Announcements 9/29/10

Three days to register your clicker. Exam starts Saturday Exam review session: Thurs, 8-9:30 pm,

room C460 Reduced Tutorial Lab hours on Saturday

due to General Conference: only open 12 – 2pm.

Increased Testing Center hours on Saturday due to G.C.: it opens at 8 am instead of the usual 10 am.

Thought Question A gas in contact with a thermal reservoir undergoes an

isothermal expansion. The gas and the thermal reservoir are isolated from the rest of the universe. Which of the following is true?

a. The entropy of both the gas and the reservoir will increase.

b. The entropy of both the gas and the reservoir will decrease.

c. The entropy of both the gas and the reservoir will not change.

d. The entropy of the gas will go up, the entropy of the reservoir will go down, and the total entropy of the system will not change.

e. The entropy of the gas will go up, the entropy of the reservoir will go down, and the total entropy of the system will increase.

Reading quiz Did you spend at least 10-15 mins

looking over Dr Colton’s “What is Entropy?” handout?

a. Yesb. No

Did you spend at least 5-10 mins reading the book section 22.8?

a. Yesb. No

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. System in macrostate with most microstates system in macrostate with largest S

System and Reservoir System: E1 Reservoir: E2

Etot = E1 + E2 (const. volume so no work)

0 = dE1 + dE2

Want to maximize S: take dS/dE1, set = 0

1 21 1

1 2

1 1

1 2

1 2

1 2

1 2

dS dS S

dE dE

dS dS

dE dE

dS dS

dE dE

dS dS

dE dE

Temperature dS/dE is the same for two systems in thermal

contact! Temperature is the same for two systems in

thermal contact!a. dS/dE has units of 1/K…

This “works” if the constant is chosen properly:

1dS

dE T

S = kB ln(#microstates)

dQdS

T

Compare to

We are assuming no work, so dE=dQ (First Law)

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…

1

1

# 2

# 2A

B

P microstatesof for case A

P microstatesof for caseB

2

2

1

1

# 2

# 2

A B

B B

S kA

S kB

P microstatesof for case A e

P microstatesof for caseB e

Math…2

2

1

1

# 2

# 2

A B

B B

S kA

S kB

P microstatesof for case A e

P microstatesof for caseB e

2 2 1

22 1

12

2

( )

( )

( )

(samewith )

A A

A

A

B

S S E E

dSS E E

dEE

S ET

S

1

1

/1

/1

A B

B B

E k TA

E k TB

P e

P e

Result:

“of”, not “times”

The Boltzmann Factor

/~ BE k TP e

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?

TheBoltzmannFactor

Maxwell-Boltzmann Velocity Distribution

E = ½mv2

What’s probability of having speed 5 vs speed 10?

Multiplicities (Number of states with speed v) ~ v2

Tkmv B

ev/

2

12

2

1

1

/1

/1

A B

B B

E k TA

E k TB

P e

P e

2

2

1/2 2

11

/1 2 2

A B

B B

mv k T

A A

mv k TBB

P v e

Pv e

Maxwell-Boltzmann Velocity Distribution

The result:

Tkmv B

ev/

2

12

2

2

2

1/2 2

1/2 2

0

(speed v)B

B

mv k T

mv k T

v eP

v e dv

Exactly the equation given for the velocity distribution in your textbook! (after you do the integral in the denominator with, e.g. Mathematica)