Problem 3.32. A cylinder contains one liter of air at room temperature (300 K) and atmospheric pressure (105 N/m²). At one end of the cylinder is a massless piston, whose surface area is 0.01 m². Suppose that you push the piston in very suddenly, exerting a force of 2000 N. The piston moves only one millimeter, before it is stopped by an immovable barrier of some sort. (a) How much work have you done on this system? (b) How much heat has been added to the gas? (c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the energy of the gas increase? (d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).