Thermodynamics and Statistical Mechanics

[Raz91]

Prove that the following two statements of the second law are equivalent (each one can be derived from the other):

(a) “It is impossible for heat to flow spontaneously from a colder to a warmer body without causing other changes”
(b) “It is impossible to convert all the heat taken from a body of uniform temperature into work without causing other changes”

(hint: one can prove the equivalence of the two statements by showing that if one is violated then so is the other and vice versa.)


The attempt at a solution

I have no idea how to approach this question mathematically...

 

[jfizzix]

If it were possible for heat to spontaneously flow from cold to hot, then one could convert all heat into work with 100 percent efficiency because any waste heat could be made to flow spontaneously back into the high temperature reservoir to be recycled into work. Whatever waste heat is left can again be recycled over and over again until all heat is converted into work.

If it were possible to convert heat into work with 100 percent efficiency, that work could be used to drive a heat pump causing heat to flow from a low temperature reservoir to a high temperature reservoir, even the same one driving the heat engine. Coupling the heat pump to the same temperature reservoirs as the heat engine would result in a spontaneous flow of heat from the low temperature reservoir to the high temperature reservoir.