Development of 2nd Law Of Thermodynamics

[Andy Resnick]

I'm almost ready to get to Clausius. But in order to do that, I need to write Carnot's results in a more analytical form. Rather than copy down Clapeyron's attempt, I'll do it more literally; that is, a literal 'translation' of Carnot's statements into a mathematically consistent form. Because the LaTex editor is limited (for example, I can't put explicit limits on integrals), I'll use text to explain any missing information. I'll also use different letters than the usual to emphasize this.

First, the work 'L' done in a process:

$$L = L(P) = \int p(t) \dot{V}dt = \int p((V(t),T(t)) \dot{V}dt = \int p(V,T) dV$$

Where the second integral uses an equation of state and selects as state variable the volume V and temperature T. Getting to the last integral is straightforward, but now instead of a process occurring over a time interval, it occurs over a *path* in the V-T 'state space'.

Similarly, the heat added C is defined as:

$$C = C(P) = \int Q(t) dt = \int [\Lambda (V, T) dV + K(V,T) dT]$$

This also uses an equation of state. The second integral expresses the heating Q(t) as a function of the latent heat $\Lambda$ and specific heat K (these are at constant volume- I omitted the subscript 'V'). Specifically,

$$Q = \Lambda (V,T) \dot{V} + K (V,T) \dot{T}$$

Here's an important point- because Q is continuous, the latent heat and specific heat can also be written as

$$\Lambda = \frac{\partial H}{\partial V}, K = \frac{\partial H}{\partial T}$$

Where H is an unknown function (call it the 'heat function') H(T,V).

Ok, so now Carnot's axiom/claim can be written down:

1) the work extracted from a Carnot cycle depends on the temperatures of the hot and cold reserviors, and the heat added:

$$L(P_{c}) = G(T^{+}, T^{-}, C(P_{c}))$$

furthermore, the function G is seperable:

$$G(T^{+}, T^{-}, C(P_{c})) = F(T^{+}, T^{-})C(P_{c})$$

Carnot goes further, and unnecessarily restricts F. However, we do not require such a strong restriction. Instead, all we require is F(T+, T-) > 0 if T+ > T- > 0, and F(T,T) = 0

We now derive what is known as the Carnot-Clapeyron theorem:

Begin with L for a Carnot *cycle*- a closed loop in p- V (or V-T) space, consisting of 2 isotherms and 2 adiabats:

$$L(P_{c}) = \int dpdV = \int\frac{\partial p(V,T)}{\partial T} dVdT$$

Also, using the defintion of Q,

$$C (P_{c}) = \int \Lambda (V,T) dV$$

And so, Carnot's axiom (1) takes the form:

$$\int dpdV = \int\frac{\partial p(V,T)}{\partial T} dVdT = F(T^{+},T^{-})\int \Lambda (V,T^{+}) dV$$

Now I do some hand-waving. A critical point is that the function F is *not* a state function- it is independent of both material and process: that's the reason it's not in the integral. Historically, instead of F, people used $\mu$, and future researchers (Joule, Clausius, etc) called $\mu (T)$ 'Carnot's function'. Carnot's function can be written as dF/dT, but the way it was done back then, F(T+,T-) was assumed to be of the form F(T+) - F(T-). For infinitesimal changes in temperature, there is no conflict. For finite differences in temperature, it's a little more complex and the two results do not agree. So in the interest of not having a googleplex of different symbols, I'll just switch to Carnot's function and not worry what the specific form of it is in terms of F.

Just remember that from Carnot until the Joule-Thompson papers in 1852 and 1853 establishing the absolute temperature scale, people tried to measure the function $\mu$.

Converting F(T+,T-) to Carnot's function $\mu(T)$, we get the heat added for a Carnot cycle is:

$$C(P_{c}) = \frac{1}{\mu(T)}\int\frac{\partial p(V,T)}{\partial T} dV$$

This is the Carnot-Clapeyron theorem. For an infinitesimal process,$\mu\Lambda = \partial p/ \partial T$.

One final note (for now): putting the above special case of an ideal gas equation of state p/T = R/V into the expression for the heat:

$$C(P_{c}) = \frac{R}{\mu(T)}\int \frac{dV}{V} = \frac{R}{\mu(T)} log \frac{dV}{V}$$

And finally, on dimensional grounds, since L/C is a constant bearing the units of work/heat, that constant (called J, since Joule measured it) and Carnot's function $\mu$ are related by:

$$\mu =\frac{J}{T}$$

This may give you some insight as to what the heat function H and Carnot function 'really' are.

Now, we are ready for Clausius. For, after Clapeyron (1834) came Duhamel (1837,1838), Mayer (1842)- who is oddly credited with discovering the first law- and a preliminary experimental work by Joule (1845) that everyone panned. None of these had sufficient mathematical skill to properly translate Carnot's work (the stuff above is a correct 'translation', which came from Truesdell). Kelvin, in an early paper (1849) coined the term 'thermodynamics' but that's about it. Clausius wrote his first paper in 1850.

Some of it is here:

http://www.humanthermodynamics.com/Clausius.html#anchor_116

but I couldn't find a free on-line version of the whole thing. EDIT: that URL points to Clausus's later papers in 1854. We aren't there, yet!

EDIT 2: I did find an online version of all six papers: http://books.google.com/books?id=8LIEAAAAYAAJ

 

[Andy Resnick]

Clausius's first paper, published in 1850, is the next real contribution to thermodynamics. Unfortunately, just like Carnot, Clausius' mathematical skills are inadequate. His writing is very awkward, but the results are correct.

After 15 pages of "mathematical introduction', in which he treats differentials as fractions, he opens with the following:

"The steam engine having furnished us with a means of converting heat into a motive power, and our thoughts being thereby led to regard a certain quantity of work as an equivalent for the amount of heat expended in it's production, the idea of establishing theoretically some fixed relation between a quantity of heat and a quantity of work which it can possibly produce, from which relation conclusions regarding the nature of heat itself might be deduced, naturally presents itself".

Recall that the Caloric theory was in vogue- heat was modeled as a fluid. In essence, there were two kinds of state functions used to describe materials: one, a mechanical state function L (defined above), and another, C (also defined above). There was no reason to relate one to the other in any way. 30 years prior to Clausius, Carnot's paper made the assumption that heat can be used to generate work, not by the *consumption* of 'caloric', but by the *transport* of heat from a hot to a cold reserviour. Carnot claimed that assuming heat is consumed in the process of generating work would deny the the entire (caloric) theory of heat (which, ironically, is what Carnot proved).

Clausius responds:

"I am not, however, sure that [Carnot's] assertion is sufficiently established by experiment. [...] still other facts render it exceedingly possible that a loss occurs. [...] it is almost impossible to explain the ascension of temperature brought about by friction otherwise than by assuming an actual increase of heat."

He then places Joule's work in this context: mechanical work can produce heat, and so it is reasonable to assume that heat can also be lost, and in so doing, create work.

Clausius discussed experimental work by Holtzmann, Mayer, Regnault, and the subsequent analysis by Thomson (later Kelvin). Recall, the caloric theory *required* heat to be consumed, and so Carnot (thought he) was asserting a concept in opposition to existing theory.

Clausius justifies his claim that the principle of Carnot is not in opposition to caloric theory by saying

"On a nearer view of the case, we find the the new theory is opposed, not to the real fundamental principle of Carnot, but to the addition no heat is lost; for it is quite possible that in the production of work both may take place at the same time; a certain portion of heat may be consumed and a further portion transmitted from a warm body to a cold one; and both portions may stand in a certain definite relation to the quantity of work produced."

Clausius states Carnot's assertion this way ('Clausius's maxim'):

"In all cases where work is produced by heat, a quantity of heat proportional to the work is consumed; and inversely, by the expenditure of a like quantity of work, the same amount of heat be produced."

Mathematically, this is exactly the same as L = F*C, which we wrote above. It may have seemed (prior to Carnot) that we have 2 types of equations of state: one mechanical in origin (p, V) and other thermal in origin ($\Lambda$, K). One relates the mechanical work to material properties, another relates heat to material properties. At that time, there was no reason to think they should be related (and even to this day, 'thermal expansion' is often presented without any mention of the work evolved due to the pV change: 'free expansion').

Clausius then spends some time talking about how heat and work are interconvertible in the case of steam. Next he says something that (to me) is truly remarkable:

"Let a certain quantity of permanent gas, say a unit of weight be , given. To determine its present condition, three quantities are necessary; [p, V, T]. These quantities stand to each other in a relation of mutual dependence, which, by a union of the laws of Mariotte and Gay-Lussac, is expressed in the following equation: (in modern notation) pV = R(a+T). [...] It must be remarked, that Regnault has recently proved [...] that this law is not in all strictness correct."

No sooner does he say this, then quote experimental data by Magnus, Regnault and write:

"a = 273"

Consider this: Clausius just wrote down the absolute thermodynamic temperature scale. Clausius does not attach any importance to the factor 'a'. It is a total accident we do not use "degrees Clausius".

Clausius then goes on to establish cycles (using p-V diagrams) and calculates the work done in a Carnot cycle *for an ideal gas* as (again, in modern notation) L = R dT dV/V, in agreement with results I posted earlier. Now he wants to calculate the heat consumed during a cycle, and obtains the following result:

$$\frac{\partial p}{\partial T} =J (\frac{\partial \Lambda}{\partial T}-\frac{\partial K}{\partial V})$$

This is his maxim above: L/C = J (he used '1/A'), and substituting in the definition of L and C for a cycle (written above). This is a central result of thermodynamics; it is a relation between a mechanical property and thermal properties of substances. This relation is not a result from 'ideal gases' (although Clausius did in fact use them, this result is without his restrictions), all materials obey this equation of state, and thus this expression is a statement *restricting* the physically allowed behavior of material bodies.

Clausius then considers the *maximum* amount of work than can be produced by heat. He writes that the maximum may be represented as T/$\mu$. Thus he sets the experimental program to determine Carnot's function mu, and finds that $\mu$ does not depend on the material used- either permanent gases or vapor.

At the same time Clausius's paper is presented, Rankine presents (completely independently) a similar paper. This paper is of note because Rankine presents a *molecular theory of heat*. That is, he presents a theory that claims to show the origin of heat from molecular properties. His theory of 'molecular vortices' is not worth discussing, but using his molecular model, he obtains a 'constitutive function U(V,T) with the following property:

$$(T-T_{0})\dot{U} = Q$$.

This should look familiar; although Rankine did not interpret U in that way- he only stated U depends "on molecular forces. The only case in which it can be calculated is that of a perfect gas." However, to end on a positive note, we summarize the two papers as:

1) Clausius constructed the thermodynamics of ideal gases and discovered the concept of internal energy.
2) Rankine obtained the basic constitutive relations of the thermodynamics of fluids and expressed them using a function that looks essentially like entropy.

I can keep going if there's still interest; next is Rankine's second paper, calculating the efficiency of a Carnot cycle and Joule&Kelvin's paper defining the absolute temperature scale.

 

[James A. Putnam]

Andy Resnick,

I appreciate your approach to teaching theoretical physics. You are a valuable resource and, I am sure, a very good instructor.

In message #2 you said in reply to wcg1989's question:

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

I thought your response was the kind of insightful answer that I have not seen in physics forums:

"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."

Iblis, in message #14, replied that: "This statement is false." The statement he quoted is your first paragraph given above.

He also stated, in that same message, that:

"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."

This answer was, in my opinion, an example of skipping past Clausius' derivation without explaining its meaning. In message #39, I meant to compliment you and your explanation.

You responded in part in message #43:

"Asking what entropy 'is' (or what temperature 'is'), is very similar to asking (in the context of mechanics) what mass 'is'. In mechanics, we can't answer very much beyond 'mass is how much stuff there is'. ... There are many unsolved problems in Thermodynamics (limit of small size is an obvious one, wetting is another), but relating entropy change and time is not one of them. "

The reference to mass was not an answer for either entropy or mass. The ending of your statement '...but relating entropy change and time is not one of them.' I thought skipped past the absorption of heat part of Clausius' derivation.

I responded in message #44:

"Absorbing energy under equilibrium conditions requires time. As thermodynamic entropy changes so does something that relates to time. Why do you say that a process that requires time does not relate to time? Perhaps I do not understand your point. How does thermodynamic entropy change without relating it to a change in time?"

You responded in message #48:

"This is one of the notions I hope to disabuse you of. Maybe you are thinking of steady-state conditions, but at equilibrium, there can be *no* absorption. That's part of the definition of equilibrium!"

The T in the definition of thermodynamic entropy represents thermal equilibrium conditions. There is no fluctuation in temperature included. Yet energy in transit is included in the Q. Energy is absorbed over time under conditions of thermal equilibrium. I am not arguing that this theoretical ideal condition is possible in the real world. I am saying that Clausius did discover something of fundamental importance and it is not yet explained to this day.

By the way, mass is not stuff and temperature is not how hot something is. That is trading words for words. Both mass and temperature are indefinable properties with indefinable units of measurement. Both still await clear physical explanations. That is what I think. I respect your position here, your expert knowledge, and teaching talent. If you still think that my input is unhelpful, then I will drop it.

James

 

[Andy Resnick]

Thanks for the kind words. You are asking some very broad, complex, questions, and I'm not sure I can easily answer them (which is a reflection on me, not the question!). A factual correction first: wcg1989 did not ask "What’s the point, since the 2nd law can be derived by statistical mechanics?", that was me asking the question (rhetorically).

My motivation on this thread (and other related ones) is simply to *educate*. For whatever reason, thermodynamics has been treated badly by introductory physics textbooks, and this unfortunately carries forward to intermediate-level treatments. So there's this cycle of a student learning to recite half-truths, then turning around and repeating the half-truths to their students. A simple example is the half-truth that thermodynamics only applies to reversible, infinitesimal Carnot cycles of ideal gases- and that Kelvin calculated the efficiency of a Carnot cycle. Once that is accepted as dogma (because that's what's in all of the introductory Physics textbooks, and so that is what you are told), it's no wonder that the student, upon exposure to statistical mechanics, comes the believe that SM 'underlies' thermodynamics, because (among other reasons), SM uses atoms. And we all know matter is made of atoms.

Since (hopefully) some of the people here on PF will one day become faculty themselves and have a duty to teach future students, I hope that by providing a clear logical development of thermodynamics I can demonstrate thermodynamics is quite independent of SM, and should be considered on it's own merits. Is there overlap? Certainly. But they ask *different questions*.

Ok, now to the question of time in thermodynamics- the definition of L and C were first given an explicit time dependence, but then (on the same line), the time dependence became *implicit*- L and C are defined along a path in p-V space. Yes, moving along the process takes time! In the limit of an infinitesimal process (or cycle), this fact can be ignored. You have correctly identified a *major* conceptual stumbling block for many students! I'm trying to keep the time dependence as explicit as possible, by not restricting the discussion to infinitesimal cycles. Note, I have nowhere used the terms 'equilibrium', 'reversible', 'infinitesimal', etc. in the analysis of the original papers. Similarly, there is no need to invoke the (common) statement that 'adiabatic processes are very fast, so there is no heat transfer'.

What I have been doing in my (overly) long posts is a mix of original source material and a modern translation. Specifically, rather than use the mathematics contained in the original material- the occasional correct statement that was restricted (mostly) to ideal gases, hence the origin of the half-truth that 'thermodynamics only deals with ideal gases'- the mathematics have been updated and re-written in a way that was unknown to the original authors. The way I have been writing the equations, they are true for *all* materials- quantum dots and black holes, glasses and sandpiles. And all processes as well: stellar collapse and life.

You also raise the question about 'understanding' things: 'understanding' mass, for example. You are correct, it's easy to trade one set of words for another: 'mass is energy', for example. Rather than veer off into metaphysics, let me use this example:

There's a lot of research on water- the physical properties, the colligative properties of aqueous solutions, the molecular behavior, etc. etc. So it would not be wrong to say 'we don't understand water'. But that's not really true, either- I don't know where you live, but here I have indoor plumbing. The plumber *certainly* understands how to move water around, not just my house, but the distribution and collection system for the whole city. A plumber understands water just fine. Will any result coming out of molecular researches of water change building codes? Not likely. So I could be justified in claiming a plumber understands water better than I do!

Lastly, and this is more of a reflection on my limited notation, but 'T' refers to the temperature of the system. Thermodynamics takes 'T' to be a irreducible property of a system- it is a given, not a derived quantity. As we will soon see, (and Rankine has just showed), the 'temperature' is somehow related to the heating and the change in entropy. To me, this is much more appealing than saying 'temperature is a form of kinetic energy of atoms'. Which is true only for ideal gases obeying a Maxwell-Boltzmann distribution.

Does that help?

 

[James A. Putnam]

Andy Resnick,

I think your approach is great. I have modern physics resource books, but I seek out older books to see what was taught long periods ago. For the question of thermodynamic entropy I bought, off the Internet, 'Heat and Thermodynamics' by Mark Zemansky 1943. That is as far back as I have gone for this subject. Your historical laced account is a valuable easy resource for the readers like myself.

Two questions:

Is the use of the Carnot cycle in the introductory derivation of thermodynamic entropy misleading in anyway? Does its use accurately produce the mathematical basis of thermodynamic entropy?

Since the derivation includes only long established macroscopic type properties, though they relate to the internal state, shouldn't it be expected that thermodynamic entropy should be explainable as a similar macroscopic property.

Corrections are welcome.

James

 

[DrDu]

James, I think the Carnot cycle does lead in deed to an essentially correct description of phenomenological thermodynamics, however it is still a very engineering approach. However a very modern and carefull axiomatization which takes into account the many improvements made since then is available here:
http://arxiv.org/PS_cache/cond-mat/pdf/9708/9708200v2.pdf

Lieb is quite famous for his contributions in e.g. statistical mechanics and density functional theory.

 

[James A. Putnam]

DrDu,

Thank you for that link. It will take some extra time to respond. It is not only the length of 100 pages, it is making sure that I understand.

James

 

[Andy Resnick]

Dr. Du- that is a good link and I look forward to reading it.

James- IMO, it's not the introduction of a Carnot cycle that makes the treatment misleading, it's the breezy way differential quantities are treated, tying together 'reversible' with 'slow' and 'infinitesimal'. That, combined with statements like 'heat is not an exact differential', makes the theory appear cobbled together by lunatics.

To summarize where we are now (in 1850), we had Carnot's original statement:

"It is impossible for a heat engine to have done positive work yet have restored to the furnace all the heat it previously absorbed from it and have withdrawn from the refrigerator all the heat previously emitted to it.", but no quantitative statements regarding the maximum efficiency of (any) cycle.

Then, we had Kelvin, in 1849 write:

"A perfect thermo-dynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency; if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced."

This statement is made in the context of the caloric theory- heat is only a kind of 'force' or 'energy'; hence heat and work are universally and uniformly interconvertible in all circumstances."

This contradicts Carnot who noted the amount of work that can be produced is also dependent on the *temperatures* the heat is extracted and returned (specifically, the temperature difference). Joule's experiment showed work can be converted to heat.

Then Clausius, in his first paper, noted "In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced." From this, Clausius constructed the thermodynamics of ideal gases.

Rankine also developed a theory of heat based on molecular vortices, and developed a constitutive function (the 'heat function') which is what we now call 'entropy'.

Before proceeding, it's worthwhile to again mention the amazing lack of quantitation in these papers- sloppy reasoning, fuzzy mathematics, poor data. Papers like these would *never* be published today in reputable journals.

Also, it's worthwhile pointing out the many inconsistencies between the historical record we have been 'generating' here and, for example, wikipedia articles. I have no aim to correct other accounts; I am simply presenting the original works in the order they have been published.

Next up- Rankine's second paper in which he calculates the efficiency of a Carnot cycle, and Kelvin's paper establishing the absolute temperature scale. The concept of entropy is interwoven with these reports, but the word 'entropy', and all that it implies, has not yet been invented.

 

[Andy Resnick]

Rankine's second paper, in 1851, was presented as a 'fifth section' appended onto his earlier report. In it, he states:

"Carnot was the first to assert the law, that the ratio of the maximum mechanical effect, to the whole heat expended in an expansive machine, is a function solely of the two temperatures at which the heat is respectively received and emitted, and is independent of the nature of the working substance. But his investigations not being based on the principle of the dynamical convertibility of heat, involve the fallacy that power can be produced out of nothing.
The merit of combining Carnot's *Law*, as it is termed, with the convertibility of heat and power, belongs to Mr Clausius and Professor William Thomson; and in the shape into which they have brought it, it may be stated thus:-
The maximum proportion of heat converted into expansive power by any machine, is a function solely of the temperatures at which heat is received and emitted by the working substance; which function, for each pair of temperatures, is the same for all substances in nature."
[...]
"I have now come to the conclusions,-First: That Carnot's Law is not an independent principle in the theory of heat; but it is deducible, as a consequence, from the equations of the mutual conversion of heat and expansive power, as given in the First Section of this paper.
Secondly: That the function of the temperatures of reception and emission, which expresses the maximum ratio of the heat converted into power to the total heat received by the working body, is the ratio of the difference of those temperatures, to the absolute temperature of reception diminished by the constant, which I have called [k] and which must, as I have shewn in the Introduction, be the same for all substances, in order than molecular equilibrium be possible."

All Rankine is really saying is that

$$\frac{C^{-}}{C^{+}} = \frac{T^{-}-T_{0}}{T^{+}-T_{0}}$$,
and
$$L = J [C^{+} - C^{-}]$$.

Now here's the thing: T0. Rankine even calls it an 'absolute temperature'. Other than Rankine's first paper, where he defines T0 as "the temperature corresponding to the absolute privation of heat", I can find *no other* reference to an absolute temperature scale. Not in Kelvin's first paper, nor Joule's, nor any other. So, first we had Clausius toss off "pV = R(T + 273), and now we have T0 expressly written as an 'absolute zero'. Stunning, really.

combining the two expressions above, Rankine obtains

$$\frac{L}{JC^{+}} = \frac{T^{+}-T^{-}}{T^{+}-T_{0}}$$

Rankine then takes T0/T as being 0, to "the nearest approximation we can at present make", and thus obtains

C-/C+ = T-/T+ and L/JC+ = 1- T-/T+.

There's more: recall, Rankine showed

$$(T-T0)\dot{H} = Q$$

From this, the two isothermal 'arms' of the Carnot cycle have equal and opposite net gains of heat $(C = (T-T0)\Delta H)$. That is, we can state for a Carnot cycle:

The working body gives to the refrigerator exactly the amount of entropy it has received from the furnace.

Even though Rankine did not realize what he had derived, he should at least get credit for calculating the efficiency of an arbitrary fluid body undergoing a Carnot cycle.

Now Kelvin. In 1851, he published another paper on the theory of heat. He sets for himself the task:

"1) To show what modifications of the conclusions arrived at by Carnot, and by others who have followed his peculiar mode of reasoning regarding the motive power of heat, must be made when the hypothesis of the dynamical theory, contrary as it is to Carnot's fundamental hypothesis, is adopted.
2) To point out the significance in the dynamical theory, of the numerical results deduced from Regnault's observations on steam [...] and to show that by taking these numbers [...] in connexion with Joule's mechanical equivalent of a thermal unit, a complete theory of the motive power of heat, within the temperature limits of the experimental data, is obtained.
3) To point out some remarkable relations connecting the physical roperties of all substances, established by reasoning analogous to that of Carnot, but founded on the contrary principle of the dynamic theory."

"The whole theory of the motive power of heatis founded on the two following propositions, due respectively to Joule, and to Carnot and Clausius.

PROP I. (Joule).- When equal quantities of mechanical effect are produced by any means whatever from purely thermal sources, or lost in purely thermal effects, equal quantities of heat are put out of existence or are generated.

PROP II. (Carnot and Clausius).- If an engine be such that, when it is worked backwards, the physical and mechanical agencies in every part of its motions are all reversed, it produces as much mechanical effect as can be produced by any thermodynamic engine, with the same temperatures of source and refrigerator, from a given quantity of heat."

Now, Prop II seems to assert that a Carnot cycle is not required. But, it's not clear what he really meant by 'reversible', since there was no atomic theory yet. Using the theory of calorimetry (the thermal equation of state I have written previously), his statement is essentially Carnot's axiom, but written mysteriously. In any event, Kelvin then asserts a new axiom:

"It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects"

Why 'inanimate material' is required, one cannot say. Truesdell supposes it is to preserve the idea of divine miracles. In any case, it's yet another version of the Second Law. Kelvin then goes on to say that Clausius' results require

"It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature".

Kelvin then goes on to derive the Carnot-Clapeyron theorem and remarks:

"The very remarkable theorem that (dp/dT)/$\Lambda$ must be the same for all substances at the same temperature, was first given by Carnot, and demonstrated by him, according to the principles he adopted [...] Hence all of Carnot's conclusions, and all conclusions dervied by others from his theory, which depend merely on [the Carnot -Clapeyron theorem] require no modification when the dynamical theory is adopted."

Kelvin the (re)computes the efficiency of an ideal engine:

"We may suppose the engine to consist of an infinite number of perfect engines, each working within an infinitely small range of temperature, and arranged in a series of which the source of the first is the given source, the refrigerator of the last the given refrigerator, and the refrigerator of each intermediate engine is the source of that which follows it in the series."

He needed to do this because the previous derivations were so crude. His results do not differ from what we have shown already. However, he also obtains the interesting result:

$$L/JC = 1-exp(-1/J \int\mu dT)$$, and notes that

"Thus we see that, although the full equivalent of mechanical effect cannot be obtained even by means of a perfect engine, yet when the actual source of heat is at a high enough temperature above the surrounding objects, we may get more and more nearly the whole of the admitted heat converted into mechanical effect, by simply increasing the effective range of temperature in the engine."

Again, Kelvin comes close to defining 'entropy', but does not. This paper is also notable for the resultant fight over priority between Clausius, Joule, and Kelvin regarding the efficiency calculation. Never mind that Rankine did it first.

A brief note: Reech, in a 211 page paper submitted to the French mathematical journal Comptes Rendus, uses formal mathematical reasoning to derive Carnot's results in full generality, starting from first principles. That is, he derives nearly *all* of the results we have presented here so far, all by himself- that is, he was at least 100 years ahead of the competition. It's not clear if anyone actually read his paper before 1970.

In 1853, Kelvin and Joule present a measurement of Carnot's function, and in so doing, define the absolute temperature scale. That's next...

Comments? Hopefully, I am showing how the concept of 'entropy' remained hidden, although there are clues that *something* is placing a limit on engine efficiency (or equivalently, *something* is maximized during allowable thermodynamic processes)

 

[oopsCarol]

I'm expecting such interesting articles

 

[Andy Resnick]

James, I think the Carnot cycle does lead in deed to an essentially correct description of phenomenological thermodynamics, however it is still a very engineering approach. However a very modern and carefull axiomatization which takes into account the many improvements made since then is available here:
http://arxiv.org/PS_cache/cond-mat/pdf/9708/9708200v2.pdf

Lieb is quite famous for his contributions in e.g. statistical mechanics and density functional theory.

This is an interesting paper, and I am enjoying reading it. I'm less sure about the heavy reliance on 'equilibrium'. It's one thing to say an equilibrium state is possible (or must exist), but it's another to demand a system actually be in equilibrium.

The reason I don't like bringing equilibrium into the discussion is simply that I can't think of any real system that is in equilibrium. How long do you want to wait? I suppose it's possible to rescue the notion by claiming dynamics at a time scale T >>> observation time 't' means the system is 'essentially' at equilibrium, but then systems without a single time scale (granular systems, glassy systems, living systems, etc.) don't clearly apply.

I'm still reading the paper, so maybe they address this. What do you think?

 

[James A. Putnam]

I read through the paper once. I think it does not advance fundamental understanding. The important answers that we need were, I thought, not there. The approach taken is one that I do not prefer. However, I found parts of it difficult to follow. I need to study it more. Also, I am not an expert. I look forward to learning what Andy thinks about it. That would be helpful for me.

James

 

[DrDu]

i would like to bring in an interesting historical aspect which is formulated in "Friedrich Hund, Geschichte der physikalischen Begriffe, Bibliographisches Institut AG, Mannheim, 1972"(History of the physical concepts), a very interesting book. I don't know whether it has been translated into English language. Friedrich Hund, after whom e.g. some rules in atomic physics are named points out that Carnot believed that his "caloricum" corresponds to the heat measured in a calorimeter and only later it became clear that it is a distinct quantity which we nowadays call entropy. I think it is interesting that also entropy itself can be seen as a quantitative measure of the qualitative concept of heat and not only heat Q as we nowadays do.

 

[DrDu]

Andy, of course the concept equilibrium is an idealization, but I have to add that, at least for not too low temperatures one which can be excellently approximated experimentally.
E.g. the water in the bottle on my table will be in thermal equilibrium with its surrounding. I think the concept is unproblematic as long as one considers only either static systems or the starting and end points of an otherwise arbitrary thermodynamic process. E.g. I can consider some drops of gasoline to be in equilibrium before filling them into a vessel and I also can consider the gas mixture in that vessel to be in equilibrium after a week, although in the meantime, there has happened a considerably irreversible process.
I think more is not assumed in the paper by Lieb and Yngvason.

In the work of Caratheodory, on whose considerations this article is based, all the quasistatic adiabatic processes starting from one point constitute a hypersurface (the isentrope) which separates the adiabatically accessible from the inaccessible states. These set of all isentropes give the space of equilibrium states the appearance of an onion and we may introduce any monotonous labelling of these hypersurfaces as an empirical entropy.
Lieb and Ingvason go beyond this by not basing their construction of entropy on postulates about that surface and its hypothetical properties (e.g. smoothness) but on some reasonable assumptions on the accessibility of equilibrium states from each other for compound systems.

 

[James A. Putnam]

I will stick my non-expert neck out and offer some of my thoughts about the paper. I write in a less formal manner. I prefer to write so that others can more easily understand, plus, I do not know all the technical language anyway. I will appreciate any corrective explanations. I am still studying the paper, but, I tend to see things from a different perspective. So, I may be missing its meaning.

I see the paper as being analogous to reverse engineering. Equilibrium and entropy are assumed as givens. When the authors assume conditions of equilibrium, then they have already introduced temperature into their analysis. It is there from the beginning. It is represented by the presence of equilibrium. The scale used to quantify temperature is introduced later, but the scale is not the property of temperature. Equilibrium is the condition of constant temperature. Different conditions of equilibrium represent different temperatures. They represent conditions of relative hot and cold. The practice of avoiding using these words and the mathematical symbol of T for temperature is not sufficient for saying that they are absent in the analysis until derived later. Changes in levels of equilibrium, no matter how accomplished, introduce the flow of energy as a given. The analysis is formal and axiomatic, very logical. I prefer relying upon the original measureable properties. Entropy is not explained in a mechanical sense. It is not described as a measurable property analogous to heat, temperature, pressure or volume.

I still expect that it should be explanable as a macroscopic classical style property. I think that its mathematical expression does contain contradiction (Which I believe is Andy's point). Heat is energy in transit while equilibrium appears to exclude energy in transit. I think that this contradiction, or apparent contradiction, is what needs to be resolved, and, thermodynamic entropy will then be revealed. Another way of looking at this is to say that: When we finally understand what temperature is, then we will quickly understand what thermodynamic entropy is.

James

 

[Andy Resnick]

<snip>

In the work of Caratheodory, on whose considerations this article is based, all the quasistatic adiabatic processes starting from one point constitute a hypersurface (the isentrope) which separates the adiabatically accessible from the inaccessible states. These set of all isentropes give the space of equilibrium states the appearance of an onion and we may introduce any monotonous labelling of these hypersurfaces as an empirical entropy.
Lieb and Ingvason go beyond this by not basing their construction of entropy on postulates about that surface and its hypothetical properties (e.g. smoothness) but on some reasonable assumptions on the accessibility of equilibrium states from each other for compound systems.

That's one aspect of the approach I really like- because in truth, by formulating thermodynamics in terms of the ordering relation '<', the focus is on the *process*, not on the states themselves. I haven't finished the paper, but already I see some interesting ideas: for example, if the entropy must indeed be concave everywhere (p. 32), then there is a maximum S(t) for the universe, and then for time > t_m, the entropy must decrease. Of course, perhaps I misunderstand.

 

[Andy Resnick]

not sure why this was here... problems posting #88...

 

[Andy Resnick]

Note- the LaTex translator has been giving me headaches, and so some of the expressions are not displayed correctly...

We have traced the development of thermodynamics beginning with Carnot's initial paper in 1824 through to 1851, Rankine's second paper. In the intervening 30 years, there really has been only 3 quantitative statements:

L = JC (Work is intercovertible with heat)
$$\frac{\partial p}{\partial T} = J(\frac{\partial \Lambda}{\partial T}-\frac{\partial K}{\partial V})$$ The Carnot-Clapeyron theorem
L/JC+ = 1-T-/T+ (efficiency of a Carnot cycle)We have also seen a few statements that can be called "the second law of thermodynamics", but these were really just prohibitions against certain types of processes. But Rankine invented a function that appears to be 'entropy'.

Now to Kelvin's absolute temperature scale. Although it may be confusing, I am going to use only the symbol 'T' for temperature (except in one place).

Until now, "temperature" was simply something that was measured with a thermometer. To be sure, there were quantitative notions of 'hot' and 'cold', but for example, in Carnot's treatment, the reservoirs were assumed to have a constant temperature. Thermometers were usually constant-volume gas thermometers, but in general, there were really only two 'standard' temperatures: the freezing and boiling points of water. Apparently, nobody wondered what a thermometer was actually measuring.

In 1848, Kelvin wrote
"The determination of temperature has long been recognized as a problem of the greatest importance in physical science. It has accordingly been made a subject of most careful attention, and especially in late years, of very elaborate and refined experimental researches; and we are thus at present in possession of as complete a practical solution of the problem as can be desired, even for the most accurate investigations. The theory of thermometry is however as yet far from being in so satisfactory a state. The principle to be followed in constructing a thermometric scale might at first sight seem to be obvious,as it might appear that a perfect thermometer would indicate equal additions of heat, as corresponding to equal elevations of temperature, estimated by the numered divisions of its scale. It is however now recognized (from the variations in specific heat of bodies) as an experimentally demonstrated fact that thermometry under this condition is impossible,and we are left without any principle on which to found an absolute thermometric scale.
Next in importance to the primary establishment of an absolute scale, independently of the properties of any particular kind of matter, is the fixing upon an arbitrary system of thermometry, according to which results of observations made by different experimenters, in various positions and circumstances, may be exactly compared."

Kelvin then notes that Regnault's data- various thermometers filled with different gases- show that different thermometers can be reliably compared and are consistent, at bottom there is a reference to a specific substance against which others are compared. Kelvin recognized that Carnot's theory provides a material-independent means to estimate "the value of a degree... by the [maximum] mechanical effect to be obtained from the descent of a unit of heat through it..."

"The characteristic property of the scale which I now propose is, that all degrees have the same value; that is, that a unit of heat descending from a body A at the temperature T of this scale, to a body B at the temperature (T-1), would give out the same mechanical effect, whatever be the number T. This may justly be termed an absolute scale, since its characteristic is quite independent of the physical properties of any specific substance."

This is not as simple as it seems- recall that heat originating at high temperatures can do more work than heat originating at low temepratures. But there is some idea that 'temperature' should be related to how efficiently heat converted to work: not unreasonable, since work is a p-V function and that's how thermometers operated (and many still do).

Kelvin then uses Regnault's data on the latent heat of steam to generate a table of conversion factors from an air thermometer to his new scale- without presenting any equations or reasoning.

However, we already obtained an expression for the work done by a Carnot cycle:

$$L = \int(\mu dT) C^{+}$$.

Using Kelvin's requirement above for absolute temperatures $\tau$,

$$L = (\tau^{+}-\tau^{-})C^{+}$$

That is, the work a unit of heat can do for a single degree change is independent of the temperature. Obviously, $\tau = \int(\mu dT)$. Again, Clausius wrote 'pV=R(273+T), and Rankine defined a T0 as being the temperature when no heat is present. But as of now, there was no explicit connection made between Rankine's T0, Kelvin's absolute temperature scale, and Clausius's equation of state.

Kelvin's next paper on the subject, in 1853, defines a (new) "Absolute Temperature" T = J/$\mu$ this way:

"That Carnot's function (which is the same of all bodies at the same temperature), or any arbitrary function of Carnot's function, may be defined as temeprature, and is therefore the foundation of the absolute system of thermometry. We may now adopt this suggestion with great advantage, since we have found Carnot's functions varies very nearly in the inverse ratio of what has been called "temperature from the zero of the air thermometer", [...] and we may define temperature simply as the reciprocal of Carnot's function."

Note- we can relate T and $\tau$ by $\tau$ = J log T + const. Here is an important point: one of the absolute temperatures is allowed to be *negative*, the other is not. Specifically, $\tau$ may be negative (depending on the sign of $\mu$) but T can never be negative. But *both* $\tau$ and T satisfy the requirement to be material-independent temperature scales.

Kelvin then (again) provides a table comparing temperatures obtained by an air thermometer with T, and in most cases the diffrences are small. That is, while T was entirely different than any empirical temperature scale in common use, the degrees according to an air thermometer are very nearly proportional to absolute degrees in this newly defined scale. *This is what he did, with Joule:

In an 1853 paper with Joule, the two undertook an experiment to determine "the value of $\mu$, Carnot's function.". This was done using Joule's apparatus (slightly modified)- there a schematic here:

http://www.chemistry.mcmaster.ca/~ayers/chem2PA3/labs/2PA36.pdf

The apparatus was modified, replacing each pressure vessels by a long spiral tube. P1 is the pressure in the first spiral, P2 the pressure in the second. The question is, what is the quantity of heat produced?

"Let -C be the total quantity of heat emitted from the portion of the tube containing the orifice, and the second spiral, during the passage of a volume V1 through the first spiral, or of an equivalent volume V2 though the parts of the second where the temperature is sensibly T. This will consist of two parts; one (positive) the heat produced by fluid friction, and the other (negative) the heat emitted by the portion of the fluid which passes from one side to the other of the orifice, in virtue of it's expansion."

The amount of work "which will be lost as external mechanical effect, and will go to generate thermal vis viva" is given by

$$L = \int p dV + p_{1}V_{1} - p_{2}V_{2}$$

And the quantity of heat produced is simply L/J. *Subtracting "the amount previously found to be absorbed when the mechanical effect is all external" yields:

$$-C = 1/J[ \int p dV + p_{1}V_{1} - p_{2}V_{2}]- 1/\mu \int \frac{\partial p}{\partial T}dV$$.

That last bit was from the Carnot-Clapeyron theorem. It is essential to note that this is the first attempt to treat by thermodynamic theory a process that is neither isothermal nor adiabatic nor cyclic and also the first attempt to take account of an 'irreversible' effect.

So, the goal was to measure

$$\frac{J}{\mu} =\frac{JC+\int p dV + p_{1}V_{1} - p_{2}V_{2}}{\int \frac{\partial p}{\partial T}dV}$$

They adopt Rankine's equation of state:

pV = AT+ B+ (C+D/T+G/T^2)Phi/V

where A, B, C, D, and G are all constants to be determined by experiment, and Phi is a volume corresponding to a standard density. Kelvin and Joule then claim Regnault has some 'new data' (never presented) on the coefficient of expansion a, which when combined with Rankine's equation of state gives

a = 0.00365343 + 0.000011575 Phi/V, or

1/T0 = 0.00365343 and T0 = 273.72, if V = infinity.

Now, it has been remarked (by many) that Kelvin's absolute scale is based on ideal gases. This is only partly correct: in as much as Rankine's equation of state was used, this is true. But T was derived using Carnot's function, which is independent of the material. And besides, there are *at least* two entirely equivalent thermodynamic absolute temperature scales T and $\tau$. However, because T gives values close to those already in use, it becomes the preferred scale, with a physical meaning given by Rankine.

 

[DrDu]

Andy, I only now found time to have a look at your post No. 72. There you define K and L in terms of a heat function H. However, as Q is path dependent, it cannot be brought to the form of a total differential.

 

[Andy Resnick]

That's not correct- see post #71. Q can indeed be written as a total differential, using calorimetry. I have also been using Q as the 'heating', and the total heat absorbed or emitted during aprocess as 'C'.

Or do I misunderstand your point?

 

[DrDu]

Actually I meant post #71, not #72. If I specialize your considerations to a quasi-static process I find
C=int dU+pdV; Hence K=C_v= partial U/partial T|_V, the heat capacity at constant volume
and Lambda=T (partial p/ partial T)_V .
These two quantities cannot be written as a differential of a common function of T and V.

 

[Andy Resnick]

Are you referring to the work by Caratheodory? I am not that familiar with it (yet), but what you are saying sounds vaguely familiar... something about Pfaffian forms?

 

[DrDu]

Dear Andy, no, I am not referring to Caratheodory here.
I just use that Q=dU/dt +p dV/dt so that the integral over time, C, becomes
int dU+pdV, then I use that
dU(T,V)=(partial U/partial T)_V dT +(partial U/partial V)_T dV
using some standard expressions for partial derivatives one shows that
(partial U/partial V)_T =(partial U/partial V)_S+(partial U/partial S)_V (partial S/partial V)_T=p+T(partial p/partial T)_V
The last step uses the definitions of p and T as derivatives of U with respect to V and S respectively and a Maxwell equation to express the derivative of S with respect to V as a derivative of p with respect to T.

If your statement that Lambda=partial H /\partial V and K=partial H/partial T were true, than the right hand side of your Carnot Clapeyron equation would be zero given that the derivatives with respect to V and T commute.

 

[Andy Resnick]

I see what you mean.. I think I found the source of the error (it's mine), but I'm not sure how to reconcile the two results.

First, there's the Carnot-Clapeyron theorem:

$$\frac{\partial p}{\partial T}= \mu\Lambda$$,

But there's also this result from Clausius:

$$\frac{\partial p}{\partial T}= J(\frac{\partial \Lambda}{\partial T}-\frac{\partial K}{\partial V}$$

I don't see how they can be the same. I think Clausius's result comes from considering a complete cycle, while the Carnot-Clapeyron theorem comes from a single process, but I can't really see why there are two different results like this.

Using Clausius' result, you will obtain the correct expressions for the potentials that you wrote down:

$$J\Lambda - p = \frac{\partial U}{\partial V}$$

and
$$JK = \frac{\partial U}{\partial T}$$

EDIT- the LaTex engine is really starting to irritate me... These are not displaying properly, even though the code is correct. Sorry for any confusion...

 

[DrDu]

what is mu?
Did you include spaces between the \partial and the following p or T or whatever?

Ok, so mu is basically 1/T. Then both formulas seem to be correct, at least for a reversible process.

 

[Andy Resnick]

Yeah, I think I get it now:

$$\frac{\partial p}{\partial T} = J(\frac{\partial \Lambda}{\partial T}-\frac{\partial K}{\partial V})$$

Is a constitutive expression valid for anybody undergoing any *process*, while

$$\frac{\partial p}{\partial T} =\mu \Lambda}$$

is valid for anybody undergoing a Carnot cycle. There's some other implications as well, but those are secondary.

Now I understand what Caratheory's work was asking: we have assumed that any *real* cyclic process can be "constructed" out of differential elements (infinitesimal Carnot cycles). Caratheodory showed that given any neighborhood of points in p-V space, there are *always* at least two points that *cannot* be joined by an adiabat. Meaning, an infinitesimal Carnot cycle *cannot* be constructed in some neighborhood (since two of the points can't be connected), meaning finite cycles may not be able to be constructed out of Carnot cycles...

But I wonder if Caratheodory's results should instead be interpreted as a positive statement about the structure of the state space. For example, a torus has paths that cannot be shrunk to a point while a sphere does not. So our mental image of (how the P-V-T volume is projected onto) the p-V surface is insufficient- it's not a simply connected smooth surface, but something more complicated- an arbitrary loop on the surface cannot always be contracted to an infinitesimal loop.

Obviously, that's pure conjecture on my part... maybe it's time for a literature search.

 

[DrDu]

It is hard for me to judge when these relations are valid in general, mainly because I don't know the precise definitions of Lambda and K in irreversible situations where you claim (?) them to be defined, too. For me, this would be the historically interesting point when they started to realize that their expressions for heating are path dependent.

Caratheodory does not consider Carnot processes. It is not that two points cannot be joined by an adiabat, but that this adiabat can be run only in one direction. E.g. in as far as rubbing my hands together can be regarded as an adiabatic process, it is possible to rise the temperature of my palms, but not to cool them down.

 

[Andy Resnick]

Your first question is the easy one: the specific heat K of a material relates how the heating leads to a change in temperature.

The latent heat (lambda) is defined as "the quantity of heat which must be communicated to a body in a given state to convert it into another state without changing it's temperature". Typically the volume changes, but the pressure or the 'state' (melting/freezing/etc) can change as well. Without latent heat, a heat engine would not be possible.

I'm not sure I follow you on the second question- IIRC, Caratheodory's theorem relates to the inaccessibility of adiabatic processes to go from one arbitrary state to another arbitrary state. But I'm not as familiar with his work as I should be.

 

[Andy Resnick]

Ok, I feel it's time to (sort of) wrap up this thread with a discussion of Clausius 4th, 5th, and 6th papers (http://www.humanthermodynamics.com/Clausius.html#anchor_116), in which he obtains the analytical expression for entropy we are all exposed to in introductory classes. This will conclude my analysis of the historical literature; we have covered quite a bit of material already!

First I'd like to thank everyone who participated in making this thread a useful one- I am taking a lot of the material here and turning it into some decent lecture notes.

Ok- in 1854 Clausius opens his fourth paper with

"In my memoir “On the Moving Force of Heat, &c.”, I have shown that the theorem of the equivalence of heat and work, and Carnot’s theorem, are not mutually exclusive, by that, by a small modification of the latter, which does not affect its principle, they can be brought into accordance. With the exception of this indispensable change, I allowed the theorem of Carnot to retain its original form, my chief objection then being, by the application to the two theorems to special cases, to arrive at conclusions which, according as they involved known or unknown properties of bodies, might suitably serve as proofs of the truth of the theorems, or as examples of their fecundity.

Clausius doesn't cite work by anyone else (here or anywhere). He first states his 'first theorem':

"Mechanical work may be transformed into heat, and conversely heat into work, the magnitude of the one being always proportional to that of the other."

But we know that this is incomplete: the amount of work that heat may produce in a process is also proportional to the temperature; in a cycle, the temperature *difference* between hot and cold. Clausius continues:

"The forces which here enter into consideration may be divided into two classes: those which the atoms of a body exert upon each other, and which depend, of course, upon the nature of the body, and those which arise from the foreign influences to which the body may be exposed. According to these two classes of forces which have to be overcome, of which the latter are subject to essentially different laws, I have divided the work done by heat into interior and exterior work.

He writes this as (in modern notation)

$$\dot{E} = JQ - p\dot{V}$$

Clausius then considers P = P(V, T) and E = E(V, T), and obtains

$$J \Lambda - p = \frac{\partial E}{\partial V}$$
$$JK = \frac{\partial E}{\partial T}$$

And so his 'first theorem' is what we have seen several times already:

$$\frac{\partial p}{\partial T} = J(\frac{\partial \Lambda}{\partial T}-\frac{\partial K}{\partial V})$$

Next, he states a 'second theorem':

"The theorem, as hitherto used, may be enunciated in some such manner as the following:

In all cases where a quantity of heat is converted into work, and where the body effecting this transformation ultimately returns to its original condition, another quantity of heat must necessarily be transferred from a warmer to a colder body; and the magnitude of the last quantity of heat, in relation to the first, depends only upon the temperature of the bodies between which heat passes, and not upon the nature of the body effecting the transformation.

"In deducing this theorem, however, a process is contemplated which is too simple a character; for only two bodies losing or receiving heat are employed, and it is tacitly assumed that one of the two bodies between which the transformation of heat takes place is the source of the heat which is converted into work. Now by previously assuming, in this manner, a particular temperature of the heat converted into work, the influence which a change of this temperature has upon the relation between the two quantities of heat remains concealed, and therefore the theorem in the above form is incomplete.

[...]

"Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.

"Everything we know concerning the interchange of heat between two bodies of different temperature confirms this; for heat everywhere manifests a tendency to equalize differences of temperature, and therefore to pass in contrary direction, i.e. from a warmer to colder bodies. Without further explanation, therefore, the truth of this principle will be granted."

Consider what Clausius just asserted: a certain process is forbidden to occur under any circumstance, and he does not offer any quantitative proof.

Clausius continues:

"On considering the results of such processes more closely, we find that in one and the same process heat may be carried from a colder to warmer body and another quantity of heat transferred from a warmer to a colder body without any other permanent change occurring. In this case we have not a simple transmission of heat from a colder to a warmer body, or an ascending transmission of heat, as it may be called, but two connected transmission of opposite characters, one ascending and the other descending, which compensate each other. It may, moreover, happen that instead of a descending transmission of heat accompanying, in the one and the same process, the ascending transmission, another permanent change may occur which has the peculiarity of not being reversible without either becoming replaced by a new permanent change of a similar kind, or producing a descending transmission of heat. In this case the ascending transmission of heat may be said to be accompanied, not immediately, but immediately, by a descending one, and the permanent change which replaces the latter may be regarded as a compensation for the ascending transmission.

"Now it is to these compensations that our principle refers; and with the aid of this conception the principle may be also expressed thus: an uncompensated transmission of heat from a colder to a warmer body can never occur. "

(emphasis mine)

Again, Clausius make a definitive claim that a certian process can *never* occur, but does not provide any justification. However, he does go on to calculate something useful:

"If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat C of the temperature t from work, has the equivalence-value:

C/T

"and the passage of the quantity of heat Q from the temperature t1 to the temperature t2, has the equivalence-value:

C(1/T2-1/T1)

"wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.

That's confusing! He said T is a *function* of the temperature, not "the" temperature. As it happens (luckily for Clausius), his function coincides with Kelvin's definition of the absolute temperature T =J/$\mu$. But for now, 'T' is not 'temperature'.

Clausius then analyzes a series of thermal reservoirs and the flow of heat from the first to the final. This is simply $\sum \frac{C}{T}$. Passing to the continuum limit, and considering a cyclic process, Clausis obtains:

"The equation:
$$N = \int \frac{dC}{T} = 0$$
is the analytical expression, for all reversible cyclical processes, of the second fundamental theorem in the mechanical theory of heat."

Clausius then considers irreversible processes:

"If we represent the transformations which occur in a cyclical process by these expressions, the relation existing between them can be stated in a simple and definite manner. If the cyclical process is reversible, the transformations which occur therein must be partly positive and partly negative, and the equivalence-values of the positive transformations must be together equal to those of the negative transformations, so that the algebraic sum of all the equivalence-values become = 0. If the cyclical process is not reversible, the equivalence values of the positive and negative transformations are not necessarily equal, but they can only differ in such a way that the positive transformations predominate. The theorem respecting the equivalence-values of the transformations may accordingly be stated thus: The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing."

Clausius then simply writes, for "every cyclical process which is in any way possible." (not just reversible):

$$\int\frac{dC}{T} \geq 0$$.

Hopefully, you can see why there has been so much confusion about what the entropy 'really is'. Specifically, Clausius has made a series of vague statements in order to write a formula, presented without proof or derivation. There really is no logical foundation to Clausius' work.

Skipping ahead to Clausius' 9th paper, he picks up where he left off:

"The other magnitude to be here noticed is connected with the second fundamental theorem, and is contained in equation (IIa). In fact if, as equation (IIa) asserts, the integral:

$$\int\frac{dC}{T}$$.

"Vanishes whenever the body, starting from any initial condition, returns thereto after its passage through any other conditions, then the expression dC/T under the sign integration must be the complete differential of a magnitude which depends only on the present existing condition of the body, and not upon the way by which t reached the latter. Denoting his magnitude by S, we can write

dS = dC/T

"or, if we conceive this equation to be integrated for any reversible process whereby this body can pass from the selected initial condition to its present one, and denote at the same time by So the value which the magnitude S has in that initial condition,

$$S = S_{0} + \int\frac{dC}{T}$$

"This equation is to be used in the same way for determining S as equation (58) was for defining U. The physical meaning of S has already been discussed in the Sixth Memoir.

"we obtain the equation:

$$S - S_{0} = \int\frac{dC}{T}$$

"We might call S the transformation content of the body, just as we termed the magnitude U its thermal and ergonal content. But as I hold it to be better terms for important magnitudes from the ancient languages, so that they may be adopted unchanged in all modern languages, I propose to call the magnitude S the entropy of the body, from the Greek word τροπη, transformation. I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied their physical meanings, that a certain similarity in designation appears to be desirable.
[...]

"For the present I will confine myself to the statement of one result. If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat:

1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum."

Here (finally) are the laws of thermodynamics written down in a form that most of us have seen. I also want to note that I have read the 6th paper as closely as I could, I cannot guess what Clausius meant by "The physical meaning of S has already been discussed in the Sixth Memoir." I believe he meant "uncompensated" processes, but I wouldn't exactly call that a 'physical meaning'.

So, we have covered the development of Thermodynamics from 1822 (Carnot's initial report) through 1865 (Clausius's naming of 'entropy' and two laws of thermodynamics). Sadly, the field did not progress much over the next 100 years; as a result, many textbooks (most of which were first written in the 1950s and 1960s) relay the subject as it was written 100 years previously; that is without any coherent logic and mathematical structure. Fortunately (for us) during the past 20 years or so, the foundational elements of thermodynamics (heat, temperature, entropy) are being re-examined and refined, and there have been several excellent reviews posted on this thread.

 

[James A. Putnam]

Thank you Andy Resnick for posting your recap of the early development of thermodynamics. It has been a great help to me. I enjoy getting to the bottom, or beginning, of scientific ideas. I point out the special nature of the beginning, because, even though the researchers may have struggled to find their scientific bearings and eventually produce new additions to scientific learning, it is the reasons why they began their search that interest me most. Then their trains of thought also become important. Some early ideas may actually be better than some latter day ideas. Anyway, I appreciate their contributions and also your presentation about them. This has been a very enjoyable thread.

James

 

[James A. Putnam]

Andy Resnick,

I have one last question. I won't debate your response. You are the expert. I think it would be helpful to me to know: How do you choose to explain the meaning of thermodynamic entropy? In other words, if I were in your class and asked: What do you think is the latest, best explanation for thermodynamic entropy? How would you respond?

James

 

[Andy Resnick]

I'm no expert-seriously.

I like to think of entropy simply as 'energy unavailable to perform useful work'. I don't know if that's the latest, greatest definition, but it seems to be the most broadly applicable.

 

[James A. Putnam]

Dear Andy Resnick,

If you are really 'I'm no expert-seriously.', then can I rescind my promise to not debate your response? I won't extend this discussion any further unless you concur. You have been very generous in contributing your knowledge and time to this thread. Even more valuable, from my perspective, is that you are direct in your answers and honest in their quality. If you do not know, you are willing to let that be known. If you do know, you are very meticulous in showing the details about what you do know. You are a valuable resource. I do not want to be a bother to you. If you think this thread is finished, then it is finished for me also.

James

 

[James A. Putnam]

Andy Resnick,

Adding to my previous message: I do not want to burden you or anyone else with questions that would probably be repetitive. So, I will just repeat that I appreciate the detail that you have provided in your messages. I have printed all of them off and have included them in my binder under the subject of thermodynamic entropy. I have no further questions. Thank you for your time.

James

 

[Andy Resnick]

Dear Andy Resnick,

If you are really 'I'm no expert-seriously.', then can I rescind my promise to not debate your response? I won't extend this discussion any further unless you concur. You have been very generous in contributing your knowledge and time to this thread. Even more valuable, from my perspective, is that you are direct in your answers and honest in their quality. If you do not know, you are willing to let that be known. If you do know, you are very meticulous in showing the details about what you do know. You are a valuable resource. I do not want to be a bother to you. If you think this thread is finished, then it is finished for me also.

James

James,

Thanks for the kind words. It's not for me to declare a thread 'finished'- if you are still interested in continuing the discussion, then by all means- continue!

All I had meant was, I set myself the task of 'translating' for a modern audience some of the early works on thermodynamics. That task is complete. That does not mean there's nothing left to discuss :)

For example, I've started discussing the structure of the p-V surface with some mathematician colleagues here (CSU), as I feel certain issues raised on this thread merit additional consideration. I can't comment on those discussions yet as they are too preliminary for PF.

In any event, you seem eager to continue our discussion, so please continue!