Expansion of ideal gas into vacuum, connected to heat reservoir I have a mono-atomic ideal gas, expanding from a smaller volume V1 to a larger volume V2, inside a piston. If the expansion is done slowly, so the process is reversible, I understand how to calculate the work done, etc., for a system that is either thermally isolated or connected to a reservoir.
However, I'm having some difficulty understanding how an irreversible process of expansion takes place, if the gas just expands from V1 to V2 due to the removal of a partition connecting to the gas to a vacuum of volume V2-V1. If there is no heat reservoir, I understand there is no work done, so the final and initial temperatures are the same, with increasing entropy. We do not worry about the intermediate states, as our normal rules don't apply anyway.
What happens if during the expansion of the gas into the vacuum, the system is connected to a reservoir, maintained at the same temperature T as the initial temperature of the gas? Does it make sense to say the presence of the reservoir makes no difference to the previous case? The way I see it, there is still no heat transfer, so the system's behaviour is essentially identical to normal free expansion. Is this accurate?
 A: 
The way I see it, there is still no heat transfer, so the system's
behaviour is essentially identical to normal free expansion. Is this
accurate?

There is no net heat transfer if the uninsulated vessel is immersed in an ideal thermal reservoir whose temperature is the same as the gas before expanding.
Although during the expansion there will be temperature gradients in the gas, reflecting the effects of internal expansion, recompression and viscous friction heating, resulting in heat transfers across the boundary in both directions, after the expansion the gas will once again be in thermal equilibrium with the reservoir. So overall, $\Delta T=0$. Since the internal energy of an ideal gas depends only on temperature, $\Delta U=0$. Finally since there is no boundary work, from the first law the overall net heat transfer between the gas and reservoir, $Q=0$.
So the end result is free expansion in a vessel in contact with a thermal reservoir of the same temperature is equivalent to free expansion in an insulated vessel.
Hope this helps.
