in order to dissolve the greatest amount of sugar in a cup of water, you should
To dissolve the greatest amount of sugar in a cup of water, you need to move the reaction equilibrium. According to the Le Chatelier's principle, the increase in temperature favors the reaction that is endothermic.
As in the majority, the reactions of dissolving the solid matter in liquid are endothermic, heating of the cup will favor the dissolution of sugar.
Concerning the extent of dispersion of sugar (crushed into the small pieces or they put it just in large ones), one should be attentive that this dispersion affects only the kinetics of the processes, not thermodynamics. That means, whether the sugar is in small or big pieces, the amount of sugar finally dissolved will be the same. However, the small pieces of sugar will dissolve faster.
Therefore, using sugar crushed into small pieces, and heating the water, or using sugar in large pieces, and heating the water - both will help to dissolve the greatest amount of sugar in a cup of water.
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