Problem 54 - Chemistry and Chemical Reactivity

Question

Making homemade ice cream is one of life's great pleasures. Fresh milk and cream, sugar, and flavorings are churned in a bucket suspended in an ice-water mixture, the freezing point of which has been lowered by adding rock salt. One manufacturer of home ice cream freezers recommends adding \(2.50 \mathrm{lb}(1130 \mathrm{g})\) of rock salt \((\mathrm{NaCl})\) to 16.0 lb of ice \((7250 \mathrm{g})\) in a 4 -qt freezer. For the solution when this mixture melts, calculate the following: (a) the weight percent of \(\mathrm{NaCl}\) (b) the mole fraction of \(\mathrm{NaCl}\) (c) the molality of the solution

Step-by-Step Solution

Answer

The weight percent of NaCl is 13.49%, the mole fraction is 0.046, and the molality is 2.67 mol/kg.

Step 1: Calculate Weight Percent of NaCl

The weight percent of a component is defined as the mass of the component divided by the total mass of all components in the mixture, multiplied by 100. First, add the masses of ice and rock salt. Hence, total mass is: \[ 1130 \text{ g (salt)} + 7250 \text{ g (ice)} = 8380 \text{ g} \].Now, calculate the weight percent of NaCl by using the formula: \[ \text{Weight percent of NaCl} = \left( \frac{1130 \text{ g}}{8380 \text{ g}} \right) \times 100 \approx 13.49\% \].

Step 2: Calculate the Mole Fraction of NaCl

To find the mole fraction, start by calculating the number of moles of NaCl and H₂O. Molar mass of NaCl is \(58.44 \text{ g/mol}\) and for H₂O is \(18.02 \text{ g/mol}\).First, calculate moles of NaCl: \[ \text{Moles of NaCl} = \frac{1130 \text{ g}}{58.44 \text{ g/mol}} \approx 19.34 \text{ moles} \].Next, calculate the moles of water (H₂O):\[ \text{Moles of H}_2\text{O} = \frac{7250 \text{ g}}{18.02 \text{ g/mol}} \approx 402.89 \text{ moles} \].Finally, calculate the mole fraction of NaCl:\[ \text{Mole fraction of NaCl} = \frac{19.34 \text{ moles}}{19.34 \text{ moles} + 402.89 \text{ moles}} \approx 0.046 \].

Step 3: Calculate the Molality of the Solution

Molality is defined as moles of solute per kilogram of solvent. Use the number of moles of NaCl from Step 2: \[ 19.34 \text{ moles of NaCl} \].The mass of the solvent (water) is \(7250 \text{ g} = 7.25 \text{ kg}\).Thus, the molality of the solution is:\[ \text{Molality} = \frac{19.34 \text{ moles}}{7.25 \text{ kg}} \approx 2.67 \text{ mol/kg} \].

Key Concepts

Weight Percent of NaCl

When making homemade ice cream, it's essential to understand how rock salt lowers the freezing point of ice. This process is crucial for churning the mixture into a smooth, creamy treat. One way to quantify the amount of salt in the ice mixture is by calculating the weight percent. Weight percent helps us understand the concentration of a substance in a mixture. Specifically, it represents the mass of a particular component divided by the total mass of all components, then multiplied by 100 to convert the result into a percentage.

In our ice cream example, the total mass of the salt and ice mixture is 8380 grams, with 1130 grams being salt. To find the weight percent of NaCl, use the formula:

• Divide the mass of salt (1130 g) by the total mass (8380 g).
• Multiply the result by 100.

This gives approximately 13.49% weight percent. This value tells us that 13.49% of the combined mass of ice and salt is from the salt itself.

Mole Fraction of NaCl

The mole fraction is a way to express the concentration of a particular substance in a solution. It's defined as the ratio of the moles of one component to the total moles of all components in the solution. Calculating the mole fraction offers insight into the solution's composition at a molecular level. To determine the mole fraction of NaCl, we need to find the number of moles of both NaCl and the water (H₂O) in our system.

First, calculate the moles of NaCl using its molar mass (58.44 g/mol). For 1130 grams of NaCl, we have about 19.34 moles. Next, we find the moles of water using its molar mass (18.02 g/mol), giving us approximately 402.89 moles for 7250 grams of water. The mole fraction of NaCl is calculated by dividing the moles of NaCl by the total moles of both components:

• Moles of NaCl: 19.34
• Moles of water: 402.89
• Mole fraction of NaCl: \( \frac{19.34}{19.34 + 402.89} \approx 0.046 \)

This indicates the proportion of NaCl relative to the total number of moles in the solution is about 4.6%.

Molality of the Solution

Molality is an important concept when dealing with solutions, particularly because it is temperature-independent. It's defined as the moles of solute per kilogram of solvent. Unlike molarity, which can change with temperature due to volume fluctuations, molality remains constant because it's based on mass. This makes it a useful measure in temperature-sensitive processes like freezing point depression.

To calculate the molality of the NaCl and water mixture, we use the previously calculated moles of NaCl (19.34 moles) and the mass of the solvent, which is the water in our case. The mass of water is 7250 grams, which converts to 7.25 kilograms. The formula for molality is:

• Molality (m) = \( \frac{19.34 \text{ moles}}{7.25 \text{ kg}} \approx 2.67 \text{ mol/kg} \)

This means the solution has a molality of 2.67 mol/kg, which tells us there are 2.67 moles of NaCl per kilogram of water. Knowing the molality is crucial in understanding how the salt affects the freezing point of the ice, allowing us to successfully churn the ice cream.