Two gases A and B are filled at the same pressure in separate...

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

Options

A $\frac{4}{3}$
B $\frac{3}{4}$
C $\frac{2}{\sqrt{3}}$
D $\frac{\sqrt{3}}{2}$

Correct answer

B. $\frac{3}{4}$

Step-by-step solution

According to question Pressure is constant

P_A=P_B

\begin{aligned} & \mathrm{Q}_{\mathrm{A}}=\mathrm{Q}_{\mathrm{B}} \\ & \Delta \mathrm{U}_{\mathrm{A}}=\mathrm{U}_{\mathrm{B}} \end{aligned}

So according to I law

\begin{aligned} & \mathrm{Q}=\Delta \mathrm{U}+\mathrm{w} \\ & \mathrm{w}_1=\mathrm{W}_2 \\ & \mathrm{P}_1 \Delta \mathrm{~V}_1=\mathrm{P}_2 \Delta \mathrm{~V}_2 \\ & \left(\pi \mathrm{r}_{\mathrm{A}}^2\right) \mathrm{x}_{\mathrm{B}}=\left(\pi \mathrm{r}_{\mathrm{B}}^2\right) \mathrm{x}_{\mathrm{A}} \\ & \frac{\mathrm{r}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{B}}}=\sqrt{\frac{\mathrm{x}_{\mathrm{B}}}{\mathrm{x}_{\mathrm{A}}}}=\sqrt{\frac{9}{16}}=\frac{3}{4} \end{aligned}$$

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