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
VnRSSS compsursursur
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