QJEEAdv22Ph7f3a: JEE Advanced 2022 Physics - Class 11 JEEAdv22Ph7f3a Solution: A bubble has surface tension . The ideal gas inside the bubble…

Question

Verified Solution • 2025–26 Session moderate

A bubble has surface tension

S

. The ideal gas inside the bubble has ratio of specific heats

\gamma=

\frac{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?

A If the surface of the bubble is a perfect heat insulator, then $\left(\frac{r_{1}}{r_{2}}\right)^{5}=\frac{P_{a 2}+\frac{2 S}{r_{2}}}{P_{a 1}+\frac{2 S}{r_{1}}}$ .
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 $\left(\frac{r_{1}}{r_{2}}\right)^{3}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$ .
D If the surface of the bubble is a perfect heat insulator, then $\left(\frac{T_{2}}{T_{1}}\right)^{\frac{5}{2}}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$ .

Accepted Answer

For adiabatic process:

\frac{\mathrm{P}_1}{\mathrm{P}_2}=\left(\frac{\mathrm{V}_2}{\mathrm{~V}_1}\right)^\gamma

\Rightarrow \mathrm{P}_1 \mathrm{~V}_1^\gamma=\mathrm{P}_2 \mathrm{~V}_2^\gamma

\Rightarrow

\left(\mathrm{P}_{a 1}+\frac{4 \mathrm{~S}}{\gamma_1}\right)\left(\frac{4}{3} \pi r_1^3\right)^{5 / 3}

=\left(\mathrm{P}_{a 2}+\frac{4 \mathrm{~S}}{r_2}\right)\left(\frac{4}{3} \pi r_2^3\right)^{5 / 3}

\Rightarrow

{(\frac{r_{1}}{r_{2}})}^{5} = \frac{P_{a 2} + 4 S / r_{2}}{P_{a 1} + 4 S / r_{1}}

Also,

\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}

\frac{T_{2}}{T_{1}} = \frac{{(P_{a 2} + 4 S / r_{2})}^{2 / 5}}{{(P_{a 2} + \frac{4 S}{r_{1}})}^{2 / 5}}

At same temperature,

\mathrm{P}_1 \mathrm{~V}_1=\mathrm{P}_2 \mathrm{~V}_2

\Rightarrow

\left(\mathrm{P}_{a 1}+\frac{4 \mathrm{~S}}{r_1}\right)\left(\frac{4}{3} \pi r_1^3\right)=\left(\mathrm{P}_{a 2}+\frac{4 \mathrm{~S}}{r_2}\right)\left(\frac{4}{3} \pi r_2^3\right)

\Rightarrow

\left(\frac{r_1}{r_2}\right)^3=\frac{\mathrm{P}_{a_2}+4 \mathrm{~S} / r_2}{\mathrm{P}_{a_1}+4 \mathrm{~S} / r_1}

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