A hot air balloon climbs because the air inside becomes less dense than the air outside. That small density gap, created by only a modest temperature difference, sets up buoyancy strong enough to lift several tons of fabric, basket, fuel, and passengers.
The mechanism is basic thermodynamics. Heat increases the kinetic energy of gas molecules, so the same mass of air expands and occupies a larger volume. In a fixed envelope, this means the balloon holds fewer kilograms of air per cubic meter. According to Archimedes principle, the surrounding cooler air exerts an upward buoyant force equal to the weight of the displaced air. If the total weight of the balloon system is lower than that displaced weight, the net force points upward.
The numbers are less dramatic than the effect. A modest rise in temperature inside the envelope can cut the internal air density by several percent. Multiply that reduction by tens of thousands of cubic meters of volume, and the missing mass of air can correspond to several tons. The balloon trades a small input of thermal energy for a large lifting capacity because it leverages gravitational potential energy stored in the displaced, cooler atmosphere.
Control comes from managing temperature and therefore density. Burning fuel increases internal temperature, reducing density and increasing lift. Letting the air cool increases density and reduces lift, allowing descent. Within this narrow band of temperature differences, fluid mechanics and gravity quietly cooperate to move heavy human cargo through the sky.
