physics i entropy: reversibility, disorder, and information prof. wan, xin [email protected] xinwan
TRANSCRIPT
Physics IPhysics I
Entropy: Reversibility, Entropy: Reversibility, Disorder, and InformationDisorder, and Information
Prof. WAN, Xin
[email protected]://zimp.zju.edu.cn/~xinwan/
1st & 2nd Laws of Thermodynamics1st & 2nd Laws of Thermodynamics
The 1st law specifies that we cannot get more energy out of a cyclic process by work than the amount of energy we put in.
The 2nd law states that we cannot break even because we must put more energy in, at the higher temperature, than the net amount of energy we get out by work.
WQU
h
c
h
c
h T
T
Q
Q
Q
W 11 carnot
Efficiency of a Carnot EngineEfficiency of a Carnot Engine
All Carnot engines operating between the same two temperatures have the same efficiency.
An EqualityAn Equality
Now putting in the proper signs,
0c
c
h
h
T
Q
T
Q
0CycleCarnot
T
dQ
negativepositive
A Sum of Carnot CyclesA Sum of Carnot Cycles
0,
,
,
, i ic
ic
ih
ih
T
Q
T
Q
0C T
dQ
V
Padiabats Th,i
Tc,i
Any reversible process can be approximated by a sum of Carnot cycles, hence
Clausius Definition of EntropyClausius Definition of Entropy
Entropy is a state function, the change in entropy during a process depends only on the end points and is independent of the actual path followed.
T
dQdS reversible
012,21, 21
CCC
dSdSdS
21,12,21,
12
221 CCC
dSdSdSSS1
2
C1
C2
Return to Inexact DifferentialReturn to Inexact Differential
12ln)2,2(
)2,1(
)2,1(
)1,1(
dy
y
xdx
0lnln),( fyxyxf
2ln21)2,2(
)1,2(
)1,2(
)1,1(
dy
y
xdx
dyy
xdxdg Assume
Note: is an exact differential.
Integrating factor
y
dy
x
dx
x
dgdf
Digression on Multivariate CalculusDigression on Multivariate Calculus
Heat is path dependent.
Therefore, 1/T is really the integrating factor for the differential form of heat. Now we can recast the 1st law of thermodynamics as
Entropy is also a state function, as is the internal energy or volume.
PdVdUdQ
PdVTdSdU
Entropy of an Ideal Gas (1 mole)Entropy of an Ideal Gas (1 mole)
TfR
TCTU molV 2
)(
000 lnln),(
V
VR
T
TCSVTS mol
V
V
RTVTp ),(
V
RdV
T
dTCpdVdU
TdS
molV
1
Integrating from (T0,V0) to (T, V)
Carnot’s TheoremCarnot’s Theorem
No real heat engine operating between two energy reservoirs can be more efficient than Carnot’s engine operating between the same two reservoirs.
0''
1'
'1'
c
c
h
h
h
c
h
c
T
Q
T
Q
T
T
Q
Qe
0dS
What does this mean? Still, for any engine in a cycle (S is a state function!)
negativepositive
Counting the Heat Baths inCounting the Heat Baths in
h
hh T
QS
'
0 dSSgas
c
cc T
QS
'
after a cycle
Q'h > 0
Q'c < 0
0'
0'
c
c
h
hcgash T
Q
T
QSSSS
Counting the Heat Baths inCounting the Heat Baths in
h
hh T
QS
'
0 dSSgas
c
cc T
QS
'
after a cycle
Q'h > 0
Q'c < 0
The total entropy of an isolated system that undergoes a change can never decrease.
Example 1: Clausius StatementExample 1: Clausius Statement
hh T
QS
cc T
QS
0ch
ch T
Q
T
QSSS
Irreversible!
Example 3: Mixing WaterExample 3: Mixing Water
TA
A
TB
BQTA TB T T
Q
TA< TB
BA
BBAA
B
A
B
A
mm
TmTmT
TT
TT
m
m
AAA TTcmQTT :
TTcmQTT BBB :
Example 3: Mixing WaterExample 3: Mixing Water
Irreversible!
TA
A
TB
BQTA TB T T
Q
TA< TB
0ln2
BABA TT
TcmSSS
0ln:
AA
T
T
AAA T
TcmT
dTcmSTT
A
2/, BABA TTTmmm
0ln:
BB
T
T
BBB T
TcmT
dTcmSTT
B
For simplicity, assume
We can only calculate S with a reversible process! In this case, we replace the free expansion by the isothermal process with the same initial and final states.
Example 4: Free ExpansionExample 4: Free Expansion
00 SWQU
0ln
i
fV
V
V
V
V
V VV
nRV
nRdV
T
PdV
T
dQS
f
i
f
i
f
i
?
Irreversible!
The Second Law in terms of EntropyThe Second Law in terms of Entropy
The total entropy of an isolated system that undergoes a change can never decrease.
– If the process is irreversible, then the total entropy of an isolated system always increases.
– In a reversible process, the total entropy of an isolated system remains constant.
The change in entropy of the Universe must be greater than zero for an irreversible process and equal to zero for a reversible process.
0 UniverseS
Order versus DisorderOrder versus Disorder
Isolated systems tend toward disorder and that entropy is a measure of this disorder.
Ordered: all molecules on the left side
Disordered: molecules on the left and right
Macrostate versus MicrostateMacrostate versus Microstate
Each of the microstates is equally probable. Ordered microstate to be very unlikely because random
motions tend to distribute molecules uniformly. There are many more disordered microstates than ordere
d microstates. A macrostate corresponding to a large number of equival
ent disordered microstates is much more probable than a macrostate corresponding to a small number of equivalent ordered microstates.
How much more probable?
Entropy: A Measure of DisorderEntropy: A Measure of Disorder
2lnln Bi
fB NkV
VNkS
WTkS B ln
N
m
ff V
VW
N
m
ii V
VW
N
i
f
i
f
VV
WW
We assume that each molecule occupies some microscopic volume Vm.
suggesting (Boltzmann)
Let’s Play CardsLet’s Play Cards
Imagine shuffling a deck of playing cards:
– Systems have a natural tendency to become more and more disordered.
Disorder almost always increases is that disordered states hugely outnumber highly ordered states, such that the system inevitably settles down in one of the more disordered states.
Information and EntropyInformation and Entropy
(1927) Bell Labs, Ralph Hartley– Measure for information in a message
– Logarithm: 8 bit = 28 = 256 different numbers
(1940) Bell Labs, Claude Shannon– “A mathematical theory of communication”
– Probability of a particular message
But there is no information.
You are not winning the lottery.
Information and EntropyInformation and Entropy
(1927) Bell Labs, Ralph Hartley– Measure for information in a message
– Logarithm: 8 bit = 28 = 256 different numbers
(1940) Bell Labs, Claude Shannon– “A mathematical theory of communication”
– Probability of a particular messageOkay, you are going to win the lottery.
Now that’s something.
Information and EntropyInformation and Entropy
(1927) Bell Labs, Ralph Hartley– Measure for information in a message
– Logarithm: 8 bit = 28 = 256 different numbers
(1940) Bell Labs, Claude Shannon– “A mathematical theory of communication”
– Probability of a particular message
– Information ~ - log (probability) ~ negative entropy
i
ii PPS loginfomation
It is already in use under that name. … and besides, it will give you great edge in debates because nobody really knows what entropy is anyway.
---- John von Neumann
Maxwell’s DemonMaxwell’s Demon
To determine whether to let a molecule through, the demon must acquire information about the state of the molecule. However well prepared, the demon will eventually run out of information storage space and must begin to erase the information it has previously gathered. Erasing information is a thermodynamically irreversible process that increases the entropy of a system.
Landauer’s Principle & VerificationLandauer’s Principle & Verification
Computation needs to involve heat dissipation only when you do something irreversible with the information.
Lutz group (2012)
693.02ln Tk
Q
B
Homework (for the 2nd Law)Homework (for the 2nd Law)
CHAP. 24 Exercises 25, 30, 35 (P565) 4, 8 (P566)
HomeworkHomework
Reading (downloadable from my website): – Charles Bennett and Rolf Landauer, The fund
amental physical limits of computation.– Antoine Bérut et al., Experimental verification
of Landauer’s principle linking information and thermodynamics, Nature (2012).
– Seth Lloyd, Ultimate physical limits to computation, Nature (2000).
Dare to adventure where you have not been!