In a free expansion, gas quickly expands into a vacuum. No heat transferred and no work is done, because the gas does not displace anything.

According to the First Law, this means:   dE = 0   →   dT = 0.

On a PV diagram it is meaningful to plot the endpoints only. Because the temperature is constant, the connection between the initial and final states is:   P_iV_i = P_fV_f