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Recall,

$PV = nRT$

You're keeping the temperature of the air within the balloon higher than the temperature of the air outside of the balloon.

$=> n/V = rho propto 1/T$

In English that means we can prove based on the , that and temperature of gases are inversely proportional.

To be even simpler, hot air will rise in cool air because its density is less. Hence, buoyancy.

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