What mass of salt (NaCl) should you add to 1.00 L of water in an ice cream maker to make a solution that freezes at -10.0 °C? Assume complete dissociation of the NaCl and density of 1.00 g/mL for water.

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Identify the formula for freezing point depression: \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the change in freezing point, \( i \) is the van't Hoff factor, \( K_f \) is the cryoscopic constant, and \( m \) is the molality.

Determine the change in freezing point: \( \Delta T_f = 0.0 \text{ °C} - (-10.0 \text{ °C}) = 10.0 \text{ °C} \).

For NaCl, the van't Hoff factor \( i \) is 2 because it dissociates into two ions: Na\(^+\) and Cl\(^-\).

Use the known value of \( K_f \) for water, which is 1.86 \text{ °C kg/mol}. Substitute the values into the formula: \( 10.0 = 2 \cdot 1.86 \cdot m \) to solve for molality \( m \).

Convert molality to mass: Use the definition of molality \( m = \frac{\text{moles of solute}}{\text{kg of solvent}} \) and the molar mass of NaCl (58.44 g/mol) to find the mass of NaCl needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Freezing Point Depression

Freezing point depression is a colligative property that describes how the freezing point of a solvent decreases when a solute is added. The extent of this depression depends on the number of solute particles in the solution, not their identity. For ionic compounds like NaCl, which dissociates into two ions (Na+ and Cl-), the effect is doubled, making it crucial for calculating the required mass of solute to achieve a desired freezing point.

Colligative Properties

Colligative properties are properties of solutions that depend on the ratio of solute particles to solvent molecules, rather than the type of solute. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure. Understanding these properties is essential for determining how much solute to add to achieve specific physical changes in the solvent, such as lowering the freezing point of water.

Molarity and Density

Molarity is a measure of concentration defined as the number of moles of solute per liter of solution. In this context, knowing the density of water (1.00 g/mL) allows for the conversion between volume and mass, which is necessary for calculating how much NaCl to add. Since the problem involves a specific volume of water, understanding how to relate mass, volume, and molarity is key to solving the question.

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