A bubble has surface tension S . The ideal gas inside the bubble...

When,

$\begin{cases}Pressure & & Radius & & Temperature \\ {P}_{a1} & \to & {r}_{1} & \to & {T}_{1} \\ {P}_{a2} & \to & {r}_{2} & \to & {T}_{2}\end{cases}$

For adiabatic process

${P}_{1}{V}_{1}^{\gamma} = {P}_{2}{V}_{2}^{\gamma}$

\left({P}_{a1} + \dfrac{4T}{{r}_{1}}\right)\left(\dfrac{4}{3}\pi{r}_{1}^{3}\right)\dfrac{5}{3} = \left({P}_{a2} + \dfrac{4T}{{r}_{2}}\right)\left(\dfrac{4}{3}\pi{r}_{2}^{3}\right)\dfrac{5}{3}$</p><p>${\left(\dfrac{{r}_{1}}{{r}_{2}}\right)}^{5} = \dfrac{\left({P}_{a2} + \dfrac{4T}{{r}_{2}}\right)}{\left({P}_{a1} + \dfrac{4T}{{r}_{1}}\right)}

Therefore, option \left(A\right)$is incorrect.</p><p>Now,</p><p>${T}_{1}{V}_{1}^{\gamma - 1} = {T}_{2}{V}_{2}^{\gamma - 1}$</p><p>$\dfrac{{T}_{2}}{{T}_{1}} = {\left(\dfrac{{V}_{1}}{{V}_{2}}\right)}^{\gamma - 1} = {\left(\dfrac{{r}_{1}}{{r}_{2}}\right)}^{3\left(\dfrac{2}{3}\right)}

\left(\dfrac{{T}_{2}}{{T}_{1}}\right) = $\left(\dfrac{{P}_{a2} + \dfrac{4T}{{r}_{2}}}{{P}_{a1} + \dfrac{4T}{{r}_{1}}}\right)\dfrac{2}{5}$

Therefore, option \left(D\right)$is correct.</p><p>For option \left(B\right)$ Total internal energy $+$ surface energy will not be same as work done by gas will be there. Option \left(B\right)$is incorrect.</p><p>For option \left(C\right)$ change in temperature is negligible. Therefore, the process is isothermal.

${P}_{1}{V}_{1} = {P}_{2}{V}_{2}$

\left({P}_{a1} + \dfrac{4T}{{r}_{1}}\right)\left(\dfrac{4}{3}\pi{r}_{1}^{3}\right) = \left({P}_{a2} + \dfrac{4T}{{r}_{2}}\right)$\left(\dfrac{4}{3}\pi{r}_{2}^{3}\right)$

${\left(\dfrac{{r}_{1}}{{r}_{2}}\right)}^{3} = \dfrac{\left({P}_{a2} + \dfrac{4T}{{r}_{2}}\right)}{\left({P}_{a1} + \dfrac{4T}{{r}_{1}}\right)}$

Option \left(C\right) is correct.