Problem 8 - Chemistry and Chemical Reactivity

Question

You want to prepare an aqueous solution of ethylene glycol,\(\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH}\), in which the mole fraction of solute is \(0.125 .\) What mass of ethylene glycol, in grams, should you combine with 955 g of water? What is the molality of the solution?

Step-by-Step Solution

Answer

469.9 g of ethylene glycol is needed. The molality is 7.93 mol/kg.

Step 1: Understand Mole Fraction

The mole fraction of solute is given as 0.125. This means that for every 1 mole of total solution (water + ethylene glycol), 0.125 moles are ethylene glycol.

Step 2: Calculate Moles of Water

The molar mass of water is 18.02 g/mol. To find the moles of water, divide the mass of water by its molar mass: \[ \text{moles of water} = \frac{955 \text{ g}}{18.02 \text{ g/mol}} \approx 53.00 \text{ moles} \]

Step 3: Find Total Moles in Solution

The total moles in the solution for a mole fraction of 0.125 is determined by using the formula: \[ \text{Mole fraction of solute} = \frac{\text{moles of ethylene glycol}}{\text{moles of ethylene glycol} + \text{moles of water}} \] Let \(n\) be the moles of ethylene glycol. Then: \[ 0.125 = \frac{n}{n + 53.00} \]

Step 4: Solve for Moles of Ethylene Glycol

Rearrange the equation from Step 3 and solve for \(n\):\[ 0.125(n + 53.00) = n \] \[ 0.125n + 6.625 = n \] \[ 6.625 = n - 0.125n \] \[ 6.625 = 0.875n \] \[ n \approx \frac{6.625}{0.875} \approx 7.57 \text{ moles} \]

Step 5: Calculate Mass of Ethylene Glycol

The molar mass of ethylene glycol is 62.07 g/mol. Calculate the mass using the formula: \[ \text{mass} = \text{moles} \times \text{molar mass} = 7.57 \text{ moles} \times 62.07 \text{ g/mol} \approx 469.9 \text{ g} \]

Step 6: Determine Molality of the Solution

Molality \((m)\) is expressed as moles of solute per kilogram of solvent. Calculate it using: \[ m = \frac{\text{moles of ethylene glycol}}{\text{mass of water in kg}} = \frac{7.57 \text{ moles}}{0.955 \text{ kg}} \approx 7.93 \text{ mol/kg} \]

Key Concepts

Molality

Molality is a measurement of concentration. It's used to express how much solute is present in a given amount of solvent. Unlike molarity, which involves the total volume of the solution, molality focuses only on the mass of the solvent. This can be particularly useful in scenarios where temperature changes, as molality is not affected by temperature fluctuations due to its dependence on mass, not volume.

Molality \(m\) is calculated using the formula:

• \( m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \)

To find molality, you simply divide the number of moles of the solute by the mass of the solvent in kilograms. In our example, we've calculated molality as approximately 7.93 mol/kg, based on the moles of ethylene glycol and the mass of water used. This tells us how concentrated the solution is.

Aqueous Solution

An aqueous solution is a solution where water is the solvent. Water's unique properties make it an excellent solvent for many substances, including ethylene glycol. It is polar, which allows it to dissolve ionic compounds and other polar molecules effectively. This means that substances like ethylene glycol can mix well with water, creating a uniform solution.

Aqueous solutions are critical in both nature and industry. They are part of processes such as digestion, cellular respiration, and various chemical reactions. In this exercise, water is the solvent, and ethylene glycol is the solute, creating the right environment for studying the properties and concentrations in solutions.

• Water's high heat capacity also makes it useful in moderating temperature changes.
• Understanding how water interacts with solutes is key in forecasting outcomes of reactions and processes.

Ethylene Glycol

Ethylene glycol is a type of alcohol with the chemical formula \(\text{HOCH}_2\text{CH}_2\text{OH}\). It's a colorless, odorless liquid that is widely used as an antifreeze and in the manufacture of polyester fibers and resins. Due to its ability to lower the freezing point and raise the boiling point of water, it's used in applications related to temperature control.

In an aqueous solution, ethylene glycol acts as the solute. When mixed with water, it forms hydrogen bonds, which allows it to integrate well. This helps keep the solution uniform and stable.

• It's important to calculate the correct amount of ethylene glycol to achieve the desired molality and concentration.
• In our case, approximately 469.9 grams of ethylene glycol is needed to achieve the solution's specifications.

Understanding the properties of ethylene glycol helps in a range of applications from industrial manufacturing to everyday antifreeze solutions.