Why is PV work negative?

In thermodynamics, PV work refers to the work done by or on a gas during a process that involves a change in volume. The formula for this work is given by:
W=−PΔVW = - P \Delta V
where:
• WW is the work done,
• PP is the pressure,
• ΔV\Delta V is the change in volume.
Why is PV work negative?
The sign of the PV work depends on the direction of the volume change:
1. Expansion of the gas (Positive Volume Change):
o When a gas expands, its volume increases (ΔV>0\Delta V > 0). In this case, the gas does work on its surroundings because it pushes against the container's walls.
o According to the formula W=−PΔVW = - P \Delta V, since ΔV>0\Delta V > 0, the work done by the gas will be negative. This is because the system (the gas) is doing work on the surroundings, which means energy is leaving the system.
2. Compression of the gas (Negative Volume Change):
o When a gas is compressed, its volume decreases (ΔV<0\Delta V < 0). In this case, work is done on the gas by the surroundings, and the gas absorbs this work.
o Since ΔV<0\Delta V < 0, the work done on the gas will be positive because energy is entering the system.
Interpretation:
• Negative work (W<0W < 0) occurs when the gas expands and does work on the surroundings. The system loses energy in the form of work.
• Positive work (W>0W > 0) occurs when the gas is compressed and work is done on it, meaning the system gains energy.
Thus, PV work is typically considered negative in expansion because the gas is doing work on the surroundings. Conversely, it is positive in compression because work is being done on the gas.