Pressure–volume diagrams and work

A pressure–volume diagram, or $$p$$–$$V$$ diagram, shows pressure $$p$$ on the vertical axis and volume $$V$$ on the horizontal axis. A point on the diagram represents a state of the system: it gives its pressure and volume at that state.

A line or curve shows how pressure and volume change during a process. Moving to the right means the system expands; moving to the left means it is compressed. An arrow on the path shows the direction of the process.

Work on a pressure–volume diagram

The work done by the system during a volume change is represented by the area between the process curve and the volume axis, over the interval between the initial and final volumes. Work done by the system is positive for expansion and negative for compression.

At constant pressure, the path is horizontal and the area is a rectangle. The work done by the system is:

For expansion, $$\Delta V$$ is positive, so $$W$$ is positive. For compression, $$\Delta V$$ is negative, so $$W$$ is negative.

If pressure changes during the process, the area beneath the curve is not generally a rectangle. The work can be estimated by dividing the area into narrow strips, estimating their areas and adding them. The direction of the volume change determines the sign.

Work in a cyclic process

In a cyclic process, the system returns to its initial state, so its path forms a closed loop on the diagram. The magnitude of the net work done by the system in one cycle is the area enclosed by the loop.

With work done by the system taken as positive:

• A clockwise cycle gives positive net work: expansion occurs at a higher pressure than compression.
• An anticlockwise cycle gives negative net work: compression occurs at a higher pressure than expansion.