Transcript Physics 207: Lecture 2 Notes

Lecture 26
Goals:
• Chapters 18, entropy and second law of thermodynamics
• Chapter 19, heat engines and refrigerators
• No lab this week.
Physics 207: Lecture 27, Pg 1
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 2
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 3
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 4
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 5
very unlikely
 The probability for each particle to be on the left half is ½.
probability=(1/2)N
Physics 207: Lecture 27, Pg 6
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 7
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 8
Reversible vs Irreversible
 Based 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 9
Heat Engines
Pressure
 Turning heat into work: Industrial revolution.
f
i
Volume
Physics 207: Lecture 27, Pg 10
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 reservoir
QC: The amount of heat transferred to/from cold reservoir
Physics 207: Lecture 27, Pg 11
Energy-transfer diagram
Hot reservoir
QH
cyclic system
ΔEsystem=0
QC
Wout
Wout=QH-QC
Cold reservoir
Physics 207: Lecture 27, Pg 12
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 13

What is the largest thermal
efficiency that a heat engine
can have?
A)  =2

B)  =1
C)  =1/2
D)  =0
What is the lowest thermal efficiency that a heat
engine can have?
A)  =1/2
B)  =0
C)  =-1/2
D)  =-1
Physics 207: Lecture 27, Pg 14
Refrigerators
 Devices that uses work to transfer heat from a colder object to a
hotter object.
Hot reservoir
Win+QC=QH
QH
Win
QC
K=QC/Win
Cold reservoir
Physics 207: Lecture 27, Pg 15
Is perfect engine possible?
Hot reservoir
QH1
Wout
QH2
QH
=
Win
QC
QC
Cold reservoir
Physics 207: Lecture 27, Pg 16
Turbines: Brayton Cycle
Physics 207: Lecture 27, Pg 17

Which of the following processes would have the
largest work output per cycle?
A)
B)
C)
P
P
V
P
V
V
Physics 207: Lecture 27, Pg 18
Internal combustion engine: gasoline engine
 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
(Adiabats)
Physics 207: Lecture 27, Pg 19
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 20
The Carnot Engine

Carnot showed that the
thermal efficiency of a
Carnot engine is:
Tcold
Carnotcycle  1
Thot
 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.
Physics 207: Lecture 27, Pg 21