I have just started learning thermodynamics and the concept of entropy confuses me.
Suppose I have a gas in a cylindrical container fitted with a piston. I take it through an adiabatic irreversible process to an other state. There is certainly some entropy change. But when I take it through an adiabatic reversible process to the same state as I did in the irreversible process, then the entropy change would be zero, because $Q_\text{reversible}$ in this case is zero. Now since entropy is a state function, $\Delta S$ should be the same in the two cases, but it is not.
Where am I going wrong?
Best Answer
What you are saying is correct: a reversible and adiabatic process between two states, $A$ and $B$, does not change the entropy either of the system or of its environment (surroundings). An irreversible and adiabatic process between two states, $A'$ and $B'$, increases the system's entropy. The two statements are reconciled by noting that if $A=A'$ then $B\ne B'$ and vice versa.
Another way of stating the same, is that an irreversible and adiabatic cycle is impossible. In fact, this statement is almost equivalent to Caratheodory's axiom (a standard formulation of the 2nd law), namely, that in any neighborhood of any state there are states inaccessible via a purely adiabatic process.
The apparent one-sidedness of this, is a verbalization of the increase of entropy function, whose existence is a mathematical consequence of the same. You can sense the physical intuition of an entropy increase as a manifestation of the excess work expended to compensate for the irreversibility of the process to reach a certain state.