A vessel of volume  $V_0$ contains an ideal gas at pressure $P_0$  and temperature T. Gas is continuously pumped out of this vessel at a constant volume-rate $\frac{dV}{dT}=r,$  keeping the temperature constant. The pressure of the gas being taken out equals the pressure inside the vessel. The time taken to reduce the pressure to half of initial pressure is $t=\frac{V_0\mathcal{l}nx}{r}$  ‘x’ is ___

Figure shows the pressure versus volume graph corresponding to $n$  moles of an ideal  monatomic gas for a cyclic process ABCDA. Find the efficiency of the cycle.
Network done by the gas during the cyclic process is given by area enclosed by cyclic  process
  $$\begin{gathered} W=(2P_0-P_0)(2V_0-V_0) \\ =P_0{V}_0 \end{gathered}$$
  
For the isochoric process A to b the heat supplied is given by,
  $$\begin{gathered} \Delta Q=\Delta U+\Delta W \\ =n{C}_v\Delta T+0 \\ =n\left(\frac{3R}{2}\right)\frac{V_0(2P_0-P_0)}{nR} \\ =\frac{3V}{2}{P}_0{V}_0 \end{gathered}$$
   
Temperature in respective states A, B, C, D is given by
 $T_A=\frac{P_0{V}_0}{nR},T_B=\frac{2P_0{V}_0}{nR},T_C=\frac{4P_0{V}_0}{nR},T_D=\frac{2P_0{V}_0}{nR}$
For the isobaric process B to C the heat absorbed will be given by
 $$\begin{gathered} \Delta Q_2=n{C}_p\Delta T \\ =n\left(\frac{5R}{2}\right)\frac{2P_0(2V_0-V_0)}{nR} \end{gathered}$$
 
For the processes C to D and D to A, the heat is rejected from the system.
Efficiency $\left(\eta \right)$ of the cycle $=\frac{Networkdoenpercycle}{Totalheatgivenpercycle}$