Most movement of particles (such as air)
happens due to diffusion. If you put sugar or salt
into water, it will eventually dissolve into the
whole volume of water. In this example, it's
because there is originally a different
concentration at one point than another (that is,
there's a lot of sugar sitting on the bottom of
the cup, and none at the top). This creates a
gradient, a difference in concentration in this
case, which can do work (like move particles!).
Hot air rising happens for similar reasons. Hot
air is less dense than cold air. When the air gets
hot, its density decreases, and now there's more
air molecules in the colder zone than in the hot
zone. There are more complicated things happening,
but for this simple reason alone, we might expect
hot air to rise for the same reason that objects
less dense than water will float in water.
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