ch20 young freedman1
TRANSCRIPT
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Adiabatic Processes for an Ideal Gas
dU dW PdV
By definition, we have dQ=0
for any adiabatic process,
Then, from 1st Law, we have,
adiabatic expansion Tdrops
Isothermal1
PV
Adiabatic process1
PV
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Adiabatic Processes for an Ideal Gas
Now, we use the trick that for an ideal gas, dUis the same for allprocesses with the same dT = Tf Ti. As stated previously, we can
calculate dU using,
vdU nC dT
Then 1st law gives,
v
nRT
nC dT PdV dV V
(In the last step, we used
Ideal Gas Law: PV=nRT)
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Adiabatic Processes for an Ideal Gas
Rearrange terms, we have,
0v
dT R dV
T C V
Using the relations for the molar specific heats,
1 1p v p
v v v
C C CR
C C C
( 1) 0dT dV
T V
Then, we have,
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Adiabatic Processes for an Ideal Gas
Integrating this equation, we have,
1
( 1) constant
ln ( 1) ln constant
ln constant
dT dV
T V
T V
TV
1 constantTV
Using the Ideal Gas Law again, we can replace Twith ,PV
nR
1 constantPV
VnR
constantPV
(alternate form)
Note: Thas to be in K.
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Work in an Adiabatic Process (Ideal Gas)
( 0)dU dW dQ We know that,
Using the same trick on dU, we can calculate the work done
in an adiabatic process if we know the changes in state
variables.
vdW dU nC dT
2 1( )vW nC T T
2 2 1 1v
CW PV PV
R or
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insulation
Adiabatic Processes: 2 examples
Quasi-static adiabatic expansion:
Expanding gas push piston up work isdone by gas W > 0 U< 0 (energyflows out of gas)
For an Ideal Gas, Uis a function of Tonly, So, U< 0 also implies T< 0
(temperature drops!)
Adiabatic free expansion (non-quasi-static): Gas expands into vacuum no work done W=0
Adiabatic Q = 0
1st law gives U= 0
Uremains unchanged and Tis a constant!
Dfire piston
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Examples
Fire Piston (demo) Example 19.68 (Comparison of processes)
Fire Piston History
http://en.wikipedia.org/wiki/Fire_piston
Fire piston calculationshttp://complex.gmu.edu/www-phys/phys262/soln/fire_piston.pdf
Example 19.68 calculationshttp://complex.gmu.edu/www-phys/phys262/soln/ex19.66.pdf
a) Isothermal
b) Adiabatic
c) Isobaric
2VGiven initial state 1 1,P V final
3 diff ways: isotherms
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Chapter 20: The 2nd
Law ofThermodynamics
Preferential Direction inThermodynamic Processes
Heat Engine and Efficiency
The 2ndLaw of
Thermodynamics The Carnot Cycle (the most
efficient heat engine)
Entropy Entropy and Disorder
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Preferred Direction of NaturalProcesses
Processes notobserved in nature:Example 1:
Ball absorbing heat energyfrom surrounding
Then, converts it into mechanicalenergy and starts to bounce and roll
Note: energy is conserved (1st Law is NOT violated): heat mechanical eng.
BUT, we dont observe this process in nature while the reverse does!
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Disorder and ThermodynamicProcesses
The2nd Law of Thermodynamics
is the physicalprinciple which will delineate the preferred directionof natural processes.
We will also see that
The direction ofpreferred natural
processes
The degree ofrandomness (disorder) of
the resulting state
All natural processes in isolation will tend toward
the state of disorder !
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The 2ndLaw of Thermodynamics
Historically, there are more than one but equivalentway to state the 2ndLaw:
To address the condition in example #2, here is theClausius Statementon the 2ndLaw:
It is impossible for any process to have as its sole
result the transfer of heat from a cooler to a hotterbody.
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The 2ndLaw of Thermodynamics
There is also the Kelvin-Plancks Statement:
It is impossible for any system to undergo a cyclicprocess in which it absorbs heat from a reservoir at agiven temperature and converts the heat completelyinto mechanical work.
This implies that all heat engines have limited efficiency !(efficiency of real mechanical engines ~ 15 to 40%)
To understand this form of the 2ndLaw, we need to look at a toy model:
heat engine
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Heat Engines Definition: A device that converts a
given amount of heatintomechanical energy.
All heat engines carry some working
substance thru a cyclic process:
Engine releases residual heat to cold
reservoir at TC
Mechanical work is done by engine
Engine absorbs heat from hotreservoir at TH
Dstering
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Work Done by an Heat Engine
The engine works in a cyclicprocess,0U
1st Law gives,0netU Q W
netQ W
where, net H C H C Q Q Q Q Q
explicit signs for heats
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Efficiency for a Heat Engine
Thermal Efficiency e is defined as the ratio of themechanical energy output to the heat energy input,
H
We
Q
what you get out
what you put in
Substituting W=QH+QC, we have
e H C
H H
Q QW
Q Q
1 1C C
H H
Q Q
Q Q
using explicit signs here
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A perfect (100% efficient) heat engine
|QH|
1perfect
e
A perfect heat engine means 100%efficiency (e=1). This means that
1 1 means 0C CH
Qe Q
Q
All heat absorbed from reservoir THisconverted into mechanical work W.No residual heat is released back.
The Kelvin-Plancks statement of the 2ndLaw does notallow this !
1realistice
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2ndLaw, Disorder, & Available Energy
Two Forms of Energy in any Thermal Process:
Internal Energy Macroscopic Mechanical Energy
In the Kinetic-Molecular Model,
this consists of the KE and PEassociated with all the randomlymoving microscopic molecules.
The pistons motion in an
automobile engine results from thecoordinatedmacroscopic motionof the molecules.
(One typically cannot control theindividual random motions of allthese molecules.)
(Energy associated with thiscoordinated [ordered] motion can
be used for useful work.)
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2ndLaw, Disorder, & Available Energy
In a natural process (a block sliding to a stop),
e.g. vf
slightly warmerdue to friction
stopped
The coordinated motion of the block is converted into the KE & PE of theslightly more agitated random motions of the molecules in the block.
Macroscopic Mechanical Energy (KE of the block) is converted intoInternal Energy through heat exchanges as a result of friction.
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2ndLaw, Disorder, & Available Energy
Now consider the reverse direction it is unlikely that one can coordinateALL the randomly moving molecules in a concerted fashion. In otherwords, one typically cannotconvert the internal energy of a systemcompletelyback to macroscopic mechanical energy.
However, this does not mean that internal energy is not accessible. AnHeatEngine is exactly the machine that can perform this conversion but onlypartially.
The 2nd Law of Thermodynamics is basicallya statement limiting the availability of internalenergy for useful mechanical work.
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The Carnot Cycle (The Most EfficientHeat Engine)
A reversible cycle described by Sadi Carnot in1824.
The Carnot Theorem gives the theoretical limit
to the thermal efficiency of any heat engine. The Carnot cycle consists of:
Two reversible isothermal processes
Two reversible adiabatic processes
An Ideal Gas as its working substance