Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey...

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Thermodynamics Thermodynamics and the Gibbs and the Gibbs Paradox Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne

Transcript of Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey...

Page 1: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

ThermodynamiThermodynamics and the cs and the

Gibbs ParadoxGibbs Paradox

Presented by: Chua Hui Ying Grace

Goh Ying Ying

Ng Gek Puey Yvonne

Page 2: Thermodynamics and the Gibbs 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

Page 3: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 4: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Unravel the Unravel the mystery of The mystery of The Gibbs ParadoxGibbs Paradox

Page 5: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

The mixing of The mixing of non-identical gasesnon-identical gases

Page 6: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Shows obvious increase in entropy (disorder)

Page 7: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

The mixing of identical The mixing of identical gasesgases

Page 8: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Shows zero increase in entropy as action is reversible

Page 9: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Compare the two scenarios of mixing and we realize that……

Page 10: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 11: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 12: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 13: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Entropy

Smax

SimilarityS=0

Z=0 Z = 1

Page 14: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 15: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 16: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 17: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 18: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Entropy

Smax

Similarity

S=0

Z=0 Z = 1

Page 19: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 20: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 21: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

(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

Page 22: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

(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.

Page 23: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

Do you have any idea what the

picture is all about?

Page 24: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.
Page 25: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 26: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

(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

Page 27: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 28: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

The mixing of identical The mixing of identical gases (revisited)gases (revisited)

Page 29: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.
Page 30: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 31: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 32: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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

Page 33: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 34: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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.

Page 35: Thermodynamics and the Gibbs Paradox Presented by: Chua Hui Ying Grace Goh Ying Ying Ng Gek Puey Yvonne.

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