One mole of an ideal monatomic gas at 300 K and volume 5.0 L undergoes a free expansion into an evacuated chamber so that its volume doubles. The container is perfectly insulated, and no external work is done on the surroundings during the expansion. Afterward the gas is allowed to return to its original volume and temperature by a slow, reversible, isothermal compression in thermal contact with a 300 K reservoir.
(1) Derive an expression for the entropy change of the gas during the free expansion, then evaluate its numerical value.
(2) Determine the entropy change of the surroundings during the free expansion. Hence find the total entropy change of the universe for this process.
(3) For the subsequent reversible isothermal compression back to the original state, calculate (i) the heat exchanged with the reservoir and (ii) the entropy changes of the gas, the reservoir, and the universe.
(4) Explain how the results for the two processes together illustrate the second law of thermodynamics.