Assuming that no heat enters or leaves the gas, and that it expands by pushing against an external gas, or a moving wall or piston, then there is no entropy change in any case.
If there is a "free expansion" into a larger volume with no heat addition, and no work done against an outside gas or piston, then internal energy and temperature remain the same while entropy increases. The entropy gain will depend upon the ratio of initial to final volumes, and it will not matter if the expansion was done in two stages.
You allow a gas to expand freely from volume V to volume 2V. Later you allow the gas to expand freely from volume 2V to volume 3V. Is the net entropy change for these two expansions greater than, less than, or equal to the entropy change that would occur if you allowed the gas to expand freely from volume V directly to volume 3V? Explain.
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sgfs\d
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