Two gases $A$ and $B$ are filled at the same pressure in separate cylinders with movable pistons of

Question

Two gases $A$ and $B$ are filled at the same pressure in separate cylinders with movable pistons of radius $r_A$ and $r_B$, respectively. On supplying an equal amount of heat to both the systems reversibly under constant pressure, the pistons of gas $A$ and $B$ are displaced by 16 cm and 9 cm , respectively. If the change in their internal energy is the same, then the ratio $\frac{r_A}{r_B}$ is equal to

Professor PC Thomas Classes · Answer

(d) : Given, pressure, $P=$ constant $\Rightarrow P_A=P_B$ Also, $\Delta U_A=\Delta U_B$ From first law of thermodynamics $ \ \begin{aligned} & \Delta Q_A=\Delta U_A+\Delta W_A \ \Rightarrow \quad & \Delta Q_A=\Delta U_A+P_A(16) 	imes A_A \ & \Delta Q_B=\Delta U_B+\Delta W_B \ \Rightarrow & \Delta Q_B=\Delta U_B+P_B(9) 	imes A_B \end{aligned} $ \ Given $\Delta Q_A=\Delta Q_B$ $ \ \begin{aligned} & \Rightarrow \Delta U_A+P_A(16) A_A=\Delta U_B+P_B(9) 	imes A_B \ & \Rightarrow 16 	imes \pi 	imes r_A^2=9 	imes \pi 	imes r_B^2 \ & \Rightarrow\left(\frac{r_A}{r_B}ight)^2=\frac{9}{16} \Rightarrow \frac{r_A}{r_B}=\frac{3}{4} \end{aligned} $