In free expansion the system is thermally isolated so $\Delta Q = 0$ throughout the process. This implies that process is adiabatic and follows its equation:
$$P_1V_1^\gamma = P_2 V_2^\gamma$$
But since inital and final temperature is same
$$P_1 V_1 = P_2 V_2$$
Both equations can only hold when $V_1 = V_2$. But that is clearly false. How do I resolve this paradox?
Best Answer
The equation
$$PV^{γ}=Constant$$
Or
$$P_{1}V_{1}^{γ}=P_{2}V_{2}^{γ}$$
Is for a reversible adiabatic process. The free expansion is an irreversible process.
$$P_{1}V_{1}=P_{2}V_{2}$$
Only defines the end points at equilibrium for an ideal gas. It is not the same as
$$PV=constant$$
which describes an isothermal process, the path between the end points. The path between the end points is not defined for a free expansion.
Bottom line: A free expansion in an insulated system is adiabatic, but it is not reversible adiabatic.
Hope this helps.