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Is there a way to detect whether sugar is completely dissolved in a cup of water other than looking at the bottom of the cup?

I was wondering if there is a test or something that can be carried out to determine whether sugar is fully dissolved.

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    $\begingroup$ You could filter it to remove any undissolved sugar. $\endgroup$ – bon May 26 '15 at 18:38
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The sugar will keep dissolving until it reaches it's saturation concentration of around $\pu{900 g/L}$, which is of course horribly sweet, thick and undrinkable as a beverage. Assuming you don't completely saturate the water with sugar in this way, the undissolved sugar will continue to dissolve so long as there is still any remaining in the glass.

A common way of measuring the dissolved sugar concentration is by refractometry, as described below (note that home refractometers can be purchased for as little as $30 US):

A refractometer is a laboratory or field device for the measurement of an index of refraction (refractometry). The index of refraction is calculated from Snell's law while for mixtures, the index of refraction can be calculated from the composition of the material using several mixing rules such as the Gladstone–Dale relation and Lorentz–Lorenz equation.
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In homebrewing, a refractometer is used to measure the specific gravity before fermentation to determine the amount of fermentable sugars which will potentially be converted to alcohol.

So, in your case a "homebrewing" type of refractometer is probably your best bet. As long as sugar remains in solution, it will continue to dissolve until it reaches saturation. This also means it's refractive index and specific gravity will continue to increase. Once this increase is no longer observed, then the sugar is no longer dissolving, meaning you've either reached saturation or all of the sugar has dissolved. Again, saturation would mean a thick, honey-like solution. Thorough stirring throughout the measurement process is a must.

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