In the following $p-V$ diagram the equation of state along the curved path is given by $(V-2)^2=4 a

Question

In the following $p-V$ diagram the equation of state along the curved path is given by $(V-2)^2=4 a p$ where $a$ is a constant. The total work done in the closed path is

Professor PC Thomas Classes · Answer

By mapping the thermodynamic cycle defined by the parabolic curve $(V-2)^2=4ap$ and its bounding axes on the $P-V$ diagram, the work done corresponds to the enclosed area. Evaluating the integral $\oint P dV$ for the bounded parabolic segment mathematically yields an enclosed area equal to $\frac{1}{3a}$. Assuming a counter-clockwise process, the net work done by the gas is negative, resulting in $-\frac{1}{3a}$.