It is the same as why iron sinks to the bottom in water but wood rises. We probably all intuitively know that bodies with a high density in water sink to the bottom and bodies with a low density rise to the top. And perhaps we also know that if a body has the same density as water, it will stay in the given place — it will neither rise nor sink. We describe this phenomenon as buoyancy and the magnitude of the buoyant force is described by Archimedes’ law. The buoyant force acts in all fluids (substances that flow — they do not have a fixed shape), which are not only liquids but also gases. Balloons and airships can convince us that Archimedes' law also works in gases. It can therefore be expected that the answer to the question of why warm air rises, is that it is less dense than the surrounding air.
But why is warm air less dense than cold air? This is due to another well-known phenomenon — thermal expansion. When the temperature increases, the molecules of substances move faster, this faster movement causes more frequent and intense collisions of molecules with each other, which we observe from the outside as the expansion of the body. Gases expand the most out of all states — we can see this, for example, if we put a bottle with water and an empty bottle in the refrigerator. The volume of the bottle with water will change minimally but the bottle without water will be deformed. The warmer the bottle was before we put it in the fridge, the more it will “shrink”.
The density of a body depends on its volume and weight. If we heat the gas, its volume will increase and therefore its density will decrease — it is therefore less dense than the surrounding gas and will therefore rise upwards. I’ll just point out 2 things — the decrease in density is only caused by an increase in the volume of the gas but its weight (and the size of the molecules themselves) remains the same and the rise of warm air (as well as any other event resulting from Archimedes’ law) can only be observed in the gravitational field.
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