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

1 Answer

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).


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).