To determine the total mass of [tex]\( \text{KNO}_3 \)[/tex] that must be dissolved in 50.0 grams of [tex]\( \text{H}_2O \)[/tex] at [tex]\( 60^{\circ} \text{C} \)[/tex] to make a saturated solution, follow these steps:
1. Identify the given solubility information: We are provided with the solubility of [tex]\( \text{KNO}_3 \)[/tex] at [tex]\( 60^{\circ} \text{C} \)[/tex], which is 106 grams per 100 grams of water.
2. Interpret the solubility: This means that 106 grams of [tex]\( \text{KNO}_3 \)[/tex] can dissolve in 100 grams of water at [tex]\( 60^{\circ} \text{C} \)[/tex].
3. Scale down the solubility to the given amount of water: Since we only have 50.0 grams of [tex]\( \text{H}_2O \)[/tex], which is half of 100 grams, the amount of [tex]\( \text{KNO}_3 \)[/tex] that can be dissolved will also be half of 106 grams.
4. Calculate the mass of [tex]\( \text{KNO}_3 \)[/tex] that will dissolve:
[tex]\[
\text{Mass of } \text{KNO}_3 = \left( \frac{106 \, \text{grams}}{100 \, \text{grams of water}} \right) \times 50.0 \, \text{grams of water}
\][/tex]
Simplifying this calculation:
[tex]\[
\text{Mass of } \text{KNO}_3 = 53.0 \, \text{grams}
\][/tex]
Thus, the total mass of [tex]\( \text{KNO}_3 \)[/tex] that must be dissolved in 50.0 grams of [tex]\( \text{H}_2O \)[/tex] at [tex]\( 60^{\circ} \text{C} \)[/tex] to make a saturated solution is [tex]\( \boxed{53 \, \text{grams}} \)[/tex].
