Question

Potassium sulfate has a solubility of \(15 \mathrm{~g} / 100 \mathrm{~g}\) water at \(40^{\circ} \mathrm{C}\). A solution is prepared by adding \(39.0 \mathrm{~g}\) of potassium sulfate to \(225 \mathrm{~g}\) of water, carefully heating the solution, and cooling it to \(40^{\circ} \mathrm{C}\). A homogeneous solution is obtained. Is this solution saturated, unsaturated, or supersaturated? The beaker is shaken, and precipitation occurs. How many grams of potassium sulfate would you expect to crystallize out?

Short answer

Based on the calculations, the solution of potassium sulfate in water is determined to be supersaturated. When the beaker is shaken and precipitation occurs, 5.25 g of potassium sulfate will crystallize out.

Step-by-step solution

Calculate the maximum mass of potassium sulfate that can dissolve in given water mass at 40°C

Using the solubility of potassium sulfate provided (\(15 \mathrm{~g} / 100 \mathrm{~g}\) water at \(40^{\circ} \mathrm{C}\)), we can find how much potassium sulfate can dissolve in the given \(225 \mathrm{~g}\) of water: $$\text{Mass of dissolved potassium sulfate} = \frac{15 \mathrm{~g} }{100 \mathrm{~g} }\times 225 \mathrm{~g} = 33.75 \mathrm{~g}$$

Determine the type of solution

Compare the mass of dissolved potassium sulfate (33.75 g) with the mass of potassium sulfate added to the solution (39.0 g). If the amount of added potassium sulfate is less than or equal to the mass of dissolved potassium sulfate, then the solution is either saturated (if equal) or unsaturated (if less than). If the mass of potassium sulfate added is greater than the mass of dissolved potassium sulfate, then the solution is supersaturated. Since \(39.0 \mathrm{~g}\) (added) > \(33.75 \mathrm{~g}\) (dissolved), the solution is supersaturated.

Calculate the mass of potassium sulfate that would crystallize out

When the beaker is shaken, and precipitation occurs, the excess potassium sulfate will crystallize out. We need to find the difference between the mass of potassium sulfate added and the maximum mass of potassium sulfate that can dissolve at 40°C. $$\text{Mass of crystallized potassium sulfate} = 39.0 \mathrm{~g} - 33.75 \mathrm{~g} = 5.25 \mathrm{~g}$$ So, when precipitation occurs, we can expect 5.25 g of potassium sulfate to crystallize out.