Work in a cycle

Problem

In a closed cycle in the $p\text{-}V$ plane $\Delta U = 0$ (because $U$ is a state variable). What does the area enclosed by the cycle represent? Distinguish the clockwise case from the anticlockwise one.

Solution

Meaning of the area. In a cycle the gas returns to the initial state, so $\Delta U = 0$. The net work done in one full loop is the cyclic integral

$L = \oint p\,\dd V$

which geometrically is the area enclosed by the curve in the $p\text{-}V$ plane.

Clockwise cycle. The expansion stretch (at higher pressure) lies above the compression stretch: the work done by the gas exceeds that done on it, so $L>0$. The system produces useful work: it is a heat engine.

Anticlockwise cycle. The compression happens at a higher pressure than the expansion: net work is done on the gas, so $L<0$. The system absorbs work to transfer heat: it is a refrigerator or a heat pump.

Energy balance. Applying the first law to the whole cycle, with $\Delta U = 0$:

$Q_\text{net} = \Delta U + L = L$

so all the net heat exchanged is converted into net work (or vice versa).

$\ev{\text{area}=L_\text{net};\ \text{clockwise} \to L>0,\ \text{anticlockwise} \to L<0}$

Links

Topics: Thermodynamics Concepts: Thermodynamic processes Skills: Solving a thermodynamic cycle Methods: P-V and ln P - ln V diagram Objects: Heat engine