Problem 98 - Chemistry The Science in Context

Question

Making Ice Cream A mixture of table salt and ice is used to chill the contents of hand-operated ice-cream makers. What is the melting point of a mixture of 2.00 lb of \(\mathrm{NaCl}\) and 12.00 lb of ice if exactly half of the ice melts? Assume that all the NaCl dissolves in the melted ice and that the van 't Hoff factor for the resulting solution is 1.44

Step-by-Step Solution

Answer

Answer: The melting point of the NaCl and ice mixture is -15.31 °C.

Step 1: Calculate the molality of the salt solution

To calculate the molality (m) of the salt solution, we need to find the number of moles of NaCl and divide it by the kilograms of solvent (in this case, melted ice). The mass of NaCl is 2.00 lb, which we will convert to grams first. 2.00 lb * 453.59 g/lb = 907.18 g Now calculate the moles of NaCl: Moles of NaCl = mass / molar mass Moles of NaCl = 907.18 g / (58.44 g/mol) = 15.52 mol Since half of the ice melts, the mass of the melted ice is the same as the mass of the remaining ice. 6.00 lb of ice melts = 6.00 * 453.59 g/lb = 2721.54 g Convert the mass of the melted ice to kilograms: 2721.54 g * (1 kg / 1000 g) = 2.72154 kg Now, we can calculate the molality of the salt solution: Molality (m) = moles of solute / kg of solvent m = 15.52 mol / 2.72154 kg = 5.70 mol/kg

Step 2: Calculate the freezing point depression

Now that we have the molality, we can use the freezing point depression formula, ΔT_f = iK_fm, where i is the van 't Hoff factor, K_f is the cryoscopic constant for water (1.86 °C/m), and m is the molality. ΔT_f = (1.44) (1.86 °C/m) (5.70 mol/kg) = 15.31 °C

Step 3: Determine the new freezing point

Finally, we need to find the new freezing point by subtracting the freezing point depression from the original freezing point of ice. The normal freezing point of ice is 0 °C. New freezing point = original freezing point - ΔT_f New freezing point = 0 °C - 15.31 °C = -15.31 °C The melting point of the NaCl and ice mixture is -15.31 °C.

Key Concepts

Molality

Molality is a measure of the concentration of a solute in a solution, which is expressed as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the volume of solution, molality is only dependent on the mass of the solvent, making it unaffected by temperature changes. This characteristic is particularly useful when dealing with solutions at different temperatures.

To calculate molality, we use the formula:
\[ \text{Molality (m)} = \frac{\text{moles of solute}}{\text{kg of solvent}} \]
When calculating molality for the ice cream making process, it's important to convert the mass of NaCl and ice from pounds to grams (or kilograms), as the standard SI unit for molality is moles per kilogram. This conversion is crucial to ensure the accuracy of the calculation.

van 't Hoff Factor

Colligative properties are characteristics of solutions that depend on the number of solute particles present, rather than the nature of the solute itself. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.

For students grappling with these concepts, it's important to understand that colligative properties are fundamentally related to how solute particles affect the overall behavior of the solvent. They can be understood by considering how the presence of solute particles interferes with the normal processes that occur in pure solvents, such as ice forming in pure water.

To delve deeper:

• Freezing Point Depression: When a solute is dissolved in a solvent, the freezing point of the solution is lower than that of the pure solvent. This is due to the solute particles disrupting the solvent's ability to form a solid structure at the usual freezing point.
• Boiling Point Elevation: The presence of a solute raises the boiling point of the solution because the solute particles require the solvent to possess more kinetic energy to enter the vapor phase.
• Vapor Pressure Lowering: A solution has a lower vapor pressure than the pure solvent because the solute particles occupy surface space in the liquid, hindering the evaporation of solvent particles.
• Osmotic Pressure: Osmotic pressure arises from the tendency of solvent particles to move across a semipermeable membrane to balance concentrations, which is influenced by the number of solute particles.

Understanding colligative properties is essential for explaining why mixing salt and ice can lower the mixture's freezing point, which is an integral part of making ice cream with the method discussed in the exercise.