# Isothermal Gas Expansion, a Reversible or Irreversible process?

Consider the following situation. A certain quantity of ideal monatomic gas (say one mole) is confined in a cylinder by a piston and is maintained at constant temperature (say $$T_0$$) by thermal contact with a heat reservoir. Then the gas slowly expands from $$V_1$$ to $$V_2$$ while being held at the same temperature $$T_0$$.

Question: Is this process reversible or irreversible?

Attempt: When the gas expands, the temperature must decrease, so the heat reservoir gives energy to the gas so the gas is maintained to the same temperature, right? Then If we do work on the gas so that it returns to the initial volume $$V_1$$, we know that due to $$\Delta T=0$$, then $$\Delta U=0$$, right? So, the work done on the gas is going to transform itself to heat that would be absorbed by the heat reservoir. My question is: how do we know if the heat given by the heat reservoir is the same that the heat absorbed by itself? If this is true, then I guess the process will be reversible, but if it's not true, does the process would be irreversible? Or it does not matter due that we have a heat reservoir?

Put another way, work $W$ is given by $W = \int_{V_1}^{V_2} p_{ext} dV$, where $p_{ext}$ is the external pressure. So you cannot determine the work or the temperature change unless you can do this integral, which you can't, because you do not know if the internal and external pressures are even related.