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Generally speaking at least for solids such as table salt (NaCl) for example, if you cannot see the solid, it is probably dissolved. Therefore, a good way to test the solulbility of a substance,would be to keep adding the solid to the solution until the solid is no longer being dissolved so that you could see the solid phase present. The presence of a solid phase in a solution that is in equilibrium suggests that the solution is saturated with respect to that solid. This method might work with NaCl, but for other solids, it might take quite a long time for the solution to come to equilibrium, so it might not be very practical. You can calculate the solubility of different subsances. For a salt where the reaction is the dissasociation of two or more ions:
NaCl -> Na + Cl
(Na)(Cl)=K(sp)
Basically, at equilibrium the product of the activities of Na and Cl will equal a constand K(sp), which can be looked up in almost any chemistry text. For back of the envelope calculations, you can use molar concentrations (moles/liter) instead of activities. The constant will be for a specific temperature (I believe 25 deg C), so this calculation will be approximate. In this case the mole ratio of Na to Cl is one, so if you solve the following equation for x:
x^2=K(sp)
this molar concentration will be the solubility for NaCl.
I hope this helps. You could probably find a more complete explanation on the web quite easily.
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