Why is it that, in order to attain reversibility, the temperature difference between the system and the surroundings must be infinitesimally small, I understand that it has to be quasi-static, but why can't there be a difference in temperature?
When I say the total entropy I mean the combined entropy of the system + surroundings, and I mean at every point on the reversible path, not only the initial state.
Just to be clear, my question is Why do reversible processes be like this? I'm not asking why if there is no temperature difference that there would be no change in the entropy of the universe, that is understood, but why the reversible process itself has to keep a difference in temperature between the system and surroundings that tends to zero.
Best Answer
Why is it that, in order to attain reversibility, the temperature difference between the system and the surroundings must be infinitesimally small, I understand that it has to be quasi-static, but why can't there be a difference in temperature?
Let there be two bodies, A and B, which are so massive that a transfer of heat between the two does not change their temperatures. For example, the ocean (body A) and atmospheric air (body B). Let the temperature of body A (the ocean) be $T_H$ (higher temperature) and body B (the atmosphere) be $T_L$ (lower temperature). Let there be a transfer of heat $Q$ occur between body A and B. Then the entropy changes are:
Body A
$$\Delta S=\frac{-Q}{T_H}$$
Body B
$$\Delta S=\frac{+Q}{T_L}$$
The total entropy change is
$$\Delta S_{Tot}=\frac{-Q}{T_H}+\frac{+Q}{T_L}$$
You can see that for all $T_{H}>T_{L}$, $\Delta S_{Tot}>0$
The only way for the total entropy change to be zero is if the temperature difference between the two bodies is zero. But then, of course, there would be no heat transfer. Bottom line, all real processes are irreversible. A reversible process is an ideal but not attainable, process.
Hope this helps.