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A spherical bubble of surface tension S encloses an ideal gas with ratio of specific heats gamma = 5/3. The bubble stays spherical and is exposed to the atmosphere. At atmospheric pressure Pₐ1 the bubble has radius r₁ and enclosed-gas temperature T₁; at atmospheric pressure Pₐ2 they become r₂ and T₂. Which statement(s) is/are correct? (A) If the bubble surface is a perfect heat insulator, (r₁/r₂)⁵ = (Pₐ2 + 2S/r₂)/(Pₐ1 + 2S/r₁). (B) If the surface is a perfect heat insulator, the total internal energy of the bubble (gas plus surface energy) does not change with external pressure. (C) If the surface is a perfect heat conductor and ambient temperature change is negligible, (r₁/r₂)³ = (Pₐ2 + 4S/r₂)/(Pₐ1 + 4S/r₁). (D) If the surface is a perfect heat insulator, (T₂/T₁)^(5/2) = (Pₐ2 + 4S/r₂)/(Pₐ1 + 4S/r₁).
- (A) If the surface of the bubble is a perfect heat insulator, then (r₁/r₂)⁵ = (Pₐ2 + 2S/r₂)/(Pₐ1 + 2S/r₁).
- (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.
- (C) If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then (r₁/r₂)³ = (Pₐ2 + 4S/r₂)/(Pₐ1 + 4S/r₁).
- (D) If the surface of the bubble is a perfect heat insulator, then (T₂/T₁)⁵/2 = (Pₐ2 + 4S/r₂)/(Pₐ1 + 4S/r₁).
Correct answer: (C) If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then (r₁/r₂)³ = (Pₐ2 + 4S/r₂)/(Pₐ1 + 4S/r₁).
Solution
This is JEE Advanced 2022. For a soap bubble the gas pressure is P = Pₐ + 4S/r. Isothermal (perfect conductor): P V = const gives (Pₐ1 + 4S/r1) r1³ = (Pₐ2 + 4S/r2) r2³, i.e. (r1/r2)³ = (Pₐ2 + 4S/r2)/(Pₐ1 + 4S/r1) -> statement (C) is correct. Statement (A) wrongly uses 2S/r and exponent 5, so it is incorrect. (B) is incorrect because the surface energy and gas internal energy do change. (D) uses the wrong temperature exponent. The accepted answer is (C) (the official key marks C, and depending on version C alone).
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