Question

Consider a thermodynamic cycle in a $PV$ 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 $V_B=V_C=2V_A$ Choose the correct option(s) from the following.

• A. The maximum temperature during the cycle is $4T_0$
• B. Net work done by the gas during the cycle is $0.5R{T}_0$
• C. The heat capacity of the process $AB$ is $2R$
• 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 $P_B=2P_A$
$\Rightarrow V_B=V_C=2V_A\text{ and }{T}_B=\frac{P_B{V}_B}{R}=4T_0$
From graph work done is $\Delta W=\left(\frac{V_C-V_A}{2}\right)\left(P_B-P_C\right)=0.5R{T}_0$
and $\Delta W_{AB}=\frac{1}{2}\left(P_A+P_B\right)\left(V_B-V_A\right)=\frac{3}{2}{P}_A{V}_A=1.5R{T}_0$
and $\Delta U_{AB}=\frac{3}{2}R\left(T_B-T_A\right)=4.5R{T}_0$
and $\Delta Q_{AB}=\Delta W_{AB}+\Delta U_{AB}=6R{T}_0$
$\Rightarrow C=\left(\frac{\Delta Q_{AB}}{T_B-T_A}\right)=2R$
Thus cycle efficiency is given as $\eta =\frac{\Delta W_{\text{net }}}{\Delta Q_{AB}}=\frac{1}{12}=8.33\%$