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Download Ch18 The Micro/Macro Connection
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講者: 許永昌 老師 1 Contents Molecular Speeds and Collisions Pressure Temperature Thermal energy and Specific heat Thermal interaction and Heat Irreversible Processes & the 2nd Law of thermodynamics. 2 The aim of this chapter Understand Collision Average translational Pressure Temperature Energy transfer kinetic energy Microscopic energies Molecular basis Probability Thermal energy Ideal-gas law Specific Heat Heat and Thermal equilibrium Entropy 3 Molecule Speeds (請預讀P542) Changing the (1) or changing to a (2) changes the does distribution. P v dv Probability where e Boltzmann distribution , but it of the * v 2 dv states 1 1 and v mv 2 . k BT 2 4 Mean Free Path (請預讀P543) The average speed of N2 at 20oC is about 500 m/s ~0.1 s (of course not) ~10m There must be some collisions happened. Reference: http://en.wikipedia.org/wiki/Diffusion 5 Mean Free Path (continue) N coll N / V Vcyl N / V 2r 2 L area assume that all other molecules are fixed L 1 N coll 4 N / V r 2 More careful calculation (all molecules move) 1 4 2 N / V r 2 6 Stop to Think What would happen in the room if the molecules of the gas were not moving? What would happen in an isolated room if the molecular collisions were not perfectly elastic? 7 Pressure in a Gas (請預讀P544~P545) Objects: A wall whose normal is x direction. Molecules in the left hand side of this wall. Condition: perfectly elastic. J wall on molecule 2mvx xˆ = Favg N t J i i i N / V Av ix t * p vi vix * 2mvix xˆ + +… velocity: vi # of particles collide with the wall: Ni t i t Favg N N P 2mvix2 p vi vix A V i V t N mv p v V m v 2 ix i i where p v is the probability of a particle whose velocity is v , 2 x avg ,if p vi p vi . p v 1 i i 1, and x 0, x0 x0 , respectively. 8 The Root-Mean-Square Speed (請預讀P545~P546) vrms v 2 v 2 avg avg vi2 p vi i vix2 p vi viy2 p vi viz2 p vi i vx2 avg 3 vx2 v y2 i avg vz2 i avg avg assume p Rvi p vi , R Rotation Group 9 Stop to Think & Exercise What are the definitions of 1. rms speed 2. Average speed 3. Average velocity What are the benefits of the definition of rms speed? : 2 particles: v=3êx+êy, 3 particles: v=2êx+2êy, 4 particles: v=2êy. Find: (1) rms speed (2) average speed (3) average velocity. 10 Homework Student workbook: 18.5, 18.6 11 Temperature (請預讀P546~P548) Microscopic avg Macroscopic T. Average translational kinetic energy: avg(½mv2)avg. Kinetic Theory: PV=Nm(vx2)avg. (v2)avg=3(vx2)avg. Ideal-gas Law: PV=NkBT. 3 2 We get avg k BT 12 Temperature (continue) avg 3 k BT 2 For a gas, this thing we call measures the . This concept of temperature also gives meaning to as the temperature at which avg=0 and all molecular motion ceases. 13 Homework Student workbook: 18.10 14 Thermal Energy for monatomic Gases (請預讀P549~P550) Eth=Kmicro+Umicro. For monatomic gases: Eth=Kmicro. Eth=Navg=3/2NkBT=3/2nRT. Owing to the 1st Law of thermodynamics, Eth QV nCV T We get CV=3/2R=12.5 J/mol K. Q: How about other systems? 15 The Equipartition Theorem (請預讀 P550~P551) The thermal energy of a system of particles is equally divided among . For a system of N particles at temperature T, the energy stored in each mode (each degree of freedom) is ½NkBT. It is not proved here. To prove it, you need the concepts of Probability States Phase space Boltzmann distribution Example: Solid (for high enough temperature) Dulong-Petit law Detail: Solid State Physics. It has 6 degrees of freedom Eth=6*N*½kBT C=3R=25.0 J/mol K~6.00 cal/mol K. *Solid State Physics, Ashcroft/Mermin, P463 16 Specific Heat of diatomic molecules (請預讀P552~P553) micro 2 2 L L2x L 1 1 1 1 1 y 2 2 2 z MvCM , x MvCM , y MvCM , z v122 k r122 2 2 2 2 I zz 2 I yy 2 I xx 2 2 avg Why? rotational kinetic energy vibrational energy A: In quantum mechanics, 1. <Li>=nћ 2. Discrete energy levels for bounded states. 17 Additional Remark (補充) For two particles system 1 1 m1v12 m2v22 2 2 1 1 2 MvCM v122 , where M m m , v v v , m1m2 . 1 2 12 2 1 2 2 M 2 K tot <rot,z>~ mH=1.66*10-27 kg, r~3.7*10-11 m, ћ=1.05*10-34 Js, kB=1.38*10-23 J/K Teff~180K (n=1) 1 Lz 1 n2 2 1 k BTeff 2 2 Iz 2 2mH r 2 18 Thermal Interactions and Heat (請預讀P554~P555) System 1 Th System 2 Heat Tc Thermal interaction Thermal Equilibrium: Collisions Thermal Equilibrium: (1)avg= (2)avg. T1=T2. Energy can transfer Energy can transfer from 2 to 1: from 2 to 1: Probability 19 Exercise Conditions: System 1: 4.00 mol N2 at T1=27oC. System 2: 1.00 mol H2 at T2=327oC. 3 s.f. Find: Thermal energies Tf =? Heat transfer=? 20 Homework Student Workbook: 18.13, 18.15, 18.16 21 Irreversible Processes and the 2nd Law of Thermodynamics (請預讀P556~P558) (irreversible) (reversible) One particle Probability Many particles Go to reach Equilibrium. 22 Order, Disorder and Entropy (請預 讀P558~P560) Scientists and engineers use a state variable called entropy to that a macroscopic state . The second Law of thermodynamics: The entropy of an . The entropy either increases, until the system reaches equilibrium, or, if the system began in equilibrium, stays the same. Th Tc spontaneously. (Heat) 23 Homework Student Workbook: 18.18 Student Textbook: 15, 65 製作Terms and Notation的卡片,以方便自我練習。 24