue in detail in thenext section. 4.5 / Distinguishability and Evaluation of the Partition Function137 4.5Distinguishability and Evaluation of the Partition Function Wenowwish to tum our attention to the problem of evaluating the canonical and grand canonical partition functionforsystems of atoms,molecules,or other particles.As discussed in Chapter 2, particles commonly encountered in real systems fall into one of two categories: fermions and bosons. For bosons, there are no restrictions on the manner in which thesystem of particles distributes itself over available energy microstates. The behavior of fermionsislimited in that no twoidentical fermionscanoccupy thesameparticle energy microstate. Analysis offermions gives rise to Fermi-Dirac statistics, whereas consideration of bosons leads to Bose-Einstein statistics. Toapplytheensemblestatisticalframeworktoasystemof Nparticles,wemust first determinethecompleteset of energymicrostatesaccessibletothesystem.If thesystem isgovernedbyquantumtheory,thiswouldgenerallyrequirethesolutionof theN -body SchrOdingerequation,whichisavirtuallyimpossibletask.Fortunately,therearemany systems for which the N -body Hamiltonian operator can be written as a sum of the individual Hamiltonians.Thetotal energy canthen bewrittenasa sum of individual energies forall Nparticles: N H = LHi. (4.148) i=1 Themostcommonexampleof asystemthatconformstothisbehaviorisanidealgas. Another exampleisthedecomposition of theHamiltonian of a polyatomicmoleculeinto thecontributions for its various degrees of freedom: H =Htranslation+ Hrotation+ Hvibration+ Helectronic. (4.149) Formanysystemsamenabletoanalysisbyclassicalmechanicsandquantummechanics treatments,H,byaproperselectionof variables,canbewrittenasasumof individual terms.Pseudoparticlesaresometimesassociatedwithtermsinthissumevenif thereare norealparticles.Thesearesometimes alsocalled quasiparticles.Theseinclude photons, phollons. plasmons. magnons, androtons.For a systems of many distinguishable particles that behave this way,thetotal system energy can be computed as N A ""AH= ~H particle i . (4.150) i=1 If thesystem is held at fixedVand T, thecanonical partition function is given by 00 Q(N, V, T)= Le-E//kBT =L(4.151) /=0 wherea, b,c.. ..designatethedifferentmoleculesandE: j ~indicatesenergymicrostate j ~forparticlea,E: j ~indicates energy microstatej ~forparticleb,etc.Notethatwehave elected,atleastinitially,towritetherelationforthepartitionfunctionasasumoverall microstatesratherthanoverenergylevels.Thenotationj ~ ,j ~ ,j , ~ ,... inthesummation denotes a summation over all possible combinations of these indices. Equation (4.151) can 1384I Other Ensemble Formulations berewritten as 000000 Q """'-/ / kB T"""'-/ / kBT"""'-/ / kB T a h....(4.152) =0j: =0 Note that we have cast this relation as a product of summations over all the energy microstates for each particle in the system. If the degeneracy of each energy level for particle a, b, c,... is gja'gjb'g}c'... respectively, we can write Eq. (4.151) as the product of summations over all energy levels for theparticles: 000000 Q= Lgjae-EJa/kBTLgjbe-EJa/kBTLgj,e-j,/kBT.... (4.153)

Each summation in the above relation is, in fact,a partition function for a specific particle. Sincewewillmostoftenapplythisconcepttomolecules,wewilldefineamolecular partition junction qas 00 q= Lgje-j/kBT. (4.154) j=O The above definition is written as a sum over the energy levels of the particle or molecule. When it is useful, wecan also write it in terms of a summation over energy microstates: 00 q= Le-//kBT. j'=O With this definition we can write therelation (4.153) forQas Q= qaqbqc... qN(for Nparticles), (4.155) (4.156) wherethesubscriptsdenotetheparticlesforwhichthemolecularpartitionfunctionis defined.Notethat thisimplicitly assumes that themolecules aredistinguishable.If theN molecules or other particles are all of same type, this relation reduces to Q(N, V, T)=[q(V, T)]N.(4.157) If the particles are of two types, a and b,and thesystem contains N aparticles of type aand Nbof typeb, therelationforQbecomes (4.158) If wenowspecificallyconsidermoleculesforwhichthemolecularHamiltoniancan beapproximatedbyasumof Hamiltoniansforthevariousmodesof energystoragein themolecule,bythesamelineof reasoningoutlinedabove,wecanbreak themolecular pal1ition function into factors associated with each mode of energy storage, (4.159) where qlr,qrot,qvib, and qeare the partition functions for translational, rotational, vibrational, and electronic energy storage, respectively.These al'edefined as 00 q _"""' g'e-",j/kBT Ir.}, (4.160) j=O 4.5 I Distinguishability and Evaluation of the Partition Function139 00 q _"" g_e-e,o,.dkBT rot- ~rot.}, (4.161) j=O 00 L e/kT q 'b- g'b'e- "b./B VI- VI.J' (4.162) j=O 00 q _"" ge-e,.j/kBT e- ~e.,. (4.163) j=O Notethat theabovedefinitionsarewrittenassummations over energylevels.Equivalent definitions in terms of asummation over energy microstates also apply. In this manner, not only can we reduce an N -particle problem to a one-paIticle problem, butwecanreduceit furthertotheindividualenergystoragemodesof asingleparticle. While this is an attractive result, there is one major problem with it:Atoms and molecules, in general,arenot distinguishable.One exception is asystem of atoms in a crystal lattice. Because themean position of each atom is fixedin space, each paIticle is distinguishable. The results obtained above for distinguishable paIticles are therefore directly applicable to solid crystals, and we will explore how we can apply them to solid crystals in a later section. Inaliquidor gas,theparticlesarefreetomoveaboutwithinthesystemaIldanyof theparticlesmay occupyagivenlocation.For agivenmicrostate of asystemcontaining afluid,theparticles havespecific positions,velocities,and internal energy microstates.If weexchanged two particles of thesamespecies, themicrostateswouldbedifferent if the particles weredistinguishable,but wewould not beableto tell them apart if theparticles areindistinguishable.If wetreatedtheparticlesasdistinguishablewheninfacttheyare not,wewould overcount thenumber of microstates.Thus, the relation betweenQand the molecular partition function obtained for distinguishable particles is not correct for systems containing a gas or liquid. Without ameanstolinkthecanonical partitionfunctionto energy storageinindistin-guishableparticlesormolecules,thecanonicalensembleformulationof statisticalther-modynamicswouldbeoflittlevalueasananalysistool.Fortunately,thereisafairly straightforward way to establish such a link.In our analysis of the canonical ensemble, we derived the followingrelation,which links thepartition functionQtotheinternal energy, entropy, and temperature. su - =-+lnQ. kBkBT (4.41 ) InChapter2wealsoderivedthefollowingequationforasysteminamicrocanonical ensemble that contains a binary mixture of distinguishable particles: (2.51 ) Comparing Eqs.(4.41) and (2.51), it is clear that both can bevalid for a system containing a binary mixture of distinguishable particles only if (4.164) 1404/ Other Ensemble Formulations The definitions of thepartition functions 00 Za=Lg;e-,/kBT, ;=0 00 Zb=Lgje-edkBT j=O (2.50a) (2.50b) are identical to that for the molecular partition functionsdefined by Eq.(4.154).Note that botharesummationsoverallaccessibleparticleormolecularenergylevels.It follows directly that Zb=qb Substituting Eqs.(4.165) and (4.166) into Eq.(4.164)and solving for Q yields Q=q;:" q:h(for distinguishable particles). ( 4.165) (4.166) (4.167) ThisresultisidenticaltoEq.(4.158)obtained earlier in thissectionbywritingthetotal system energy in thecanonical partition functionasthesum of individual energies forall particles in the system. InChapter 2 we also derived the following relation, which isvalid for indistinguishable bosons or fermions in the limit of dilute occupancy: SU{Z,,}{Zb} -=--+N"ln- +Nbln- +Na+Nb. kBkBTNaNb (2.78) Comparing Eqs.(4.41)and (2.78), it is clear that both can bevalid for a system containing a binary mixture of indistinguishable particles only if InQ=NaIn{ ~ : }+NbIn{ ~ : }+Na+Nb. (4.168) Rearranging theright side, Eq.(4.168) can bewritten in the form InQ=NaInZa+NbInZb - (NaInNa- Na) - (NbInNb- Nb).(4.169) Using Stirling's approximation in reverse, the terms in parentheses on the right side ofEq. (4.169)aregiven by NbInNb- Nb=InNb!. Substituting theright sides of Eqs.(4.170) and(4.171) into (4.169) yields InQ=NaIn Za+NbInZb-lnNa! -lnNb!. (4.170) (4.171) (4.172) Since Za=qaand Zb=qb, we can substitute and solve Eq. (4.172) to obtain the following relation forQ: (for indistinguishable particles). in thelimit of dilute occupancy (4.173) 4.5 I Distinguishability and Evaluation of the Partition FUllction141 Thus,byrequiringconsistencybetweentheresultsforthemicrocanonical ensembleand the canonical ensemble we have derived a relation between the molecular partition function fordistinguishablebosonsandforindistinguishablebosonsorfermionsinthelimitof dilute occupancy. Furthermore, if we compare Eq.(4.167) for distinguishable particles with Eq.(4.173), we note that Q for the binary mixture of indistinguishable particles in the high temperaturelimit islower by a factor of l/(Na!Nb!).This reflects thereduced number of distinguishable system microstates when theparticles are indistinguishable. It should be transparent from the form of Eq.(4.173) that if the system contains r species, the relation forQbecomes rN Q = r r ~ i=1Ni! (for indistinguishable particles) in thelimit of dilute occupancy, (4.174) whereqiandNiarethemolecularpartitionfunctionandnumber of particlesfortheith species. The importance of Eq.(4.173) cannot be overstated. It provides the link between molec-ular energy storage characteristics and thesystem partition function.Wenowhavea clear sequenceofstepstodeterminesystemthermodynamicpropertiesfromparticleenergy storage characteristics in the high temperature limit: (1)Determinethepartition functionsfortheindividual modes of energy storagefor each species of molecule or other particle. (2)Constructthemolecularpartitionfunctionforeachspeciesasqrnolecular= qtrqrotqvibqe' (3)Construct thesystem canonical partition function using Eq.(4.173)(or (4.174. (4)UsetherelationsderivedinSection4.2todeterminethermodynamicproperties fromthe canonical partition functionQ. It shouldbenotedthatEq.(4.173)andthemethodologydefinedinthestepsaboveare applicable if thesystem meets the followingcriteria: (1)The particles areindistinguishable. (2)The total system energy can be written as thesum of the energies associated with each particle. (3)The system exhibits dilute occupancy. Chapter5of thistextwillbedevotedtoapplyingthemethodologydefinedinthesteps above to systems containing different ideal gases. In closing this section, we take on the task of defining more precisely the high temperature limit in which dilute occupancy is attained. Based on the quantum analysis of a single proticle in abox, inChapter1 wefoundthat at moderateto highparticleenergies,thenumber of translational states with energy less than a specified valueE:was given by (4.175) If the mean translational energy of a particle in a system is(E:),then n (e)is a good estimate of thenumberof microstatesaccessibletotheparticle.Furthermore,inSection2.6we showed that in a system of particles that store energy only by translation, the mean energy 1424I Other Ensemble Formulations Table4.1Assessment oJ the DiluteOccupancy Criterion Jor Several System Types FluidT(K) Liquid helium4 Gaseous helium4 Gaseous helium20 Gaseous helium100 Liquid neon27 Gaseous neon100 Liquid argon86 Gaseous argon86 Electrons insodium300 per particle is related tothe temperature as (c)=~ k B T . Dilute occupancy implies that n,N. 6N(112)3/2 1l'V12mkBT 1.6 0.10 0.0020 4.0 x10-5 0.011 3.0 x10-6 5.1x10-4 2.0 x10-6 1,500 (4.176) Takingc=(3/2)kBTinEq.(4.l75)andcombiningEqs.(4.175)and(4.176)yieldsthe following criterion for dilute occupancy: 6N(h2 )3/2 - 1. lTV12mkBT (4.177) This condition is favored by high molecular mass, high temperature, and low particle density (N IV).It turns out that this condition is satisfied forall but thelightest molecules atlow temperatures (see the typical values in Table 4.1). Strictly speaking, the criterion described above was derived for systems containing par-ticles that store energy by translation only. However, the results are also reasonably accurate for most polyatomic molecules, since translational states usually account for much of their energy.Inany case,additional internalstoragemodes only increase thenumber of acces-siblemicrostatesandsoif asystemsatisfiestheinequalitycondition(4.177),additional internal modes will only further dilute the occupancy. Thus (4.177) is a sufficient condition for dilute occupancy aslong as the particles are freeto translate within thesystem. At high system temperatures, the mean energy per particle will be high and the separation of the quantum energy states will be small compared to the energy of the particle. As a result, theenergycan beconsidered tobea continuousvariable,and thebehavior of thesystem is accurately predicted by classical mechanics models.At high temperatures, the statistical behavior of such systems can also betreated by thestatistical mechanics model developed by Boltzmann, which is based on a classical mechanics view of energy storage in particles. For thatreason,inthelimit of diluteoccupancy,theparticles aresaid toobey Boltzmann statistics (for indistinguishable particles). Because their original formulation was based on aclassicalmechanicsviewof energystorage,theBoltzmannstatisticsbehaviorathigh Exercises143 temperatureisalsosometimescalledtheclassicallimit.Further discussionof ensemble theory and thediluteoccupancy limit can befound in References [2]-[5]. Exercises 4.1The followingrelation definesan approximate partition function for a pure densegas: I(27rfflkT)3N120 Q(N, V, T)=N!112B(V- Nb)NeaN-/nBT. In this relation, a and b are different constants for each molecular species. From this partition function,derivean equation of state of the formP=P (V,N, T). 4.2For asingle-speciessystemwecan takeNb=0,dropILbasa parameter,and dropthea subscritpts onNaand ILd'Equation (4.125) then becomes . aILV.T Use this result to derive Eq. (4.145) for a single-species system held at constant V, IL,and T. 4.3Show that Eq.(4.143) can bewritten in the form wherev isthemolar specific volumeinm3/kmol. 4.4Showthatforasingle-speciessystemheldatconstantV, N,andT,thevarianceof the energy isgiven by 4.5As noted in Chapter 3, for a single-component system, thevan der Waals equation of state can bewritten intheform 4.6 ,- 3D,- Iv;' whereP,=PIPe' T,=TITe,andv,=vlve,withPe, Te,andvebeingthepressure, temperature,andvolumeperkmolatthecriticalpoint.Forv,=I,plotthevariationof (O'N I N)(PcveN I NAkB Te)1/2withT,predicted bythevander Waalsequationof statefor I :5Tr:52.What isthe physical interpretation forthebehavior of thefluidasT,--+I? (a)At constant Tand V, the chemical potentials for a binary mixture are functions of the number of particles of each species: Show that thisimplies that

ONbV.T.NG 1444 I Other Ensemble Formulations (b)Abinary mixtureof twomonatomic ideal gases aand b hastemperatureof 300 K,a pressure of 50 kPa, and a concentration of species a of one part per trillion (i.e., a mole fraction of 10-12).Youtake a small sample of the binary mixture at one instant of time that hasavolumeof onecubicmillimeter.Assumingthatthefractionaluncertainty in themeasured concentration from this sample isequal to O'Nal Na,use theresults of part (a) to assess the accuracy of bulk concentration measurements from a sample this small. 4.7DeriveEq.(4.126) forthevarianceof N= Na+ Nbinabinarymixtureof twoparticle species aand b. 4.8For 0.018kgof steamat100C,usethevander Waalsequationtoestimatehow farthe pressuremust riseabovetheequilibrium saturation pressure before O'N INisof order one. 4.9For 0.00 I kmol of liquid nitrogen at120 K, use the van der Waals equation to estimate how far the pressure must fall below the equilibrium saturation pressure before O'N I Nis of order one. 4.10For saturated neon vapor at 27.1K and atmospheric pressure, estimate the probability that a heterophase fluctuation will raisethe density to that of saturated liquid in a spherical region withadiameter of 5 nm.Notethat neonisamonatomicsubstancewith amolecular mass of 20.18 kg/kmol and asaturated liquid density of 1,205 kg/m3 at 27.1K. 4.11Evaluatetheparameter 6N(h2)3/2 1l'V12mkBT forsteamat100Candassessthevalidityof thediluteoccupancyassumptionforthis system. 4.12In a gas of protons themean kinetic energy of each proton is1.04 x10-19 J.The mass of a proton is1.66x10-27 kg.Evaluate thedilute occupancy assumption for this system. 4.13Evaluatetheparameter 6N(h2)3/2 1l'V12mkBT forelectronsinacopperwireatroomtemperatureandassessthevalidityof thedilute occupancy assumption forthissystem. References [I]Gibbs, J.W.,Elementary Principles inStatistical Mechanics, reprinted by Ox Bow Press, Woodbridge, CT,1981. [2]Hill, T.L., An Introductionto Statistical Thermodynamics, Dover Publications, New York,1986. [3]Kittel, C.and Kroemer, H., Thermal Physics, 2nd ed., W.H.Freeman and Company, New York, 1980. [4]McQuarrie,D.A., Statistical Mechanics,Harper and Row,New York,1976. [5]Robertson, H.S., Statistical Thermophysics, Prentice-Hall, Englewood Cliffs, NJ,1993. CHAPTER5 Ideal Gases InChapter 5 weapply thestatistical theory developed in earlier chapters toideal gases.Weconsiderabinarymixtureof twogasspeciesandnotethatpurecomponent relationscanberecoveredbysettingthenumber of particlesof onespeciestozero.The necessary conditions for chemical equilibrium in reacting gas mixtures arealso examined. 5.1Energy Storage and the Molecular Partition Function We now wish to tum our attention to the problem of evaluating the partition function for a system containing a pure ideal gas. In doing so we will make use of the methodology de-veloped in Section 4.5 that relates the canonical partition function to the molecular partition function for indistinguishable particles under conditions of dilute occupancy. The major task in applying this analysis to systems of ideal gas molecules is to derive relations for the molec-ular paI1ition function from available information about energy storage in the molecules. Ingeneral, energy can bestored in a molecule as translational,rotational,or vibrational motion,and asa result of electronic or nuclear transitions.For thegases considered in this chapter, these energy storage mechanisms will be assumed to be independent. We therefore expect that we can decomposethe Hamiltonian for a polyatomic molecule into its various degrees of freedom Ii ~Ii translational+ Ii rotational+ Ii vibrational+ Ii electronic+ Ii nuclear (5.1) and,asaresult,themolecularpartitionfunctioncanbewrittenastheproduct of factors associated with each of the energy storage modes, (5.2) Toevaluateeachof thefactorsontheright sideof Eq.(5.2),wewillanalyze each of the energy storage modes that are active for a given type of molecule. Wewill begin in the next section by considering monatomic gases. We then consider more complicated diatomic and polyatomic molecules in subsequent sections. 5.2Ideal Monatomic Gases Wenowwillapplytheanalysistoolsdevelopedinthepreviouschaptertoan idealmonatomicgas."Ideal"impliesthatthegasisdiluteenoughthatintermolecular interactionscan beneglected except fortheir effect duringverybrief collisionprocesses betweenmolecules.Thisisgenerally trueformonatomicgasesatpressuresbelowafew atmospheres and temperatures greater than 20e. The number of independent energy storage modes is sometimes referred to as the number of degrees offreedom for the molecule. A monatomic gas atom has translational, electronic, and nuclear energy storage modes. To a very good approximation, the translational, nuclear, 145 1465 I Ideal Gases and electronic Hamiltonians are separable. Based on the results obtained in Section 4.S, this implies that (S.3) TheTranslational Partition Function In Section 2.6, we determined the partition function for a particle that stores energy by translation alone.The definition of thepartition functionZ"in that section isidentical to the definition ofqtf: 00 q _"" g'e-,,,,,/kBT t r - ~tf.1. (S.4) ;=0 It follows directly from the analysis presented in Section 2.6 that thetranslational partition function for a monatomic gas is given by (2rrmkBT)3/2 qtr=h2 V. Theabove relation can bewritten in theform V qtr=A3' where ( h2)1/2 A=2rrmkBT (S.S) (S.6) (S.7) The factor A that occurs in the translational partition function has units of length. The usual interpretation of Ais based on the following.The mean translational energy of a particle is computed as (S.8) With a little rearranging. it is easy to show that () _kT2(aOnqtf) etT- BaT. (S.9) Substituting theabove relation (S.S)for qtryields (etr)=( ~ )kBT. (S.10) Since etr=p2/2m, wherepistheparticle momentum. theaveragemomentum ispropor-tional to (mkBT)I/2. Thus A is essentially hlp. which is equal to the de Broglie wavelength associatedwiththethermalmotionof theparticle.Asaresult,Aistermedthethermal deBrogliewavelength.Notealsothattheconditionforthevalidityof diluteoccupancy, Eq.(4.177), can be stated in terms of Aas (S .11) 5.2 I Ideal Monatomic Gases147 Sincethefraction on theright side isclose to one(0.72), the criterion can be more simply stated as A3 VIN1. (5.12) This implies that the thermal deBroglie wavelength must besmall compared to themean volumeper molecule in thesystem containment.Notethat thisissimilar tothecondition that quantum effects decrease as the de Broglie wavelength becomes small compared to the physical system dimensions. Example 5.1Calculate thethermal deBroglie wavelength and assess theassumption of dilute occupancy for nitrogen gasat 300 K and101kPa. SolutionThe mass of a nitrogen molecule (N2)is 4.65x10-26 kg.Substituting into Eq.(5.7), we obtain ( 112)1/2((6.63X10-34)2)1/2 A=2rrmkBT=2rr(4.65x10-26)(1.38x10-23)300 =1.91x10-11 m. Treating thegasasideal, we can evaluate VI Nusing the ideal gas law VkBT NP Substituting thesystem temperature and pressure yields V(1.38x10-23)300 == 4.10X10-26 m3 (per molecule). N101,000 It followsfrom these results that A3 (1.91x10-11)3-7 -- == 1.70x10. VIN4.10x10-26 This ratio being much less than one implies that the dilute occupancy idealization is appro-priate for thissystem. The Electronic Partition Function Forouranalysishere,wewritetheelectronicpartitionfunctionasasumover energy levels: 00 qe=L ge.;e-e,.,/kBT,(5.13) ;=0 wherege,;isthedegeneracyandCe,;istheenergyof theith electroniclevel.Wefixthe arbitraryzeroof theenergysuchthatce,O=O.Indoingsoweeffectivelymeasureall electronic energy levels relative to the ground state: (5.14) 1485 I Ideal Gases where!">.ce,iOistheenergy of electroniclevel irelativetothelowest energy level(ground state), These !">.ce,iOterms are typically of the order of electron volts, In general, !">.ce,iO/ kB T is quite large at ordinary temperatures and only the first term in the summation is significantly different from zero, Consequently, we need retain only the first term in most cases, In a few cases,suchasforhalogen gases,thesecond and thirdlevelsare closetotheground state, For such cases, thesecond- and third-level telms should also be retained, For all monatomic gases, the electronic energy levels and degeneracies areknown from measurementsand/or quantum theorymodels,Hence,wewritetheelectronic partition in the form (5,15) Possibleinclusionof additionalhigher energyleveltermsneedstobeconsidered onlyat very small changes !">.ce,iOor at extremely high temperatures, The Nuclear Partition Function Because nuclear energy levels are separated by millions of electron volts, at room temperature only the first term in a summation for qnuclneed be considered: (5.16) where gnucl,Ois the degeneracy of thelowest energy leveL Properties Assemblingtheparts,wehavethefollowingrelationforthemolecular partition functionfor a monatomic ideal gas: 3/2 (2rrmkBT)(-M/kT) =h2 Vge,o+ ge.1 e,.10Bgnucl,O'(5.17) Applying Eq,(4,174) we obtain the followingrelation for the canonical partition function: _12rrmkBT-D.e,lO/kBT [ 3/21N Q- N!(h2)V(ge,o+ge.1e )gnucLO (5,18) Wewillusuallyfindthatitisappropriatetotakegnucl,O= 1,Wewillthereforedosoin evaluating theproperties of thegas,It followsdirectly fromour analysisof thecanonical ensemble that U=kBT2--- =-NkBT + ---='---e,'-----------'e,'---'-------;-_:-;---;;,-(a(ln Q))3Ngl!">.cloe-D.e,.lO/kBT aTNY2ge,o+ ge.1 e-D.e,.IO/kBT' (5,19) (ann Q)) S=kBlnQ +kBT--aTNY { (2rrmkBT )3/2ve5/2} = NkBIn2-- + Se, hN (5.20) 5.3 I Ideal Diatomic Gases149 where (5.21) Equation (5.20) isreferred to asthe Sacku/'-Tetrode equation. Wealso can obtain a relation for thepressure P= kBT(a(lnQ) aTN.T NkBT V (5.22) which, not surprisingly, is the ideal gas equation of state. Note that inclusion of the electronic storage terms does not affect this equation of state. 5.3Ideal Diatomic Gases Asanextstep,wewillconsideranidealgascomposedof diatomicmolecules. Inaddition totranslational and electronic degrees of freedom,diatomic molecules possess vibrationalandrotationaldegreesof freedomaswell.Diatomicmoleculesrepresentthe next step upwardincomplexity aboveamonatomicgas.Many technologicallyandenvi-ronmentally important gases are diatomic. Examples include H2,N2,O2,CO, NO, Cl2,and HCl. A general procedure to determine the energy levels of a diatomic molecule would beto set up the Schrodinger equation for the two nuclei and n electrons and solve it for the set of eigenvalues of the diatomic molecule. Exact application of this method has been done for H2, but it becomes increasingly difficult asthenumber of electrons in themolecule increases. Approximate methodologies have been developed to circumvent difficulties associated with fullsolution of theSchrodinger equation.One way of determining electron orbit