1. 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

2. 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

3. 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)

4. 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

5. 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

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 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?

8. 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

9. 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

10. 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)

11. 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)

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

13. 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?

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

15. 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)