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Why does hot air rise?

Consider a small volume in air, we will assume that this volume contains hot air while the surrounding air is cold. The forces acting on this gas are - gravity acting downwards, pressure of gas acting upwards. Gravity is same for everything but acceleration caused by pressure depends on density. If this volume element is in pressure equilibrium with the surrounding gas, as it must be as the time taken to reach pressure eq. is generally very short, its density must be less than the surrounding gas. A less dense gas is accelerated faster and hence it rises in comparison with the surrounding cold air. For cold air the pressure neatly cancels the force due to gravity which gives us the stable atmosphere. As this packet of hot air rises, it expands to maintain pressure equilibrium with its surroundings and cools down.

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