Lecture 18 Lecture 18 ——The Canonical Ensemble The Canonical Ensemble Chapter 6, Chapter 6, Wednesday February 20Wednesday February 20thth•Rotational energy levels in diatomic molecules•Vibrational energy levels in diatomic molecules•More on the equipartition theoremReading: Reading: All of chapter 5 (pages 91 All of chapter 5 (pages 91 --123)123)Homework 5 due next Friday (22nd)Homework 5 due next Friday (22nd)Homework assignments available on web pageHomework assignments available on web pageAssigned problems, Assigned problems, Ch. 5Ch. 5: 8, 14, 16, 18, 22: 8, 14, 16, 18, 22Rotational energy levels for diatomic moleculesRotational energy levels for diatomic molecules()21221llllIglε=+=+=I = momentof inertial = 0, 1, 2... is angular momentum quantum numberCO2I2HI HCl H2θR(K) 0.56 0.053 9.4 15.3 88Vibrational energy levels for diatomic moleculesVibrational energy levels for diatomic molecules()12nnεω=+=ω= naturalfrequency ofvibrationn = 0, 1, 2... (harmonic quantum number)I2F2HCl H2θV(K) 309 1280 4300 6330ωSpecific heat at constant pressure for HSpecific heat at constant pressure for H22CP(J.mol−1.K−1)ω52R72R92RHH22boilsboilsTranslationTranslationCCPP= = CCVV+ + nRnRMore on the equipartition theoremMore on the equipartition theoremClassical uncertainty:V(x)V = ∞ V = 0V = ∞xx = LWhere is the particle?Where is the particle?WW= 9= 9More on the equipartition theoremMore on the equipartition theoremClassical uncertainty:V(x)V = ∞ V = 0V = ∞xx = LWhere is the particle?Where is the particle?WW= 18= 18More on the equipartition theoremMore on the equipartition theoremClassical uncertainty:V(x)V = ∞ V = 0V = ∞xx = LWhere is the particle?Where is the particle?WW= 36= 36More on the equipartition theoremMore on the equipartition theoremClassical uncertainty:V(x)V = ∞ V = 0V = ∞xx = LWhere is the particle?Where is the particle?WW= = ∞∞SS= = ∞∞More on the equipartition theorem: phase spaceMore on the equipartition theorem: phase spacexpxdxdpxArea hCell:(x,px)More on the equipartition theorem: phase spaceMore on the equipartition theorem: phase spacexpxdxdpxArea hCell:(x,px)More on the equipartition theorem: phase spaceMore on the equipartition theorem: phase spacexpxdxdpxArea hCell:(x,px),xxxpdxdpdWh=More on the equipartition theorem: phase spaceMore on the equipartition theorem: phase space33,33xyzrpdxdydzdp dp dpdrdpdWhh==GGIn 3D:dxdpxdxdpx= hUncertainty relation:Examples of degrees of freedom:Examples of degrees of freedom:()()221122221122222322211,22112211221212average, or r.m.s. valueLC B BHO B Btrans x y z Brot dia x y B BECV Li kT kTEk x mv kT kTE mvvv kTEIkTkTωω=+=+=+
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