Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Lecture 38: Review for 3Lecture 38: Review for 3rdrd Test TestChapters 8-10, Wednesday April 16Chapters 8-10, Wednesday April 16thth•Finish Ch. 10: Heat capacity of Bose gas•Quiz•ReviewReading: Reading: Chapters 8-10 (pages 160 - 207)Chapters 8-10 (pages 160 - 207)Homework 10 due Homework 10 due todaytoday at 5pm at 5pmAssigned problems, Assigned problems, Ch. 10Ch. 10: 4, 6, 8, 10, : 4, 6, 8, 10, 1818Exam 3 on Friday (18th), Chs 8-10Exam 3 on Friday (18th), Chs 8-10Fermi/Bose distribution functionsFermi/Bose distribution functions( )( )( ) /( ) /1( )11( )1BBFk k TBk k Tn kef kee me m--=+=-( )/Bk Te m-kn0 1 2 3 4-4-20 T lnTTemperature (T/Tc)TT-dependence of chemical potential-dependence of chemical potential( )( ) /1( )1BBk k Tf kee m-=-22 / 33.31B ck T nm= �hHeat capacity of a Bose gasHeat capacity of a Bose gasT 3/2BE CondensationBE Condensation3/ 203/ 203/ 21154ccV BcTN N NTTN NTTC NkT� �- =� �� �� �� �� �= -� �� �� �� ��� �=� �� �Review of Review of Chapters 8-10Chapters 8-10Black-body spectrumBlack-body spectrumBlack-body spectrumBlack-body spectrumWien's displacement LawWien's displacement Lawmm TT = constant = 2.898 = constant = 2.898 ×× 10 1033 m.K, or m.K, or mm  TT11 Found empirically by Joseph Stefan (1879); later calculated by Boltzmann = 5.6705 × 108 W.m2.K4.Stefan-Boltzmann LawStefan-Boltzmann LawPower per unit area radiated by black-bodyPower per unit area radiated by black-bodyRR = = TT 44Rayleigh-Jeans equationRayleigh-Jeans equationuu(())dd = (# modes in cavity in range = (# modes in cavity in range dd) ) × × (average energy of modes)(average energy of modes)Define spectral energy distribution such that u()d is the fraction of energy per unit volume in the cavity with wavelengths in the range  to  + d.Density of modesDensity of modes( )148; Average modeenergyBn U k Tpll= =( ) ( ) ( )148BBk Tu n U n k Tpl l ll= = =Wien, Rayleigh-Jeans and Planck Wien, Rayleigh-Jeans and Planck distributionsdistributions( ) ( ) ( )( )/RJ W P/4 5 58 8; ;1BTBhc k Tk T e hcu u ueb llp pl l ll l l-= � =-Maxwell-Boltzmann statisticsMaxwell-Boltzmann statisticsDefine energy distribution function:( ) ( ) ( )0exp / ,such that 1Bf A k T f de e e e�= - =�Then,( )0 0exp( / )B Bf d A k T d k Te e e e e e e� �= = - =� �This is simply the result that Rayleigh and others used, i.e. the average energy of a classical harmonic oscillator is kBT, regardless of its frequency.Planck Planck postulatedpostulated that the energies of harmonic oscillators could that the energies of harmonic oscillators could only take on discrete values equal to multiples of a fundamental only take on discrete values equal to multiples of a fundamental energy energy  = = hh, where , where  is the frequency of the harmonic oscillator, is the frequency of the harmonic oscillator, i.e.i.e. 0, 0, , 2, 2, 3, 3, , etc.etc.......Then,Then,UUnn = = nnnhnh = = 0, 1, 2...0, 1, 2...Where Where nn is the number of modes excited with frequency is the number of modes excited with frequency . Although . Although Planck knew of no physical reason for doing this, he is credited with Planck knew of no physical reason for doing this, he is credited with the birth of quantum mechanics.the birth of quantum mechanics.The new quantum statisticsThe new quantum statistics( ) ( )exp / exp /n n B Bf A U k T A nh k Tn= - = -Replace the continuous integrals with a discrete sums:( )0 0exp /n n Bn nU U f nh A nh k Tn n� �= == = � -� �( )0 0exp / 1n Bn nf A nh k Tn� �= == - =� �Solving these equations together, one obtains:( ) ( ) ( )/exp / 1 exp / 1 exp / 1B B Bh hcUk T h k T hc k Te n le n l= = =- - -Multiplying by D(), to give....( )58( )exp / 1Bhcuhc k Tpll l=-This is Planck's lawThis is Planck's lawEinstein and Debye models of vibrations in Einstein and Debye models of vibrations in solidssolidsAtom3N modes of vibration; N = number of atomsEinstein model of vibrations in solidsEinstein model of vibrations in solids•Assumes 3Assumes 3NN independent modes of vibration independent modes of vibration•All modes have same frequency, All modes have same frequency, EEE is the only freeparameter in the model( )2/2/31E BE Bk TEV Bk TBeC Nkk Tewww� �=� �� �-hhh( )2 4 33 31/ 322, as 056 /BVDB D Dk T VC Tss N Vkpw pq w= �==hhs is average sound speed; the only freeparameter in modelDebye model of vibrations in solidsDebye model of vibrations in solidsReminder: The combined 1Reminder: The combined 1stst and 2 and 2ndnd LawsLaws, , ,, , ,; ;V N S N S VV N S N S VdU TdS PdV dNU U UdS dV dNS V NU U UT PS V Nmm= - +� � �� � � � � �= + +� � � � � �� � �� � � � � �� � �� � � � � �� = - = =� � � � � �� � �� � � � � �If more than one type of particle (If more than one type of particle (constituentconstituent) ) is added, thenis added, then1, ,1mj jjjjS V NkdU TdS PdV dNUk m jNmm== - +� ��= = � �� �� ��� ��Microcanonical ensembleMicrocanonical ensembleE, V and N fixedS = kB lnW(E,V,N)Canonical ensembleCanonical ensembleT, V and N fixedF = kBT lnZ(T,V,N)Grand canonical ensembleGrand canonical ensembleT, V and  fixed = kBT ln (T,V,)Grand canonical ensembleGrand canonical ensembleGrand partition function:( )( )/. .j j BE N k Tj j jje i e E E Nmm- -� �X = � -� ��( )( )( )/ //i i B i i Bj j BE N k T E N k TiE N k Tje epem mm- - - -- -= =X�Grand canonical ensembleGrand canonical ensembleGrand potential:( )lnG Bk T U N TS F Nm mF =- X = - - = -Gd PdV Nd SdTm� F =- - -( )( )( ),,,,,,lnlnlnBGVVBGTTBGT VT Vk TST Tk TPV Vk TNmmmmm m� X� ��F� �=- =� �� �� �� �� �� X� ��F� �=- =� �� �� �� �� �� X� �� ��F=- =� �� �� �� �� �0 1 2 3 4-4-20 T lnTTemperature (T/TQ)TT-dependence of chemical potential-dependence of chemical potential3/ 23 2ln12degeneracyBQBQDnk Tgnmk Tngml p� �=� �� �� �� �= =� �� �=hFor