General Physics IGeneral Physics I

New Lecture 27: Carnot Cycle, The New Lecture 27: Carnot Cycle, The 2nd Law, Entropy and Information2nd Law, Entropy and Information

Prof. WAN, Xin

xinwan@zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

Carnot’s EngineCarnot’s Engine

Efficiency of a Carnot EngineEfficiency of a Carnot Engine

isothermal process

adiabatic process

Efficiency of a Carnot EngineEfficiency of a Carnot Engine

Efficiency of a Carnot EngineEfficiency of a Carnot Engine

All Carnot engines operating between the same two temperatures have the same efficiency.

Carnot CycleCarnot Cycle

✦ The efficiency of the Carnot cycle only depends on the temperatures Tc & Th.

✦ The efficiency approaches 1 when Tc/Th 0, or Tc 0.

✦ T is the absolute energy scale, in units of K.

✦ Carnot cycle is a reversible cycle – quasistatic with no dissipation.

Reverse Carnot CycleReverse Carnot Cycle

Carnot’s TheoremCarnot’s Theorem

No real heat engine operating between two energy reservoirs can be more efficient than my engine operating between the same two reservoirs.

-- Sadi Carnot

What if not?What if not?

+ = ?

hcarnot

h Q

W

Q

W

'

''If we suppose

Match Work FirstMatch Work First

If we suppose

+ =

0'1''

'1'

''

carnoth

h

hhhh Q

W

Q

Q

WQQ

W

WQ

hcarnot

h Q

W

Q

W

'

''

W

W '

Clausius StatementClausius Statement

✦ It is impossible to construct a cyclical machine whose sole effect is the continuous transfer of energy from one object to another object at a higher temperature without the input of energy by work.

No perfect refrigerator!

Match Heat at the Cold EndMatch Heat at the Cold End

If we suppose

+ =

01

'11'

'

'1'

''

carnoth

c

h

h

chh

c

ch Q

Q

Q

Q

QQQ

Q

QQ

hcarnot

h Q

W

Q

W

'

''

c

c

Q

Q'

Kelvin StatementKelvin Statement

✦ It is impossible to construct a heat engine that, operating in a cycle, produces no effect other than the absorption of energy from a reservoir and the performance of an equal amount of work.

No perfect heat engine!

Laws in Plain LanguageLaws in Plain Language

✦ The 1st and 2nd laws of thermodynamics can be summarized as follows:

– The first law specifies that we cannot get more energy out of a cyclic process by work than the amount of energy we put in.

– The second law states that we cannot break even because we must put more energy in, at the higher temperature, than the net amount of energy we get out by work.

An EqualityAn Equality

Now putting in the proper signs,

0c

c

h

h

T

Q

T

Q

0CycleCarnot

T

dQ

negativepositive

A Sum of Carnot CyclesA Sum of Carnot Cycles

0C TdQ

V

Padiabats Th,i

Tc,i

Any reversible process can be approximated by a sum of Carnot cycles, hence

0,

,

,

, i ic

ic

ih

ih

T

Q

T

Q

Clausius Definition of EntropyClausius Definition of Entropy

✦ Entropy is a state function, the change in entropy during a process depends only on the end points and is independent of the actual path followed.

T

dQdS reversible

012,21, 21

CCC

dSdSdS

21,12,21,

12

221 CCC

dSdSdSSS1

2

C1

C2

Return to Inexact DifferentialReturn to Inexact Differential

12ln)2,2(

)2,1(

)2,1(

)1,1(

dy

y

xdx

0lnln),( fyxyxf

2ln21)2,2(

)1,2(

)1,2(

)1,1(

dy

y

xdx

dyy

xdxdg Assume

Note: is an exact differential.

Integrating factor

y

dy

x

dx

x

dgdf

The Second Law in terms of EntropyThe Second Law in terms of Entropy

✦ The total entropy of an isolated system that undergoes a change can never decrease.

– If the process is irreversible, then the total entropy of an isolated system always increases.

– In a reversible process, the total entropy of an isolated system remains constant.

✦ The change in entropy of the Universe must be greater than zero for an irreversible process and equal to zero for a reversible process.

0 UniverseS

Example 1: Clausius StatementExample 1: Clausius Statement

hh T

QS

cc T

QS

0ch

ch T

Q

T

QSSS

Irreversible!

Example 2: Kelvin StatementExample 2: Kelvin Statement

0T

QS

Irreversible!

We can only calculate S with a reversible process! In this case, we replace the free expansion by the isothermal process with the same initial and final states.

Example 3: Free ExpansionExample 3: Free Expansion

00 SWQU

0ln

i

fV

V

V

V

V

V VV

nRV

nRdV

T

PdV

T

dQS

f

i

f

i

f

i

?

Irreversible!

Entropy: A Measure of DisorderEntropy: A Measure of Disorder

2lnln Bi

fB NkV

VNkS

WkS B ln

N

m

ff V

VW

N

m

ii V

VW

N

i

f

i

f

VV

WW

We assume that each molecule occupies some microscopic volume Vm.

suggesting (Boltzmann)

Order versus DisorderOrder versus Disorder

✦ Isolated systems tend toward disorder and that entropy is a measure of this disorder.

Ordered: all molecules on the left side

Disordered: molecules on the left and right

Landauer’s Principle & VerificationLandauer’s Principle & Verification

✦ Computation needs to involve heat dissipation only when you do something irreversible with the information.

✦ Lutz group (2012)

693.02ln Tk

Q

B

Information and EntropyInformation and Entropy

✦ (1927) Bell Labs, Ralph Hartley– Measure for information in a message

– Logarithm: 8 bit = 28 = 256 different numbers

✦ (1940) Bell Labs, Claude Shannon– “A mathematical theory of communication”

– Probability of a particular message

But there is no information.

You are not winning the lottery.

Information and EntropyInformation and Entropy

✦ (1927) Bell Labs, Ralph Hartley– Measure for information in a message

– Logarithm: 8 bit = 28 = 256 different numbers

✦ (1940) Bell Labs, Claude Shannon– “A mathematical theory of communication”

– Probability of a particular messageOkay, you are going to win the lottery.Now that’s

something.

Information and EntropyInformation and Entropy

✦ (1927) Bell Labs, Ralph Hartley– Measure for information in a message

– Logarithm: 8 bit = 28 = 256 different numbers

✦ (1940) Bell Labs, Claude Shannon– “A mathematical theory of communication”

– Probability of a particular message

– Information ~ - log (probability) ~ negative entropy

i

ii PPS loginfomation

For Those Who Are InterestedFor Those Who Are Interested

✦ Reading (downloadable from my website): – Charles Bennett and Rolf Landauer, The

fundamental physical limits of computation.

– Antoine Bérut et al., Experimental verification of Landauer’s principle linking information and thermodynamics, Nature (2012).

– Seth Lloyd, Ultimate physical limits to computation, Nature (2000).

Dare to adventure where you have not been!