Pentcho Valev

2017-02-12 00:48:40 UTC

Systems converting heat into work at constant temperature, e.g. that of the surroundings, are commonplace, but, if they can do this CYCLICALLY, they violate the second law of thermodynamics. In other words, they contradict the Kelvin-Planck statement of the second law of thermodynamics:

"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 possibble after all? By building one and showing 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 yet it could still be a genuine perpetual-motion machine of the second kind.

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:

Loading Image...

We assume that the system does work slowly, virtually reversibly. 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:

dF1/dX2 = dF2/dX1 /3/

where "d" should be the partial derivative symbol - when X2 varies, X1 is fixed and vice versa.

In terms of 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 dF1/dX2 and dF2/dX1 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. dF1/dX2 is positive and dF2/dX1 negative - the second law of thermodynamics is false and will have to be abandoned.

Consider the so-called "chemical springs". There are two types of macroscopic contractile polymers which on acidification (decreasing the pH of the system) contract and can lift a weight:

http://www.google.com/patents/US5520672

"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."

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1367611/pdf/biophysj00645-0017.pdf

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:

http://pubs.acs.org/doi/abs/10.1021/jp972167t

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

dFu / dLk = dFk / dLu

where Fu>0 and Fk>0 are work-producing forces of contraction. The values of the partial derivatives dFu/dLk and dFk/dLu 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, dFu/dLk is positive. In contrast, as U is being stretched (Lu increases), it absorbs protons, the pH increases and Fk must decrease. Therefore, dFk/dLu is negative. One partial derivative is positive, the other negative: this shows that the second law of thermodynamics is false.

Pentcho Valev

"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 possibble after all? By building one and showing 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 yet it could still be a genuine perpetual-motion machine of the second kind.

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:

Loading Image...

We assume that the system does work slowly, virtually reversibly. 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:

dF1/dX2 = dF2/dX1 /3/

where "d" should be the partial derivative symbol - when X2 varies, X1 is fixed and vice versa.

In terms of 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 dF1/dX2 and dF2/dX1 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. dF1/dX2 is positive and dF2/dX1 negative - the second law of thermodynamics is false and will have to be abandoned.

Consider the so-called "chemical springs". There are two types of macroscopic contractile polymers which on acidification (decreasing the pH of the system) contract and can lift a weight:

http://www.google.com/patents/US5520672

"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."

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1367611/pdf/biophysj00645-0017.pdf

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:

http://pubs.acs.org/doi/abs/10.1021/jp972167t

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

dFu / dLk = dFk / dLu

where Fu>0 and Fk>0 are work-producing forces of contraction. The values of the partial derivatives dFu/dLk and dFk/dLu 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, dFu/dLk is positive. In contrast, as U is being stretched (Lu increases), it absorbs protons, the pH increases and Fk must decrease. Therefore, dFk/dLu is negative. One partial derivative is positive, the other negative: this shows that the second law of thermodynamics is false.

Pentcho Valev