A bubble has surface tension $$S$$. The ideal gas inside the bubble has ratio of specific heats $$\gamma = \dfrac{5}{3}$$. 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?

For a thin soap bubble in air, the excess pressure due to surface tension is $$\dfrac{4S}{r}$$ (two surfaces of the film). Hence at any instant

$$P = P_a + \dfrac{4S}{r}$$

where $$P_a$$ is the atmospheric pressure, $$P$$ the gas pressure inside the bubble and $$r$$ the instantaneous radius.

Case 1: Surface is a perfect heat insulator (adiabatic process)

The enclosed gas obeys the adiabatic relation

$$PV^{\gamma}= \text{constant}$$

with $$\gamma = \dfrac{5}{3}$$ and $$V=\dfrac{4\pi r^{3}}{3} \;\;(\therefore V\propto r^{3})$$.

Therefore

$$\bigl(P_a+\dfrac{4S}{r}\bigr)\;r^{3\gamma}= \text{constant}$$

$$\Rightarrow\; \bigl(P_{a1}+\dfrac{4S}{r_1}\bigr)\,r_1^{5} =\bigl(P_{a2}+\dfrac{4S}{r_2}\bigr)\,r_2^{5}$$

$$\Rightarrow\; \left(\dfrac{r_1}{r_2}\right)^{5}= \dfrac{P_{a2}+ \dfrac{4S}{r_2}}{P_{a1}+ \dfrac{4S}{r_1}}$$

• Option A uses $$2S/r$$ instead of $$4S/r$$, so Option A is wrong.

Next, relate temperature using the ideal-gas law $$PV = nRT$$.

Combining $$PV = nRT$$ with $$PV^{\gamma}= \text{constant}$$ gives the standard result for an adiabatic change

$$TV^{\gamma-1}= \text{constant}$$

Since $$\gamma-1=\dfrac{2}{3}$$, we get $$T r^{2}= \text{constant}$$, i.e.

$$\dfrac{T_2}{T_1}= \left(\dfrac{r_1}{r_2}\right)^{2}$$

Raising both sides to the power $$\dfrac{5}{2}$$ and substituting the radius relation derived earlier:

$$\left(\dfrac{T_2}{T_1}\right)^{5/2}= \left(\dfrac{r_1}{r_2}\right)^{5}= \dfrac{P_{a2}+ \dfrac{4S}{r_2}}{P_{a1}+ \dfrac{4S}{r_1}}$$

This is exactly the statement in Option D, so Option D is correct.

The total energy of the system is

$$U_{\text{total}} = U_{\text{gas}} + E_{\text{surface}} = nC_{V}T + 4\pi r^{2}S$$

Both $$T$$ and $$r$$ change during the adiabatic compression/expansion, hence $$U_{\text{total}}$$ is not constant. Therefore Option B is wrong.

Case 2: Surface is a perfect heat conductor (isothermal process)

If heat flows freely through the film while the surrounding temperature remains nearly unchanged, the gas undergoes an isothermal change so that $$PV=\text{constant}.$$ Using $$V\propto r^{3}$$:

$$\bigl(P_a+\dfrac{4S}{r}\bigr)\,r^{3}= \text{constant}$$

$$\Rightarrow\; \left(\dfrac{r_1}{r_2}\right)^{3}= \dfrac{P_{a2}+ \dfrac{4S}{r_2}}{P_{a1}+ \dfrac{4S}{r_1}}$$

This matches Option C exactly, so Option C is correct.

Hence the correct statements are:
Option C and Option D.