JEE Advanced 2022 · Paper 2 · Physics Q11

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PhysicsMultiple correct+4 / -2Question 29 of 54

A bubble has surface tension $S$ . The ideal gas inside the bubble has ratio of specific heats $\gamma = \dfrac53$ . 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?

If the surface of the bubble is a perfect heat insulator, then

\left(\dfrac{r_1}{r_2}\right)^5 = \dfrac{P_{a2} + \frac{2S}{r_2}}{P_{a1} + \frac{2S}{r_1}}

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.

If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then

\left(\dfrac{r_1}{r_2}\right)^3 = \dfrac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}

If the surface of the bubble is a perfect heat insulator, then

\left(\dfrac{T_2}{T_1}\right)^{\frac52} = \dfrac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}

Answer and worked solution

Official answer: C, D

Gas pressure. A bubble has two surfaces, so the gas pressure is $P=P_a+\frac{4S}r$ , and the volume is proportional to $r^3$ .

Insulating surface (adiabatic).

$PV^\gamma$ is constant with $3\gamma=5$ , so $\left(\dfrac{r_1}{r_2}\right)^5=\dfrac{P_{a2}+4S/r_2}{P_{a1}+4S/r_1}$ . Option (A) uses $2S/r$ , so it is false.
$T^\gamma P^{1-\gamma}$ is constant, so $P\propto T^{\gamma/(\gamma-1)}=T^{5/2}$ . This gives (D). True.
The atmosphere does work on the bubble when $P_a$ changes, so the total energy, surface energy included, changes. (B) is false.

Conducting surface, constant temperature. $PV$ is constant, so
$\left(\frac{r_1}{r_2}\right)^3=\frac{P_{a2}+4S/r_2}{P_{a1}+4S/r_1}.$
(C) is true.

(C), (D).

Topics:Surface tension Thermodynamic processes

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JEE Advanced 2022 · Paper 2, Physics question 11