Chapter 2

The Second Law

Why does Q (heat energy) go from high temperature to low temperature?

coldcold hothot

Q flow

Thermodynamics explains the direction of time.The Big Bang

32 NkBT 1

2 mv 2

Hot objects are faster so they are more quick to move to the cold side.

BUT in a solid objects aren’t actually moving from one side to the other. ? . ? . ?

Why does Q (heat energy) go from high temperature to low temperature?

coldcold hothot

Q flow

Let’s look at how probability tells us which way energy should flow.

How many ways can energy be arranged?

Which arrangements are most likely?EE

EEEE

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EE

EE

EEEE

EE

EE

EE

EE

EE

EE EE EE

EE

EEEEEE

EE EE

ENTROPY

Simple ProbabilityIn the mid-1960s, the Adams gum company acquired American Chicle and introduced a new slogan for Trident: "4 out of 5 Dentists surveyed would recommend sugarless gum to their patients who chew gum." The phrase became strongly associated with the Trident brand.

Some New Terms• Mutually Exclusive

– Outcomes of events have a single possibility, others are excluded• A coin flip is heads or tails, not both

• Collectively Exhaustive– The full set of propositions or outcomes

• Heads and tails are all possible outcomes

• Independent– An event or outcome does not depend of previous or future events or

outcomes• A previous heads does not determine the next coin flip

• Multiplicity– The number of ways to get a particular outcome

• W or W is typically used as the variable

• Conditional Probability– An outcome depends on a previous event

• Drawing colored balls from a bag

Distributions Continued• Discrete

– Coin flips• List the macrostates, probability, microstates, & multiplicity

One coin

Two coins

Three coins

Distributions Continued• Discrete

– Four coin flip

How can we predict what will happen?

How can we talk about outcomes in percentages?

Distributions Continued• Discrete

– Four coin flip

Pascal’s Triangle

Paramagnets

B = 0

B ≠ 0

(N ) N!

N !N !

N!

N !(N N )!

Cards

)!(!

!)(

nNn

NN

Harmonic Oscillators – Einstein Solid

)( 21 nEn

U 12 kx 2

Schrodinger Eqn. solutions for energy are

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

Einstein Solid

3,2,1,0

3

A

A

n

NTotal energy

0

1

Oscillator #1 #2 #3 W

Einstein Solid

3,2,1,0

3

A

A

n

NTotal energy

0

1

2

0 0 0

Oscillator #1 #2 #3 W

100

010

001

Einstein Solid

Total energy

3

Oscillator #1 #2 #3 W

3,2,1,0

3

A

A

n

N

Einstein Solid

!)!1(

)!1()(

nN

nNn

Einstein Solids in Thermal Equilibrium

1

3

3

3

B

B

A

A

n

N

n

N

Einstein Solids in Thermal Equilibrium

1

3

3

3

B

B

A

A

n

N

n

N

n

iBA

BA

innP

0

)()(

Einstein Solids in Thermal Equilibrium

Einstein Solids in Thermal Equilibrium

3059

P70/3059 = 0.023350/3059 = 0.114825/3059 = 0.2701100/3059 = 0.360714/3059 = 0.233

1.000sum

Einstein Solids in Thermal Equilibrium

Normalized probability for an energy arrangement

n

iBA

BA

innP

0

)()(

For NA = NB = 100 oscillatorsP(nA=30) = P(nA=70) = 0.004 = 0.4%

Einstein Solids in Thermal Equilibrium

Normalized probability for an energy arrangement

n

iBA

BA

innP

0

)()(

For NA = NB = 100 oscillatorsP(nA=30) = P(nA=70) = 0.004 = 0.4%

Numbers• Small

– 1, 5, 10, 235, etc.• Large

– 1010, 1023, 10114, etc.• Very Large

The Universe: 1018 s old (Big Bang)~1080 atoms

Computer Numbers and the 2nd Law• 64 bit floating point numbers

– 52 bit mantissa– 11 bit exponent– 1 bit sign

The Second Law

The second law of thermodynamics: "You can't even break even, except on a very cold day."

Energy will "flow" until the state of maximum multiplicity is obtained.

S = kB ln (W)

Very Large NumbersStirling’s Approximation

Very Large NumbersStirling’s Approximation

n

nxn

e

nndxexn 2!

0

See appendix B

A more rigorous derivation comes from

Multiplicity of a Large Einstein Solid

n

nxn

e

nndxexn 2!

0

Multiplicity of a Large Einstein Solid

Multiplicity of a Large Einstein SolidHigh temperature limit: Assume n >> N

TNknE Bf

n 221 )(

N

N

en

Multiplicity of a Large Einstein SolidHigh temperature limit: Assume n >> N

N

N

en

This can’t be plotted for very large systems 25

23

23

10

102

10

total

B

A

n

N

N

N

N

en

ln)ln( Neither can this.

N

enN ln)ln( But this can.

Multiplicity of a Large Einstein SolidHigh temperature limit: Assume n >> N

A

AAA N

enN ln)ln(

Multiplicity of a Large Einstein SolidHigh temperature limit: Assume n >> N

B

ABB N

nneN

)(ln)ln(

Multiplicity of a Large Einstein SolidHigh temperature limit: Assume n >> N

B

AB

A

AABABA N

nneN

N

enN

)(lnln)ln()ln()ln(

W is a Gaussian Function

NA=NB=100n=500

NN

B

B

N

A

Atot N

en

N

en

N

enBA 2

2

2s Nnw 2

222222

maxln2

nx

nxn NNNN

Ne eee

W is a Gaussian Function

Multiplicity of a Monatomic Ideal Gas

z

yx

A container of monatomic gas. How can we describe the ways to arrange the atoms and the energy they contain?

Multiplicity of a Monatomic Ideal Gas

z

yx

In 2D momentum space, constant energy is defined by a circle.

Multiplicity of a Monatomic Ideal Gas

z

yx

Multiplicity of a Monatomic Ideal Gas

z

yx

Multiplicity of a Monatomic Ideal Gas

z

yx

Multiplicity of a Monatomic Ideal Gas

z

yx

Multiplicity of an Ideal GasWhat is the relative probability of a gas taking the full volume of its container to the probability of taking half the volume of its container?

Multiplicity of an Ideal Gas

A 1

NA

VANA

h3NA

23NA

2

3NA2 ! 2mUA

3NA2

B 1

NB

VBNB

h3NB

23NB

2

3NB2 ! 2mUB

3NB2

Multiplicity of an Ideal GasExchanges possibleNA NB : Diffusive EquilibriumVA VB : Pressure EquilibriumUA UB : Thermal Equilibrium

Utotal

2 3N2

Vtotal

2 N

Entropy

S kB ln()

S NkB lnV

N

4mU

3Nh2

32

5

2

Entropy – Ideal Gas

Creating Entropy

Free Expansion

W = 0 because there is nothing to push against in a vacuum.

Q = 0 because this is an adiabatic process, and insulated from the surroundings.

DU = Q + W = 0.

BUT there is a volume change!

Entropy of Mixing

S NkB lnV

N

4mU

3Nh2

32

5

2

Entropy of Mixing

S NkB lnV

N

4mU

3Nh2

32

5

2

Entropy – Einstein Solid

Entropy

Entropy

W vs. ln(W)

NA=NB=1000n=1000

NA=NB=1023

n=1024

Creating Entropy

Free Expansion

Entropy

• Experiment• Flip n=10 coins• Count and record number of heads, nH• Repeat N=1000 times• Create a histogram from 0 to 10 of nH

• Computer Simulation• Generate n=10 random 1 or 0 (heads or tails)• 1 1 1 0 0 1 1 1 0 1

• Count and record number of heads, nH• Repeat N=1000 times• Create a histogram from 0 to 10 of nH

• Fit a gaussian to the data• My results

• x0 = 4.93 ± 0.04• = 2.27 ± 0.09