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A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular forces. Where a solid is the same, but the atoms have much less freedom of movement.

Even if the water is still, its molecules are still moving against each other.

My question is: If you take a solid metal and create a fine dust and mix it with water, will the water molecules move the solid powder all the time?

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The short answer to this question is: yes.

The movement of particles in suspension like the powder in water would be a perfect example of Brownian motion as Mithoron already said. The particles are being kicked around by the water molecules, which are in constant motion as you already said.

This is also true for small airborne particles that collide with fast moving gas molecules.

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