In ice-cream making, the ingredients are kept below \(0.0^{\circ} \mathrm{C}\) in an ice-salt bath. (a) Assuming that \(\mathrm{NaCl}\) dissolves completely and forms an ideal solution, what mass of it is needed to lower the melting point of \(5.5 \mathrm{~kg}\) of ice to \(-5.0^{\circ} \mathrm{C} ?\) (b) Given the same assumptions as in part (a), what mass of \(\mathrm{CaCl}_{2}\) is needed?

Colligative properties are those properties of solutions that depend on the number of dissolved particles but not on their chemical identity. One of the key colligative properties is freezing point depression. When a solute is added to a solvent, it lowers the temperature at which the solvent freezes.
The formula to determine the freezing point depression is given by \ \[ \Delta T_f = K_f \times m \] where \( K_f \) is the cryoscopic constant (freezing point depression constant) and \( m\) is the molality of the solution.
In the context of making ice-cream, adding substances like salt (\( \mathrm{NaCl} \) or \( \mathrm{CaCl}_2 \)) to the ice lowers its freezing point, thereby allowing us to achieve a temperature lower than 0°C. This is crucial for the consistency of the ice cream.

Molality is a measure of the concentration of a solution, given in moles of solute per kilogram of solvent. It is used in freezing point depression calculations because it remains constant regardless of temperature changes.
To calculate the molality (\( m \)), you use the formula \ \[ m = \frac{ \text{moles of solute} }{ \text{kilograms of solvent} } \]For the given problem, let's assume we need to lower the temperature of 5.5 kg of ice by 5°C. Since we're using NaCl, which dissociates completely in water to give 2 particles (\( \mathrm{Na^+} \) and \( \mathrm{Cl^-} \)). This i value of NaCl is 2.

• Given:\( \Delta T_f = 5^{\circ}\mathrm{C}\) , \( K_f = 1.86 \ \text{°C kg/mol}\)
• Molality can then be calculated by rearranging the equation: \ \[ m = \frac{\Delta T_f}{i \times K_f} \]
  After calculating the molality, we can find out the required mass of NaCl.

An ideal solution is one where the enthalpy of mixing is zero and the components adhere to Raoult's Law across all concentrations. This assumption is simplified but useful for calculations.
For the given problem, we assume that \( \mathrm{NaCl} \) or \( \mathrm{CaCl}_2 \) dissolves completely without any interactions that alter the properties outside of the ideal behavior. This means the substance dissociates completely, contributing its full effect to colligation and thus, the freezing point depression.
For NaCl, the calculation would involve:
Step 1: Determine molality from freezing point depression provided.
Step 2: Calculate moles of solute from molality.
Step 3: Find the mass from the number of moles using the molar mass.Similarly, calculate for \( \mathrm{CaCl}_2 \) considering it dissociates into 3 particles (1 \( \mathrm{Ca^{2+}} \) and 2 \( \mathrm{Cl^-} \)) to achieve the desired lowering of the freezing point.
This approach offers a straightforward way to determine the quantities needed in practical applications like freezing point depression in ice-cream making!