So far in my thermodynamics lecture course, my understanding of the laws of thermodynamics is that the first law is about the conservation of energy, the second law says entropy must always increase or stay the same which apparently results in the fact you can never achieve 100% efficiency of heat engines, unless at $T = 0\,\mathrm K$, and the last law says that you can't get to $T= 0\,\mathrm K$.
I have never explicitly seen why the fact that entropy must always increase or stay the same results in the prevention of achieving 100% efficiency. The only proof I have is showing the Carnot cycle is the most efficient and that is only 100% efficient if the cold reservoir is at absolute zero, which it can not be at.
Is there any way to work from the statement: $\Delta S \geq 0$ (for any process in a closed system), to some result which says you can not achieve 100% efficiency?
Best Answer
If you can convert all of the heat to work, you're reducing entropy by definition ($\Delta S = \frac{Q}{T}$ , If $Q<0$ then $\Delta S < 0$).
If you allow yourself to let some heat flow into somewhere cold (heating something up instead of using all of the heat to work) you raise the entropy in the cold substance enough to let you not defy the second law, and the rest can go to useful work.