ISOTHERMAL HEAT ENGINES AND THE SECOND LAW

Consider a macromolecule which, when the concentration of hydrogen ions in the solution increases, reversibly contracts anf lifts a weight:

http://www.ncbi.nlm.nih.gov/pmc/arti...00645-0017.pdf
POLYELECTROLYTES AND THEIR BIOLOGICAL INTERACTIONS, A. KATCHALSKY, pp. 13-15: "Let the polymolecule be a negatively charged polyacid in a stretched state and have a length L. Now let us add to the molecule a mineral acid to provide hydrogen ions to combine with the ionized carboxylate groups and transform them into undissociated carboxylic groups according to the reaction RCOO- + H+ = RCOOH. By means of this reaction, the electrostatic repulsion which kept the macromolecule in a highly stretched state vanishes and instead the Brownian motion and intramolecular attraction cause a coiling up of the polymeric chains. Upon coiling, the polymolecule contracts and lifts the attached weight through a distance deltaL. On lifting the weight, mechanical work f*deltaL was performed... (...) FIGURE 4: Polyacid gel in sodium hydroxide solution: expanded. Polyacid gel in acid solution: contracted; weight is lifted."

If the conditions are isothermal, the work-producing force of contraction, F, is a function of L, the length of the macromolecule, and C, the concentration of hydrogen ions in the solution: F=F(L, C).

Let L be fixed, and let the system be closed, that is, only energy can be exchanged with the surroundings. We wish to somehow increase C in the expectation that F will increase as well and a greater weight will be lifted in a subsequent step. Can C be increased if the system is closed and the conditions are isothermal? Yes if we have another reversible heat engine in the system, e.g. another contractile polymer, which, by exchanging work with us, releases precious hydrogen ions in the solution.

Our closed system is more complex now: We have TWO macromolecules with two work-producing forces, F1 and F2, which both increase as C increases. It is easy to see that, since the system is closed and the conditions are isothermal, the variable C can be eliminated so F1 and F2 are functions of the two lengths only:

F1 = F1(L1, L2); F2 = F2(L1, L2)

The following cycle is conceivable:

1. The two contractile macromolecules are initially stretched (ready to do work for us). Then we let one of them contract and lift a weight, isothermally and reversibly (that is, in this step the force of contraction F1 does work for us). As the macromolecule contracts, it RELEASES hydrogen ions - C increases and F2 increases as well (L2 remains fixed).

2. We let the second macromolecule contract and lift a weight, isothermally and reversibly (that is, in this second step the force of contraction F2, increased in the first step, does work for us). The second macromolecule differs from the first in that, as it contracts, it ABSORBS hydrogen ions and C decreases.

3. We stretch the first macromolecule, isothermally and reversibly, and restore its initial length (that is, in this third step we do work on the system). Since F1 was weakened in the second step, the work we lose in this third step is less than the work we gained in the first step. This (first) macromolecule ABSORBS hydrogen ions on stretching so C decreases in this third step.

4. We stretch the second macromolecule, isothermally and reversibly, and restore its initial length (that is, in this fourth step we do work on the system). Since F2 was weakened in the third step, the work we lose in this fourth step is less than the work we gained in the second step. The initial state of the system is restored and the cycle can be repeated.

In a more rigorous treatment it can be shown that the partial derivative equation:

(dF1/dL2)_L1 = (dF2/dL1)_L2

is a consequence of the second law of thermodynamics (Kelvin's version). Accordingly, if the partial derivatives (dF1/dL2)_L1 and (dF2/dL1)_L2 are shown to be unequal (e.g. one is positive and the other negative), then heat CAN, cyclically and isothermally, be converted into work, in violation to the second law of thermodynamics.

In the description of the four-step cycle the first macromolecule was expected to ABSORB hydrogen ions on stretching and RELEASE hydrogen ions when it contracts, while the second macromolecule was expected to behave in the opposite manner. Is that realistic? Yes it is. The macromolecules studied by Katchalsky (see the reference above) RELEASE protons on stretching but contractile polymers designed by D. Urry ABSORB protons on stretching:

https://data.epo.org/publication-ser...9&iepatch=.pdf
Dan Urry (pp. 14-15): "When the pH is lowered (that is, on raising the chemical potential, mu, 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. This is chemomechanical transduction, also called mechanochemical coupling. The mechanism by which this occurs is called a hydration-mediated apolar-polar repulsion free energy and is characterized by the equation 0(dmu/df)_n; that is, the change in chemical potential with respect to force at constant matrix composition is a negative quantity. Such matrices take up protons on stretching, i.e., stretching exposes more hydrophobic groups to water which makes the COO- moieties energetically less favored. This is quite distinct from the charge-charge repulsion mechanism for mechanochemical coupling of the type where (dmu/df)_n0 and where stretching of such matrices causes the release of protons."

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: "In short, stretching causes an uptake of protons."

Pentcho Valev