Lecture 17 Lecture 17 ——The Canonical Ensemble The Canonical Ensemble Chapter 6, Chapter 6, Monday February 18Monday February 18thth•Single-particle in a box (quantum mechanics)•3D particle in a box•Factorizing the partition function•Equipartition theorem•Rotational energy levels in diatomic moleculesReading: 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, 22Quantum mechanicsQuantum mechanicsThe oneThe one--dimensional, timedimensional, time--independent Schrindependent Schröödinger equation:dinger equation:()()() ()2222xVx x xmxφφεφ∂−+=∂=Kinetic energyKinetic energyPotential energyPotential energyTotalTotalenergyenergy2ˆˆ.. ;2xxpKE p imx∂==−∂=()2222Vmxφεφ∂−=−∂=For a region of constant potential:A single particle in a oneA single particle in a one--dimensional boxdimensional boxV(x)V = ∞ V = 0V = ∞xx = LsinnnxALπφ⎛⎞=⎜⎟⎝⎠0φ=0φ=222222nnnmLπεα⎛⎞==⎜⎟⎝⎠=The threeThe three--dimensional, timedimensional, time--independent Schrindependent Schröödinger equation:dinger equation:()()()()22,, ,, ,, ,,2xyz V xyz xyz xyzmφφεφ−∇ + ==2222222xyz∂∂∂∇= + +∂∂∂A single particle in a threeA single particle in a three--dimensional boxdimensional box()12 3123, , sin sin sin , ,inx ny nzxyzA nnnLLLπππφ⎛⎞⎛ ⎞⎛⎞==⎜⎟⎜ ⎟⎜⎟⎝⎠⎝ ⎠⎝⎠()2222 212 32, 1,2,3...2ninn n nmLπε⎛⎞=++=⎜⎟⎝⎠=Factorizing the partition functionFactorizing the partition function()12322 2trans 1 2 3111expnnnZnn nγ∞∞∞===⎡⎤=−++⎣⎦∑∑∑222312123222312123222312trans11111112300 0nnnnnnnnnnnnnnnZeeeeeeedn edn ednγγγγγγγγγ∞∞∞−−−===∞∞∞−−−===∞∞ ∞−−−=⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠∑∑∑∑∑∑∫∫∫Factorizing the partition functionFactorizing the partition function222312123222312123222312trans11111112300 03/3/2233/223 222nnnnnnnnnnnnnnnBBDZeeeeeeedn edn ednmk T V mk LLTγγγγγγγγγπλ π∞∞∞−−−===∞∞∞−−−===∞∞ ∞−−−=⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎛⎞⎛⎞===⎜⎟⎜⎟⎝⎠⎝⎠∑∑∑∑∑∑∫∫∫==2Equipartition theoremEquipartition theorem222312123trans 1 2 3111nnnnnnZeee ZZZγγγ∞∞∞−−−=====××∑∑∑If the energy can be written as a sum of independent terms, then the partition function can be written as a product of the partition functions due to each contribution to the energy.() ()123 123ln lnNNNZZZZ Z N ZZZ=×× ⇒ =()123ln ln lnBFNk T Z Z Z=− + +free energy may be written as a sum. It is in this way that each degree of freedom ends up contributing 1/2kBto the heat capacity.() ( )123 1 2 3ln ln ln lnBBFkT ZZZ kT Z Z Z=− =− + +Also,Rotational 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 H2θR(K) 309 1280 4300
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