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Is the atmospheric pressure the only pressure acting on the exterior walls of a balloon keep the air/gas inside the ballon at equilibrium? Or the walls of balloon being elastic, do exert pressure on the gas inside but its magnitude is too low compared to that of atmospheric pressure?

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I vote for your second iterpretation: the elastic tension of the balloon takes part in the pressure balance.

Just consider a very strong balloon, it will hold gas inside even in a vacuum condition.

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