Question 403: Properties of Matter

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?

Accepted Answer

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) $
$$ \Rightarrow $$ $\left(\frac{r_1}{r_2}\right)^3=\frac{\mathrm{P}_{a_2}+4 \mathrm{~S} / r_2}{\mathrm{P}_{a_1}+4 \mathrm{~S} / r_1}$

Answer

If the surface of the bubble is a perfect heat insulator, then $\left(\frac{r_{1}}{r_{2}}\right)^{5}=\frac{P_{a 2}+\frac{2 S}{r_{2}}}{P_{a 1}+\frac{2 S}{r_{1}}}$.

Answer

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.

Answer

If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then $\left(\frac{r_{1}}{r_{2}}\right)^{3}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.

Answer

If the surface of the bubble is a perfect heat insulator, then $\left(\frac{T_{2}}{T_{1}}\right)^{\frac{5}{2}}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.

Properties of Matter

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