Question

In mole a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes.
$A\to B$ : Isothermal expansion at temperature T so that the volume is doubled from $V_1$ to $V_2=2V_1$ and pressure changes from $P_1$ to $P_2$
$B\to C$ : Isobaric compression at pressure $P_2$ to initial volume $V_1$.
$C\to A$ : Isochoric change leading to change of pressure from $P_2$ to $P_1$.
Total workdone in the complete cycle ABCA is

• A. 0
• B. $nRT\left(\ln 2+\frac{1}{2}\right)$
• C. $nRTln2$
• D. $nRT\left(\ln 2-\frac{1}{2}\right)$

Answer 1

Answer: (D)

$W_{\text{Isothermal }}=nRTln\left(\frac{v_2}{v_1}\right)$
$W_{\text{Isobaric }}=P\Delta V=nR\Delta T$
$W_{\text{Isochoric }}=0$

$W_1=nRT\ln \left(\frac{2V}{V}\right)=nRT\ln 2$
$W_2=nR\left(\frac{T}{2}-T\right)=-nR\frac{T}{2}$
$W_3=0$
$\Rightarrow W_{net}=W_1+W_2+W_3$
$W_{net}=nRT\left(\ln 2-\frac{1}{2}\right)$