When is PV (pressure-volume) work negative?
Step 1: Pressure-volume work
Consider an ideal gas having a definite mass of n moles in a cylinder fitted with a movable piston. Let be the area of the cross-section of the cylinder. Let the gas expand from to against constant external pressure exerted on the piston. Due to this expansion, the piston moves upward by a distance . Hence, the work done in the system is negative.
Step 2: Derivation
We know that pressure is the force per area, this can be written as
Rearranging the equation, we get
Substitute equation (2) in equation (1).
But, is equal to ; therefore, the equation becomes
Here,
is the change in displacement.
is the change in volume.
is the initial volume.
is the final volume.
Step 3: Explanation
Here, the negative () sign indicates that the system has lost energy.
Therefore, whenever a gas expanded against that external pressure, certain energy must be transferred to the environment. As a result, the negative work reduces the total energy of the gas.