The entropy change for an irreversible process in a closed system can sometimes be determined by calculating the entropy change for a reversible process in a system that is not closed (that is, a system that exchanges heat with something else). If the initial and final states are the same in both cases, the change in entropy must also be the same.

Consider an ideal gas that experiences a slow isothermal expansion. This is a reversible process - a slow isothermal compression would return the system to its initial state. Knowing the initial and final states for the expansion, we can calculate the heat transferred. Since everything takes place at one temperature the change in entropy is easy to find.

This is a reversible process, but it is not a closed system because heat is added - that's why the entropy goes up.

Let's say a free expansion takes place instead of an isothermal expansion, but the system ends up in the same final state. This is an irreversible process, and it's a closed system because no heat is exchanged. There's no way to calculate an entropy change directly, but we know it has to be the same as that for the isothermal expansion.