Thermodynamics: Dismal Swamp of Obscurity

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Clifford Truesdell, The Tragicomical History of Thermodynamics, 1822-1854, p. 6: "Finally, I confess to a heartfelt hope - very slender but tough - that even some thermodynamicists of the old tribe will study this book, master the contents, and so share in my discovery: Thermodynamics need never have been the Dismal Swamp of Obscurity that from the first it was and that today in common instruction it is; in consequence, it need not so remain." [...] p. 333: "Clausius' verbal statement of the "Second Law" makes no sense, for "some other change connected therewith" introduces two new and unexplained concepts: "other change" and "connection" of changes. Neither of these finds any place in Clausius' formal structure. All that remains is a Mosaic prohibition. A century of philosophers and journalists have acclaimed this commandment; a century of mathematicians have shuddered and averted their eyes from the unclean." https://www.amazon.com/Tragicomical-Thermodynamics-1822-1854-Mathematics-Physical/dp/1461394465

Truesdell is exaggerating perhaps? See this:

Wikipedia: "The second law of thermodynamics states that the total entropy can only increase over time for an isolated system, meaning a system which neither energy nor matter can enter or leave. The total entropy can remain constant in ideal cases where the system is in a steady state (equilibrium) or undergoing a reversible process." https://en.wikipedia.org/wiki/Second_law_of_thermodynamics

There is an omission that makes the above formulation of the second law meaningless (not even wrong). The entropy increase is only defined for processes that BEGIN AND END IN EQUILIBRIUM STATES. Scientists who study different processes and claim that the entropy increases simply don't know what they are talking about.

Let us imagine that the condition

BEGIN AND END IN EQUILIBRIUM STATES

is resuscitated and strictly obeyed. Will there be much improvement? No, because processes in an isolated system that begin and end in equilibrium states don't exist:

Jos Uffink, Bluff your Way in the Second Law of Thermodynamics, p. 4: "Even deliberate attempts at careful formulation of the Second Law sometimes end up in a paradox. One sometimes finds a formulation which admits that thermodynamics aims only at the description of systems in equilibrium states, and that, strictly speaking, a system does not always have an entropy during a process. The Second Law, in this view, refers to processes of an isolated system that begin and end in equilibrium states and says that the entropy of the final state is never less than that of the initial state (Sklar 1974, p. 381). The problem is here that, by definition, states of equilibrium remain unchanged in the course of time, unless the system is acted upon. Thus, an increase of entropy occurs only if the system is disturbed, i.e. when it is not isolated." http://philsci-archive.pitt.edu/313/

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Jos Uffink, Bluff your way in the Second Law of Thermodynamics: "I therefore argue for the view that the second law has nothing to do with the arrow of time. [...] Before one can claim that acquaintance with the Second Law is as indispensable to a cultural education as Macbeth or Hamlet, it should obviously be clear what this law states. This question is surprisingly difficult. The Second Law made its appearance in physics around 1850, but a half century later it was already surrounded by so much confusion that the British Association for the Advancement of Science decided to appoint a special committee with the task of providing clarity about the meaning of this law. However, its final report (Bryan 1891) did not settle the issue. Half a century later, the physicist/philosopher Bridgman still complained that there are almost as many formulations of the second law as there have been discussions of it. And even today, the Second Law remains so obscure that it continues to attract new efforts at clarification." http://philsci-archive.pitt.edu/313/1/engtot.pdf

As Clifford Truesdell suggests, the confusion started with Clausius's 1850 idiotic argument - later formulations of the second law of thermodynamics have all been defective. However previous formulations - those of Carnot - were both clear and correct. The simplest one is this:

"A cold body is necessary"

That is, heat cannot be cyclically converted into work unless a hot body, source of heat, and a cold body, receiver of heat, are available. The problem is that in 1824 Carnot deduced "A cold body is necessary" from a postulate that eventually turned out to be false:

Carnot's (false) postulate: Heat is an indestructible substance (caloric) that cannot be converted into work by the heat engine.

Unpublished notes written in the period 1824-1832 reveal that, after realizing that his postulate was false, Carnot found "A cold body is necessary" implausible:

Sadi Carnot, REFLECTIONS ON THE MOTIVE POWER OF HEAT, p. 225: "Heat is simply motive power, or rather motion which has changed form. It is a movement among the particles of bodies. Wherever there is destruction of motive power there is, at the same time, production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power." p. 222: "Could a motion (that of radiating heat) produce matter (caloric)? No, undoubtedly; it can only produce a motion. Heat is then the result of a motion. Then it is plain that it could be produced by the consumption of motive power, and that it could produce this power. All the other phenomena - composition and decomposition of bodies, passage to the gaseous state, specific heat, equilibrium of heat, its more or less easy transmission, its constancy in experiments with the calorimeter - could be explained by this hypothesis. But it would be DIFFICULT TO EXPLAIN WHY, IN THE DEVELOPMENT OF MOTIVE POWER BY HEAT, A COLD BODY IS NECESSARY; why, in consuming the heat of a warm body, motion cannot be produced."
http://www.nd.edu/~powers/ame.20231/carnot1897.pdf

Generally, a cold body is not necessary, that is, the second law of thermodynamics is false. The cold body is only TECHNOLOGICALLY necessary - such heat engines are fast-working and powerful. Heat engines working under isothermal conditions (in the absence of a cold body) are commonplace but are too slow and impuissant to be of any technological importance. Except, perhaps, for the case where water is placed in an electric field - the non-conservative force (pressure) that emerges seems to be able to convert ambient heat into work quite vigorously:

"However, in experiments in which a capacitor is submerged in a dielectric liquid the force per unit area exerted by one plate on another is observed to decrease... [...] This apparent paradox can be explained by taking into account the DIFFERENCE IN LIQUID PRESSURE in the field filled space between the plates and the field free region outside the capacitor."
http://farside.ph.utexas.edu/teaching/jk1/lectures/node46.html

"Liquid Dielectric Capacitor" http://www.youtube.com/watch?v=T6KAH1JpdPg

"The Formation of the Floating Water Bridge including electric breakdowns"
https://www.youtube.com/watch?v=17UD1goTFhQ

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Two isothermal heat engines violating the second law of thermodynamics:

1. pH-sensitive polymer shown in fig. 4 in Katchalsky's article:

A. Katchalsky, POLYELECTROLYTES AND THEIR BIOLOGICAL INTERACTIONS, p. 15, Figure 4: "Polyacid gel in sodium hydroxide solution: expanded. Polyacid gel in acid solution: contracted; weight is lifted."
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1367611/pdf/biophysj00645-0017.pdf

Mineral acid (hydrogen ions, H+) is added to the system and "the polymolecule contracts and lifts the attached weight through a distance ΔL". Then added H+ can be removed and the macromolecule resumes its initial stretched state, ready to lift another weight. The net work involved in adding and removing hydrogen ions, if the process is carried out quasi-statically, is virtually zero, while the net work extracted from contracting and stretching is obviously positive - the system is cyclically lifting weights at the expense of heat absorbed from the surroundings.

2. pH-Responsive Hydrogel Composite Artificial Muscle https://www.youtube.com/watch?v=JGn2a21FvLM

The system is analogous, the difference is that contraction occurs at high pH. See detailed explanation here:

M P M Dicker et al, Hydrogel core flexible matrix composite (H-FMC) actuators: theory and preliminary modelling http://iopscience.iop.org/article/10.1088/0964-1726/23/9/095021/meta

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"Entropy was discovered when it was noticed to be a quantity that behaves as a function of state, as a consequence of the second law of thermodynamics." https://en.wikipedia.org/wiki/Entropy

It was Clausius who "noticed" that the entropy is a state function, but was he correct? Here is the story:

If you define the entropy S as a quantity that obeys the equation dS=dQrev/T, you will find that, so defined, the entropy is a state function FOR AN IDEAL GAS. Clausius was very impressed by this statefunctionness and decided to prove that the entropy (so defined) is a state function for ANY system. So "Entropy is a state function" became a fundamental theorem in thermodynamics. Clausius deduced it from the assumption that any cycle can be disintegrated into small Carnot cycles, and nowadays this deduction remains the only justification of "Entropy is a state function":

"Carnot Cycles: S is a State Function. Any reversible cycle can be thought of as a collection of Carnot cycles - this approximation becomes exact as cycles become infinitesimal. Entropy change around an individual cycle is zero. Sum of entropy changes over all cycles is zero."
http://mutuslab.cs.uwindsor.ca/schurko/introphyschem/lectures/240_l10.pdf

"Entropy Changes in Arbitrary Cycles. What if we have a process which occurs in a cycle other than the Carnot cycle, e.g., the cycle depicted in Fig. 3. If entropy is a state function, cyclic integral of dS = 0, no matter what the nature of the cycle. In order to see that this is true, break up the cycle into sub-cycles, each of which is a Carnot cycle, as shown in Fig. 3. If we apply Eq. (7) to each piece, and add the results, we get zero for the sum." http://ronispc.chem.mcgill.ca/ronis/chem213/hnd8.pdf

The assumption on which "Entropy is a state function" is based - that any cycle can be subdivided into small Carnot cycles - is obviously false. An isothermal cycle CANNOT be subdivided into small Carnot cycles. A cycle involving the action of conservative forces CANNOT be subdivided into small Carnot cycles.

Conclusion: The belief that the entropy is a state function is totally unjustified. Any time scientists use the term "entropy", they don't know what they are talking about.

"My greatest concern was what to call it. I thought of calling it 'information', but the word was overly used, so I decided to call it 'uncertainty'. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, 'You should call it entropy, for two reasons: In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage."
https://en.wikipedia.org/wiki/History_of_entropy

Jos Uffink, Bluff your way in the Second Law of Thermodynamics: "I therefore argue for the view that the second law has nothing to do with the arrow of time. [...] This summary leads to the question whether it is fruitful to see irreversibility or time-asymmetry as the essence of the second law. Is it not more straightforward, in view of the unargued statements of Kelvin, the bold claims of Clausius and the strained attempts of Planck, to give up this idea? I believe that Ehrenfest-Afanassjewa was right in her verdict that the discussion about the arrow of time as expressed in the second law of the thermodynamics is actually a RED HERRING." http://philsci-archive.pitt.edu/313/1/engtot.pdf

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The version of the second law of thermodynamics stated as "Entropy always increases" (a version which, according to A. Eddington, holds "the supreme position among the laws of Nature") is in fact a theorem deduced by Clausius in 1865:

Jos Uffink, Bluff your Way in the Second Law of Thermodynamics, p. 37: "Hence we obtain: THE ENTROPY PRINCIPLE (Clausius' version) For every nicht umkehrbar [irreversible] process in an adiabatically isolated system which begins and ends in an equilibrium state, the entropy of the final state is greater than or equal to that of the initial state. For every umkehrbar [reversible] process in an adiabatical system, the entropy of the final state is equal to that of the initial state." http://philsci-archive.pitt.edu/archive/00000313/

Clausius' deduction was based on three postulates:

Postulate 1 (implicit): The entropy is a state function.

Postulate 2: Clausius' inequality (formula 10 on p. 33 in Uffink's paper) is correct.

Postulate 3: Any irreversible process can be closed by a reversible process to become a cycle.

All the three postulates remain totally unjustified even nowadays. Postulate 1 can easily be disproved by considering cycles (heat engines) converting heat into work in ISOTHERMAL conditions. Postulate 3 is almost obviously false:

Uffink, p.39: "A more important objection, it seems to me, is that Clausius bases his conclusion that the entropy increases in a nicht umkehrbar [irreversible] process on the assumption that such a process can be closed by an umkehrbar [reversible] process to become a cycle. This is essential for the definition of the entropy difference between the initial and final states. But the assumption is far from obvious for a system more complex than an ideal gas, or for states far from equilibrium, or for processes other than the simple exchange of heat and work. Thus, the generalisation to all transformations occurring in Nature is somewhat rash."

Note that, even if Clausius's theorem were correct (it is not), it only holds for "an adiabatically isolated system which begins and ends in an equilibrium state". This means that (even if Clausius's theorem were correct) all applications of "Entropy always increases" to processes which do not begin and end in equilibrium would be still unjustified!

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