Problem 88 - Chemistry: An Atoms-Focused Approach

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 Ib 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 new melting point of the mixture is 257.81 K (or -15.34°C in Celsius).

Step 1: Convert the masses to moles

First, we need to convert the given masses of NaCl (2.00 lb) and ice (12.00 lb) to moles. To do this, we'll use the molar masses of NaCl and water (H₂O). Molar mass of NaCl = 58.44 g/mol Molar mass of H₂O = 18.02 g/mol We also need to convert pounds to grams. 1 lb = 453.592 g. Mass of NaCl in grams = 2.00 lb * 453.592 g/lb = 907.185 g Mass of water in grams = 12.00 lb * 453.592 g/lb = 5443.104 g Now, we can find the moles: Moles of NaCl = 907.185 g / 58.44 g/mol = 15.52 mol Moles of H₂O = 5443.104 g / 18.02 g/mol = 302.15 mol

Step 2: Determine the molality of the solution

Molality (m) is defined as the moles of solute per kilogram of solvent. In this case, the solute is NaCl, and the solvent is water (H₂O). Since exactly half of the ice melts, the mass of water in the solution will be half of the initial mass of ice: Mass of melted ice = 0.5 * 5443.104 g = 2721.552 g Now we can calculate the molality: Molality (m) = moles of solute / mass of solvent(kg) m = 15.52 mol / (2721.552 g / 1000) = 5.70 mol/kg

Step 3: Calculate the freezing point depression

To calculate the freezing point depression (ΔTf), we use the following formula: ΔTf = Kf * m * i Where: - Kf is the cryoscopic constant for water, 1.86 K·kg/mol - m is the molality of the solution (calculated in Step 2) - i is the van 't Hoff factor, which is given as 1.44 in the problem. ΔTf = 1.86 K·kg/mol * 5.70 mol/kg * 1.44 = 15.34 K

Step 4: Find the new melting point of the mixture

Now that we've calculated the freezing point depression, we can find the new melting point of the mixture. The normal melting point of ice (pure H₂O) is 0°C (273.15 K). We subtract the freezing point depression from the normal melting point: New melting point = Normal melting point - ΔTf New melting point = 273.15 K - 15.34 K = 257.81 K The melting point of the mixture of NaCl and ice, when half of the ice melts, is 257.81 K (or -15.34°C in Celsius).

Key Concepts

Molality

Molality is a concentration measurement often used in freezing point depression problems, like the case of making ice cream with salt and ice. It indicates the number of moles of solute dissolved in a kilogram of solvent. In our exercise, sodium chloride (\(\mathrm{NaCl}\)) acts as the solute and water (ice) is the solvent.

To calculate molality (\(m\)), we used the formula:

• Molality (\(m\)) = moles of solute / mass of solvent in kilograms

Here, once the ice melts partially, it turns into water weighing 2721.552 grams (or 2.721552 kg), dissolved by 15.52 moles of \(\mathrm{NaCl}\), resulting in a molality of 5.70 mol/kg.

Molality is temperature-independent, which makes it very useful for scenarios involving temperature changes, like calculating freezing point depression.

Van 't Hoff Factor

The Van 't Hoff factor (\(i\)) represents how many particles a solute dissociates into in a solution. For ideal ionic compounds like \(\mathrm{NaCl}\), it would normally dissociate into two ions: \(\mathrm{Na^+}\) and \(\mathrm{Cl^-}\). However, due to interactions between ions, the factor is often slightly different in reality.

In this ice-cream-making scenario, the Van 't Hoff factor was given as 1.44. This indicates that the presence of ionic interactions slightly lowers the maximum theoretical dissociation of \(\mathrm{NaCl}\).

The Van 't Hoff factor is essential in calculating the freezing point depression, as it accounts for the effect of multiple particles resulting from dissociation, which influences how much the temperature of the solution's melting point changes.

Cryoscopic Constant

The cryoscopic constant (\(K_f\)) is a property of the solvent, showing how much the freezing point drops per molal concentration of solute. Different solvents have unique cryoscopic constants.

For this particular exercise, water is the solvent, having a known cryoscopic constant of 1.86 K·kg/mol. This constant helps determine how much the solution's temperature will drop due to solute presence.

We used the equation:

• Freezing point depression (\(\Delta T_f\)) = \(K_f \times m \times i\)

Substituting in the given values, we computed the change in freezing point, aiding us in finding the new freezing point once the solute is dissolved.

Melting Point

The melting point is the temperature at which a solid turns into a liquid, which for pure water (ice) is ordinarily 0°C (273.15 K).

In scenarios involving solutes like table salt, the melting point lowers — a phenomenon known as freezing point depression.

By applying the calculated depression (\(\Delta T_f = 15.34 \text{ K}\)) to the normal melting point of water, we determined that the new melting point of the salted ice mixture is 257.81 K (\(-15.34 \degree \text{C}\)).

Understanding how solutes affect the melting point of solvent solutions is crucial for processes such as making ice cream, where lower temperatures facilitate its formation without traditional freezing methods.