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 salt. One manufacturer of home ice cream freezers recommends adding \(2.50 \mathrm{lb}(1130 \mathrm{g})\) of salt \((\mathrm{NaCl})\) to \(16.0 \mathrm{lb}\) of ice \((7250 \mathrm{g})\) in a \(4-\mathrm{qt}\) 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

Short answer

(a) 13.48% NaCl; (b) mole fraction of NaCl is 0.0458; (c) molality is 2.67 mol/kg.

Step-by-step solution

Calculate Total Mass of Solution

The total mass of the solution is the sum of the mass of the salt and the mass of the ice. So, we have: \[\text{Total mass} = 1130 \text{ g of NaCl} + 7250 \text{ g of ice} = 8380 \text{ g}\]

Calculate Weight Percent of NaCl

Weight percent is calculated as the mass of the solute divided by the total mass of the solution, multiplied by 100. \[\%\text{ by weight of NaCl} = \left( \frac{1130 \text{ g}}{8380 \text{ g}} \right) \times 100 = 13.48\%\]

Molar Mass Calculation

The molar mass of NaCl is the sum of the atomic masses of sodium (Na) and chlorine (Cl). \[\text{Molar mass of NaCl} = 22.99 \text{ g/mol (Na)} + 35.45 \text{ g/mol (Cl)} = 58.44 \text{ g/mol}\]

Calculate Moles of NaCl

Using the molar mass calculated in Step 3, we find the moles of NaCl. \[\text{Moles of NaCl} = \frac{1130 \text{ g}}{58.44 \text{ g/mol}} \approx 19.34 \text{ mol}\]

Calculate Moles of Water

Assume that water is entirely from melted ice, with a molar mass of 18.02 g/mol. \[\text{Moles of water} = \frac{7250 \text{ g}}{18.02 \text{ g/mol}} \approx 402.89 \text{ mol}\]

Calculate Mole Fraction of NaCl

The mole fraction is the ratio of the moles of the solute to the total moles in the solution (solute + solvent). \[X_{\text{NaCl}} = \frac{19.34 \text{ mol of NaCl}}{19.34 \text{ mol} + 402.89 \text{ mol}} = 0.0458\]

Calculate Molality of the Solution

Molality is calculated as the moles of solute per kilogram of the solvent (water in this case). \[m = \frac{19.34 \text{ mol of NaCl}}{7.250 \text{ kg of water}} \approx 2.67 \text{ mol/kg}\]

Key concepts

Weight Percent

Weight percent is a way of expressing the concentration of a component in a mixture. It's particularly useful for solid-state mixtures. This measurement tells us how much of the total weight of the solution is made up by the solute, which is the substance dissolved in the solvent.

• To calculate weight percent, use the formula: \( ext{Weight percent} = \left( \frac{\text{mass of solute}}{\text{total mass of solution}} \right) \times 100 \).
• In the context of the ice cream exercise, the solute is salt (\( \text{NaCl} \)) and the solution consists of melted salt and water mixture. The weight percent gives an insight into how salty the water becomes as the mixture melts.
• Using 1130 g for \( \text{NaCl} \) and 8380 g for the total mass, the weight percent of salt is calculated to be approximately 13.48%.

Weight percent is important for understanding not only concentrations but also various practical applications, such as how much salt is needed to optimally lower the freezing point in homemade ice cream mixtures.

Mole Fraction

The mole fraction is another way to express the concentration of a component in a mixture. It is the ratio of the moles of one component to the total number of moles of all components in the mixture. This measure of concentration is important in thermodynamic calculations.

• The formula to determine the mole fraction of a component is: \( X_{\text{component}} = \frac{\text{moles of component}}{\text{total moles in solution}} \).
• For the ice cream salt solution, we calculate the moles of \( \text{NaCl} \) at 19.34 mol and the moles of water at 402.89 mol. The total moles then become the sum of these two values.
• The mole fraction of \( \text{NaCl} \) in the mixture is 0.0458, meaning a small portion of the total mole content is salt, which is typical for realistic solutions requiring solute dispersion.

Mole fraction is particularly helpful because it remains constant regardless of temperature or pressure, unlike some other measures of concentration.

Molality

Molality is a concentration term that measures the moles of a solute per kilogram of solvent. Unlike molarity, which depends on volume, molality remains unchanged with fluctuations in temperature and pressure, providing a stable basis for various calculations.

• The formula to calculate molality \( m \) is: \( m = \frac{\text{moles of solute}}{\text{mass of solvent in kilograms}} \).
• In the exercise, we have 19.34 mol of \( \text{NaCl} \) dissolved in 7.250 kg of water, resulting in a molality of approximately 2.67 mol/kg.
• Molality is critical for colligative property calculations, like boiling point elevation and freezing point depression, where temperature changes do not alter the concentration measurements.

Molality is exclusive to scenarios where calculations reliant on temperature and pressure independence are needed, such as in the design of antifreeze solutions or homemade ice cream making.

Freezing Point Depression

Freezing point depression is a colligative property, meaning it depends on the number of particles in a solution rather than their nature. When a solute is added to a solvent, the freezing point of the resulting solution is lower than that of the pure solvent.

• This phenomenon occurs because the presence of solute particles disrupts the formation of the orderly solid structure of ice, requiring a lower temperature to achieve the phase transition.
• The magnitude of freezing point depression can be calculated using the formula: \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the freezing point depression, \( i \) is the van't Hoff factor (number of particles the solute breaks into), \( K_f \) is the freezing point depression constant, and \( m \) is the molality of the solution.
• In making ice cream, salt (\( \text{NaCl} \)) is added to the ice-water mixture to lower its freezing point, thus ensuring a colder environment around the mixture. This is crucial for the proper crystallization and texture of ice cream.

Understanding freezing point depression has applications across various fields, from culinary activities to industrial processes where precise temperature control is required.