An Easily Refutable Version of the Second Law of Thermodynamics
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 illustration. We assume that the system does work UNDER ISOTHERMAL CONDITIONS (that is, the system converts heat absorbed from the surroundings into work but operates so slowly, virtually reversibly, that the temperature of both the system and the surroundings remains unchanged). The work done by this system on the surroundings is:
dW = dW1 dW2 = F1dX1 F2dX2 /2/
Is W a function of X1 and X2? If yes, the second law of thermodynamics (Kelvin's version) is obeyed - at the end of the (isothermal) 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 the system with two work-producing forces which does work under isothermal conditions, the second law now states:
SECOND LAW OF THERMODYNAMICS (KELVIN'S 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, for instance, 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:
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"
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: "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.