lecture 18 second law of thermodynamics. carnot's cycle
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Lecture 18Second law of
thermodynamics. Carnot cycle. Absolute zero.
ACT: Leaving the fridge open
If you leave the door of your fridge open, you will get a heart-stopping electricity bill, but you will also:
A. Freeze the kitchen
B. Warm up the kitchen
H C C Q Q W Q
Now the fridge and the kitchen are one system. We are taking QC out of this system and dumping QH into it. Overall, heat is added!
Second law of thermodynamics
It is impossible for any process to have as its sole result the transfer of heat from a cooler to a hotter body (“refrigerator” or Clausius statement)
i.e.,
It is impossible to build a 100%-efficient heat engine (e = 1)
It is impossible for any system to undergo a process in which it absorbs heat form a reservoir at a single temperature and convert the heat completely into mechanical work, with the system ending in the same state as it began (“engine” or Kelvin-Plank statement)
Or
i.e.,
It is impossible to build a workless refrigerator (K ∞)
Clausius Kelvin-Plank
100%-efficient engineWorkless refrigerator
Kelvin-Plank Clausius
100%-efficient engine
Workless refrigerator
The Carnot cycle
Cycle made with reversible isothermal and adiabatic processes.
A
B
D
C
ACT: Carnot cycle
Where in the cycle is heat absorbed? 1
2
4
3
A. At point 1
B. Between 1 and 2
C. Between 4 and 1
To absorb heat, we need a process between two states. 1 is a state.
No heat transfer between 4 and 1 (adiabatic)
Between 1 and 2, temperature is alwaysTH
1 2
1 2 1 2
0
0
U
Q W
Heat exchange in Carnot’s cycle
1 2
21 2 1 2 H
1
0
ln 0
U
VQ W nRT
V
1
2
4
3
3 4
43 4 3 4 C
3
0
ln 0
U
VQ W nRT
V
1 1H 1 C 4T V T V 1 1
H 4 3
C 1 2
T V V
T V V
4 3
1 2
V V
V V1 1
H 2 C 3T V T V
2H
H 1 2 1
C 3 4 4C
3
ln
ln
VnRT
Q Q V
Q Q VnRT
V
32
1 4
VV
V V
H
C
T
T H H
C C
Q T
Q T
Carnot efficiency
H
We
Q C
H
1T
T C H
H
Q Q
Q
C
H
1Q
Q C
CarnotH
1T
eT
Carnot’s cycle is completely reversible. Run backwards, it is a Carnot refrigerator
CC C
H CH C
QQ TK
T TW Q Q
C
CarnotH C
TK
T T
In-class example: Carnot engine
A Carnot engine can operate between different sets of reservoirs.
1: TH = 80°C, TC = 10°C
2: TH = 10°C, TC = −50°C
3: TH = − 60°C, TC = −100°C
Rank their efficiencies (largest to smallest).
A. 1,2,3
B. 3,2,1
C. 3,1,2
D. 2,1,3
E. 2,3,1
1: TH = 353K, TC = 283K
2: TH = 283K, TC = 223K
3: TH = 213K, TC = 173KC
1
2831 0.198
353e C
CarnotH
1T
eT
2
2231 0.212
283e
3
1731 0.188
213e
Carnot is the ideal cycle
The Carnot cycle is completely reversible. So let’s use it as a refrigerator, and couple it to a hypothetical superengine with eSE > eCarnot.
Carnot Super engine
QH
QC
QH + Δ
WΔ
QC
TC
TH
Δ
Δ
TC
TH
Superengine produces more work than Carnot for these temperatures
Required to balance energy in SE
This is equivalent to this!
Impossi
ble
No engine can be more efficient than a Carnot engine operating between the same two temperatures.
Absolute zero
The colder you try to go, the less efficient the refrigerator gets:
Since heat leaks will not disappear as the object is cooled, you need more cooling power the colder it gets.The integral of the power required diverges as T 0.
Therefore you cannot cool a system to absolute zero
The best refrigerator you can get (Carnot) has performance
CCarnot
H C
TK
T T
C
CCarnot 0
H C
0T
TK
T T
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