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Kelvin's version of the second law of thermodynamics: It is impossible
to perform a cyclic process with no other result than that heat is
absorbed from a reservoir, and work is performed.

THEOREM: Kelvin's version of the second law is true if and only if,
whenever two INTERACTING heat engines absorb heat from a reservoir
(the surroundings) and perform reversible work, the following equality
of partial derivatives holds:

(dF1 / dX2)_X1 = (dF2 / dX1)_X2

where "d" is the sign for partial derivative, F1 and F2 are work-
producing forces and X1 and X2 are the respective displacements. If
the two partial derivatives are not equal, the second law is false.

Consider INTERACTING "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://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"

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

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)_Lu  =  (dFk / dLu)_Lk

where Fu>0 and Fk>0 are work-producing forces of contraction. The
values of the partial derivatives (dFu/dLk)_Lu  and  (dFk/dLu)_Lk 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)_Lu is positive. In contrast, as U is being stretched (Lu
increases), it absorbs protons, the pH increases and Fk must decrease.
Therefore, (dFk/dLu)_Lk is negative.

One partial derivative is positive, the other negative: this proves
that the second law of thermodynamics is false.

Pentcho Valev
pvalev@yahoo.com