Work Done in Volume Changes

When a gas held in a cylinder with a movable piston (Figure 8.21) expands quasi-statically, pushing the piston outward by a small distance $dx$, the small work it does is $dW=F\,dx$; since the gas exerts force $F=PA$ on the piston (area $A$), and $A\,dx=dV$ is the corresponding small volume change, this gives $dW=P\,dV.$ For a finite change from $V_i$ to $V_f$, the total work done by the gas is $W=\int_{V_i}^{V_f}P\,dV.$ If the gas expands, $W$ is positive (work done BY the gas); if compressed, $W$ is negative (work done ON the gas). Note carefully that $P$ inside this integral need not be constant -- evaluating it in general requires first expressing $P$ as a function of $V$ (and $T$) via the equation of state, which is only strictly valid when the process is quasi-static. Exampl …