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JEE Advanced Physics Thermodynamics 2022 JEE Advanced 2022 (Paper 2)

JEE Advanced Physics Question (2022) — Solution

Question

A bubble has surface tension S . The ideal gas inside the bubble has ratio of specific heats γ = 5 3 . The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is P a 1 , 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 a 2 , 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?

Options

  1. A. If the surface of the bubble is a perfect heat insulator, then  r 1 r 2 5 = P a 2 + 2 S r 2 P a 1 + 2 S r 1
  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, the  r 1 r 2 3 = P a 2 + 4 S r 2 P a 1 + 4 S r 1
  4. D. If the surface of the bubble is a perfect heat insulator, then  T 2 T 1 5 2 = P a 2 + 4 S r 2 P a 1 + 4 S r 1

Answer

D. If the surface of the bubble is a perfect heat insulator, then  T 2 T 1 5 2 = P a 2 + 4 S r 2 P a 1 + 4 S r 1

Step-by-step solution

When,  Pressure Radius Temperature P a 1 → r 1 → T 1 P a 2 → r 2 → T 2 For adiabatic process P 1 V 1 γ = P 2 V 2 γ P a 1 + 4 T r 1 4 3 π r 1 3 5 3 = P a 2 + 4 T r 2 4 3 π r 2 3 5 3 r 1 r 2 5 = P a 2 + 4 T r 2 P a 1 + 4 T r 1 Therefore, option A  is incorrect. Now, T 1 V 1 γ - 1 = T 2 V 2 γ - 1 T 2 T 1 = V 1 V 2 γ - 1 = r 1 r 2 3 2 3 T 2 T 1 = P a 2 + 4 T r 2 P a 1 + 4 T r 1 2 5 Therefore, option  D  is correct. For option B  Total internal energy +  surface energy will not be same as work done by gas will be there. Option B  is incorrect. For option  C  change in temperature is negligible. Therefore, the process is isothermal. P 1 V 1 = P 2 V 2 P a 1 + 4 T r 1 4 3 π r 1 3 = P a 2 + 4 T r 2 4 3 π r 2 3 r 1 r 2 3 = P a 2 + 4 T r 2 P a 1 + 4 T r 1 Option  C  is correct.

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