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Physics 207: Lecture 27, Pg 1
Lecture 26Goals:Goals:
• Chapters 18, entropy and second law of thermodynamicsChapters 18, entropy and second law of thermodynamics
• Chapter 19, heat engines and refrigeratorsChapter 19, heat engines and refrigerators
• No lab this week.
Physics 207: Lecture 27, Pg 2
Equipartition theorem
Things are more complicated when energy can be stored in other degrees of freedom of the system.
monatomic gas: translation
solids: translation+potential energy
diatomic molecules: translation+vibrations+rotations
Physics 207: Lecture 27, Pg 3
Equipartition theorem
The thermal energy is equally divided among all possible energy modes (degrees of freedom). The average thermal energy is (1/2)kBT for each degree of freedom.
εavg=(3/2) kBT (monatomic gas)
εavg=(6/2) kBT (solids)
εavg=(5/2) kBT (diatomic molecules)
Note that if we have N particles:
Eth=(3/2)N kBT =(3/2)nRT (monatomic gas)
Eth=(6/2)N kBT =(6/2)nRT (solids)
Eth=(5/2)N kBT =(5/2)nRT (diatomic molecules)
Physics 207: Lecture 27, Pg 4
Specific heat
Molar specific heats can be directly inferred from the thermal energy.
Eth=(6/2)N kBT =(6/2)nRT (solid)
ΔEth=(6/2)nRΔT=nCΔT
C=3R (solid)
The specific heat for a diatomic gas will be larger than the specific heat of a monatomic gas:
Cdiatomic=Cmonatomic+R
Physics 207: Lecture 27, Pg 5
Second Law and Entropy
A perfume bottle breaks in the corner of a room. After some time, what would you expect?
A) B)
Physics 207: Lecture 27, Pg 6
very unlikely
probability=(1/2)N
The probability for each particle to be on the left half is ½.
Physics 207: Lecture 27, Pg 7
Second Law of thermodynamics
The entropy of an isolated system never decreases. It can only increase, or in equilibrium, remain constant.
The laws of probability dictate that a system will evolve towards the most probable and most random macroscopic state
Thermal energy is spontaneously transferred from a hotter system to a colder system.
Physics 207: Lecture 27, Pg 8
Reversible vs Irreversible
The following conditions should be met to make a process perfectly reversible:
1. Any mechanical interactions taking place in the process should be frictionless.
2. Any thermal interactions taking place in the process should occur across infinitesimal temperature or pressure gradients (i.e. the system should always be close to equilibrium.)
Physics 207: Lecture 27, Pg 9
Reversible vs IrreversibleBased on the above comments, which of the following
processes is not reversible?
A. Lowering a frictionless piston in a cylinder by placing a bag of sand on top of the piston.
B. Lifting the piston described in the previous statement by removing one tiny grain of sand at a time.
Physics 207: Lecture 27, Pg 10
Heat Engines
Turning heat into work: Industrial revolution.
Volume
Pre
ssur
e
i
f
Physics 207: Lecture 27, Pg 11
Key concepts
Work done by the system:
Wsystem=-Wexternal
Energy reservoir: An object that interacts with the system that is sufficiently large such that its temperature is almost constant.
QH: The amount of heat transferred to/from hot reservoirQC: The amount of heat transferred to/from cold reservoir
Physics 207: Lecture 27, Pg 12
Energy-transfer diagram
Hot reservoir
Cold reservoir
QH
QC
Wout
cyclic systemΔEsystem=0 Wout=QH-QC
Physics 207: Lecture 27, Pg 13
Thermal efficiency
For practical reasons, we would like an engine to do the maximum amount of work with the minimum amount of fuel. We can measure the performance of a heat engine in terms of its thermal efficiency η (lowercase Greek eta), defined as
We can also write the thermal efficiency as
Physics 207: Lecture 27, Pg 14
What is the largest thermal efficiency that a heat engine can have?
A) η=2 B)η=1 C) η=1/2 D) η=0
What is the lowest thermal efficiency that a heat engine can have?
A) η=1/2 B)η=0 C) η=-1/2 D) η=-1
Physics 207: Lecture 27, Pg 15
Refrigerators Devices that uses work to transfer heat from a colder object to a
hotter object.
Hot reservoir
Cold reservoir
QH
Win
Win+QC=QH
K=QC/WinQC
Physics 207: Lecture 27, Pg 16
Is perfect engine possible?
Hot reservoir
Cold reservoir
QH1
Wout Win
QH2
QC
=QC
QH
Physics 207: Lecture 27, Pg 18
Which of the following processes would have the largest work output per cycle?
V
P
V V
P PA) B) C)
Physics 207: Lecture 27, Pg 19
Internal combustion engine: gasoline engine
(Adiabats)
A gasoline engine utilizes the Otto cycle, in which fuel and air are mixed before entering the combustion chamber and are then ignited by a spark plug.
Otto Cycle
Physics 207: Lecture 27, Pg 20
The best thermal engine ever, the Carnot engine A perfectly reversible engine (a Carnot engine) can be
operated either as a heat engine or a refrigerator between the same two energy reservoirs, by reversing the cycle and with no other changes.
Physics 207: Lecture 27, Pg 21
The Carnot Engine
All real engines are less efficient than the Carnot engine because they operate irreversibly due to the path and friction as they complete a cycle in a brief time period.
Carnot showed that the Carnot showed that the thermal efficiency of a thermal efficiency of a Carnot engine is:Carnot engine is:
hot
coldcycleCarnot T
T1η