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For instance, if there is cup of water saturated with a salt to the point where any more added salt will just float around in the solution, are those undissolved salt molecules in equilibrium with the dissolved ones? For instance, do the undissolved ones become partially dissolved for a moment, while the dissolved ones swap places with them?

Or does this behave in such a manner that whatever salts were dissolved first to reach saturation are the ones that will stay dissolved, in spite of other added salts?

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Yes, they enter into a constant equilibrium. Just as we write equilibrium expressions for insoluble salts as equilibrium expressions, like $\ce{CaF2} \leftrightharpoons \ce{Ca}^{2+} + \ce{2F-}$, when you reach the saturation point, solute particles are in dynamic equilibrium between aqueous and solid phase.

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Yes. At the solution saturation point there is a dynamic equilibrium between the solid and solvated solutes.

From the IUPAC Gold Book:

saturated solution: A solution which has the same concentration of a solute as one that is in equilibrium with undissolved solute at specified values of the temperature and pressure.

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As others have said, they are in equilibrium. However there is a physical/mechanical (mass transfer) barrier to this equilibrium. When a crystal is dissolving/being precipitated, all shrinkage/growth occurs at the surface of the crystal.

Once the equilibrium state is reached, the exchange occurs only at the surface of the crystal. There is no change for the material in the body of the crystal (except by diffusion, which in crystals is an extremely slow process.) For a radioactive tracer to reach the centre of the crystals, complete crystals would have to dissolve and new ones be precipitated, which is only likely to happen if there is a gradient of conditions (for example a temperature gradient) between one part of the vessel and another.

Also, if you have a layer of salt slurry with clear solution above it, the solid phase of the slurry part is only in equilibrium with the liquid phase in the slurry part. If you add a radioactive tracer to the clear solution, it will be quickly incorporated into the surface of the crystals in the top layer of the slurry, but take a long time to reach (by liquid diffusion) the bottom part of the slurry. And it may never make it into the middle of any of the crystals.

PS I am a chemist turned chemical engineer, and for quantitative design of dissolution/precipitation process, this distinction is important.

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