Pressure and Volume in an Adiabatic Process

Free expansion of a gas is an adiabatic process. However, there are few differences between free expansion and adiabatic expansion. During free expansion, no work is done, and there is no change in internal energy. But, for an adiabatic expansion, work is done, and there is a change in internal energy. During an adiabatic process, the relation between the pressure and volume is obtained from the condition for the adiabatic process, that is, 

However, for a free expansion process, this condition does not hold even if the expansion is adiabatic. Also, in a free expansion process, the gas is in an equilibrium state only at the initial and final points. Thus, on a pV diagram, only those initial and final points can be plotted, not the expansion itself. In addition to this, the initial and final temperature for an adiabatic process are related by the expression,

But for the free expansion, there is no change in internal energy; hence, the initial and final temperatures are equal. Thus, instead of the above equation, we have

From the expression for temperature, the initial and final states of a free expansion lie on the same isotherm in a pV diagram. Now, if the gas is assumed to be ideal, then, as there is no change in the temperature, there can be no change in the product of pressure and volume. Hence for the free expansion, the pressure and volume can be expressed as,