Question

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 $\left(\frac{T_2}{T_1}\right)^{\frac{5}{2}}=\frac{P_{a 2}+\frac{4 S}{r_2}}{P_{a 1}+\frac{4 S}{r_1}}$.

Answer 1

Answer: (C), (D)

$P_{ gas }=P_{ a }+\frac{4 S}{r}$
$PV ^{\gamma}=$ constant [adiabatic process]
$\left( Pa _1+\frac{4 S }{ r _1}\right)\left(\frac{4}{3} \pi r _1^3\right)^{5 / 3}=\left( P _{ a _2}+\frac{4 S }{ r _2}\right)\left(\frac{4}{3} \pi r _2^3\right)^{5 / 3}$
$ \frac{r_1^3}{r_2^3}=\left(\frac{P_{a_2}+\frac{4 S}{r_2}}{P_{a_1}+\frac{4 S}{r_1}}\right) $
$ P^{1-y} T^y=$ constant
$ \left(P_{a_2}+\frac{4 S}{r_2}\right)^{1-5 / 3} T_2^{5 / 3}=\left(P_{a_1}+\frac{4 S}{r_1}\right)^{1-5 / 3} T_1^{5 / 3} $
$ \left(\frac{T_2}{T_1}\right)^{5 / 3}=\left(\frac{P_{a_1}+\frac{4 S}{r_1}}{P_{a_2}+\frac{4 S}{r_2}}\right)^{-2 / 3} $
$ \left(\frac{T_2}{T_1}\right)^{5 / 2}=\left(\frac{P_{a_2}+\frac{4 S}{r_2}}{P_{a_1}+\frac{4 S}{r_1}}\right)$