Heat engines convert, cyclically, thermal energy (heat) into mechanical energy (work). Here is an illustration:

http://www.chemistry.wustl.edu/~edud...es/HeatEng.jpg

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 is the work-producing force and dX0 is the respective displacement.

Let us consider a system of two heat engines doing 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 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:

http://www.youtube.com/watch?v=Y-lEuHpTS9k
"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" is the partial derivative symbol.

The two sides of /3/ have physical meanings and, even more importantly, are easily MEASURABLE. This leads to the following conclusion:

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 is negative - the second law of thermodynamics is false and will have to be abandoned.

It is easy to see that:

dF1/dX2 = dF2/dX1 = 0

when the two heat engines do not interact, and that:

dF1/dX2 is �not equal to dF2/dX1

when the two heat engines interact in some asymmetric way.

Pentcho Valev