thermodynamics - Change in entropy adiabatic expansion

I think that an adiabatic expansion of a gas should cause the entropy to increase. On the other hand we have for adiabatic processes that $dQ = 0$ and therefore $dS= 0$, which is why I thought that adiabatic processes are always isentropic. But somehow this adiabatic expansion of an ideal gas does not fit into this scheme, as it is quite obvious that this cannot be a reversible process. So how is this reconcilable with $dS=0$?

Answer

I guess you refer to the free expansion of a gas, which is an irreversible process. During free expansion, no work is done by the gas. The gas goes through states of no thermodynamic equilibrium before reaching its final state, which implies that one cannot define thermodynamic parameters as values of the gas as a whole. For example, the pressure changes locally from point to point, and the volume occupied by the gas (which is formed of particles) is not a well defined quantity. For that reason the standard equation $dS=dQ/T$ cannot be used because is not well defined. In such a case there is a change in entropy. For a calculation of this change you cah ckeck this link.