JEE Advance - Physics (2022 - Paper 2 Online - No. 11)
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?
Which of the following statement(s) is(are) correct?
If the surface of the bubble is a perfect heat insulator, then $\left(\frac{r_{1}}{r_{2}}\right)^{5}=\frac{P_{a 2}+\frac{2 S}{r_{2}}}{P_{a 1}+\frac{2 S}{r_{1}}}$.
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 $\left(\frac{r_{1}}{r_{2}}\right)^{3}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.
If the surface of the bubble is a perfect heat insulator, then $\left(\frac{T_{2}}{T_{1}}\right)^{\frac{5}{2}}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.
Explanation
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} $
$=\left(\mathrm{P}_{a 2}+\frac{4 \mathrm{~S}}{r_2}\right)\left(\frac{4}{3} \pi r_2^3\right)^{5 / 3}$
$$ \Rightarrow $$ $$ \left(\frac{r_1}{r_2}\right)^5=\frac{P_{a 2}+4 S / r_2}{P_{a 1}+4 S / r_1} $$
Also, $\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}$
$$ \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\frac{\left(\mathrm{P}_{a 2}+4 \mathrm{~S} / r_2\right)^{2 / 5}}{\left(\mathrm{P}_{a 2}+\frac{4 \mathrm{~S}}{r_1}\right)^{2 / 5}} $$
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) $
$$ \Rightarrow $$ $\left(\frac{r_1}{r_2}\right)^3=\frac{\mathrm{P}_{a_2}+4 \mathrm{~S} / r_2}{\mathrm{P}_{a_1}+4 \mathrm{~S} / r_1}$
$\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} $
$=\left(\mathrm{P}_{a 2}+\frac{4 \mathrm{~S}}{r_2}\right)\left(\frac{4}{3} \pi r_2^3\right)^{5 / 3}$
$$ \Rightarrow $$ $$ \left(\frac{r_1}{r_2}\right)^5=\frac{P_{a 2}+4 S / r_2}{P_{a 1}+4 S / r_1} $$
Also, $\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}$
$$ \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\frac{\left(\mathrm{P}_{a 2}+4 \mathrm{~S} / r_2\right)^{2 / 5}}{\left(\mathrm{P}_{a 2}+\frac{4 \mathrm{~S}}{r_1}\right)^{2 / 5}} $$
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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