ThermodynamicsHard
Question
In the following $p - V$ diagram the equation of state along the curved path is given by $(V - 2)^{2} = 4ap$ where $a$ is a constant. The total work done in the closed path is
Options
A.$- \frac{1}{a}$
B.$+ \frac{1}{3a}$
C.$\frac{1}{2a}$
D.$- \frac{1}{3a}$
Solution
$W =$ Area of parabola
$$\begin{matrix} & \ = \frac{2}{3}(\text{~}\text{Area of rectangle}\text{~}AC31\text{ }A) \\ & \ = \frac{2}{3}P_{0}(3 - 1) = \frac{4P_{0}}{3} \end{matrix}$$
When $V = 1$
$$\begin{matrix} & \ (1 - 2)^{2} = 4{aP}_{0} \\ & P_{0} = \frac{1}{4a} \\ & \text{ }W = \frac{4}{3}P_{0} = \frac{4}{3}\frac{1}{4a} = \frac{1}{3a} \\ & {\text{ }W}_{\text{gas~}} = \frac{- 1}{3a} \end{matrix}$$
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