Spring 2010PHYS 172: Modern Mechanics Lecture 23 ? Entropy: limits of the possible Read 12.1-12.3 EXAM 3 TOMORROW Time: 8:00-10:00 pm Wednesday April 14 Place: Elliott Hall Material: lectures 16-22, HW 15-22 (chapters 9-11 in the book) Equation sheet: make your own, two-sided ?letter? size page Sample exam: posted Note: no lecture on April 15 Statistical issue: Thermal energy flow Why when you add ice into your drink it becomes colder and not hotter? Statistical issue: Reversibility Movie taken and modified from: www.theclam.com/Video/JumpingIndex.asp A statistical model of solids Spring-ball model of solid How probable is certain distribution of speeds (energies)? 5 Einstein?s model of solid (1907) Each atom in a solid is connected to immovable walls 2 2 2 2 x y zp p p p= + + 2 2 2 2 x y zs s s s= + + Energy: 22 2 2 2 21 1 1 0 0 02 2 22 2 2 yx z vib s s x s y s z pp pK U k s U k s U k s U m m m ? ?? ? ? ?+ = + + + + + + + + ? ?? ? ? ?? ? ? ?? ? ? ? 3D oscillator: can separate motion into x,y,z components Each component: energy is quantized in the same way Block containing N 1D-oscillators is equivalent to a block with N/3 3D-oscillators Distributing energy: 4 quanta sk mplanckover2pi Analogy: distributing 4 dollars among 3 pockets: 1. Can have all 4$ in one pocket: 3 ways 1. Can have all 4 quanta in one oscillator - 3 ways Distributing energy: 4 quanta 3 ways: 4-0-0 quanta 6 more ways: 2-2-0 quanta 1-1-2 quanta 6 ways: 3-1-0 quanta 15 microstates: The same macrostate 3 3 3 6 The fundamental assumption of statistical mechanics Each microstate corresponding to a given macrostate is equally probable. Macrostate ? total energy Microstate ? microscopic distribution of energy Distributing energy 3 ways: 4-0-0 quanta 6 ways: 3-1-0 quanta 15 microstates: the same macrostate All microstates are equally probable 6 more ways: 2-2-0 quanta 1-1-2 quanta Interacting atoms: 4 quanta and 2 atoms q1 (# of ways 1) q2 (# of ways 2) (# of ways 1) ×(# of ways 2) 0 (1) 4 (15) 15 1 (3) 3 (10) 30 2 (6) 2 (6) 36 3 (10) 1 (3) 30 4 (15) 0 (1) 15 Which state is the most probable? Macroscopic blocks: too many atoms Need formula to calculate probabilities Total: 126 3 POCKETS 3 POCKETS 4 dollar bills Arrangement: microstates # ways ? to arrange q quanta among N one-dimensional oscillators ( ) ( ) 1 ! ! 1 ! q N q N + ?? = ? Derivation: see pp. 479 Reminder: N! = 1×2×3×?×N 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 10! = 3,628,800 20! = 8,828,514,807,271,391,232,000,000 ~ 1025 Large number of atoms Suppose there 100 quanta and 300 oscillators (100 atoms) ( ) ( ) 961 ! 1.7 10 ! 1 ! q N q N + ?? = = × ? Macroscopic block of matter: N ~ 1023 CLICKER: If there are 300 oscillators, what is the # of ways to have all 100 quanta on any one oscillator? A) 0 B) 1 C) 300 D) 30,000 E) 1.7×1076 100 shirts ? 300 pockets 100 dollars 13 Thermal equilibrium of two blocks in contact + total number of energy quanta = 100 14 Thermal equilibrium of two blocks in contact 300 300 60% 300 200 500= =+ 15 Width of the distribution Fractional width or , whichever is larger 1~ q 1~ N For large systems: 1223 1 1~ ~ ~10 6 10AN ? × The most probable is the only real possibility! sergei Microsoft PowerPoint - Lect23-Phys172s10-(12.1-12.3-Entropy).ppt