Post on 22-Dec-2015
transcript
ThermodynamiThermodynamics and the cs and the
Gibbs ParadoxGibbs Paradox
Presented by: Chua Hui Ying Grace
Goh Ying Ying
Ng Gek Puey Yvonne
OverviewOverview
The three laws of thermodynamicsThe three laws of thermodynamics The Gibbs ParadoxThe Gibbs Paradox The Resolution of the ParadoxThe Resolution of the Paradox
Gibbs / JaynesGibbs / Jaynes Von NeumannVon Neumann
Shu Kun Lin’s revolutionary ideaShu Kun Lin’s revolutionary idea ConclusionConclusion
The Three Laws of The Three Laws of ThermodynamicsThermodynamics
11stst Law Law Energy is always conservedEnergy is always conserved
22ndnd Law Law Entropy of the Universe always Entropy of the Universe always
increaseincrease 33rdrd Law Law
Entropy of a perfect crystalline Entropy of a perfect crystalline substance is taken as zero at the substance is taken as zero at the absolute temperature of 0K.absolute temperature of 0K.
Unravel the Unravel the mystery of The mystery of The Gibbs ParadoxGibbs Paradox
The mixing of The mixing of non-identical gasesnon-identical gases
Shows obvious increase in entropy (disorder)
The mixing of identical The mixing of identical gasesgases
Shows zero increase in entropy as action is reversible
Compare the two scenarios of mixing and we realize that……
To resolve the To resolve the ContradictionContradiction
Look at how people do thisLook at how people do this1.1. Gibbs /JaynesGibbs /Jaynes
2.2. Von NeumannVon Neumann
3.3. Lin Shu KunLin Shu Kun
Gibbs’ opinionGibbs’ opinion
When 2 non-identical gases mix and When 2 non-identical gases mix and entropy increase, we imply that the entropy increase, we imply that the gases can be separated and returned to gases can be separated and returned to their original statetheir original state
When 2 identical gases mix, it is When 2 identical gases mix, it is impossible to separate the two gases impossible to separate the two gases into their original state as there is no into their original state as there is no recognizable difference between the recognizable difference between the gasesgases
Gibbs’ opinion (2)Gibbs’ opinion (2)
Thus, these two cases stand on Thus, these two cases stand on different footing and should not be different footing and should not be compared with each othercompared with each other
The mixing of gases of different The mixing of gases of different kinds that resulted in the entropy kinds that resulted in the entropy change was independent of the change was independent of the nature of the gasesnature of the gases
Hence independent of the degree Hence independent of the degree of similarity between themof similarity between them
Entropy
Smax
SimilarityS=0
Z=0 Z = 1
Jaynes’ explanationJaynes’ explanation
The entropy of a macrostate is given asThe entropy of a macrostate is given as
)(log)( CWkXS
Where S(X) is the entropy associated with a chosen set of macroscopic quantities
W(C) is the phase volume occupied by all the microstates in a chosen reference class C
Jaynes’ explanation (2)Jaynes’ explanation (2)
This thermodynamic entropy This thermodynamic entropy S(X)S(X) is not a is not a property of a microstate, but of a certain property of a microstate, but of a certain reference class reference class C(X)C(X) of microstates of microstates
For entropy to always increase, we need For entropy to always increase, we need to specify the variables we want to to specify the variables we want to control and those we want to change. control and those we want to change.
Any manipulation of variables outside Any manipulation of variables outside this chosen set may cause us to see a this chosen set may cause us to see a violation of the second law.violation of the second law.
Von Neumann’s ResolutionVon Neumann’s Resolution
Makes use of the quantum Makes use of the quantum mechanical approach to the mechanical approach to the problemproblem
He derives the equationHe derives the equation 2log21log11log12
Nk
S
Where measures the degree of orthogonality, which is the degree of similarity between the gases.
Von Neumann’s Resolution Von Neumann’s Resolution (2)(2)
Hence when Hence when = 0 entropy is at its highest = 0 entropy is at its highest and when and when = 1 entropy is at its lowest = 1 entropy is at its lowest
Therefore entropy decreases continuously Therefore entropy decreases continuously with increasing similaritywith increasing similarity
Entropy
Smax
Similarity
S=0
Z=0 Z = 1
Resolving the Gibbs Paradox - Using Entropy and its revised relation with Similarity proposed by Lin Shu Kun.
• Draws a connection between information theory and entropy
• proposed that entropy increases continuously with similarity of the gases
Analyse 3 concepts!
(1) high symmetry = high similarity,
(2) entropy = information loss and
(3) similarity = information loss.
Why “entropy increases with similarity” ?
Due to Lin’s proposition that
• entropy is the degree of symmetry and
• information is the degree of non-symmetry
(1) high symmetry = high similarity
• symmetry is a measure of indistinguishability
• high symmetry contributes to high indistinguishability
similarity can be described as a continuous measure of imperfect symmetry
High Symmetry Indistinguishability High similarity
(2) entropy = information loss
an increase in entropy means an increase in
disorder.
a decrease in entropy reflects an increase in order.
A more ordered system is more highly organized
thus possesses greater information content.
Do you have any idea what the
picture is all about?
From the previous example,
• Greater entropy would result in least information registered
Higher entropy , higher information loss
Thus if the system is more ordered,
• This means lower entropy and thus less information loss.
(3) similarity = information loss.
1 Particle (n-1) particles
For a system with distinguishable particles,
Information on N particles
= different information of each particle
= N pieces of information
High similarity (high symmetry) there is greater information loss.
For a system with indistinguishable particles,
Information of N particles
= Information of 1 particle
= 1 piece of information
Concepts explained:
(1) high symmetry = high similarity
(2) entropy = information loss and
(3) similarity = information loss
After establishing the links between the various concepts,
If a system is
highly symmetrical high similarity
Greater information loss
Higher entropy
The mixing of identical The mixing of identical gases (revisited)gases (revisited)
Lin’s Resolution of the Gibbs Lin’s Resolution of the Gibbs ParadoxParadox
Compared to the non-identical gases, we have Compared to the non-identical gases, we have less information about the identical gasesless information about the identical gases
According to his theory, According to his theory, less information=higher entropyless information=higher entropy
Therefore, the mixing of gases should result in Therefore, the mixing of gases should result in an increase with entropy.an increase with entropy.
Comparing the 3 graphsComparing the 3 graphs
Entropy
Smax
Similarity
S=0
Z=0 Z = 1
Entropy
Smax
Similarity
S=0
Z=0 Z = 1 Z=0
Entropy
Smax
Similarity
S=0
Z = 1
Gibbs Von Neumann Lin
Why are there Why are there differentdifferent ways in ways in resolving the paradox?resolving the paradox?
Different ways of considering EntropyDifferent ways of considering Entropy
Lin—Static Entropy: consideration of Lin—Static Entropy: consideration of configurations of fixed particles in a configurations of fixed particles in a systemsystem
Gibbs & von Neumann—Dynamic Gibbs & von Neumann—Dynamic Entropy: dependent of the changes in Entropy: dependent of the changes in the dispersal of energy in the the dispersal of energy in the microstates of atoms and moleculesmicrostates of atoms and molecules
We cannot compare the We cannot compare the two ways of resolving the two ways of resolving the paradox!paradox!
Since Lin’s definition of entropy is Since Lin’s definition of entropy is essentially different from that of essentially different from that of Gibbs and von Neumann, it is Gibbs and von Neumann, it is unjustified to compare the two unjustified to compare the two ways of resolving the paradox.ways of resolving the paradox.
ConclusionConclusion
The Gibbs Paradox poses problem The Gibbs Paradox poses problem to the second law due to an to the second law due to an inadequate understanding of the inadequate understanding of the system involved.system involved.
Lin’s novel idea sheds new light on Lin’s novel idea sheds new light on entropy and information theory, entropy and information theory, but which also leaves conflicting but which also leaves conflicting grey areas for further exploration.grey areas for further exploration.
AcknowledgementsAcknowledgements
We would like to thank We would like to thank
Dr. Chin Wee Shong for her Dr. Chin Wee Shong for her support and guidance support and guidance throughout the semesterthroughout the semester
Dr Kuldip Singh for his kind Dr Kuldip Singh for his kind supportsupport
And all who have helped in one And all who have helped in one way or anotherway or another