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 $P_{a1}$, 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 $P_{\mathrm{a}2}$, 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 ${\left(\frac{r_1}{r_2}\right)}^5=\frac{P_{a2}+\frac{2S}{r_2}}{P_{a1}+\frac{2S}{r_1}}$
• 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.
• Correct answer
  C

  If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, the ${\left(\frac{r_1}{r_2}\right)}^3=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$
• Correct answer
  D

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

Hence, the correct answer is options: 'C', 'D'.