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Entropy Confusion

by gadje
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Andy Resnick
#37
Apr15-10, 07:24 AM
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Quote Quote by mr. vodka View Post
Hm, I've come to the realization I probably know too little to have enough insight for decent arguments.

But Andy Resnick, let me ask you one thing. SM has a rigid definition of entropy that in essence allows you to calculate the entropy of any situation (in practice, one might be lacking knowledge/skill to actually do it). But more importantly, and basically my question, how is entropy defined in classical thermodynamics? I was taught that dS = dQ/T viewed from a reversible process, but this definition is incredibly limited, so I assume there is a more general one out there that everybody has neglected to teach me. Could you tell me?

EDIT: re-reading my post, it may sound as though I am sniping, but that wasn't meant that way, it was/is an honest question
Having questions is always good :)

Entropy is defined by the second law. That may not sound helpful, but recall the second law, properly written, is a statement on the rate of entropy *production* during a process.

SM is a theory of equilibrium states, nothing more. Thus, the "definition" of entropy in SM (k ln (W)) can only correspond to thermo*statics*. In thermodynamics, the entropy is related to the maximum amount of heat it is possible to generate during a process. Specifically,

[tex]T \dot{S} \geq Q[/tex]

It's important to recall that Q is not the 'heat' but the *rate* of heating, as seen from conservation of energy:

[tex] \dot{E} = W + Q[/tex]

Similarly, W is the *rate* of work (not the partition function). In thermostatics and SM, the time derivatives are gotten rid of, and differentials added (often times very confusingly).

The problem of thermodynamics, from this point forward, is first how to determine the constitutive functionals W, S, and F (the free energy) and second, to derive thermostatics from the thermodynamic functionals previously assigned.

Let me emphasize again that just because one has a "rigid" definition of thermostatic functionals, one *cannot* extrapolate to thermodynamics. k ln W is not a 'more fundamental' definition of entropy, since the domain of validity of SM is limited to only equilibrium states.

Does this help?
Andy Resnick
#38
Apr15-10, 07:28 AM
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Quote Quote by Count Iblis View Post
Statistical mechanics as used in practice (like evaluating partition functions or doing monte carlo simulations), may not be all that useful to engineers who use thermodynamics. But that doesn't mean that the foundations of thermodynamics lie firmly within the realm of what we in theoretical physics call "statistical mechanics".

Note that statistical mechanics can also be the study of chaos theory, far out of equilibrium phenomena etc. etc.
Look, I've tried to be as explicit as possible: SM hold for equilibrium, which is thermo*statics*. Thermo*dynamics* lies outside the domain of validity of SM. Parroting the word salads of others is not going to help you when you leave the comfort of previously worked examples and venture forth on your own.
Andy Resnick
#39
Apr15-10, 07:29 AM
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Quote Quote by Studiot View Post
In what way?

<snip>
I note that your overlong post neglected to contradict my claim.
SpectraCat
#40
Apr15-10, 08:04 AM
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Quote Quote by Studiot View Post
In what way?
But all this would be at nought without the theory of the chemicals and their reactions that go into these plants.

It is often said in textbooks on physical chemistry that thermodynamics (read classical here) define what reactions are possible, but tell us nothing about the rates of these reactions. The reaction may be thermodynamically feasible, but so slow as to be unusable.

For instance glass is soluble in pure water.
The catch is that the rate of solution is measured on the geological timescale.

The mathematics of these rates is definitely the province of statistical mechanics. I amsure you will find lots of reaction rate information in the references you mention amongst others.
No, I do not think this is correct. Statistical mechanical models have certainly helped provide some insight into the microscopic phenomena that account for measured reaction rates, and in the very simplest cases, statistical mechanical models can provide accurate numerical estimates for reaction rates. Note however, that in order to do this, the shape of the entire potential energy surface for the reaction must be known (which is an intractable problem for systems larger than a few atoms), and even then one must assume a "quasi-equilibrium" between the reactants and the activated complexes at the tops of the reaction barriers. Microscopic theories of chemical kinetics are highly approximate in practice, and rely on empirical constants and constraints for accurate work.

The most fundamental equation in chemical kinetics, the Arrhenius equation, is an empirically derived formula, and the field remains firmly grounded in using empirical models for reaction rates. Chemical engineers certainly rely primarily on empirical data and models when determining things like how scaling up a chemical reactor will affect dynamic quantities like heat and material transport, and how those will then affect reaction rates. In almost all cases, the microscopic details of what is going on at the molecular scale is far too low-level and far too "fuzzy" to be worth considering.

In any case, these microscopic theories are not well-enough understood or developed to be useful to a chemical engineer in the first place. Try telling them, "I have a model that describes in detail the molecular scale processes that must be contributing to the large-scale process you are trying to model." They might say, "Great! How accurately can it predict changes in reaction rates over such-and-such a range of temperatures and pressures." If one then responded, "It does really well, and is never off by more than an order of magnitude or so.", then one would be lucky if all they do is laugh in one's face. For their purposes, such a large uncertainty is almost certainly completely untenable.
Studiot
#41
Apr15-10, 08:07 AM
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I note that your overlong post neglected to contradict my claim.
What exactly do you mean by that?
Studiot
#42
Apr15-10, 08:32 AM
P: 5,462
At first I thought to myself

Perhaps they teach a different brand of reaction kinetics in American universities from that put forward in Oxford and Cambridge in the Uk.

But no, wait, I find on Page 274 of what is probably the foremost American text on the subject of physical chemistry by Moore, a chapter entitled
'Collision theory of gas reactions', followed by statistical mechanics maths.

So I see all is OK and it is just that some at PhysicsForums have not read this book or its brothers.

For the record

The link between chemical kinetics and statistical mechanics is to do with the probabilities of two molecules meeting.
This must obviously happen for those two molecules to react chemically.

This application of SM is obviously different from the application Andy et al are describing, so can be expected to have a different appearance.

In particular they are not necessarily about equilibrium. SM can be and indeed is employed in non equilibrium situations.

Mix 1 mole of sodium hydroxide with 1 mole of hydrochloric acid.
You have a definite non equilibrium situation.

I wasn't thinking of the Arrhenius equation when I mentioned SM, but it is true you can get to it from there.
However this is not the fundamental equation of chemical kinetics; it merely describes the temperature dependance of what is known as the 'rate constant'. This, of course, is only useful with simple order reactions that have a single rate constant.
SpectraCat
#43
Apr15-10, 09:08 AM
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Quote Quote by Studiot View Post
At first I thought to myself

Perhaps they teach a different brand of reaction kinetics in American universities from that put forward in Oxford and Cambridge in the Uk.

But no, wait, I find on Page 274 of what is probably the foremost American text on the subject of physical chemistry by Moore, a chapter entitled
'Collision theory of gas reactions', followed by statistical mechanics maths.

So I see all is OK and it is just that some at PhysicsForums have not read this book or its brothers.
Nothing you wrote there provides any refutation or counter-examples for anything I have written. I assure you that my understanding of chemical kinetics and SM is just fine . Go back and read my post carefully, and you will see that I was not talking about the *existence* of microscopic theories of chemical reactions, I was talking about their *accuracy* when applied to real, non-ideal experimental systems of non-trivial complexity. Everything in that section on "collision theory of gas reactions" provides a highly-idealized conceptualization of the microscopic phenomena. This is immensely useful for building an intuitive qualitative understanding of chemical reactions, but it does not provide quantitatively correct descriptions for any but the most simple reactions.

For the record

The link between chemical kinetics and statistical mechanics is to do with the probabilities of two molecules meeting.
This must obviously happen for those two molecules to react chemically.
That is not really a "link" at all ... it is as you say, obvious, and a necessary condition. However, it says nothing at all about what happens *after* the molecules come together. What is the probability of reaction? How does it depend on the quantum states and/or relative orientation of the molecules in the collision? How does the energy flow between the internal degrees of freedom of the molecules during the reaction, and how does that affect the reaction probability? Those are the phenomena we try to understand in chemical kinetics and dynamics (at least in the U.S. ) ... and while they are intensely interesting (to me at least), they have not led to accurate general theories of chemical reactions, even for gas phase systems of moderate complexity.

In particular they are not necessarily about equilibrium. SM can be and indeed is employed in non equilibrium situations.
Please give an example (what you have written below does not qualify).

Mix 1 mole of sodium hydroxide with 1 mole of hydrochloric acid.
You have a definite non equilibrium situation.
Ok, now write down a detailed description of the microscopic chemical processes involved in that reaction from the first-principles equations of stat. mech. (don't forget to take into account the effects of the solvent molecules!) Then use those results to come up with a quantitative prediction of the reaction rate and compare it to what was observed experimentally. I am afraid that you will find this is simply not possible.

I wasn't thinking of the Arrhenius equation when I mentioned SM, but it is true you can get to it from there.

However this is not the fundamental equation of chemical kinetics; it merely describes the temperature dependance of what is known as the 'rate constant'. This, of course, is only useful with simple order reactions that have a single rate constant.
What does any of that have to do with our discussion? Show me any equation describing chemical kinetics that is both accurate, and derived from first-principles using only the microscopic properties of the chemical system (i.e. the potential energy surface). You will find that, where such equations exist, they are typically applicable only to fairly simple chemical systems (i.e. unimolecular dissociation of a gas phase molecule). I am not aware of any such equations that can be extended to general cases in the gas phase, or any that apply for chemical reactions in condensed phases, at least not without modification with empirically derived expressions or constants.
Count Iblis
#44
Apr15-10, 09:20 AM
P: 2,157
Quote Quote by Andy Resnick View Post
Look, I've tried to be as explicit as possible: SM hold for equilibrium, which is thermo*statics*. Thermo*dynamics* lies outside the domain of validity of SM. Parroting the word salads of others is not going to help you when you leave the comfort of previously worked examples and venture forth on your own.
This is very misleading. What you wrote about entropy e.g. was wrong. In statistical mechanics we do have a definition of entropy for non-equilibrium systems. Also, note that thermodyamics is strictly spreaking always "thermostatics" as F. Reif explains in detail in his book.

The second law does not define entropy at all. Rather one has a general definition of entropy applicable for general systems that does not assume thermal equilibrium. Then the second law can be derived under some assumptions (like the equal prior probability assumption and time reversibility).

When we do thermodynamics what we actually do is approach a complicated situation as closely as we can from within thermostatics. The typical undergraduate textbook method is to consider only initial and final states which are accuratley describred by thermostatics.

What engineers do in practice when considering the actual dynamics of a system in terms of fluid velocity field, temperature and pressure distribution, is also strictly speaking approaching the real problem from within thermostatics. What happens here is that you add a lot of external variables in the description of the system. So, you just use a different coarse graining level and make statistical assumptions about those desgrees of freedom that you do not keep in your description.

Those assumptions do not have to be that it is in equilibrium, you can e.g. describe the situation using some Boltzmann distribution function whose evolution is given by a collision integral. But it should be clear that whenever you describe a system consisting of 10^23 degrees of freedom formally in terms of formulas that can be described in, say, 100 bits, you are necessarily making statistical assumptions about all those degrees of freedom.
Andy Resnick
#45
Apr15-10, 09:25 AM
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Quote Quote by Studiot View Post
What exactly do you mean by that?
You claimed "The whole of chemical reaction dynamics, and therefore chemical engineering, is built on statistical mechanics. Only the simplest reactions have linear dynamics."

I provided 3 major chemical engineering references, and SM is nowhere to be found in them. Now you provide a text on physical chemistry, and make some more claims (which I cannot substantiate; amazon does not allow me to see the table of contents). However, other texts on physical chemistry that I can see the TOC (Atkins is one, Silbey is another) again have *nothing* on SM.

Now you claim SM can be applied to nonequilibrium situations, and again, do not provide a reference. You do not see how kinetics is a linearized situation.

Just as I can't make you eat your vegetables, I can't make you learn. I can, however, try and prevent you from spreading misinformation to others.
Andy Resnick
#46
Apr15-10, 09:27 AM
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Quote Quote by Count Iblis View Post
<snip> In statistical mechanics we do have a definition of entropy for non-equilibrium systems. <snip>
I would be very interested in seeing this; I have not seen it before. Can you supply a reference?
Andy Resnick
#47
Apr15-10, 09:37 AM
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Quote Quote by Count Iblis View Post
<snip>

When we do thermodynamics what we actually do is approach a complicated situation as closely as we can from within thermostatics.
Are you kidding me? This is a joke, right?

Quote Quote by Count Iblis View Post
<snip>What engineers do in practice when considering the actual dynamics of a system in terms of fluid velocity field, temperature and pressure distribution, is also strictly speaking approaching the real problem from within thermostatics.
Can you provide a reference for this outlandish statement?

Quote Quote by Count Iblis View Post
Those assumptions do not have to be that it is in equilibrium, you can e.g. describe the situation using some Boltzmann distribution function whose evolution is given by a collision integral. But it should be clear that whenever you describe a system consisting of 10^23 degrees of freedom formally in terms of formulas that can be described in, say, 100 bits, you are necessarily making statistical assumptions about all those degrees of freedom.
You are completely changing the subject here. I never claimed one cannot describe a large number of discrete particles using statistical methods; I claim that a statistical description does *not* underlie thermodynamics. SM is *not* 'more fundamental' than thermodynamics. Thermodynamics does not 'emerge' from SM.
Studiot
#48
Apr15-10, 10:43 AM
P: 5,462
The most fundamental equation in chemical kinetics, the Arrhenius equation, is an empirically derived formula,
I wasn't thinking of the Arrhenius equation when I mentioned SM,
What does any of that have to do with our discussion?
Please give an example (what you have written below does not qualify).
I cannot hold a rational discussion with someone who introduces something, then appears to deny it's relevance.

Or appears to deny that if you mix some of the (nearly) stongest acid with some of the (nearly) strongest alkali you have a non equilibrium situation.
Studiot
#49
Apr15-10, 10:49 AM
P: 5,462
Just as I can't make you eat your vegetables, I can't make you learn. I can, however, try and prevent you from spreading misinformation to others.
I can't imagine why you are intent on being so downright rude, especially when most of what I say supports you.

However cast first the beam from thine own eye.

However, other texts on physical chemistry that I can see the TOC (Atkins is one, Silbey is another) again have *nothing* on SM.
1) I specified American textbooks, not UK ones.

2) Atkins is an excellent book: My copy has two chapters about Statistical Thermodynamics, Ch19 entitled the concepts and Ch20 entitled the machinery.

3) I do not know Silbey so cannot comment.

4) I did provide a gentle comment on your bibliography in reply to your question, which is more than you did for mine when I provided an example of a non equilibrium situation in the chemical reaction.

Do you deny that standard chemical reaction kinetics can be applied to this reaction?
SpectraCat
#50
Apr15-10, 12:26 PM
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Quote Quote by Studiot View Post
I cannot hold a rational discussion with someone who introduces something, then appears to deny it's relevance.
As I explained in context when I introduced it, the Arrhenius equation was derived empirically, rather than from first principles. That was my point, and you failed to address it.

Or appears to deny that if you mix some of the (nearly) stongest acid with some of the (nearly) strongest alkali you have a non equilibrium situation.
What are you talking about? How could you possibly get that from what I wrote? Of course it's not at equilibrium! I challenged you to use stat mech to derive the rate constant.

My point is that, while chemical kinetics can be rationalized qualitatively in terms of stat mech, you can't work the other way and derive quantitatively correct expressions from first principles...empirical adjustments are required.
Count Iblis
#51
Apr15-10, 01:11 PM
P: 2,157
Quote Quote by Andy Resnick View Post
Are you kidding me? This is a joke, right?



Can you provide a reference for this outlandish statement?



You are completely changing the subject here. I never claimed one cannot describe a large number of discrete particles using statistical methods; I claim that a statistical description does *not* underlie thermodynamics. SM is *not* 'more fundamental' than thermodynamics. Thermodynamics does not 'emerge' from SM.
I suspect we are using different definitions of SM, and what it means for X to be the foundation of Y. My definitions have nothing to do with how scientists and engineers work in the subject areas in practice. What I'm saying is similar to the statement that physics is the foundation of chemistry and biology. This does not imply that chemist use the Schrödinger equation to calculate the properties of molecules. The fact that they don't doesn't mean that physics is not a foundation of chemistry.
Studiot
#52
Apr15-10, 01:52 PM
P: 5,462
SpectraCat

What are you talking about? How could you possibly get that from what I wrote? Of course it's not at equilibrium! I challenged you to use stat mech to derive the rate constant.
OK well I stated that SM methods are also applicable to non equilibrium situations and offered an example

In particular they are not necessarily about equilibrium. SM can be and indeed is employed in non equilibrium situations.

Mix 1 mole of sodium hydroxide with 1 mole of hydrochloric acid.
You have a definite non equilibrium situation
Your response was to state that my example was inadequate (did not qualify) and to ask for, I presume another one.

Please give an example (what you have written below does not qualify).
Now addressing the second part of your statement above.
Why do you keep harping on about the 'rate constant' when I presume you know that it applies strictly to equilibrium situations?
Are you actually asking me to set up and solve the relevant differential equations (which involve this constant) that do lead to the actual rate of reaction?


I wonder if, as you yourself observed so well in your post #29 that we are working on different interpretations of Statistical Mechanics?

Taking Andy’s definition form his post #31 (I agree with this)

Statistical Mechanics: (classical or quantum) mechanics of a large number of discrete particles.
I think my example qualifies as it is the average action of some 1024 molecules.

This is not about partition functions, energy surfaces or whatever – though we could discuss those.
Andy Resnick
#53
Apr15-10, 02:13 PM
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Quote Quote by Count Iblis View Post
I suspect we are using different definitions of SM, and what it means for X to be the foundation of Y. My definitions have nothing to do with how scientists and engineers work in the subject areas in practice. What I'm saying is similar to the statement that physics is the foundation of chemistry and biology. This does not imply that chemist use the Schrödinger equation to calculate the properties of molecules. The fact that they don't doesn't mean that physics is not a foundation of chemistry.
That's all well and good to have an idiosyncratic point of view...which you do not divulge except for an opaque metaphor... but it makes meaningful communication impossible.
Andy Resnick
#54
Apr15-10, 02:16 PM
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Quote Quote by Studiot View Post
SpectraCat



OK well I stated that SM methods are also applicable to non equilibrium situations and offered an example

<snip>
That was a poor example. To wit; you offered an example from *kinetics*. Kinetics is steady-state.

The reason steady-state can be modeled in SM is because it can be transformed to appear like *equilibrium*. Kinetics is a linearized theory- steady-state conditions are a linear condition.

You never responded to my example: calculate Q(t) for the OP's problem.


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