JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Fluid Mechanics

1. (A)If the surface of the bubble is a perfect heat insulator, then ${\left(\frac{r_1}{r_2}\right)}^5=\frac{P_{a2}+\frac{2S}{r_2}}{P_{a1}+\frac{2S}{r_1}}$
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, then ${\left(\frac{r_1}{r_2}\right)}^3=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$.✓ Correct
4. (D)If the surface of the bubble is a perfect heat insulator, then ${\left(\frac{T_2}{T_1}\right)}^{\frac{5}{2}}=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$.✓ Correct

Answer: (C), (D)

Solution

For a soap bubble, the pressure inside is $P_{\text{gas}}=P_a+\frac{4S}{r}$ (two surfaces).

(A) For adiabatic process: $P{V}^{\gamma }=\text{const}$, giving ${\left(\frac{r_1}{r_2}\right)}^{3\gamma }=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$. With $\gamma =\frac{5}{3}$, this gives ${\left(\frac{r_1}{r_2}\right)}^5$, but the pressure should be $\frac{4S}{r}$, not $\frac{2S}{r}$. Incorrect.

(B) In an adiabatic process, internal energy changes. Incorrect.

(C) For isothermal process: $PV=\text{const}$, giving ${\left(\frac{r_1}{r_2}\right)}^3=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$. Correct.

(D) For adiabatic: $P^{1-\gamma }{T}^{\gamma }=\text{const}$, giving ${\left(\frac{T_2}{T_1}\right)}^{5/3}={\left(\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}\right)}^{2/3}$, which simplifies to ${\left(\frac{T_2}{T_1}\right)}^{5/2}=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$. Correct.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · One or more correct options