Hot Air Balloons and Buoyancy

Why do hot air balloons float in the air?

Hot air balloons float in the air due to the difference in density between the hot air inside the balloon and the cold air outside. How can we calculate the density of hot air inside the balloon, its total weight, the magnitude of the buoyant force, and the minimum temperature required for the balloon to float?

Explanation:

Hot air balloons float in the air due to the difference in density between the hot air inside the balloon and the cold air outside. When the air inside the balloon is heated, it becomes less dense than the surrounding air, causing the balloon to rise.

1) Density of Hot Air Inside the Balloon

The density of the hot air inside the balloon, rhoh, can be expressed in terms of Th, Tc, and rhoc using the ideal gas law:
rhoh = (rhoc * Tc) / Th

2) Total Weight of the Balloon

The total weight of the balloon, W, can be calculated by adding the weight of the balloon fabric, basket, and other equipment to the weight of the hot air:
W = mb * g + (V1 + V2) * rhoh * g

3) Magnitude of the Buoyant Force

The magnitude of the buoyant force, FB, on the balloon can be calculated using the density of the cold air, rhoc, the volume of the hot air, V1, and the volume of the basket and equipment, V2:
FB = (V1 + V2) * rhoc * g

4) Minimum Temperature for the Balloon to Float

The minimum temperature, Tmin, of the hot air inside the balloon for it to float can be determined by setting the weight of the balloon plus the hot air equal to the buoyant force:
mb * g + (V1 + V2) * rhoh * g = (V1 + V2) * rhoc * g