Hot air balloon thermodynamics

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In hot air balloons, the air inside remains at atmospheric pressure due to a hole at the bottom, and the envelope's volume is effectively constant. Heating the air increases its temperature, which raises the kinetic energy of the remaining molecules, leading to decreased density. Although the balloon expands slightly when heated, the volume of air within the envelope remains constant, contributing to buoyancy. The buoyancy effect arises because the heated air inside the balloon is less dense than the cooler surrounding atmosphere. Thus, heating the air results in fewer molecules, maintaining pressure while allowing the balloon to rise.
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In hot air balloons the air inside the envelope stays at atmospheric pressure because of the hole in the bottom and the volume of the envelope is essentially constant. Yet in order to raise the balloon it must be heated which increases the its temperature. If the temperature increases how does the pressure and volume stay constant?

My guess is that the number of molecules of air in the balloon decreases (decreasing the density). The pressure stays the same because the increased temperature increases the kinetic energy of the remaining molecules.

What do you think? I'm not sure if my guess is correct.
 
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What makes you think the volume is constant? When you heat the air the balloon gets bigger.
 
Volume within the balloon is constant, but due to the higher temperature, the density is less inside the balloon than the surrounding cooler atmosphere, resulting in bouyancy effect.
 
...so yes when you heat the air you reduce the number of molecules in the balloon.
 

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