In an adiabatic free expansion, a gas is allowed diffuse into a vacuum. The initial and final states of the system are the same as with an isothermal expansion. However, in an isothermal expansion, heat flows from the surroundings to the system, so the entropy lost to the surroundings compensates the entropy gained by the system and the change is reversible. An adiabatic free expansion is not reversible, however, because total entropy in the universe increases.
An isothermal expansion occurs when a piston is slowly expanded at constant temperature. To maintain constant temperature, thermal equilibrium must be maintained with a heat sink in contact with the piston throughout the expansion. Entropy increases in the system during the isothermal expansion as heat flows in. Because the hot sink is the same temperature as the piston, it experiences an equal decrease of entropy in losing heat to the piston. The overall entropy of the universe is unchanged in this nonspontaneous, reversible heat flow. (Do you remember from the 1st law why heat must flow in to the piston during the isothermal expansion?)
A good, traditional conceptual problem is to compare the isothermal expansion discussed above to another kind of thermodynamic transformation called an adiabatic free expansion. The initial state of an adiabatic free expansion consists of a canister in which a membrane divides an area containing a gas from the second portion of the canister which is empty. When the membrane is pierced, the gas spontaneously diffuses to fill the container.
The initial and final states of the system in an adiabatic free expansion are identitical to the initial and final states of an isothermal expansion. What is interesting about an adiabatic free expansion, though, is that while the initial and final states of the system are the same as with the isothermal expansion, the initial and final states of the surroundings are not the same as with the isothermal expansion. While the isothermal expansion is reversible, the adiabatic free expansion is irreversible. The internal energy change is zero (no work is performed and no heat flows in during the free expansion) but, like the isothermal expansion, the entropy of the canister has increased, but unlike the isothermal expansion, a compensating decrease in the entropy of the surroundings did not occur.
While entropy did not increase in the universe for the isothermal expansion, it did increase in the universe for the adiabatic free expansion. Think about it for a minute, and you will intuitively realize that the while an isothermal expansion is reversible, an adiabatic free expansion is not, though the entropy changes within the system are equivalent for the two transformations. When you decide whether an event is spontaneous, you have to think about the entropy of the universe, which includes both the system and the surroundings.