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 compression 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

Answer: (C)

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

Here Compression is done only in process $BCD$,

So,

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

=Area of rectangle $DGHB-$ Area of arc $BCD$,

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

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

$=\frac{(12-\pi )P_0{V}_0}{8}$ = Compressive work done (Work done on the system)

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