How much borax should I dissolve in hot water in order to ensure it won't return to its crystalline state as the solution cools back down? Will a small enough amount stay dissolved as it cools?

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This is a complicated question, because the answer might sound vague. The answer is really however much you can put into solution at the current temperature.

You seem to be aware of the solubility curve, in which the higher the temperature of a solvent, the more solute you can dissolve...

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This is a complicated question, because the answer might sound vague. The answer is really however much you can put into solution at the current temperature.

You seem to be aware of the solubility curve, in which the higher the temperature of a solvent, the more solute you can dissolve into it. For borax, I was able to find a graph with an experimental solubility curve for borax at this website.

Let's say you have a solution at 100 degrees Celsius, and you want to still have a clear solution when you drop the temperature to 40 degrees. If you have less than 60 grams per liter, then according to this data, you will still have a clear solution with no falling out of solution. More than that, and you run the risk of some dropping to the floor of your beaker.

This being said, there is a property of solutions known as supersaturation, where a solution has more solute than it should be able to hold. This can be achieved by warming a solution, bringing it to saturation, and then letting the liquid cool. If there is no structure for a crystal to form in the beaker, the solute will stay in solution as it cools. As soon as a structure is presented, such as a seed crystal, all of the excess solute will bond to the seed crystal until the solution reaches its saturation point again. This guy demonstrates a supersaturated solution of sodium acetate, but borax is capable of the same reaction.

Less solute will work for sure, but you can still supersaturate if you must.

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