Question

What volume of ethylene glycol \(\left(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{O}_{2}\right)\), a nonelectrolyte, must be added to \(15.0 \mathrm{~L}\) water to produce an antifreeze solution with a freezing point \(-25.0^{\circ} \mathrm{C} ?\) What is the boiling point of this solution? (The density of ethylene glycol is \(1.11 \mathrm{~g} / \mathrm{cm}^{3}\), and the density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3} .\) )

Short answer

To create an antifreeze solution with a freezing point of -25.0°C, you must add approximately 181.4 mL of ethylene glycol to 15.0 L of water. The boiling point of this solution is about 106.88°C.

Step-by-step solution

Calculate the molality of the solution

We know that the freezing point depression is related to the molality of the solution through the following formula: ΔTf = Kf × m Where ΔTf is the freezing point depression, Kf is the freezing point depression constant for water, and m is the molality of the solution. The freezing point of pure water is at 0.0°C, and we want the new freezing point to be -25.0°C. Therefore, ΔTf = 25.0°C and Kf = 1.86 °C kg/mol (for water). Now we can calculate the molality (m): m = ΔTf / Kf m = 25.0°C / 1.86 °C kg/mol m = \(13.44 \frac{\text{mol}}{\text{kg}}\)

Calculate the mass of ethylene glycol

To find the mass of ethylene glycol, we'll need to convert the 15.0 L water to mass by using the density of water: mass of water = volume × density mass of water = 15.0 L × 1.00 g/mL × 1000 mL/L × 1 kg/1000 g mass of water = 15.0 kg Now we can use the mass of water and the molality we found in Step 1: mass of ethylene glycol = molality × mass of water mass of ethylene glycol = \(13.44 \frac{\text{mol}}{\text{kg}}\) × 15.0 kg mass of ethylene glycol ≈ 201.6 g

Calculate the volume of ethylene glycol

We are now able to calculate the volume of ethylene glycol needed. We will use its density to convert mass to volume: volume of ethylene glycol = mass / density volume of ethylene glycol = 201.6 g / (1.11 g/mL) volume of ethylene glycol ≈ 181.4 mL

Calculate the boiling point elevation of the solution

Now, let's find the boiling point of the solution. The boiling point elevation formula is: ΔTb = Kb × m Where ΔTb is the boiling point elevation, Kb is the boiling point elevation constant for water, and m is the molality of the solution. For water, Kb = 0.512 °C kg/mol. We can now calculate the boiling point elevation: ΔTb = Kb × m ΔTb = 0.512 °C kg/mol × \(13.44 \frac{\text{mol}}{\text{kg}}\) ΔTb ≈ 6.88°C

Calculate the boiling point of the solution

The boiling point of pure water is 100.0°C. To find the boiling point of the solution, we will add the boiling point elevation to the boiling point of pure water: boiling point of solution = boiling point of pure water + ΔTb boiling point of solution = 100.0°C + 6.88°C boiling point of solution ≈ 106.88°C So, it is required to add 181.4 mL of ethylene glycol to 15.0 L of water to produce an antifreeze solution with a freezing point of -25.0°C. The boiling point of this solution is approximately 106.88°C.

Key concepts

Freezing Point Depression

The concept of freezing point depression explains why adding a solute, like ethylene glycol (a nonelectrolyte), to a solvent such as water lowers the freezing point of the solvent. This principle is particularly useful in making antifreeze solutions for vehicles. The change in the freezing point is calculated using the formula:

• ΔTf = Kf × m

where ΔTf is the freezing point depression, Kf is the freezing point depression constant (1.86 °C kg/mol for water), and m is the molality of the solution. Here, the goal is to lower the freezing point of water from 0°C to -25°C. By substituting these values into the formula, you can calculate the necessary molality of the solution, which is about 13.44 mol/kg. This understanding of freezing point depression is crucial for designing solutions that resist freezing in cold climates.

Boiling Point Elevation

Boiling point elevation is another colligative property. It occurs when a solute is added to a solvent, causing the boiling point of the solvent to increase. Similar to freezing point depression, boiling point elevation depends on the molality of the solution. The formula used is:

• ΔTb = Kb × m

where ΔTb is the elevation in boiling point, Kb is the boiling point elevation constant (0.512 °C kg/mol for water), and m is molality. For our antifreeze solution, the molality calculated previously is used to determine a boiling point elevation of approximately 6.88°C. Therefore, the boiling point of the solvent increases from 100.0°C to about 106.88°C. Ensuring a higher boiling point is valuable in preventing engine fluids from boiling over in hot conditions.

Molality

Molality is a measure of the concentration of a solute in a solvent. It is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the volume of the solution, molality is based on the mass of the solvent, making it independent of temperature changes. This independence is particularly beneficial in calculating colligative properties like freezing point depression and boiling point elevation. In the given problem, we determined the molality to be 13.44 mol/kg by using the freezing point depression formula, thus allowing us to further calculate other colligative properties of the solution. Understanding molality helps students to relate how different the concentration measurements are and when to use each.

Nonelectrolyte

Nonelectrolytes are substances that do not dissociate into ions in a solution, which means they do not conduct electricity. Ethylene glycol, a common antifreeze compound, is an example of a nonelectrolyte. When used in solute calculations, nonelectrolytes simplify the equations for colligative properties since they are expected to affect only the number of particles in solution without any additional electrical effects. In the exercise, it’s important to note that ethylene glycol’s presence does not split into ions like a salt would, hence affecting only the physical properties of the water. This aspect significantly impacts the measurements for freezing point depression and boiling point elevation, as the solution's behavior is solely dependent on particle concentration.

Solution Density

Solution density is the measure of how much mass is contained in a given volume of a solution, often expressed in grams per milliliter (g/mL). In calculations involving colligative properties, knowing the densities of both the solute and solvent is important to determine the precise amounts needed to achieve the desired properties. In the given problem, the density of ethylene glycol is 1.11 g/mL, while the density of water is 1.00 g/mL. These values allowed for the conversion from mass to volume when calculating how much ethylene glycol is necessary to reach the targeted freezing point and boiling point adjustments. Mastering these calculations is essential for precision in creating solutions suitable for specific purposes, such as antifreeze mixtures.