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

Question

marallocajcyb · Accepted Answer

A gas does not exert any energy during a free expansion into a vacuum because there is no resistance to the gas' expansion, i.e.

P = 0

and therefore the quantity

- P Δ V = 0

.
To say that

W = - P_{atm} Δ V

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

Δ V

is positive, as the final volume is greater than the initial volume

(Δ V = 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 = P Δ V

, then it is from the perspective of the surroundings (around the gas), and not the system (the gas).

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

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