Question

An ideal gas is expanded isothermally from a volume $V_1$ to volume
$V_2$ and then compressed adiabatically to original volume $V_1$. The initial pressure is $P_1$ and the final pressure iss $P_3$. If the net work done is W, then

• A. $P_3>P_1,W>0$
• B. $P_3<P_1,W<0$
• C. P_3 > P_1, W < 0
• D. P_3 = P_1, W = 0

Answer 1

The correct option is C

P_3 > P_1, W < 0

For an isothermal process: PV = constant and for an adiabatic process:
$P{V}^{\gamma }$ = constant, where $\gamma$ is the ratio of the two specific heats $\left(\frac{C_p}{C_v}\right)$ of the gas. When a gas is compressed from a volume V to a volume $(V-\Delta V)$, the increase in pressure is
$(\Delta P{)}_{adia}=\frac{\gamma \Delta VP}{V}$ for an adiabatic compression and $(\Delta P{)}_{iso}=\frac{\Delta VP}{V}$ for isothermal compression.
Hence $P_3$ will be greater than $P_1$, Therefore, the P - V diagrams of isothermal expansion from $V_2,P_{2}to{V}_1,P_3$ are as shown in Fig. 17.19. Let $W_{1}and{W}_2$ be the work done in isothermal expansion and adiabatic compression respectively. Therefore, net work done is
$W=W_1+(-W_2)=W_1-W_2$
Now, the area under the adiabatic curve is more than that under the isothermal curve. Hence $W_2>W_1$. Therefore, $W<0$.