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 redb. 50 red, 50 greenc. red, red, green, redd. 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 factorb. Einstein factorc. Fermi factord. Maxwell factore. 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

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 also the same for two systems

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

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

A B

B B

S k

S k

e

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”

Read on your own time

The Boltzmann Factor

/(beinginstatewith energy ) ~ BE k TP E 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?

“Boltzmann Factor”

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 with, e.g. Mathematica)