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Why do helium-filled balloons float?
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Lets start by assuming the balloon stays the same size as it rises in the air column and we'll ignore the temperature drop. The pressure and density of the gas inside the balloon remains the same, but at some point the air density outside the baloon will drop to match the density of helium inside the baloon. At that point the balloon would stop rising as the weight of the atmosphere it displaces is the same as weight of the helium filled balloon. It's like a little boat on a sea of air.
However, balloons don't stay the same size. As they float up the atmospheric pressure drops. The balloon will expand because the pressure inside the balloon is higher that the pressure outside. It still has a bouyant force on it because the weight of atmosphere it displaces is still larger than it's own weight, so it continues to go up. Outside pressure continues to drop. Balloon continues to grow. Eventually the balloon bursts.