Question

Consider a thermodynamic cycle in a $P V$ diagram shown in the figure performed on one mole of a monatomic gas. The temperature at $A$ is $T_0$ and volume at $A$ and $B$ are related as $V_B=V_C=2 V_A$ Choose the correct option(s) from the following.

• A. The maximum temperature during the cycle is $4 T_0$
• B. Net work done by the gas during the cycle is $0.5 R T_0$
• C. The heat capacity of the process $A B$ is $2 R$
• D. The efficiency of the cycle is $8.33 \%$

Answer 1

Answer: (A), (B), (C), (D)

We use $P_A V_A=R T_0$ as $P_B=2 P_A$
$\Rightarrow V_B=V_C=2 V_A \text { and } T_B=\frac{P_B V_B}{R}=4 T_0$
From graph work done is $\Delta W=\left(\frac{V_C-V_A}{2}\right)\left(P_B-P_C\right)=0.5 R T_0$
and $ \Delta W_{A B}=\frac{1}{2}\left(P_A+P_B\right)\left(V_B-V_A\right)=\frac{3}{2} P_A V_A=1.5 R T_0$
and $ \Delta U_{A B}=\frac{3}{2} R\left(T_B-T_A\right)=4.5 R T_0$
and $\Delta Q_{A B}=\Delta W_{A B}+\Delta U_{A B}=6 R T_0$
$\Rightarrow C=\left(\frac{\Delta Q_{A B}}{T_B-T_A}\right)=2 R$
Thus cycle efficiency is given as $\eta=\frac{\Delta W_{\text {net }}}{\Delta Q_{A B}}=\frac{1}{12}=8.33 \%$