A cylinder fitted with a movable, massless piston contains 2.0mol of an ideal diatomic gas initially at pressure P0, volume V0, and temperature T0. Three distinct processes bring the gas to different final equilibrium states:
• Process I: The gas expands isothermally until its volume is 2V0.
• Process II: The gas is heated at constant volume until its temperature is 2T0.
• Process III: The gas undergoes a two–step change: first an isothermal expansion from V0 to 2V0, then an isochoric heating that doubles the temperature reached at the end of the first step.
(a) For the gas alone, rank the entropy of the three final states (after each full process is complete) from lowest to highest.
(b) Provide a detailed justification for your ranking, explicitly discussing the separate contributions of volume change and temperature change to the entropy.
(c) For Process III, describe qualitatively how the entropy of the surroundings changes during each step if the surroundings remain at temperature T0. State whether the total entropy change of the universe for Process III is positive, negative, or zero, and justify your answer.
(d) Suppose the piston is now locked in place and the gas is allowed to cool spontaneously back to T0. Explain how the entropy of the gas changes during this cooling and whether the cooling process can return the gas to its initial entropy value.