UY1: Work output by gas pistons

In engines, a gas is often employed to perform work through gas expansion or pressure-volume work.

Infinitesimal work output by the gas for an infinitesimal piston displacement dy is given by:

$$\begin{aligned} dW_{\text{out}} &= F \, dy \\ &= PA \, dy \\ &= P \, dV \end{aligned}$$

, where
F is the force exerted by the gas on the piston
y is the outward piston displacement
P is gas pressure
A is piston area
V is gas volume

Conversely, infinitesimal work input to the gas is given by: dWin = -dWout

The work output for a finite volume change from Vi Vf is:

$$\begin{aligned} W_{\text{out}} &= \int\limits_{1}^{2} dW_{out} \\ &= \int\limits_{V_{1}}^{V_{2}} P \, dV \end{aligned}$$

Geometrically, this is simply the area under the PV curve that represents the piston process, integrated from start volume V1 to end volume V2

gas piston work
Work output is the area under the PV curve

If piston opens, i.e., V2 > V1, Wout > 0.

If piston closes, i.e., V2 < V1, Wout < 0.


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