Question

The solubility of \(\mathrm{KNO}_{3}\) is \(155 \mathrm{~g}\) per \(100 \mathrm{~g}\) of water at \(75^{\circ} \mathrm{C}\) and \(38.0 \mathrm{~g}\) at \(25^{\circ} \mathrm{C}\). What mass (in grams) of \(\mathrm{KNO}_{3}\) will crystallize out of solution if exactly \(100 \mathrm{~g}\) of its saturated solution at \(75^{\circ} \mathrm{C}\) is cooled to \(25^{\circ} \mathrm{C} ?\)

Short answer

22.78 grams of KNO₃ will crystallize out.

Step-by-step solution

Determine the Composition of the Saturated Solution at 75°C

A saturated solution at 75°C contains 155 g of KNO₃ in 100 g of water. Therefore, the total solution (solute + solvent) weighs 255 g (155 g of KNO₃ + 100 g of water).

Calculate the Mass of Water in the Original Solution

The total saturated solution at 75°C weighs 255 g. This includes 100 g of water, since that is how saturated solutions are defined (155 g of solute per 100 g of water).

Determine the Mass of KNO₃ in 100 g of Solution at 75°C

We need to find out how much KNO₃ is in 100 g of the solution at 75°C. Using the ratio, we have 155 g of KNO₃ in 255 g of the solution. Thus,\[\frac{155}{255} \times 100 = 60.78 \text{ g of KNO₃}\]

Determine the Solubility of KNO₃ at 25°C

The solubility at 25°C is 38 g of KNO₃ per 100 g of water. Since the amount of water remains constant (100 g), the solution can contain, at most, 38 g of KNO₃ at 25°C.

Calculate the Mass of KNO₃ that will Crystallize

Since 60.78 g of KNO₃ was originally dissolved and only 38 g can remain dissolved at 25°C, the remaining amount will crystallize:\[60.78 - 38 = 22.78 \text{ g of KNO₃}\]

Key concepts

KNO₃ crystallization

Crystallization is a fascinating process by which a solid forms from a homogeneous solution. In the case of KNO₃ (potassium nitrate) crystallization, the solute particles gather to form a crystal structure. When you cool a potassium nitrate solution, the solubility of KNO₃ decreases. Thus, excess KNO₃ that cannot remain dissolved begins to form solid crystals. In our example, when a saturated KNO₃ solution at 75°C is cooled to 25°C, the solution cannot hold as much KNO₃, leading to crystallization. Crystals are pure, solid forms often containing the original structure of the compound. This is a practical method to purify chemicals or to grow crystals for scientific purposes. The amount of KNO₃ that crystallizes can be calculated based on the difference in solubility between the two temperatures that the solution experiences.

Saturated solution

A saturated solution contains the maximum concentration of solute that can dissolve in a solvent at a given temperature. This means that any additional solute will not dissolve but instead will remain as a solid. In the context of KNO₃, when described at 75°C, it specifically means that no more than 155 g of KNO₃ can dissolve in 100 g of water. The solution at this point has reached its capacity for the dissolved solute. Understanding this concept is crucial because when the temperature drops, the solubility changes, yet the amount of water remains constant, which directly impacts how much solute can stay in solution. With the decrease in temperature from 75°C to 25°C in our example, the solution becomes oversaturated, leading to excess KNO₃ crystallizing out of the solution.

Solubility temperature dependence

The solubility of substances can vary greatly with temperature. Typically, solubility increases with temperature, meaning more solute can dissolve in the solvent at higher temperatures. For KNO₃, the solubility is significantly higher at 75°C compared to 25°C. This concept is key in determining how much solute can be held by a solution as conditions change.

• At 75°C, you can dissolve 155 g of KNO₃ in 100 g of water.
• At 25°C, only 38 g of KNO₃ can remain dissolved in the same amount of water.

When a solution transitions from a warmer to a cooler temperature, it can no longer hold as much solute in solution, leading to crystallization. Understanding this temperature-solubility relationship is vital in many fields, such as chemical manufacturing, cooking, and even environmental science.

Solubility calculation

Calculating solubility involves determining how much of a solute can be dissolved in a solvent at a specific temperature. In the given problem, we calculate how much KNO₃ remains dissolved at different temperatures and how much crystallizes out. Initially, you find out how much KNO₃ is present in the solution at the higher temperature:Using the proportion: \[\frac{\text{Mass of KNO₃}}{\text{Total Mass of Solution}} = \frac{155}{255}\]This gives you the mass of KNO₃ in a smaller mass of the same solution. Next, apply temperature-specific solubility to see what happens when the solution is cooled:

• At 75°C, calculate that 60.78 g KNO₃ can be dissolved in 100 g of solution.
• At 25°C, only 38 g of KNO₃ remains dissolved, leaving 22.78 g to crystallize out as the solution cools.

Solubility calculations help us predict changes in the state of a solution as conditions change and are crucial for many applications in chemistry and industry.