Change in entropy adiabatic expansion

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.

Irreversible free expansion of a gas in adiabatic condition is not isentropic. There is an increase of entropy. The equation of dS=dQ/T is not an accurate equation, the actual equation should be dS=dQ(reversible)/T. For dQ of irreversible the equation should be changed into the clausius inequality form which is dS> dQ (irreversible)/T. Under this clausius inequality equation, even dQ is zero in the adiabatic condition, and the right hand side equation is equal to zero, the left side dS is greater than zero which means that the dS entropy is increasing. You may want to ask why the equation should be changed to this clausius inequality, then, you should read on the derivation of this clausius inequality which is a bit of long story. To elucidate the clausius inequality briefly (and thus incompletely), we start from the concept of internal energy, U (I assume you know what is internal energy, if not the story will be longer). The equation of internal energy is U = Q (heat) + (-W, Work) (Work is flow in different direction, though not neccesarily, so negative is placed, not because of minus out the work). However, U is a state function (i assume you know what is state function), so whether the condition is reversible or irreversible, the U should be the same. But W and Q are not state function and they are condition depending. From the Carnot experiment, we know that the amount of reversible work, W(R), is greater > than the amount of irreversible work, W(IR) (I assume you had read about Carnot cycle and understand why reversible work> irreversible work). Thus, as U is a state function and reversible U(R) is equal to irreversible U(IR), U(R) = U(IR), to equate the condition in the aspect of Q and W; We must have reversible heat, Q(R) is also greater than irreversible heat Q(IR) so that when Q(R)+[-W(R)] = Q(IR)+[-W(IR)]. The original equation is dS = dQ(R)/T. As dQ(R)> dQ(IR), the equation for reversible heat and irreversible heat cannot be equal. dQ(R)/T > dQ(IR)/T; Thus, dS> dQ(IR)/T which is the clausius inequality.