Which gas law explains why heating air makes a hot-air balloon’s envelope expand?

Charles’s law explains why heating air makes a hot-air balloon’s envelope expand.

The law states that, at constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature. In simple terms, warmer air occupies more volume than the same air when cooler, provided pressure remains broadly similar.

A balloon envelope is flexible rather than rigid, so heated air can expand and fill it. In an operating balloon, the burner raises the temperature of the air inside, while the open bottom allows pressure to stay close to atmospheric pressure. The resulting difference in density helps produce lift.

Charles’s law is not the complete explanation of flight. Buoyancy, described by Archimedes’ principle, determines the upward force. Boyle’s law instead concerns the relationship between pressure and volume at constant temperature, so it is not the best answer here.