A cylinder with rigid well-insulated walls is divided into two parts by a rigid insulating wall with a small hole in it. A frictionless, insulated piston is held against the perforated partition, thus preventing the gas that is on the other side from seeping through the hole.

The gas is maintained at a pressure P_i by another frictionless insulated piston. Imagine both pistons to move simultaneously in such a way that, as the gas streams through the hole, the pressure remains at a constant value P_i on one side of the dividing wall and at a constant lower value P_f on the other side, until all the gas is forced through the hole. (Note: this process is called a throttling process.) Prove that

U_i + P_iV_i = U_f + P_fV_f.