# Bubble connected by straws

Please provide a solution. I am not able to understand the problem. What does final state mean? @Alisher @Miras @Ersultan

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It is known that the internal energy of a bubble equals U=\sigma A, where A is the surface area. In fact, this is only an approximate relation which assumes that \sigma doesn’t depend on temperature T, and, in general, this relation holds in terms of Helmholtz free energy F\equiv U-TS=\sigma A. Now, using the 1st law of thermodynamics for a bubble \delta Q = dU -\sigma dA, you may prove that

U=\left(\sigma-T\frac{d\sigma}{dT}\right)A.
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Anyone wants to help me…

oh thanks, let me read about it… and for the 2nd and 3rd parts… how will we go around solving it…? Entropy change is confusing especially, never saw it for bubbles

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“A thousand-mile journey begins with the first step.”

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ok well… in all seriousness, what does Final state mean in problem2? what answer are you getting for problem 2?

Honestly, I have no clue how to solve that. I supposed they meant that under perturbations the temperature of the system will change, thereby changing the radii of the bubbles. However, after writing equations for energy and pressure, I got an equation with the only solution R=R_0. Probably after perturbations system will be asymmetrical, but there won’t be enough equations to find the final state in this case.

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oh yeah exactly, I am struggling on this for around 2 days now… still no progress. The answers given are:

1. Proof
2. R_f \approx 2^{1 / 3} R_0+\frac{4 \sigma}{3 p_0}\left(2^{1 / 3}-1\right)
P_f=P_0+\frac{4 \sigma}{2^{1 / 3} R_0}
1. \Delta S = \frac{8 \pi R_0^2}{T}\left(1-2^{-1 / 3}\right)\left(\frac{4}{3} \sigma-b T\right)
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@Ersultan @Alisher Any progress sir?