JEE Advanced 2022 Paper 2 Q11 Physics Fluids & Surface Tension Surface Tension Medium

JEE Advanced 2022 Paper 2 · Q11 · Surface Tension

A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma = 5/3$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a1}$, the radius of the bubble is found to be $r_1$ and the temperature of the enclosed gas is $T_1$. When the atmospheric pressure is $P_{a2}$, the radius of the bubble and the temperature of the enclosed gas are $r_2$ and $T_2$, respectively. Which of the following statement(s) is(are) correct?

  1. A. If the surface of the bubble is a perfect heat insulator, then $\dfrac{r_1}{r_2} = \left(\dfrac{P_{a2} + 2S/r_2}{P_{a1} + 2S/r_1}\right)^{5/2}$.
  2. B. If the surface of the bubble is a perfect heat insulator, then the total internal energy of the bubble including its surface energy does not change with the external atmospheric pressure.
  3. C. If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then $\dfrac{r_1}{r_2} = \left(\dfrac{P_{a2} + 4S/r_2}{P_{a1} + 4S/r_1}\right)^{1/3}$.
  4. D. If the surface of the bubble is a perfect heat insulator, then $\dfrac{T_2}{T_1} = \left(\dfrac{P_{a2} + 4S/r_2}{P_{a1} + 4S/r_1}\right)^{2/5}$.
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Reveal answer + step-by-step solution

Correct answer:C, D

Solution

A soap bubble has two surfaces, so the pressure inside exceeds outside by $4S/r$ (single-film bubble would give $2S/r$). Let $P_{\text{in},i} = P_{ai} + 4S/r_i$. (C) Heat conductor (isothermal): $P_{\text{in},1} V_1 = P_{\text{in},2} V_2 \Rightarrow (P_{a1}+4S/r_1) r_1^3 = (P_{a2}+4S/r_2) r_2^3$, giving $r_1/r_2 = ((P_{a2}+4S/r_2)/(P_{a1}+4S/r_1))^{1/3}$ — correct. (D) Heat insulator (adiabatic, $\gamma = 5/3$): $P_{\text{in}}V^\gamma = $ const and $T V^{\gamma-1} = $ const, so $T \propto P^{(\gamma-1)/\gamma} = P^{2/5}$. Hence $T_2/T_1 = ((P_{a2}+4S/r_2)/(P_{a1}+4S/r_1))^{2/5}$ — correct. (A) Statement uses $2S/r$ (wrong factor for bubble) and exponent $5/2$ instead of the correct adiabatic exponent $3/5$ (since $r^3 \propto V \propto P^{-1/\gamma}$ gives $r_1/r_2 = (P_{\text{in},2}/P_{\text{in},1})^{1/(3\gamma)} = (\cdot)^{1/5}$) — incorrect. (B) For an adiabatic transformation, the gas does work against the atmosphere and against surface tension; the internal energy + surface energy does change — incorrect. Official key: (C), (D).

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