Lecture 31Lecture 31——Planck & Grand canonical ensemblePlanck & Grand canonical ensembleChapters 8/9, Monday March 31Chapters 8/9, Monday March 31stst• Einstein and Debye models of vibrations in solids• Chapter 9• Systems with variable particle number• Conditions for equilibriumReading: Reading: All of chapters 8 and 9 (pages 160 All of chapters 8 and 9 (pages 160 --207)207)Homework 8 due today at 5pm!Homework 8 due today at 5pm!Assigned problems, Assigned problems, Ch. 8Ch. 8: 2, 6, 8, 10, 12: 2, 6, 8, 10, 12Next assignment on Wed. (due Wed. 9th)Next assignment on Wed. (due Wed. 9th)Einstein and Debye models of vibrations in solidsEinstein and Debye models of vibrations in solidsAtom3N modes of vibration; N = number of atomsEinstein model of vibrations in solidsEinstein model of vibrations in solids••Assumes 3Assumes 3NNindependent modes of vibrationindependent modes of vibration••All modes have same frequency, All modes have same frequency, ωωEEωEis the only freeparameter in the modelDebye model of vibrations in solidsDebye model of vibrations in solidsWaves (phonons) travel at the speed of soundTreats lattice vibrations as elastic wavesTreats lattice vibrations as elastic waves••Distribution of frequencies (unlike Einstein), i.e. dispersion.Distribution of frequencies (unlike Einstein), i.e. dispersion.••There is a cutThere is a cut--off frequency off frequency ωωDD..Debye model of vibrations in solidsDebye model of vibrations in solids()1/326/DsNVωπ=s is average sound speed; the only freeparameter in modelDebye model of vibrations in solidsDebye model of vibrations in solids()1/326/DsNVωπ=Chapter 9: systems with variable #s of particlesChapter 9: systems with variable #s of particles1ABdNdN=−Chemical reaction:CD CD+
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