1. 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.

2. Reading quiz Which of the following was a “macrostate” listed in the marble example? a. a.1 green, 3 red b. b.50 red, 50 green c. c.red, red, green, red d. d.the number 6

3. Reading quiz Which of the following is the equation that relates the # microstates (W) to entropy (S)? a. a.S = k B e W b. b.S = k B lnW c. c.S = tan(W) d. d.S = tan -1 (W) e. e.S = W

4. Reading quiz Which of the following is the factor that tells you how likely states are to be occupied at a given temperature? a. a.Boltzmann factor b. b.Einstein factor c. c.Fermi factor d. d.Maxwell factor e. e.Plank factor

5. Microstates vs Macrostates Reminder a. a.Left microstate: part of the “royal flush” macrostate b. b.Right microstate: part of the “garbage” macrostate

6. 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?

7. MANY Dice You roll 10 23 dice with your left hand. How many microstates are there? You roll 10 23 dice with your right hand. How many microstates are there? How many microstates are there in the COMBINED system? Isn’t this ridiculous?

8. Solution: Use logarithms S = some constant  ln(#microstates) [units of J/K] a. a.Much more manageable numbers. b. b.Combining two systems: S tot = C  ln(#microstates1  #microstates2) = C  ln(#microstates1) + C  ln(#microstates2) = S 1 + S 2 c. c.2 nd Law: System in macrostate with most microstates  System in macrostate with largest S

9. System and Reservoir System: E 1 Large reservoir: E 2 E tot = E 1 + E 2 (const. volume so no work) 0 = dE 1 + dE 2 Want to maximize S: take dS/dE 1, set = 0

10. Temperature dS/dE is the same for two systems in thermal contact! Temperature is also the same for two systems in thermal contact! a. a.dS/dE has units of 1/K, so… This “works” if the constant is chosen properly: S = k B ln(#microstates) Compare to We are assuming no work, so dE=dQ (First Law)

11. Small system with 2 possible energies: E 1A vs E 1B (1 microstate each) Probability of system 1 being in state A vs state B? P 1A ~ (#microstates of system 1 having energy E 1A )  (#microstates of system 2 having energy E 2A = E – E 1A ) Let #microstates of E 1A = 1 for now. Same thing for state 1B…

12. Math… Result: “of”, not “times” Read on your own time

13. The 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 2  10 -23 J. If the temperature is 1.5 K, what is the probability the atom is in the lower state? “Boltzmann Factor”

14. Maxwell-Boltzmann Velocity Distribution E = ½mv 2 What’s probability of having speed 5 vs speed 10? Multiplicities  (Number of states with speed v) ~ v 2

15. Maxwell-Boltzmann Velocity Distribution The result:  Exactly the equation given for the velocity distribution in your textbook! (after you do the integral with, e.g. Mathematica)