The change in energy when a big drop is split into $n$ small droplets is:

The excess pressure inside the first soap bubble of radius $R_1$ is three times that inside the second soap bubble of radius $R_2$. The ratio of volumes of the first bubble to second bubble is

Under isothermal condition, two soap bubbles of radii $r_1$ and $r_2$ combine to form a single soap bubble of radius $R$. If $P$ is the external pressure and $T$ is the surface tension of the soap solution, find the expression for $T$.

$A$ bubble has surface tension $S$. The ideal gas inside the bubble has a 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 $r_1$ 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 $\left(\frac{r_1}{r_2}\right)^5 = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}$
$(B)$ 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 $\left(\frac{T_2}{T_1}\right)^{\frac{5}{2}} = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}$

$A$ big drop of radius $R$ is formed by $1000$ small droplets of water. The radius of each small droplet is:

$A$ liquid drop of diameter $D$ breaks into $27$ small drops of equal size. If the surface tension of the liquid is $\sigma$,then the change in surface energy is