Announcements 9/28/11 A word about HW 13… Exam 1 starts Saturday morning, goes until Thursday evening On Friday during class I will talk a bit about what to expect for the exam (unless we have time today). Exam review session: Friday, 4 pm, room C460  Come with questions! (HW, optional HW, old exams, etc.) Pearls Before Swine

Reading quiz Which of the following was a “macrostate” listed in the marble example? 1 green, 3 red 50 red, 50 green red, red, green, red the number 100

Reading quiz Which of the following is the equation that relates the # microstates (W) to entropy (S)? S = CV W S = kB eW S = kB lnW S = tan(W) S = tan-1(W)

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

Microstates vs Macrostates Reminder Left microstate: part of the “royal flush” macrostate Right microstate: part of the “garbage” macrostate The most common macrostates are those with the most microstates

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 in the COMBINED system? Isn’t this ridiculous?

Solution: Use logarithms S = some constant  ln(#microstates) [units of J/K] Much more manageable numbers. Combining two systems: Stot = C  ln(#microstates1  #microstates2) = C  ln(#microstates1) + C  ln(#microstates2) = S1 + S2 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

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

Small system with 2 possible energies: E1A vs E1B 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… (1 microstate each)

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

The Boltzmann Factor Prob is proportional to BF, but not equal  Must normalize: 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?

Maxwell-Boltzmann Velocity Distribution E = ½mv2 What’s probability of having speed 5 vs speed 10? Multiplicities  (Number of states with speed v) ~ v2

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