Thermodynamics and Surface Energy of a Spherical Soap Bubble

To determine the correct statement(s), we analyze the thermodynamic state of the gas inside the soap bubble under different conditions.

1. Excess Pressure Inside a Soap Bubble

A soap bubble has two free surfaces (inner and outer). Therefore, the excess pressure inside the bubble over the atmospheric pressure $P_a$ is given by: $P_{in} - P_a = \frac{4S}{r} \implies P_{in} = P_a + \frac{4S}{r}$

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2. Analysis of Option C (Isothermal Process)

If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, the temperature of the enclosed gas remains constant ($T_1 = T_2 = T$).

For an ideal gas undergoing an isothermal process: $P_{in1} V_1 = P_{in2} V_2$

Since the volume of a sphere is $V = \frac{4}{3}\pi r^3$: $\left(P_{a1} + \frac{4S}{r_1}\right) \left(\frac{4}{3}\pi r_1^3\right) = \left(P_{a2} + \frac{4S}{r_2}\right) \left(\frac{4}{3}\pi r_2^3\right)$

Rearranging terms: $\left(\frac{r_1}{r_2}\right)^3 = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}$

Thus, Option C is correct.

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3. Analysis of Options A and D (Adiabatic Process)

If the surface of the bubble is a perfect heat insulator, no heat is exchanged with the surroundings ($dQ = 0$).

For an ideal gas undergoing an adiabatic process with ratio of specific heats $\gamma = \frac{5}{3}$: $P_{in} V^{\gamma} = \text{constant} \implies P_{in} V^{5/3} = \text{constant}$

Taking the $3^{\text{rd}}$ power on both sides: $P_{in}^3 V^5 = \text{constant}$

Since $V \propto r^3$, we have $V^5 \propto r^{15}$: $P_{in}^3 (r^3)^5 = \text{constant} \implies P_{in}^3 r^{15} = \text{constant} \implies P_{in} r^5 = \text{constant}$

Applying this between states $1$ and $2$: $P_{in1} r_1^5 = P_{in2} r_2^5 \implies \left(\frac{r_1}{r_2}\right)^5 = \frac{P_{in2}}{P_{in1}} = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}$

Option A gives $\frac{2S}{r}$ instead of $\frac{4S}{r}$, so Option A is incorrect.

Now, using the ideal gas equation $P_{in} V = n R T \implies V \propto \frac{T}{P_{in}}$, we substitute $V$ into $P_{in}^3 V^5 = \text{constant}$: $P_{in}^3 \left(\frac{T}{P_{in}}\right)^5 = \text{constant} \implies \frac{T^5}{P_{in}^2} = \text{constant} \implies T^5 \propto P_{in}^2$

Taking the square root on both sides: $T^{5/2} \propto P_{in}$

Therefore: $\left(\frac{T_2}{T_1}\right)^{5/2} = \frac{P_{in2}}{P_{in1}} = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}$

Thus, Option D is correct.

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4. Analysis of Option B (Total Energy Balance)

By the First Law of Thermodynamics applied to the bubble system (gas + surface): $dE_{\text{total}} = dQ - dW_{\text{ext}}$

For an thermally insulated surface, $dQ = 0$. However, as the external atmospheric pressure changes, the volume of the bubble changes ($dV \neq 0$). The work done against the surrounding atmosphere is: $dW_{\text{ext}} = P_a dV \neq 0$

Therefore: $dE_{\text{total}} = -P_a dV \neq 0$

This means the total internal energy of the bubble changes when external pressure changes.

Thus, Option B is incorrect.

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Conclusion

The correct options are C and D.