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Lecture Lecture 66: :

Entropy and 2Entropy and 2nd nd Law of ThermodynamicsLaw of Thermodynamics

�� Can we make heat work? Can we make heat work?

�� 22nd nd Law of ThermodynamicsLaw of Thermodynamics

�� Entropy. Maximum entropy principle.Entropy. Maximum entropy principle.

�� Heat Engines. Carnot cycle.Heat Engines. Carnot cycle.

�� Computing EntropyComputing Entropy

Adiabatic

Isochoric

Isothermal

Isobaric

0

0

pVγγγγ

V

T

p

WWQQConstant QuantityConstant QuantityProcess TypeProcess Type

Summary of Processes.Summary of Processes.

Can we find a better variable?Can we find a better variable? Can we make heat work?Can we make heat work?

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StirlingStirling engine.engine.

p

V

A

B

C

D

T2

T1

V1 V2

Efficiency =Efficiency =Work in cycleWork in cycle

Absorbed HeatAbsorbed Heat

(absorbed heat)(absorbed heat)

(released heat)(released heat)

Can we do better?Can we do better?

Given amount of absorbed heat, Given amount of absorbed heat,

work depends on “shape” (=efficiency) of cycle work depends on “shape” (=efficiency) of cycle

Limitations. Limitations. Reversible/Irreversible Processes.Reversible/Irreversible Processes.

1. Glass of wine drops from the table. 1. Glass of wine drops from the table.

Pieces will never assemble back on their own. Pieces will never assemble back on their own.

2. Middle wall removed allowing gas to expand freely. 2. Middle wall removed allowing gas to expand freely.

Molecules will “Molecules will “nevernever” spontaneously return back ” spontaneously return back

to left part oto left part of container. f container.

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Limitations. Limitations. Playing with 1Playing with 1stst Law of TD.Law of TD.

1. Direction of Heat Flow1. Direction of Heat Flow

2. Consider the World Ocean. It is warm after all! 2. Consider the World Ocean. It is warm after all!

Take all its heat and convert into work.Take all its heat and convert into work.

---- still OK with 1still OK with 1st st Law of TD Law of TD

BUT!BUT! 11st st Law of TD does not forbid 2Law of TD does not forbid 2ndnd processprocess

as long as energy is conserved.as long as energy is conserved.

CommonCommon sense tells us otherwise.sense tells us otherwise.

Second Law of ThermodynamicsSecond Law of Thermodynamics

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Second Law of Thermodynamics.Second Law of Thermodynamics.

Many formulations. Consider the following two.Many formulations. Consider the following two.

1.1. SpontaneousSpontaneous (i.e. without external agent)(i.e. without external agent) transfer of heattransfer of heat

from from coldcold object to object to hothot object is impossible.object is impossible.

2. No 2. No Perpetual Motion MachinePerpetual Motion Machine of second type.of second type.

(device that realizes full (device that realizes full conversion of heat into useful work, conversion of heat into useful work,

without any “side effect”without any “side effect”, in contrast with real engines.), in contrast with real engines.)

Let us formulate this mathematically.Let us formulate this mathematically.

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Entropy.Entropy.

There exist one more There exist one more function of state = ENTROPYfunction of state = ENTROPY

•• If If TT=const : =const : ∆∆QQ==TT∆∆SS

•• ∆∆S =S = 0 for adiabatic process 0 for adiabatic process ∆∆S S ≠≠ 0 for non0 for non--adiabaticadiabatic

Note: now have 4 variables (Note: now have 4 variables (p, V, T, Sp, V, T, S). ). (any) two (any) two are stillare still sufficient to describe state of systemsufficient to describe state of system

In 1In 1st st Law of TDLaw of TD (for (for infinitesimalinfinitesimal process)process)

QQ = not = not function of statefunction of state, depends on process, depends on process

so thatso that

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REVERSIBLEREVERSIBLE

• Series of quasi-static

(equilibrium) steps

• No dissipation (e.g. no

friction)

• Examples?

IRREVERSIBLEIRREVERSIBLE

• Series of complex, non-

equilibrium steps

• Dissipation may be

present - Examples?

Any process that can be

represented as a path on a

PV diagram; e.g

• Isochoric

• Isobaric

• Isothermal

• Adiabatic

Any process that is sudden,

spontaneous, uncontrolled, or

involves dissipation

• Free expansion

• Explosion

• Heat transfer between a hot

and a cold object

• Volume changes of gas with a

piston that has friction

Reversible and Irreversible Processes. Reversible and Irreversible Processes. SummarySummary

•• Adiabatic process?Adiabatic process?

•• General process: (General process: (TTii, , VVii, , ppii) ) ----> (> (TTff, , VVff, , ppff))

What is entropy change for:What is entropy change for:

••Isothermal process?Isothermal process?

••Isochoric process?Isochoric process?

Calculating Entropy Changes.Calculating Entropy Changes. Reversible ProcessesReversible Processes

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Principle of Maximum Entropy.Principle of Maximum Entropy.

If system is NOT at thermal equilibrium, but consists of If system is NOT at thermal equilibrium, but consists of

equilibrium subsystems with equilibrium subsystems with SSii, introduce , introduce SSΣΣ= = ΣΣiiSSii

Intuitively: Intuitively: Entropy = Measure of DisorderEntropy = Measure of Disorder

With time, entropy of isolated system: increases in irreversibleWith time, entropy of isolated system: increases in irreversible process, process,

remains the same in reversible processremains the same in reversible process

In isolated system In isolated system ((fixed internal energyfixed internal energy)), state of thermal equilibrium , state of thermal equilibrium

corresponds to absolute maximum of total entropy, i.e. S=corresponds to absolute maximum of total entropy, i.e. S=((SSΣΣ))maxmax

Disorder grows!Disorder grows!

If you reach “thermal equilibrium” with untidy roommate, If you reach “thermal equilibrium” with untidy roommate,

that would correspond to maximum possible disorder.that would correspond to maximum possible disorder.

Heat engine takes some substance (e.g. gas) Heat engine takes some substance (e.g. gas)

through cyclic process during which:through cyclic process during which:

� Heat QH absorbed from hot reservoir

� Work W+ done by engine

� Heat QC dumped into cold reservoir

� “Negative” work W- done by engine to bring gas to original state.

How can we extract heat? How can we extract heat? Heat Engines.Heat Engines.

In a cyclic process, it is impossible to convert all the energy extracted from an equilibrium system into work. Some part of this energy must be transferred to a colder system.

Want engine to work repeatedlyWant engine to work repeatedly

Want cyclic process!Want cyclic process!

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Internal combustion engine.Internal combustion engine.

Typical cycle consists of 4 processesTypical cycle consists of 4 processes

a)a) Burning of fuel Burning of fuel (isothermal expansion)(isothermal expansion)

b)b) Fast expansion Fast expansion (adiabatic)(adiabatic)

c)c) Compression to original pressure Compression to original pressure (isothermal process)(isothermal process)

d)d) Compression to original temperatureCompression to original temperature

(adiabatic process)(adiabatic process)

Carnot heat engine takes gas through reversible cyclic process consisting of 4 steps:

� Isothermal expansion at T=TH

� Adiabatic expansion from T=TH to T=TC

� Isothermal compression at T=TC

� Adiabatic compression from T=TC to T=TH

11 <−=H

C

T

TK

Carnot Heat EngineCarnot Heat Engine

•• A corollary to the second law of thermodynamics:A corollary to the second law of thermodynamics:

•• NO REAL ENGINE CAN EXCEED THE EFFICIENCY NO REAL ENGINE CAN EXCEED THE EFFICIENCY OF A CARNOT ENGINE OPERATING BETWEEN THE OF A CARNOT ENGINE OPERATING BETWEEN THE SAME HEAT RESERVOIRSSAME HEAT RESERVOIRS

•• Q: can an engine (real or ideal) ever have an efficiency of 1 Q: can an engine (real or ideal) ever have an efficiency of 1 (I.e. 100 %)? Why/why not?(I.e. 100 %)? Why/why not?

Carnot heat engine and 2Carnot heat engine and 2ndnd Law of TD.Law of TD.

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•• An irreversible process is characterized by a series of nonAn irreversible process is characterized by a series of non--equilibrium states equilibrium states ---- thermodynamic parameters such as thermodynamic parameters such as T,pT,p are illare ill--defineddefined

•• So: how to calculate So: how to calculate ∆∆S?S?

•• Entropy is a state function Entropy is a state function ---- so so ∆∆S only depends on the initial and final S only depends on the initial and final equilibrium states equilibrium states ---- NOT THE PATH!NOT THE PATH!

•• Strategy: choose a reversible path connecting the initial and fiStrategy: choose a reversible path connecting the initial and final states and nal states and determine determine ∆∆S.S.

Calculating Entropy Changes.Calculating Entropy Changes. Irreversible ProcessesIrreversible Processes

2 moles of a gas in a thermally insulated box freely 2 moles of a gas in a thermally insulated box freely

expands, tripling its volume in the process. What is the expands, tripling its volume in the process. What is the

change in entropy?change in entropy?

p

V

p1, V1, T1

p2, V2, T1

•Q = 0; W = 0; so, ∆∆∆∆U = 0•Final T = initial T•Choose a reversible, isothermal path J/K3.183ln2

3ln

0

0

0

0

0

0

0

0

3

3

33

==

=

=

=

==∆

∫

∫

∫∫

R

nR

V

dV

T

nRT

V

dV

T

nRT

T

pdV

T

dQS

V

V

V

V

V

V

V

V

Sample problem. Sample problem. Free expansions.Free expansions.

Two identical 500 g copper blocks A and B of mass are Two identical 500 g copper blocks A and B of mass are

initially at 20 initially at 20 00C and 80 C and 80 00C. They are placed in thermal C. They are placed in thermal

contact and allowed to come to thermal equilibrium. contact and allowed to come to thermal equilibrium.

What is the change in entropy? (Specific heat capacity of What is the change in entropy? (Specific heat capacity of

copper = 400 J/kg. K)copper = 400 J/kg. K)

∆SA =mcdT

T293

323

∫

= mc ln323

293

= (0.5kg)(400J/kg.K)(0.098)

=19.6 J/K

∆SB = mc ln323

353

= -17.8 J/K

∆Stotal = (19.6 −17.8)J/K

= +1.8 J/K

What reversible process can we use?

Sample problem. Sample problem. Heat transfer.Heat transfer.

• Reversible/Irreversible Processes

• Entropy

• 2nd Law

• Heat Engines (Carnot)

0≥∆S

+

=∆

i

f

i

f

vV

VnR

T

TnCS lnln

HQ

WK =

H

C

T

TK −= 1

What we learnedWhat we learned

“The law that entropy always increases, holds, I think, the supreme position among the Laws of Nature. If someone points out to you that your pet theory of the Universe is in disagreement with Maxwell’s Equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse indeepest humiliation.”

--Sir Arthur Stanley Eddington,

The Nature of the Physical World (1927)

18821882--19441944