if the Earth's core provides the heat that drives plate tectonics, then what will eventually happen as the Earth's core cools down over billions of years?

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If the Earth's core cools down, then one of the first things that would happen is that plate tectonic activity would slow down. Over time, the Earth would then dry up into this ball of dry soil because the heat of the Earth's core drives water up to the surface. Since the crust would be void of majority of liquid, it would shrink by an unknown amount. 

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