Q.Derive the expression for the work done in a volume change in a thermodynamic system.

When a gas held in a cylinder with a movable piston expands quasi-statically, pushing the piston outward by a small distance $dx$, the small work done by the gas on the piston is $dW = F\,dx$. The force exerted by the gas on the piston is $F=PA$ (area $A$), and since $A\,dx = dV$ is the small change in volume, this gives $dW = P\,dV$. For a finite change of volume 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 ($V_f > V_i$), $W$ is positive (work done BY the gas); if it is compressed ($V_f<V_i$), $W$ is negative (work done ON the gas). Crucially, $P$ inside this integral need not be constant -- to evaluate it, pressure must first be expressed as a function of volume (and temperature) using the equation of state, which is only strictly valid if the pro …