Announcements 2/4/11 Exam starts Tuesday, goes until next Tuesday (late fee on last day after 5 pm) Exam review session: today, 3-4:30 pm, room C261 Come with questions! On Monday I will talk a little about what’s on the exam. Reading quiz Which of the following was a “macrostate” listed in the marble example? a. 1 green, 3 red b. 50 red, 50 green c. red, red, green, red d. the number 6 Reading quiz Which of the following is the equation that relates the # microstates (W) to entropy (S)? a. S = kB eW b. S = kB lnW c. S = tan(W) d. S = tan-1(W) e. S = W Reading quiz Which of the following is the factor that tells you how likely states are to be occupied at a given temperature? a. Boltzmann factor b. Einstein factor c. Fermi factor d. Maxwell factor e. Plank factor Microstates vs Macrostates Reminder a. Left microstate: part of the “royal flush” macrostate b. Right microstate: part of the “garbage” macrostate Dice You roll two dice. What are the microstates? (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),… How many microstates are there? How does that compare to the number of microstates for rolling one die? How many microstates if we roll 3 dice? What are the macrostates for 2 dice? (sum of numbers) What is the most likely macrostate? MANY Dice You roll 1023 dice with your left hand. How many microstates are there? You roll 1023 dice with your right hand. How many microstates are there? How many microstates are there in the COMBINED system? Isn’t this ridiculous? Solution: Use logarithms S = some constant ln(#microstates) [units of J/K] a. Much more manageable numbers. b. Combining two systems: Stot = C ln(#microstates1 #microstates2) = C ln(#microstates1) + C ln(#microstates2) = S1 + S2 c. 2nd Law: System in macrostate with most microstates System in macrostate with largest S System and Reservoir System: E1 Large reservoir: E2 Etot = E1 + E2 (const. volume so no work) 0 = dE1 + dE2 Want to maximize S: take dS/dE1, set = 0 dS d S1 S2 dE1 dE1 dS1 dS 2 dE1 dE1 dS1 dS 2 dE1 dE2 dS1 dS 2 dE1 dE2 Temperature dS/dE is the same for two systems in thermal contact! Temperature is also the same for two systems in thermal contact! a. dS/dE has units of 1/K, so… Compare to dS 1 dE T dQ dS T We are assuming no work, so dE=dQ (First Law) This “works” if the constant is chosen properly: S = kB ln(#microstates) Small system with 2 possible energies: E1A vs E1B (1 microstate each) Probability of system 1 being in state A vs state B? P1A ~ (#microstates of system 1 having energy E1A) (#microstates of system 2 having energy E2A = E – E1A) Let #microstates of E1A = 1 for now. Same thing for state 1B… P1A # microstates of 2 for case A P1B # microstates of 2 for case B e S2 A S e 2B kB kB Math… P1A # microstates of 2 for case A e S2 A S P1B # microstates of 2 for case B e 2 B kB kB S 2 A S 2 ( E E1 A ) “of”, not “times” dS 2 S2 ( E ) E1 A dE E1 A S2 ( E ) T (same with S 2 B ) E1 A / k BT P1A e Result: E /k T P1B e 1B B The Boltzmann Factor P(beinginstate with energy E) ~ e E / kBT “Boltzmann Factor” Prob ~ BF; Prob = BF/(sum of all BFs) Worked Problem: Suppose an atom has only two available energy levels, which are separated by 210-23 J. If the temperature is 1.5 K, what is the probability the atom is in the lower state? Maxwell-Boltzmann Velocity Distribution E = ½mv2 What’s probability of having speed 5 vs speed 10? P1A e E1 A / kBT E /k T P1B e 1B B v 2 1 mv 2 / k BT e 2 1 mv A2 / k BT 2 2 e P1 A v A 1 P1B mvB 2 / k BT 2 2 vB e Multiplicities (Number of states with speed v) ~ v2 Maxwell-Boltzmann Velocity Distribution The result: P(speed v) v 2 1 mv 2 / k BT e 2 1 2 mv / k BT 2 2 v e 1 2 mv / k BT 2 2 v e dv 0 Exactly the equation given for the velocity distribution in your textbook! (after you do the integral with, e.g. Mathematica)