What is the difference between an extremely thick liquid and a solid? At which point does the difference stop mattering, it at all?
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The difference between a liquid and a solid is the organization of the particles at a molecular level. There is in fact a bright-line difference between solids and liquids, a point at which liquids suddenly stop being liquid.
It’s not gradual! It is much more of a “poof” now I’m not a liquid anymore moment. Think of ice and water – water doesn’t gradually thicken until it is ice, it gets cold and then poof, it doesn’t flow at all anymore!
To describe the difference more technically, **solids have a crystal structure** – a fancy way of saying that the molecules are arranged in a repeating geometry – rows of tiny rectangles or hexagons or whatever. Solids have a fixed volume and do not fill the space they inhabit.
**Liquids do not have a crystal structure.** Liquids have molecules that are all in contact with one another, but roll past each other randomly, any old way, with no repeating patterns. Think of this like digits in pi, they never repeat, and there’s no way of predicting which one will come next. Liquids fill spaces they inhabit and have a fixed volume.
(Gasses fill the space they inhabit and fill the volume they inhabit.)
The transition between being a liquid and a solid happens as the molecules lose the energy that keeps them from forming crystals. Temperature stays constant in this transition phase, and so does the state of matter.
If this seems weird and counterintuitive – it is! Liquids are weird and frankly exotic as a state of matter. Under most possible conditions for a given material, being liquid simply isn’t a thing. Your guess in your initial post, that a substance would get gradually thicker and thicker as it cools is what happens to most stuff; it simply skips becoming a liquid or solid and becomes a very dense gas.
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