I don't understand why heat transfer from hot reservoir to the system is considered reversible in this case:
$T_{reservoir}$ = $T_{system}$ + dT
but it's considered irreversible in this case:
$T_{reservoir}$ = $T_{system}$ + ΔT
Where dT is infinitesimal difference while ΔT is finite difference in temperature between reservoir and the system.
In both cases some heat is transferred from the reservoir to the system, so it should be irreversible in both cases. What understanding am I missing here?
Best Answer
To do it reversibly, you can heat the body from $T_1$ to $T_2$ (i.e., over a finite temperature change) using an infinite sequence of constant temperature reservoirs, in which each reservoir in turn is only dT higher in temperature than the body at any time (and also only dT higher in temperature than the reservoir before it in the sequence). Each increment in heat transfer would take place with only a differential temperature driving force between the body and the current reservoir. To reverse the process, and bring both the body and the reservoirs back to their original states, you would simply contact the body with the reservoirs in the reverse sequence, in which case the reservoirs would be dT lower in temperature than the body in each step of the process). The only difference would be with regard to the very first and very last reservoirs (which could not be returned to their original states). But this would be insignificant.
In the case where the body is heated from $T_1$ to $T_2$ by bringing it into contact with a constant temperature reservoir at $T_2$ for the entire time until the body equilibrated at $T_2$, all the heat transfer would take place with a finite temperature driving force, and there would be no way to return both the body and the reservoir to their original states without causing a significant change in something else (like using other reservoirs).
A reversible process is one in which the system is only slightly removed from being at thermodynamic equilibrium throughout the change. Thus, a reversible process can be viewed as imposing a continuous sequence of thermodynamic equilibrium states.