Question

A cyclic process is shown on a $P-V$ diagram. The working fluid is an ideal gas. Four points $A,B,C$ and $D$ are marked on the $P-V$ diagram and corresponding thermodynamic co-ordinates are given. Here, the $P$ and $V$ scales are so chosen that these cycles on $P-V$ diagram appear to be circular.
Analyse these cycles on $P-V$ diagram and answer the following questions about these processes.
What is the work done during expansion in the given process?

• A. $\frac{(12+\pi )P_0{V}_0}{8}$
• B. $-\frac{(12+\pi )P_0{V}_0}{8}$
• C. $\frac{(12-\pi )P_0{V}_0}{8}$
• D. $-\frac{(12-\pi )P_0{V}_0}{8}$

Answer 1

The correct option is A $\frac{(12+\pi )P_0{V}_0}{8}$

Here Expansion is done only in process $DAB$,

So,

$W_{DAB}$ =Area under curve $DAB$.

=Area of rectangle $DGHB+$ Area of arc $DAB$,

$=\frac{3P_0{V}_0}{2}$ + $\frac{\pi }{2}\frac{P_0}{2}\frac{V_0}{2}$

$=\frac{3P_0{V}_0}{2}$ + $\frac{\pi P_0{V}_0}{8}$

$=\frac{(12+\pi )P_0{V}_0}{8}$

Hence Option $\text{A}$ os the right answer.