Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radius and , respectively. On supplying an equal amount of heat to both the systems reversibly under constant pressure, the pistons of gas A and B are displac

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Solution

Q.
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 [2025]

1 $\frac{2}{\sqrt{3}}$

2 $\frac{\sqrt{3}}{2}$

3 $\frac{4}{3}$

4 $\frac{3}{4}$

Ans.
(4)

Given, pressure, $P = constant \Rightarrow P_{A} = P_{B}$

Also, $Δ U_{A} = Δ U_{B}$

From first law of thermodynamics

$Δ Q_{A} = Δ U_{A} + Δ W_{A}$

$\Rightarrow Δ Q_{A} = Δ U_{A} + P_{A} (16) \times A_{A}$ ...(i)

$Δ Q_{B} = Δ U_{B} + Δ W_{B}$

$\Rightarrow Δ Q_{B} = Δ U_{B} + P_{B} (9) \times A_{B}$ ...(ii)

Given $Δ Q_{A} = Δ Q_{B}$

$\Rightarrow Δ U_{A} + P_{A} (16) A_{A} = Δ U_{B} + P_{B} (9) \times A_{B}$

$\Rightarrow 16 \times π \times r_{A}^{2} = 9 \times π \times r_{B}^{2}$

$\Rightarrow {(\frac{r_{A}}{r_{B}})}^{2} = \frac{9}{16} \Rightarrow \frac{r_{A}}{r_{B}} = \frac{3}{4}$

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