In 100 g of water at 0°C, table sugar has a solubility of 180 g. What will happen if you add 100 g of table sugar to 50 g of water at 0°C?

Sugar is said to have a solubility of #"180 g"# in #"100 g"# of water at #0^@"C"#. This tells you that dissolving that much sugar per #"100 g"# of water would produce a saturated solution.

A saturated solution is a solution in which the rate at which solid particles are dissolved is equal to the rate at which dissolved solute particles reform the solid.

Simply put, a saturated solution is a solution that cannot dissolve more solute than what is already dissolved.

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By comparison, in an unsaturated solution, the rate at which solid particles are dissolved exceeds the rate at which dissolved particles reform the solid, which means that the solution can dissolve more solute than what is already dissolved.

In your case, adding #"180 g"# of sugar to #"100 g"# of water will produce a saturated solution. Use this as a conversion factor to determine the solubility of sugar in #"50 g"# of water at #0^@"C"#

#50 color(red)(cancel(color(black)("g water"))) * "180 g sugar"/(100color(red)(cancel(color(black)("g water")))) = "90 g sugar"#

This means that adding #"90 g"# of sugar to #"50 g"# of water at #0^@"C"# will produce a saturated solution.

Now, you want to add #"100 g"# of sugar in #"50 g"# of water. Since you can only hope to dissolve #"90 g"# at this temperature, your resulting solution will contain undissolved sugar.

More specifically, it will contain

#m_"undissolved sugar" = overbrace("100 g")^(color(purple)("amount added")) - overbrace("90 g")^(color(blue)("amount dissolved"))#

#m_"undissolved sugar" = "10 g"#

Therefore, adding #"100 g"# of sugar to #"50 g"# of water at #0^@"C"# will produce a saturated solution that contains #"90 g"# of dissolved sugar and #"10g"# of undissolved.