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

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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

Accepted Answer

Using first law of thermodynamics$$\Delta Q = \Delta U + P\Delta V $$$\Delta Q$ is same$\Delta U$ is also same$${W_{A} = W_{B} }{	herefore(P\Delta V)_{A} = (P\Delta V)_{B} }$$$P$ is also same$${	herefore\ A_{A}d_{A} = A_{B}d_{B} }{\pi r_{A}^{2}d_{A} = \pi r_{B}^{2}d_{B} }{\frac{r_{A}}{r_{B}} = \left( \frac{dB}{dA} ight)^{\frac{1}{2}} = \left( \frac{9}{16} ight)^{\frac{1}{2}} }{= \frac{3}{4}}$$

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