Free Expansion

In a free expansion, gas is allowed to expand into a vacuum. This happens quickly, so there is no heat transferred. No work is done, because the gas does not displace anything. According to the First Law, this means that:

ΔEint = 0

There is no change in internal energy, so the temperature stays the same.

On a PV diagram all you can do is plot the end-points. The process follows a path on the diagram that is not well-defined. Because the temperature is constant, the connection between the initial and final states is:

PiVi = PfVf