Problem 25 - Chemistry and Chemical Reactivity

Question

A \(35.0-\mathrm{g}\) sample of ethylene glycol, \(\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH},\) is dissolved in \(500.0 \mathrm{g}\) of water. The vapor pressure of water at \(32^{\circ} \mathrm{C}\) is \(35.7 \mathrm{mm}\) Hg. What is the vapor pressure of the water-ethylene glycol solution at \(32^{\circ} \mathrm{C} ?\) (Ethylene glycol is nonvolatile.)

Step-by-Step Solution

Answer

The vapor pressure of the solution is 35.0 mm Hg.

Step 1: Calculate the Moles of Ethylene Glycol

The molar mass of ethylene glycol \( \mathrm{HOCH}_2\mathrm{CH}_2\mathrm{OH} \) is calculated as follows: \(2 \times 1.01 + 2 \times 12.01 + 6 \times 1.01 + 2 \times 16.00 = 62.08 \, \mathrm{g/mol}\). For a \(35.0-\mathrm{g}\) sample:\[\text{moles of ethylene glycol} = \frac{35.0 \, \mathrm{g}}{62.08 \, \mathrm{g/mol}} = 0.564 \, \text{mol}\]

Step 2: Calculate the Moles of Water

The molar mass of water \(\mathrm{H}_2\mathrm{O}\) is \(18.02 \, \mathrm{g/mol}\).For a \(500.0-\mathrm{g}\) sample:\[\text{moles of water} = \frac{500.0 \, \mathrm{g}}{18.02 \, \mathrm{g/mol}} = 27.76 \, \text{mol}\]

Step 3: Determine the Mole Fraction of Water

The mole fraction of water is defined as:\[X_\text{water} = \frac{\text{moles of water}}{\text{moles of water} + \text{moles of ethylene glycol}}\]Substitute the values:\[X_\text{water} = \frac{27.76}{27.76 + 0.564} = 0.980\]

Step 4: Apply Raoult's Law

Raoult's Law states that the vapor pressure is the product of the pure solvent's vapor pressure and its mole fraction in the solution:\[P_\text{solution} = X_\text{water} \times P_\text{water}\]Given \(P_\text{water} = 35.7 \, \mathrm{mm} \, \text{Hg}\):\[P_\text{solution} = 0.980 \times 35.7 = 35.0 \, \mathrm{mm} \, \text{Hg}\]

Key Concepts

Understanding Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is calculated by dividing the number of moles of the component by the total number of moles of all components present in the mixture. For instance, in our solution of ethylene glycol and water, we determine the mole fraction of water using its moles and the moles of ethylene glycol.

The formula is:

• \( X_{\text{water}} = \frac{\text{moles of water}}{\text{moles of water} + \text{moles of ethylene glycol}} \)

This calculation gives us a proportion that indicates how much of the solution is made up by water in this case. Understanding mole fraction is crucial because it helps in predicting how the solution will behave, especially when it comes to properties like vapor pressure.

What is Vapor Pressure?

Vapor pressure is the pressure exerted by a vapor when it is in equilibrium with its liquid form in a closed container. Essentially, it's a measure of the tendency of molecules to escape from the liquid into the gas phase.

In a solution, the presence of a solute affects the vapor pressure of the solvent. According to Raoult's Law, the vapor pressure of the solvent in the solution (\( P_{\text{solution}} \)) is equal to the vapor pressure of the pure solvent (\( P_{\text{water}} \)) multiplied by its mole fraction in the solution. This relation is expressed as:

• \( P_{\text{solution}} = X_{\text{water}} \times P_{\text{water}} \)

In the given practice problem, we've calculated the vapor pressure using the mole fraction of water and its pure vapor pressure at a specified temperature.

Characteristics of a Nonvolatile Solute

A nonvolatile solute is a substance that does not readily evaporate into a gas under existing conditions. In the context of our exercise, ethylene glycol serves as the nonvolatile solute. This means that it adds to the total number of solute particles in the water, but it does not contribute to vapor pressure since it cannot evaporate significantly at the temperature given.

The presence of a nonvolatile solute like ethylene glycol lowers the vapor pressure of a solvent compared to the pure solvent. Since the solute does not evaporate, it doesn't add to the vapor pressure, but it takes up space, reducing the number of solvent molecules at the surface that can escape into the vapor phase.

• This phenomenon is a demonstration of Raoult's Law, where the solvent's vapor pressure is primarily affected by the concentration of the solute that doesn't vaporize.

The lower vapor pressure indicates that fewer water molecules are escaping into the gas phase due to the interference of ethylene glycol in the solution.