A bubble has surface tension . The ideal gas inside the bubble has ratio of specific heats The bubble is exposed to the atmosphere and it always retains its spherical shape. [2022]

Question

A bubble has surface tension $S$ . The ideal gas inside the bubble has ratio of specific heats $γ = \frac{5}{3} .$ 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 [2022]

Which of the following statement(s) is(are) correct? [2022]

Smart Achievers · Accepted Answer

(3, 4)

$If the surface of the bubble is perfect heat insulator,$

$So process is adiabatic, so,$

$P V^{γ} = constant$

$r_{1}$

$\Rightarrow {(\frac{r_{1}}{r_{2}})}^{5} = \frac{P_{a 2} + \frac{4 s}{r_{2}}}{P_{a 1} + \frac{4 s}{r_{1}}}$

$So (1) is incorrect.$

$Also, T_{1} V_{1}^{γ - 1} = T_{2} V_{2}^{γ - 1}$

$\Rightarrow \frac{T_{2}}{T_{1}} = {(\frac{V_{1}}{V_{2}})}^{2 / 3} \Rightarrow \frac{T_{2}}{T_{1}} = {(\frac{r_{1}}{r_{2}})}^{2} \Rightarrow$ $\frac{T_{2}}{T_{1}}$ $= {(\frac{P_{a 2} + \frac{4 s}{r_{2}}}{P_{a 1} + \frac{4 s}{r_{1}}})}^{2 / 5}$

$So (4) is correct.$

$Total internal energy + surface energy will not be same as$ $work done by gas will be there$

$i.e. Δ U = - Δ W$ $(∵ Δ Q = 0)$

$So, (2) is incorrect.$

$Now, if bubble is perfect heat conductor, then temperature$ $will remain constant. So, P V = constant$

$(P_{a 1} + \frac{4 s}{r_{1}}) \frac{4}{3} π r_{1}^{3} = (P_{a 2} + \frac{4 s}{r_{2}}) \frac{4}{3} π r_{2}^{3}$

${(\frac{r_{1}}{r_{2}})}^{3} = (\frac{P_{a 2} + \frac{4 s}{r_{2}}}{P_{a 1} + \frac{4 s}{r_{1}}})$

$So (3) is correct.$

Solution

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

2 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.

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

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

Want Us to Email you About Special Offers & Updates?

Solution

1 If the surface of the bubble is a perfect heat insulator, then (r1r2)5=Pa2+2Sr2Pa1+2Sr1.

2 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.

3 If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then (r1r2)3=Pa2+4Sr2Pa1+4Sr1.

4 If the surface of the bubble is a perfect heat insulator, then (T2T1)52=Pa2+4Sr2Pa1+4Sr1.

Want Us to Email you About Special Offers & Updates?

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

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

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