Is the Second Law of Thermodynamics Particularly Controversial?
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Pentcho Valev
2017-08-10 17:55:44 UTC
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Nature: "But thermodynamics is paradoxical. The second law, which also puts limits on how efficiently heat can be converted into work - as happens in a steam engine - is particularly controversial."

"Particularly controversial" and yet there is no controversy at all - complete silence surrounds this 19th century wisdom. The second law of thermodynamics has long been under suspicion but a red herring deviating the attention to small, microscopic, quantum etc. systems has been very powerful so far:

Nature: "Second law broken. Researchers have shown for the first time that, on the level of thousands of atoms and molecules, fleeting energy increases violate the second law of thermodynamics."

The truth is that MACROSCOPIC heat engines violating the second law of thermodynamics are COMMONPLACE. But here misleading education is the problem:

"A necessary component of a heat engine, then, is that two temperatures are involved. At one stage the system is heated, at another it is cooled."

This is simply not true. There are heat engines functioning in ISOTHERMAL conditions - e.g. the work-producing force is activated by some chemical agent, not by heating.

All isothermal heat engines, except for analogs of ideal gas systems, can violate the second law of thermodynamics. Examples can be found here:

Isothermal Heat Engines Violate The Second Law of Thermodynamics

Pentcho Valev
Pentcho Valev
2017-08-11 18:21:01 UTC
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The simplest and most precise formulation of the second law of thermodynamics was given by Sadi Carnot himself:

"A cold body is necessary"

That is, heat cannot be cyclically converted into work unless there is some temperature gradient - a hot body, source of heat, and a cold body, receiver of heat, must be 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Unpublished notes written in the period 1824-1832 reveal that, after realizing that his postulate was false, Carnot found "A cold body is necessary" implausible: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."

Generally, a cold body is not necessary, that is, the second law of thermodynamics is false. The cold body is only TECHNOLOGICALLY necessary as it makes heat engines fast-working. 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 between the cathode and the anode seems to be quite vigorous:

"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."

Liquid Dielectric Capacitor

"The Formation of the Floating Water Bridge including electric breakdowns"

Pentcho Valev
Pentcho Valev
2017-08-11 21:38:40 UTC
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"The Kelvin-Planck statement (or the heat engine statement) of the second law of thermodynamics states that it is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work. This implies that it is impossible to build a heat engine that has 100% thermal efficiency."

The problem is that the Kelvin-Planck statement is practically unfalsifiable. How can one prove that the forbidden device is possible after all? By building one and demonstrating it to a jury? But there can be various technological and other reasons (that have nothing to do with the second law of thermodynamics) why the device would not work. And even if all those reasons are overcome, the jury would refuse to consider the project because the constructor is insane by definition.

Here I am going to convert the above unfalsifiable statement of the second law of thermodynamics into an easily refutable version.

For a closed system (exchanges energy but not matter with the surroundings) the first law of thermodynamics defines the internal energy change, dU, to be:

dU = dQ - dW = dQ - FdX /1/

where dQ is the heat absorbed, dW is the work done by the system on the surroundings, F>0 is the work-producing force and dX is the respective displacement.

Let us consider a system with two work-producing forces, F1 and F2 - here is an oversimplified illustration:

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We assume that the system does work quasi-statically and isothermally. The work done by this system on the surroundings is:

dW = dW1 + dW2 = F1dX1 + F2dX2 /2/

Is W a function of the displacements X1 and X2? If yes, the second law of thermodynamics (Kelvin-Planck version) is obeyed - at the end of any cycle W returns to its initial value and no net work is done on the surroundings.

The following theorem is relevant:

Theorem: W is a function of X1 and X2 if and only if the mixed partial derivatives are equal:

"Mixed Partial Derivatives"

Since F1 and F2 are in fact the first partial derivatives, the theorem can be expressed in the following way:

Theorem: W is a function of X1 and X2, that is, the second law is obeyed, if and only if:

(∂F1/∂X2)_X1 = (∂F2/∂X1)_X2 /3/

For our system with two work-producing forces, the Kelvin-Planck version of the second law now states:

EQUIVALENT TO KELVIN-PLANCK VERSION: The partial derivatives (∂F1/∂X2)_X1 and (∂F2/∂X1)_X2 are equal.

That is, if experiments show that the two sides of /3/ are equal, the second law is confirmed. If, however, experiments unambiguously show that the two sides of /3/ are not equal - e.g. (∂F2/∂X1)_X2 is positive and (∂F2/∂X1)_X2 negative - the second law of thermodynamics is false and will have to be abandoned.

Let us consider two types of macroscopic contractile polymers which on acidification (decreasing the pH of the system) contract and can lift a weight:

"When the pH is lowered (that is, on raising the chemical potential, μ, of the protons present) at the isothermal condition of 37°C, these matrices can exert forces, f, sufficient to lift weights that are a thousand times their dry weight."

POLYELECTROLYTES AND THEIR BIOLOGICAL INTERACTIONS, A. KATCHALSKY, p. 15: "FIGURE 4: Polyacid gel in sodium hydroxide solution: expanded. Polyacid gel in acid solution: contracted; weight is lifted."

Polymers designed by Urry (U) absorb protons as their length, Lu, increases, whereas polymers designed by Katchalsky (K) release protons as their length, Lk, increases. See discussion on p. 11020 in Urry's paper:

J. Phys. Chem. B, 1997, 101 (51), pp 11007 - 11028, Dan W. Urry, "Physical Chemistry of Biological Free Energy Transduction As Demonstrated by Elastic Protein-Based Polymers". p. 11020: "Stretching causes an uptake of protons" for Urry's polymers, and "stretching causes the release of protons", for Katchalsky's polymers.

Let us assume that two macroscopic polymers, one of each type (U and K) are suspended in the same system. At constant temperature, IF THE SECOND LAW IS TRUE, we must have

∂Fu/∂Lk = ∂Fk/∂Lu

where Fu>0 and Fk>0 are work-producing forces of contraction. The values of the partial derivatives ∂Fu/∂Lk and ∂Fk/∂Lu can be assessed from experimental results reported on p. 11020 in Urry's paper. As K is being stretched (Lk increases), it releases protons, the pH decreases and, accordingly, Fu must increase. Therefore, ∂Fu/∂Lk is positive. In contrast, as U is being stretched (Lu increases), it absorbs protons, the pH increases and Fk must decrease. Therefore, ∂Fk/∂Lu is negative. One partial derivative is positive, the other negative: the second law of thermodynamics is false.

Pentcho Valev