# Development of 2nd Law Of Thermodynamics

1. Apr 16, 2010

### wcg1989

I am new here and have always loved the history of science as much as the science itself. I have been intrigued by the 2nd law of thermodynamics and entropy for a long time but have also had something about its evolution that bothered me. So I'll summarize its stated development from my thermodynamics book and then ask my question.

The relevant chapters cover it this way. First, they describe it as a set of qualitative postulates. Such as Kelvin-Planck which says roughly a device cannot be constructed that raises a weight and exchanges heat with only one reservoir. Or 2nd, the Clausius statement that says a device cannot be made that has no effect other than the transfer of heat from a cold to hot body. And finally, its impossible to construct a perpetual motion machine.

Now the book describes the concept of a reversible process and factors like friction or heat transfer through a finite difference that make a process irreversible. Then it discusses a reversible Carnot cycle and proposes that a reversible engine is the most efficient.

So far all qualitative.

Then finally quantitative. The chapter states that the efficiency of the Carnot cycle is only dependent on temperature.

eff = W/QH = (QH - QL) / QH = 1 - QL/QH = f(TH, TL)

and finally says that Lord Kelvin proposed this relationship for the development of a temperature scale in a reversible process.

QH/QL = TH/TL which also equates for a Carnot cycle that eff = 1 -QL/QH = 1-TL/TH

This can be re-written as QH/TH - QL/TL = 0 and finally that is generalized as the Clausius integral where

Integral (dQ/T) = 0 for a reversible engine. And since for an irreversible engine, QLirr > QLrev, then for an irreversible engine,

Integral (dQ/T) < 0

And that is the 2nd law of thermodynamics stated in a quantitative way. Now here is the problem I have.

With the 1st law of thermodynamics, Joule demonstrated it by using a paddle wheel in a container of water and showing that a falling weight connected to the paddle would heat the container by a quantifiable amount. That demonstrated the 1st law. delta U = Q - W

Isaac Newton had developed the theory of gravity with F = GMm/d^2 and that was demonstrated because that equation can be used to derive the elliptical orbits of the planets that Kepler had described from observations a few decades earlier.

What experiment showed the 2nd Law specifically that the best efficiency of an engine was eff = 1 - TL/TH?

The equation eff = W/QH is a definition and the 1st law can be used to show that that is equivilant of eff = (QH-QL)/QH = 1-QL/QH but once that is transformed into efficiency of a reversible engine, eff = 1 - TL/TH, that is a declaration of the 2nd law.

Was it believed because it was elegant? Was there an experiment that really showed it, or as I suspect, it wasn't rigorously demonstrated because there was no good way of building a device that matched a Carnot reversible cycle at the time? It could in the 19th century, only generally be shown that as TL/TH decreased for a real engine, the efficiency got better.

I know that this is classical 2nd Law developed in the 1st half of the 19th century and that later a statistical concept of the 2nd law was formulated from which Integral (dQ/T) could be derived. But BEFORE that occurred, was the 2nd law demonstrated by experiment?

Thanks.

Last edited: Apr 16, 2010
2. Apr 16, 2010

### Andy Resnick

You are asking good questions! Your book appears to be a very typical undergraduate textbook and unfortunately it is quite incomplete.

If you can find it, Truesdell’s “The Tragicomical History of Thermodynamics, 1822-1854 (Studies in the History of Mathematics and Physical Sciences)”, is the gold-standard reference for this material. My copy is well-worn.

First, I want to focus on this:

“The relevant chapters cover it this way. First, they describe it as a set of qualitative postulates. Such as Kelvin-Planck which says roughly a device cannot be constructed that raises a weight and exchanges heat with only one reservoir. Or 2nd, the Clausius statement that says a device cannot be made that has no effect other than the transfer of heat from a cold to hot body. And finally, its impossible to construct a perpetual motion machine.

Now the book describes the concept of a reversible process and factors like friction or heat transfer through a finite difference that make a process irreversible. Then it discusses a reversible Carnot cycle and proposes that a reversible engine is the most efficient.”

Rather than give a detailed recount of the original publications from Clausius, Fourier, Reech, Duhamel, Carnot, Laplace, Kelvin, etc., let’s focus on the supposed ‘qualitative’ aspect of the body of work.

It is perhaps surprising that the first papers on heat and work had such rudimentary mathematics, but in point of fact, the paper by Carnot (in particular, but there are others) is atrocious.

This is a crucial point of fact as well: even though the original mathematical results had no rigorous foundation, the physical concepts the authors were attempting to describe mathematically are the correct concepts. That is, even though the formulas are a hideous conflagration of $$\delta , \Delta$$, d and ill-defined integrals, it is a fact (for example) that the rate of change of the internal energy of a system is the sum of the work(ing) and heat(ing).

But now we are left with all these ill-defined (in the quantitative sense) qualities like ‘energy’, ‘entropy’, ‘heat’, ‘temperature’… and this led to the next round of development: Kelvin, mostly. Kelvin corrects the mathematics and in so doing, invents the absolute temperature scale. An unfortunate side effect is that Kelvin turned thermodynamics into thermo*statics*. Every formula you presented is not a result of thermodynamics, but of thermostatics. The concepts ‘reversible’ and ‘equilibrium’ and ‘static’ are the same, and all expressed mathematically the same way.

Your book likely ends here, with this:

“I know that this is classical 2nd Law developed in the 1st half of the 19th century and that later a statistical concept of the 2nd law was formulated from which Integral (dQ/T) could be derived.”

This is unfortunate, because there is quite a bit more to the story. In brief, the study of thermodynamics was revived in the 1950s-1960s (along with continuum mechanics) and the fundamental axioms of thermo*dynamics* (note; 4 paragraphs above I wrote the 1st law in dynamical terms, not the usual E = W + Q thermostatic definition) were put on much more secure mathematical footing. Papers by Coleman (http://www.mechanics.rutgers.edu/BDC.html [Broken]) and others gave us mathematically rigorous definitions of thermodynamic variables.

What’s the point, since the 2nd law can be derived by statistical mechanics?

That sentence above reflects a fundamental misunderstanding of both statistical mechanics and thermodynamics. The arguments used to compute the entropy from axioms of statistical mechanics only hold for thermo*statics*. One essential difference between the laws of thermostatics and the laws of thermodynamics is the presence (or absence) of time. The 2nd law of thermodynamics has not been shown to be derivable from statistical methods. AFAIK, time appears nowhere in statistical mechanics, except for a single special case

Thermodynamics and continuum mechanics are correct physical models that encompass more of the universe than any other model, and many physicists have not heard of either.

How’s that? I can step you through details, but I need my books in front of me….

Last edited by a moderator: May 4, 2017
3. Apr 16, 2010

### wcg1989

That was interesting thanks. I have seen some discussions of the differences between thermostatics and thermodynamics. I have also seen a discussion on deriving the thermostatic definition from statistics, but although I get the idea, I have not rigorously grasped it.

However, I am an engineer mostly involved in testing so that is how this thread sort of came up. I have seen little in the way of someone demonstrating the classical 2nd law with a test at least one from the time period that these classical thermostatic equations were first published.

Perhaps there were no tests in the 19th century on the 2nd law, or that they were just not rigorous? I mean Joule's test on the 1st law with the weight and paddle is simplicity itself, almost seems trivial now but I know at the time it was a big deal. But how does one demonstrate the 2nd law in that time period? All I can think of is that they could show that a real engine had a greater efficiency the larger the temperature difference. But they couldn't show it rigorously or precisely because those engines were far from "reversible".

Anyway thanks for the response.

Last edited: Apr 16, 2010
4. Apr 16, 2010

### Andy Resnick

The book I mentioned steps through Carnot's paper (the 2nd law). Carnot was an engineer, and he did in fact do a measurement (I'm not sure I'd call it an 'experiment'). As I said, I need the book in front of me, and it's in my office. I can provide a description on Monday.

5. Apr 16, 2010

### wcg1989

I look forward to it. Thanks.

Edited to add:. Andy, when you are talking about the formulation of the 2nd law in terms of thermodynamics is this what you are referring to below? I copied this from Wikipedia and is a formulation of the rate of change of entropy in an open system with flow. Seems like a "dynamic" equation.

http://en.wikipedia.org/wiki/Entropy

dS/dt = mdot * Sin - mdot * Sout + Qdot /T + Sdotgen

Last edited: Apr 16, 2010
6. Apr 19, 2010

### Andy Resnick

Ok… now that I have my book, I can step through the various papers needed to establish the experimental basis of the second law. I want to say I’m glad to have to opportunity to work through this material in detail; I haven’t done this in a long time.

I figured we would start with Carnot’s (“REFLECTIONS ON THE MOTIVE POWER OF HEAT”, 1824) since it contains the very first statement of the second law. In the interest of scholarship, I’ll first write down exactly what is in the paper, and then ‘translate’ it as needed. Two things: first, remember I said the paper is atrociously written. Not incorrect, but a horrible mix of brilliant physical insight and grade-school mathematics. Second, most of what I have is written straight out of Truesdell’s book. I found an online copy of the translated original paper here: http://www.history.rochester.edu/steam/carnot/1943/

Again, I need to point out that his paper was written prior to notions of equilibrium and absolute temperatures.
First, the punchline: the data he used.

“According to the experiments of MM. Delaroche and Berard on the specific heat of gases, that of air is, for equal weights, 0.267 that of water. If, then, we take for the unit of heat the quantity necessary to raise 1 kilogram of water 1 degree; that which will be required to raise 1 kilogram of air 1 degree would have for its value 0.267. Thus the quantity of heat furnished by the body A is 0.267 units.
This is the heat capable of producing 0.000000372 units of motive power by its fall from 00.001 to zero.
For a fall a thousand times greater, for a fall of one degree, the motive power will be very nearly a thousand times the former, or 0.000372.
If, now, instead of 0.267 units of heat we employ 1000 units, the motive power produced will be expressed by the proportion 0.267/0.000372 = 1000/x , whence x = 372/267 = 1,395.
Thus 1000 units of heat passing from a body maintained at the temperature of 1 degree to another body maintained at zero would produce, in acting upon the air, 1.395 units of motive power.
[…]
Suppose the temperature of the body A 100 degrees, and that of the body B 99 degrees: the difference of the tensions will be, according to the table of M. Dalton, 26 millimetres of mercury or Om.36 head of' water.
The volume of the vapor is 1700 times that of the water. If we operate on one kilogram, that will be 1700 litres, or lme.700. Thus the value of the motive power developed is the product 1.700 x 0.36 = 0.611 units, of the kind of which we have previously made use.
The quantity of heat employed is the quantity required to turn into vapor water already heated to 1000. This quantity is found by experiment. We have found it equal to 5500, or, to speak more exactly, to 550 of our units of heat.
Thus 0.611 units of motive power result from the employment of 550 units of heat. The quantity of motive power resulting from 1000 units of beat will be given by the proportion 550/0.611 = 1000/x, whence x = 611/550 = 1.112.
Thus 1000 units of heat transported from one body kept at 100 degrees to another kept at 99 degrees will produce, acting upon vapor of water, 1.112 units of motive power.
[…]
Finally, we will show how far we are from having realized, by any means at present known, all the motive power of combustibles.
One kilogram of carbon burnt in the calorimeter furnishes a quantity of heat capable of raising one degree Centigrade about 7000 kilograms of water, that is, it furnishes 7000 units of heat according to the definition of these units given on page 88.
The greatest fall of caloric attainable is measured by the difference between the temperature produced by combustion and that of the refrigerant bodies. It is difficult to perceive any other limits to the temperature of combustion than those in which the combination between oxygen and the combustible may take place. Let us assume, however, that 10000 may be this limit, and we shall certainly be below the truth. As to the temperature of the refrigerant, let us suppose it .00 We estimated approximately (page 91) the quantity of motive power that 1000 units of heat develop between 1000and 990. We found it to be 1.112 units of power, each equal to 1 metre of water raised to a height of 1 metre.
If the motive power were proportional to the fall of caloric, if it were the same for each thermometric degree, nothing would be easier than to estimate it from 10000 to 00. Its value would be
1.112 x 1000 = 1112.
But as this law is only approximate, and as possibly it deviates much from the truth at high temperatures, we can only make a very rough estimate. We will suppose the number 1112 reduced one-half, that is, to 560. Since a kilogram of carbon produces 7000 units of heat, and since the number 560 is relatively 1000 units, it must be multiplied by 7, which gives
7 x 560 = 3920.”

Ok, to summarize so far:

1) 1000 units of heat passing from a body maintained at temperature of 1 degree to another maintained at 0 degrees would produce 1.395 units of work.
2) 1000 units of heat passing from a body maintained at the temperature of 100 degrees to another maintained at 99 degrees would produce, acting on steam, 1.112 units of work.
3) The motive power of burning 1 kg coal is 3920 units.

Recall, Carnot’s goal was to develop a more efficient steam engine, and he attempts to make his theory as general as possible. How does the data above lead the the second law of thermodynamics? Now we have the brilliant physical insight:

“According to established principles at the present time, we can compare with sufficient accuracy the motive power of heat to that of a waterfall. Each has a maximum that we cannot exceed, whatever may be, on the one hand, the machine which is acted upon by the water, and whatever, on the other hand, the substance acted upon by the heat. The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive. power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the height of its fall,* that is to say, the difference of temperature of the bodies between which the exchange of caloric is made. In the waterfall the motive power is exactly proportional to the difference of level between the higher and lower reservoirs. In the fall of caloric the motive power undoubtedly increases with the difference of temperature between the warm and the cold bodies; but we do not know whether it is proportional to this difference. We do not know, for example, whether the fall of caloric from 100 to 50 degrees furnishes more or less motive power than the fall of this same caloric from 50 to zero. It is a question which we propose to examine hereafter.
“Every change of temperature which is not due to a change of volume or to chemical action (an action that we provisionally suppose not to occur here) is necessarily due to the direct passage of the caloric from a more or less heated body to a colder body. This passage occurs mainly by the contact of bodies of different temperatures; hence such contact should be avoided as much as possible. It cannot probably be avoided entirely, but it should at least be so managed that the bodies brought in contact with each other differ as little as possible in temperature.”

After a while, Carnot concludes that the maximum work a given quantity of heat can do in a cycle is achieved when all the heat is absorbed at a high temperature and emitted at a low temperature. This is a Carnot cycle.

What Carnot does not say just what *is* maximized. Later physicists added concepts like ‘reversible cycles’, but Carnot’s reasoning is independent of the existence of these concepts. It is important to understand that much of the supporting data, here and in later papers by Joule, Kelivn, Clausius, etc., is given by measured values of the specific heat of various substances.

Then, Carnot claims:
“The production of motive power is then due in steam-engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body, that is, to its re-establishment of equilibrium-an equilibrium considered as destroyed by any cause whatever, by chemical action, such as combustion, or by any other. We shall see shortly that this principle is applicable to any machine set in motion by heat.
[…]
Wherever there exists a difference of temperature, wherever it has been possible for the equilibrium of the caloric to be re-established, it is possible to have also the production of impelling power. Steam is a means of realizing this power, but it is not the only one. All substances in nature can be employed for this purpose, all are susceptible of changes in volume, of successive contractions and dilatations, through the alternation of heat and cold. All are capable of overcoming in their changes of volume certain resistances, and of thus developing the impelling power. A solid body-a metallic bar for example- alternately heated and cooled increases and diminishes in length, and can move bodies fastened to its ends. A liquid alternately heated and cooled increases and diminishes in volume, and can overcome obstacles of greater or less size, opposed to its dilatation. An aeriform fluid is susceptible of considerable change of volume by variations of temperature. If it is enclosed in an expansible space, such as a cylinder provided with a piston, it will produce movements of great extent. Vapors of all substances capable of passing into a gaseous condition, as of alcohol, of mercury, of sulphur, etc., may fulfill the same office as vapor of water. The latter, alternately heated and cooled, would produce motive power in the shape of permanent gases, that is, without ever returning to a liquid state. Most of these substances have been proposed, many even have been tried, although up to this time perhaps without remarkable success. “

The claim is that all Carnot cycles that absorb the same amount of heat and have the same operating temperature also have the same motive power. This is crucial, as it renders his results completely independent of the design of the engine, and of the constitution of the working body. Again, later physicists add in certain equations of state or assumptions regarding the microscopic structure of matter, but those restrictions do not imply the original concept is likewise restricted. In order to justify his claim, Carnot describes the effect of the working body on it’s environment and then writes down an assumption:

“It is impossible for a heat engine to have done positive work yet have restored to the furnace all heat it previously absorbed from it and have withdrawn from the refrigerator all the heat it previously admitted to it.”

This is (historically) the first statement of the second law of thermodynamics.

Carnot’s work left many things undefined. Clayperon translated Carnot’s ideas into mathematics in 1834, but this merely magnified Carnot’s weak mathematics. Much of your book is based on Clayperon’s original essay, specifically the confusion between the general physical principle and the constitutive properties of a specific material.

That’s enough for today, I think… I have to prepare for class. Let me know what you think, what you want to discuss next, etc.

Last edited by a moderator: Apr 25, 2017
7. Apr 19, 2010

### DrDu

Beautifull stuff, Andi!
I really enjoyed reading, only missed some pop corn!
I am curious about your remark in the first post about the difference of thermostatics vs. thermostatistics. However personally, I think the insight, that thermodynamics (or thermostatics if you wish) refers to transitions between equilibrium states, was a big step forward: You can define temperature properly via the zeroth law, you drastically restrict the degrees of freedom so that you often only so become able to assign them at all. And you can define heat as the difference of the work done in an adiabatic process - which is assumed to be a state variable named U - and the heat done in the actual process operating between the same equilibrium states.

The articles by Coleman seem to refer to what is sometimes known as "rational thermodynamics", of which I don't know anything but that it is usually subsumed under linear irreversible thermodynamics. One possible criticism from a look at the papers: Coleman assumes the stress tensor to depend (among other things) on the history of absolute temperature. (There are many "We assume" in the article especially about the dependence of stress and heat on strain and temperature). But temperature is defined by the system being in equilibrium with the thermometer. How can you even define equilibrium for a time dependent process?
Furthermore the theory seems to be stream lined to discuss some rather specialized visco-elastic properties.
Don't get me wrong: I don't want to say that these papers don't contain some very interesting physics, I only doubt that they really extend on the range of thermostatics.

I'd just like to mention that there was another revival in classical thermodynamics at the beginning of the 20th century with the works of Max Planck (who did his thesis on it and, also maybe not directly, finally led him to discover quantum mechanics.) and Constantin Caratheodory, who both formulated some clever versions of the second law.

8. Apr 19, 2010

### Andy Resnick

Thanks, I appreciate that. Honestly, I enjoy discussions like this.

To your questions- I recommend you first unlearn all that stuff about 0th law, states, etc. We will add them back in as needed, but for now we don't need it. Because you are correct- defining an absolute temperature scale involved restricting systems to equilibrium systems. For over 100 years, that sufficed in Physics because of smart people that could tweak the concept to include things like 'local equilibrium' and 'detailed balance'. But now we have reached the end of those concepts; we can try to expand statistical methods to include fully dynamical systems (as many people are doing), or we can re-examine the fundamental thermodynamic concepts and re-develop them (the continuum program),

I'll be explicit- thermodynamics is currently a continuum theory only. SM is, for the purposes of this post, a different theory with different aims and axioms.

"rational thermodynamics" incorporates the axioms of thermodynamics to continuum mechanics. For example, wcg1989's formula is a balance equation, similar to those of mass, momentum, and energy. As I have posted elsewhere, a fundamental aim of continuum mechanics to is to specify the constitutive relations- the response of a material to an external force, if you like. It is an open problem, although there has been much progress in defining allowed classes of constitutive relations: materials with no memory, materials with memory, linear, nonlinear, idealized, etc. etc. Continuum mechanics is a fully axiomatized physical theory- the only one.

You are correct- the thermodynamic definition of absolute temperature currently holds only for equilibrium conditions. But it's clear that that definition needs to be updated-not just becasue of non-equilibrium considerations, but also from problems involving granular media, information theory, and relativity (Unruh radiation). Here's a paper you may find of interest:

http://www.physics.montana.edu/students/gilmore/iop_relativistic-thermo.pdf [Broken]

Last edited by a moderator: May 4, 2017
9. Apr 19, 2010

### wcg1989

Andy,

Thanks for posting that excerpt and for your commentary. There are three things that stick out to me. One is that the analogy that Carnot used with a waterfall is brilliant. Quantity of water with energy (although he didn't call it that) and height with temperature difference. Genius.

Second, what is as intriguing to me is that from the writing it appears that there were careful experiments that measured the motive power (work) based on a given amount of heat produced and a difference in temperature that it is allowed to "fall" through a process. What kind of experiments were these? How were these done and was he proposing that the results were the most amount of work that one could get based on the heat input and temperature difference.

And third, since he clearly was able to do experiments on heat and show work from it, it seems in hindsight that Joules experiment showing heat from work, shouldn't have been that big of surprise, but from further reading on Wikipedia, it was because the theory that Carnot was using called the "Caloric" was different than our concept of energy because the "Caloric" could be destroyed, not conserved.

Wayne

Edited:

I calculated the maximum possible efficiency for each experiment based on the information that you had summarized.

First just based on the work and heat input the actual efficiency for each experiment is:

Exp 1: eff = 1.395/1000 = 0.1395%
Exp 2: eff = 1.112/1000 = 0.1112%

However, the maximum possible efficiencies are based on the equation that we now know.

Max Eff = (TH-TL)/TH = 1/TH because the temperature difference (TH-TL) is 1 degree for both these experiments.

Assuming that he was using a Celsius/Kelvin scale, the maximum efficiency would be

Experiment 1:

Max eff = 1/(274.15) = 0.364%

Experiment 2:

Max eff = 1/(373.15) = 0.267%

Even if you use a Fahrenheit/Rankine scale, its:

Exp 1 :

Max eff = 1/(460.67) = 0.21%

Experiment 2:

Max eff = 1/(559.67) = 0.178%

So the experiments were definitely not the maximum efficiency possible and don't represent the efficiency of a Carnot cycle. But by measuring efficiencies and putting it in terms of heat input and temperature difference, it was instructive and opened up a whole new way of thinking about heat and work!

Last edited: Apr 20, 2010
10. Apr 20, 2010

### Studiot

I understand that Carnot subscribed to the caloric theory of heat, popular in his time.

Carnot was the only expositor of second law type statements to make a positive statement.

"Whenever a temperature difference exists, motive power can be produced."

As you have noted others stated the impossible.

11. Apr 20, 2010

### Andy Resnick

You are raising good questions- it's important to remember that there were, AFAIK, *no* experiments done on engines. The entire theory of thermodynamics (and of processes) was based solely on measurements of specific heats. I don't have the time right now to draw a detailed line between specific heat and entropy, but heuristically, the measurement answers "how much heat moves around during a given process?" This amount of heat C is given from the (then) theory of calorimetry:

C=$$\int [\Lambda_{V} (V, T) dV + K_{V} (V,T) dT]$$

Where the integral is over the path of the process, $$\Lambda_{V} (V, T)$$ is the latent heat at constant volume, $$K_{V} (V,T)$$ the specific heat at constant volume, and 'T' should not be confused with the (later) absolute temperature. Both latent and specific heat are allowed to vary with time, as well. So a measurement of K_V, plus the relevant constitutive relations for What I don't have time for is how to get the Joule-Thompson coefficient mu from this:

http://en.wikipedia.org/wiki/Joule–Thomson_effect

And that is the key function used to define absolute temperature (and entropy). I don't dispute your calculations, but again, AFAIK, there were no experiments done on heat engines at the time.

Sorry- it's "science day" here today, and I have to get my lab ready for the incoming hordes....

Last edited: Apr 20, 2010
12. Apr 20, 2010

### DrDu

I had a look on the interesting article on temperature concepts you provided, but I don't see that any of these definitions is too promising as a generalization, or even replacement, of the usual definition in terms of the zeroth law.

13. Apr 20, 2010

### wcg1989

Thanks Andy. Good luck on science day.

That was some useful insight but it seems to me that the use of measured specific heats and calculation of the caloric with the integral doesn't come up with the most efficient and therefore maximum work that is possible for a given heat input and temperature difference? Otherwise, if it did provide the most efficient work, and if Carnot had been using it, wouldn't Carnot have come up pretty easily with the equation that was later developed for his cycle?

efficiency = (TH-TL)/TH

My understanding was that that was not really expressed until later when Kelvin stated

QH/QL = TH/TL which can be used to change the efficiency from an expression of heat inputs (QH-QL)/QH to one of temperature differences (TH-TL)/TH.

14. Apr 20, 2010

### Count Iblis

This statement is false:

Also, strictly speaking there is no such thing as thermodynamics. Strictly speaking it is always thermostatics.

The entropy of a system can only be rigorously defined within the statistical framework. There is no rigorous definition of entropy within "thermodynamics".

The second law is certainly derivable form within statistical mechanics as most textbooks on the subject will point out. Of course there are assumptions here (e.g. you need to assume special initial conditions), but these assumptions underly all of statistical mechanics and thermodynamics.

15. Apr 20, 2010

### Studiot

This is potentially a very interesting thread, and I for one do not have the answer.
It would be very sad if the original question was lost in a fundamentalist wrangle over definitions that came after the time period wcg1989 is enquiring about.

16. Apr 20, 2010

### Andy Resnick

I agree- that's why it's still an open research problem :)

17. Apr 20, 2010

### Andy Resnick

That's a good question, and one I need to think about how to best answer. For now, all we know is:

1) A Carnot cycle (accepting heat at a hot furnace and rejecting it at a cold refrigerator) is the most efficient cycle,
2) All Carnot cycles operating between the same temperatures and moving the same amount of heat have the same efficiency,
3) There is an upper limit to the amount of work that can be extracted from a given amount of heat.

Let me think a bit....

18. Apr 20, 2010

### Andy Resnick

Provide a reference for this, please.

Provide a reference for this, please.

19. Apr 20, 2010

### Count Iblis

Fundamentals of Statistical and Thermal Physics by F. Reif

20. Apr 20, 2010

### Andy Resnick

What page? I have a copy.

21. Apr 20, 2010

### Count Iblis

In don't have the book with me right now. However, he constantly uses reasoning like that "all accessible states are equally likely" and then invoking Omega function to explain why entropy increases etc. etc. From the beginning of the book right until the end when Brownian motion and other non-equilibrium phenomena are discussed.

22. Apr 20, 2010

### Andy Resnick

Ok... here goes. The relevant math will be presented as summary statements, with many symbols not explicitly defined (here), and please remember that when I write 'T', 'p', 'V', 'S', it's to avoid writing the usual thermodynamic '$\theta, \varpi , \Gamma$, etc.. Time is an explicit part of this development.

First, the basic axioms of thermodynamics:
1) first law: $$\dot{E}= W + Q$$ , where the overdot means differentiation with respect to time, and W and Q are rates of working and heating.
2) second law: $$Q \leq B$$, where B is the least upper bound for heating. That is, there is a limiting rate at which a body may convert heat into energy without doing work. This statement is but one of the many, many different equivalent statements, each calling themselves 'The' second law.

Defining $$S_{t,t_0} = \int B/T dt$$, the second law becomes $$T\dot{S}\geq Q$$. Note, we *defined* the entropy S this way, without any context. The units of Entropy is not energy, but an energy per unit degree. The second law can then be integrated and written as

$$\int Q dt \leq TS_{t-t0}-\int \dot{T}S dt$$, with implicit integration limits of t_0 and t.

From those 2 axioms, we can then define adiabatic (Q = 0), isentropic ($$\dot{S} = 0$$), and isothermal ($$\dot{T}$$=0) processes.

Now, we define a process as 'reversible' if the integral formula above is an equality for all times 't'. It's also useful to define the free energy F = E-TS, in which case the second law is written as:

$$\dot{F} - W + S \dot{T} \leq 0$$

This is (nearly) the entirety of thermodynamics- the two equations written above are satisfied for any physically realizable deformable body undergoing any physically realizable process. Obviously, they are not of much help in predicting what will happen to a *specific* body undergoing a *specific* process. For that, we now require constitutive relations- we require functions, for example:

W = W(T, V)
F = F(T, V)
S = S(T,V)

I will now work two examples- first, using the 'classical equations of state', and then adding 'linear friction'.

For the classical equations of state, we have:

$$W = -p(T, V)\dot{V}$$
F = F(T, V), S = S(T, V)

The second law becomes: $$(\frac{\partial}{\partial T} F + S)\dot{T} + (\frac{\partial}{\partial V} F + p)\dot{V} \leq 0$$

Which can only be satisfied if S = -$$\frac{\partial}{\partial T} F$$ and $$p = -\frac{\partial}{\partial V}F$$.

These should look familiar, though maybe written in a 'strange' way. Now, linear friction (a quadratic term is added):

$$W = - p(T, V)\dot{V} - p_{12}(T, V) \dot{V_{1}}\dot{V_{2}}$$
F = F(T, V), S = S(T, V)

Going through the same process as above for 'ideal gases' results in the 'residual heating' $$B - Q = -p_{12}(T, V)\dot{V1}\dot{V2}$$. This is what we expect: frictional processes result in heat that is unavailable to be used to perform work (remember the inequality above, which corresponds with B-Q).

Other constitutive relations can be used (and there are hundreds), with the same procedure used to generate other thermodynamic relationships.

Ok, so much for classes of bodies. Now what about classes of processes? Again, we define:

Work done by the body: $$W = \int p \dot{V}dt$$
Heat gained by the body $$C = \int Q dt$$

using the basic constitutive relations $$J\Lambda_{V} = T \frac{\partial p}{\partial T}$$ and $$\frac{\partial}{\partial T}(\frac{\Lambda_{V}}{T})=\frac{\partial}{\partial V}(\frac{K_{V}}{T})$$, the first and second laws can be written in terms of the latent and specific heats:

$$\Lambda_{V} = T\frac{\partial S}{\partial V}, K_{V} = T \frac{\partial S}{\partial T}, J \Lambda_{V} =p+ T\frac{\partial E}{\partial V}, J K_{V} = \frac{\partial E}{\partial T}$$

J is the value Joule measured in his experiments.

And so, $$Q = T\dot{S}[/itex] and [itex]\dot{E} = JQ - p \dot{V}$$

These can be formally integrated to give $$\Delta S =\int \frac{Q}{T}dt$$ and $$\Delta E$$ = JC - W.

For a Carnot cycle, $$W/C_{abs} = J(1-T^{-}/ T^{+})$$, or $$C_{emit}/C_{abs} = T^{-}/ T^{+}$$. This is Kelvin's result.

A more general bound can also be found: $$W+ \Delta E \leq J(1-T^{-}/ T^{+}) C^{+} + T_{min}\Delta S$$.

Now, why do all this? Surely, using ideal gases and equilibrium states makes this much briefer. However, I feel that it also obscures the general nature of thermodynamics; for example, the results on linear friction and general bound on engine efficiency (and many others) are excluded by assumptions of equilibrium.

23. Apr 20, 2010

### Andy Resnick

I eagerly await an actual reference supporting your first sentence.

24. Apr 21, 2010

### wcg1989

Andy,

Thanks for the response. Interesting analysis and one that will take me some time to digest. I've been out of school for decades now, so I'll have to open up the old math closets in my brain to pull in that one.

Maybe I'm over-simplifying this but in Carnot's paper when he talked about 1.395 units of motive power (work) coming from 1000 units of heat with a 1 degree drop was that representative of an isentropic process meaning was that really the maximum work that could have been pulled from that heat with that drop in temperature?

Last edited: Apr 21, 2010
25. Apr 21, 2010

### DrDu

Strange stuff! What assumptions are implicit in your formulation of the second law? Is T the temperature of the surrounding, does it have to be equal to the temperature of the system? I don't see that there is any limit B as long as you don't specify the temperature of the surrounding.
I interprete your procedure as follows: The system under consideration is in contact with a heat bath of temperature T(t) and has a volume V(t), both time dependent.
Then however, neither E nor W nor S can be a function of (momentary) T and V as they will depend on the whole history of these parameters. This implies also that the latent and specific heats are functionals of the complete V and T history. I strongly doubt that anybody at the time of Carnot was able to determine these quantities.