1. Pressure inside the bubble

A spherical soap bubble has two surfaces, so the excess pressure is

Hence at any state,

where $P$ is the pressure of the gas inside the bubble.

So for the two states,

Also volume of the bubble is

2. Check option A: insulated bubble

If the surface is a perfect heat insulator, the process is adiabatic for the enclosed ideal gas:

Thus,

Since $V\propto r^3$,

Therefore,

which gives

But option A states

which uses $2S/r$ instead of $4S/r$.

So A is false.

3. Check option B: insulated bubble and total internal energy including surface energy

For a soap bubble, total relevant energy is

• internal energy of gas: $U$,
• surface energy: $E_s=2(4\pi r^2)S=8\pi Sr^2$.

For an adiabatic change, $\delta Q=0$. If the bubble changes quasi-statically,

where $P$ is gas pressure inside.

Now consider total energy

Then

Since

we get

Using

So total energy of gas + surface energy is not constant; it changes due to work done against atmosphere.

Hence B is false.

4. Check option C: perfectly conducting surface, atmospheric temperature unchanged

If the surface is a perfect heat conductor and atmospheric temperature change is negligible, the gas remains at constant temperature. So process is isothermal:

Since $V\propto r^3$,

Thus,

This matches option C.

So C is true.

5. Check option D: insulated bubble relation between temperatures

For adiabatic process of an ideal gas,

With $\gamma=5/3$,

Since $V\propto r^3$, we have

Hence,

So,

Therefore,

From adiabatic relation already found,

Thus,

This matches option D.

So D is true.

6. Final conclusion

Correct statements are:

These agree with the stored answer.