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When the balloon rises and the pressure on its outer surface decreases, does the inner pressure also decrease? If Boyle's law holds true in this case, then the pressure of the gas inside the balloon decreases since its volume increases. However, were that the case, does the pressure inside equal the pressure outside, i.e., is the gas pressure equal to the atmospheric pressure?

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    $\begingroup$ Why do you think that the pressure inside a rising balloon would be constant? (Hint: the envelope of the balloon is not rigid.) $\endgroup$ Commented Apr 11, 2023 at 10:57
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    $\begingroup$ Mechanical stability (aka Newton's 3rd law) means that the pressure inside the balloon is the same as outside - if this were not the case, the balloon would be changing its shape. Boyle's law also means that the volume of the balloon should increase with lower pressure (if the balloon is closed - this if often not the case for the hot air balloons.) $\endgroup$
    – Roger V.
    Commented Apr 12, 2023 at 7:53

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In equilibrium, absolute pressure inside a closed balloon is always higher than the atmospheric pressure. It needs to balance the pressure of the surrounding atmosphere plus the tension of the balloon's skin, which tries to contract it. Otherwise the balloon would expand or contract.

The skin of the balloon behaves, within some reasonable limits, according to Hooke's law: its elongation is proportional to strain. If we denote the tension inside the wall $\gamma$, this additional pressure is $\frac{2\gamma}{r}$ (from the Young–Laplace equation). Without delving into equations, this can be seen fairly easily: since force scales linearly ($\propto r$) and surface area quadratically ($\propto r^2$), this extra pressure should scale $\propto 1/r$. This is consistent with a common observation: it is difficult to start inflating a balloon, but not so much to add air to an already inflated one.

If the external pressure decreases, for instance when rising higher inside the atmosphere, the balloon will expand. Since the amount of gas inside is constant, its absolute pressure must decrease (for simplicity we assume that everything happens slowly and at a constant temperature). The force in the walls will increase, but the pressure will go down, so the internal pressure will now be closer to the external one than before.

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