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 $P_{a1}$, 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?

• 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.
• C. If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then ${\left(\frac{r_1}{r_2}\right)}^3=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$.
• D. If the surface of the bubble is a perfect heat insulator, then ${\left(\frac{T_2}{T_1}\right)}^{\frac{5}{2}}=\frac{P_{a2}+\frac{4S}{r_2}}{P_{a1}+\frac{4S}{r_1}}$.

Answer 1

Answer: (C), (D)

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