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# Accounting for Entropy Class 28.2 Objectives Qualitatively understand reversibility/irreversibility...

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Accounting for EntropyClass 28.2

Objectives

• Qualitatively understand reversibility/irreversibility• Quantitatively understand reversibility/irreversibility• Understand entropy• Perform simple calculations involving entropy• Know how to account for entropy• Quantitatively state the second law of thermodynamics

GasolineAir

CO2

H2OMotionAir turbulenceTire deformationHead lightsAir conditioningStereoHot exhaust

A “natural” process…

GasolineAir

CO2

H2OMotionAir turbulenceTire deformationHead lightsAir conditioningStereoHot exhaust

An “unnatural” process…

Although we all recognize this is impossible, it is still allowed by the first law of thermodynamics (conservation of energy).

We need another law…

The second law of thermodynamics

i.e., naturally occurring processes are directional

How do we quantify the second law of thermodynamics?

Entropy

Entropy is closely tied to…

Reversible processesIrreversible processes

– do not generate entropy– do generate entropy

A reversible process…

Frictionless pulley

If a movie of this process were run backwards, you could not tell.

An irreversible process…

If a movie of this process were run backwards, you could tell.

Imagining a movie running forwards or backwards is a useful method for thinking about reversibility, but what would we do for a process that we are not familiar with?

We need a better way to determine if a process is reversible or not.

Better Approach:

Return the system to its initial state, i.e., run a “cycle.” The more change in the surroundings, the more irreversible the process.

Universe

Surroundings

System

1 Initial state of the system

Amount of weight determines friction

System boundary

2

3

4

5

6 Final state of the system. (Same as initial state.)

Surroundings have changed. (Two weights now on the floor.)

More weight here causes more weight to be on the floor.

This weight controls the amount of irreversibility in the system.

Energy changes of this process:

Potential energy internal energy heat} }“Ordered” energy “Disordered” energy

•Potential •Kinetic•Work

•Internal energy•Heat

Observation:

Irreversibilities occur when ordered energy is converted to disordered energy.

...

. .

.

..

.

.

Steam100oC

Ice Bath0oCTime passes

This is an irreversible process. Heat will not spontaneously flow from the ice bath to regenerate the steam. (A movie run backwards would look funny.)

Copper rod

Observation:

Heat transfer from a high-temperature body to a low-temperature body is an irreversible process.

T + dTV + dV

T V

T + dTV + dV

T V

Reversible heat transfer…

Perfect insulation

Observation:

Systems with differential driving forces are reversible.

Corollary:

Systems with differential driving forces are infinitely slow.

Expander

P1 , V1 P2 , V2 V1

P1

P2

V2

WorkProduced

Irreversible

)( 122 VVPWirrev

V1

P1

P2

V2

P1 , V1 P2 , V2

SandReversible

1

2lnV

VnRTWrev

Generalized Observation:

A reversible process produces more work than an irreversible process.

Pairs Exercise #1

The initial conditions for 1 mol of air in a piston/cylinder are 5 atm and 300 K. The piston decreases the pressure to final conditions of 1 atm and 300 K.

Calculate the work (J) produced from the gas using

a.Irreversible expansion by removing a weight from the pistonb.Reversible expansion

Compressor

V1

P1

P2

V2

Required Work

Irreversible

Reversible

)( 121 VVPWirrev

V1

P1

P2

V2

1

2lnV

VnRTWrev

P1 , V1 P2 , V2

P1 , V1 P2 , V2

Sand

Generalized Observation:

A reversible process requires less work than an irreversible process.

Pairs Exercise #2

The initial conditions for 1 mol of air in a piston/cylinder are 1 atm and 300 K. The piston increases the pressure to final conditions of 5 atm and 300 K.

Calculate the work (J) required to compress the gas using

a.Irreversible compression by adding a weight to the pistonb.Reversible expansion

Note: For the reversible case, the work produced by the expansion was identical to the work required by the compression.

Generalized Observation:

A reversible process that has a given work output when run in the forward direction requires the same work input when run in the reverse direction.

Work

Heat

Work

Heat

Expansion Compression

P

V

P

VMany irreversible paths, but only one reversible path.Each path has its own work and heat.

Suppose you have 1000 Btu available at 400oF, 100oF, and 60oF. What could you do with it?

400oF: 1000 Btu 1 lb of 250-psia steam useful work

100oF: 1000 Btu home heating

60oF: 1000 Btu ambient environment

Observation: Heat flows from higher temperatures to lower temperatures, but becomes less useful as it does so.

1000 Btu

2000 Btu

3000 Btu

How can we quantify the notion that heat available at a higher temperature is more useful than heat available at a lower temperature?

The following combinations of heat and temperature may be proposed:

... ... 3223

TQTQTQT

Q

T

Q

T

Qrevrevrev

revrevrev

where Qrev indicates the heat associated with a reversible process.

Of these possibilities, Rudolf Clausius found the following term was useful

which he defined as entropy.

T

QS rev Input to the system

being studied.

System Boundary

InitialState

TSinitial

T

FinalStateQrev

TSfinal

T

QSSS rev

initialfinal State quantity Path quantity

State quantity

Rule 9, page 490: An algebraic combination of a well-defined path quantity with a state quantity is a state quantity.

This is why it is important to specify reversible path.

WorkReversible Expansion

T T

Qrev

TS1S2

1

2lnV

VnRTWQQ outrevin

1

21

2

12 lnln

V

VnR

TVV

nRT

T

QSSS rev

V1V2

Ek + Ep + U = Win - Wout + Qin - Qout

0 0 0 0 00

Energy Accounting(closed system)

initial final

Pairs Exercise #3

a. Calculate the entropy change of 1 mole of constant-temperature gas that is reversibly expanded from 1 m3 to 5 m3.

b. Calculate the entropy change of 1 mole of constant-temperature gas that is irreversibly expanded from 1 m3 to 5 m3.

The entropy of the system is a state quantity and does not depend upon the path, whether reversible or irreversible.

Observation

The entropy increases when the volume increases. In the larger volume the gas is more “disordered” so more entropy corresponds to more disorder.

Observation

WorkReversible Compression

T T

Qrev

TS1S2

2

1lnV

VnRTWQQ inrevout

2

12

1

21 lnln

V

VnR

TVV

nRT

T

QSSS rev

V1V2

initial

Ek + Ep + U = Win - Wout + Qin - Qout

0 0 0 0 00

Energy Accounting(closed system)

final

Pairs Exercise #4

a. Calculate the entropy change of 1 mole of constant-temperature gas that is reversibly compressed from 5 m3 to 1 m3.

b. Calculate the entropy change of 1 mole of constant-temperature gas that is irreversibly compressed from 5 m3 to 1 m3.

The entropy decreases when the volume decreases. In the smaller volume the gas is less “disordered” so less entropy corresponds to less disorder.

Observation

Cycle – A system that returns to the initial conditions

T S1V1 T S2V2

initial

Expansion

Compression

Pairs Exercise #5

a. Calculate the entropy change of 1 mole of constant-temperature gas that is reversibly expanded from 1 m3 to 5 m3 and then reversibly compressed from 1 m3 to 5 m3.

b.Calculate the entropy change of 1 mole of constant-temperature gas that is irreversibly expanded from 1 m3 to 5 m3 and then irreversibly compressed from 1 m3 to 5 m3.

For a cycle, the system entropy does not change, regardless of whether the path is reversible or irreversible.

Observation

Reversible Expander

Ek + Ep + U = Win - Wout + Qin - Qout

0 0 0 0 00

Energy Accounting(closed system)

1

2lnV

VnRTWQ outin

V1

P1

P2

V2

P1 , V1 P2 , V2

Sand

1

2lnV

VnRTWout

Wout

Qin

What happens from the perspective of the surroundings?

Qout (from the perspective of the water bath surroundings)

Qin,gasT

Wout

Qout,surr

T

Qin (from the perspective of the gas)

1

21

2

,,exp, ln

ln

V

VnR

TVV

nRT

T

Q

T

QS gasinsurout

sur

Negative because entropy is defined based upon heat input.Here we have output.

gasinsurout QQ ,,

Reversible Compressor

Ek + Ep + U = Win - Wout + Qin - Qout

0 0 0 0 00

Energy Accounting(closed system)

1

2lnV

VnRTWQ inout

V1

P1

P2

V2

1

2lnV

VnRTWrev

P1 , V1 P2 , V2

Sand

Win

Qout

What happens from the perspective of the surroundings?

Qin (from the perspective of the water bath surroundings)

Qout,gasT

Win

Qin,surr

T

Qout (from the perspective of the gas)

1

21

2

,,, ln

ln

V

VnR

TVV

nRT

T

Q

T

QS gasoutsurin

compsur

gasoutsurin QQ ,,

What happens to the surroundings for a cyclical reversible process?

T P2V2T P1V1

initial

Compression

Expansion

0lnln1

2

1

2,exp,

V

VnR

V

For a reversible cycle, the entropy of the surroundings does not change.

Observation

What happens to the universe for a cyclical reversible process?

T P2V2T P1V1

initial

Compression

Expansion

000 sursysuniverse SSS

For a reversible cycle, the entropy of the universe does not change.

Observation

Irreversible Expander

Ek + Ep + U = Win - Wout + Qin - Qout

0 0 0 0 00

Energy Accounting(closed system)

V1

P1

P2

V2

)( 122 VVPWout

P1 , V1 P2 , V2

TT

Wout

Qin

T

)( 122 VVPWQ outin

What happens from the perspective of the surroundings?

Qout (from the perspective of the water bath surroundings)

Qin,gasT

Wout

Qout,surr

T

Qin (from the perspective of the gas)

T

VVP

T

Q

T

QS gasinsurout

sur)( 122,,

exp,

Negative because entropy is defined based upon heat input.Here we have output.

gasinsurout QQ ,,

Irreversible Compressor

Ek + Ep + U = Win - Wout + Qin - Qout

0 0 0 0 00

Energy Accounting(closed system)

V1

P1

P2

V2

)( 121 VVPWin

P1 , V1 P2 , V2

Win

TT

Qout

T

)( 121 VVPWQ inout

What happens from the perspective of the surroundings?

Qin (from the perspective of the water bath surroundings)

Qout,gasT

Win

Qin,surr

T

Qout (from the perspective of the gas)

T

VVP

T

Q

T

QS gasoutsurin

compsur)( 121,,

,

gasoutsurin QQ ,,

What happens to the surroundings for a cyclical irreversible process?

T P2V2T P1V1

initial

Compression

Expansion

T

VVP

T

VVPSSS compsursursur

)()( 121122,exp,

0)( 12

21

T

VVPP

Positive Positive

For an irreversible cycle, the entropy of the surroundings always increases.

Observation

What happens to the universe for a cyclical irreversible process?

T P2V2T P1V1

initial

Compression

Expansion

0)(

)(0 1221

T

VVPPSSS sursysuniverse

Positive Positive

For a irreversible cycle, the entropy of the universe increases.

Observation

Restatement of the second law of thermodynamics…

0 universeS

For any process that occurs in nature,

Entropy Accounting

consgenoutin SSSSS

0

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