Question 166: Thermodynamics & KTG

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

Accepted Answer

The work done in a cyclic process represented on a $\mathrm{p}-\mathrm{V}$ diagram is equal to the area enclosed by the closed loop.
The path of the cycle is $\mathrm{A} ightarrow \mathrm{B} ightarrow \mathrm{C} ightarrow \mathrm{A}$ in an anti-clockwise direction.
In a $\mathrm{p}-\mathrm{V}$ diagram, a clockwise cycle represents positive work done by the system $(\mathrm{W}<0)$.
The equation of curve is $(V-2)^2=4$ ap $($ path $A ightarrow B ightarrow C)$.
At $\mathrm{V}=2, \mathrm{p}=0$ (Point B is on the V -axis).
At $\mathrm{V}=1$ and $\mathrm{V}=3, \mathrm{p}=\frac{(1)^2}{4 \mathrm{a}}=\frac{1}{4 \mathrm{a}}$. This is the pressure at points A and C .
The total work done W is the area under the straight line minus the area under the curve:
$$ \mathrm{W}=\int_1^3 \mathrm{pdV}-\operatorname{Area}(\mathrm{CA}) $$
Since the line CA is at $\mathrm{p}=\frac{1}{4 \mathrm{a}}$, its area is :
$$ \operatorname{Area}(\mathrm{CA})=\mathrm{p} 	imes \Delta \mathrm{V}=\frac{1}{4} \mathrm{a} 	imes(3-1)=\frac{2}{4 \mathrm{a}}=\frac{1}{2 \mathrm{a}} $$
Work done along the curve is,
$$ \int\limits_1^3 \mathrm{pdV}=\int_1^3\left[\frac{(\mathrm{~V}-2)^2}{4 \mathrm{a}}ight] \mathrm{dV}=\frac{1}{4 \mathrm{a}}\left[\frac{(\mathrm{~V}-2)^3}{3}ight]_1^3 $$
$$\Rightarrow $$ $$ \int\limits_1^3 \mathrm{pdV}=\frac{1}{4 \mathrm{a}}\left[\frac{(3-2)^3}{3}-\frac{(1-2)^3}{3}ight]=\frac{1}{4 \mathrm{a}}\left[\frac{1}{3}-\left(-\frac{1}{3}ight)ight]=\frac{1}{4 \mathrm{a}} 	imes \frac{2}{3}=\frac{1}{6 \mathrm{a}} $$
So, the net work done in the cycle is
$$ W_{	ext {net }}=\frac{1}{6 a}-\frac{1}{2 a}=\frac{1-3}{6 a}=-\frac{1}{3 a} $$
Therefore, the total work done in closed path is $-\frac{1}{3 a}$. Hence, the correct option is (D).

Answer

$+\frac{1}{3 a}$

Answer

$-\frac{1}{a}$

Answer

$\frac{1}{2 a}$

Answer

$-\frac{1}{3 a}$

Thermodynamics & KTG

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

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