Question

What is the work for the free expansion of a gas into vacuum?

Answer 1

For an expansion into a vacuum, `W=0`.

Explanation:

For a free expansion into a vacuum, a gas does no work, as there is no resistance on the gas as it expands, i.e. `P=0` and therefore the quantity `-PDeltaV=0`.

To say that `W=-P_"atm"DeltaV` is to say that the gas (the system) expands against an external pressure (applied by the surroundings), given specifically as atmospheric pressure `P_"atm"`. This is not an expansion into a vacuum, simply by definition.

If we ignore the part about expanding into a vacuum and just think about the gas expanding, the sign of `W` should be negative from the perspective of the gas.

For an expansion we know that `DeltaV` is positive, as the final volume is greater than the initial volume `(DeltaV=V_f-V_i)` and `P` is always positive.

The work done by the system should therefore be negative for a regular expansion, as work is done by the system (the gas) onto the surroundings. The opposite is true for a compression (`W > 0` for the gas).

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Note that if you define `W = PDeltaV`, then it is from the perspective of the surroundings (around the gas), and not the system (the gas).