Problem 25 - Chemistry and Chemical Reactivity

Question

A 35.0 -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 34.99 mm Hg.

Step 1: Determine Mole Fraction of Ethylene Glycol

First, calculate the number of moles of ethylene glycol. The molar mass of ethylene glycol, \( \mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH} \), is approximately \(62.07\, \mathrm{g/mol}\). Therefore, the number of moles is given by: \[ \text{Moles of ethylene glycol} = \frac{35.0\, \text{g}}{62.07\, \text{g/mol}} \approx 0.564\, \text{mol} \]

Step 2: Determine Mole of Water

Next, calculate the number of moles of water. The molar mass of water, \( \mathrm{H}_{2}\mathrm{O} \), is approximately \(18.02\, \mathrm{g/mol}\). Therefore, the number of moles is given by: \[ \text{Moles of water} = \frac{500.0\, \text{g}}{18.02\, \text{g/mol}} \approx 27.76\, \text{mol} \]

Step 3: Calculate Total Moles in Solution

Add the moles of ethylene glycol and moles of water to find the total mole count in the solution: \[ \text{Total moles} = 0.564 + 27.76 = 28.324\, \text{mol} \]

Step 4: Calculate Mole Fraction of Water

Calculate the mole fraction of water, which is the ratio of the moles of water to the total moles in the solution: \[ \text{Mole fraction of water} = \frac{27.76}{28.324} \approx 0.980\]

Step 5: Apply Raoult's Law

Raoult's Law states that the vapor pressure of a solvent over a solution (\( P_{\text{solution}} \)) is the product of the mole fraction of the solvent and the vapor pressure of the pure solvent (\( P_{0} \)). Thus,\[ P_{\text{solution}} = \chi_{\text{water}} \cdot P_{0,\text{water}} \]Plug in the known values:\[ P_{\text{solution}} = 0.980 \times 35.7\, \text{mm Hg} \approx 34.99\, \text{mm Hg} \]

Step 6: Conclusion

The vapor pressure of the water-ethylene glycol solution at \(32^{\circ} \mathrm{C}\) is \(34.99\, \text{mm Hg}\).

Key Concepts

Raoult's Law

Raoult's Law is a fundamental principle for understanding how solutions behave, especially when it comes to vapor pressure. Simply put, this law states that the vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent. So, as more solvent is mixed or diluted with other substances, the resulting mixture exerts a lower vapor pressure than the pure solvent.

The formula representing Raoult's Law is given by:

• \[ P_{\text{solution}} = \chi_{\text{solvent}} \times P_{0,\text{solvent}} \]
• \( P_{\text{solution}} \) is the vapor pressure of the solution.
• \( \chi_{\text{solvent}} \) is the mole fraction of the solvent in the solution.
• \( P_{0,\text{solvent}} \) is the vapor pressure of the pure solvent.

Raoult's Law is particularly useful for calculating the vapor pressure of a solution with nonvolatile solutes, where the solute doesn't itself evaporate. In the exercise provided, ethylene glycol is the nonvolatile solute, and water is the solvent whose vapor pressure we are interested in. By using Raoult's Law, we can find the vapor pressure of the water mixed with ethylene glycol.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. This concept is crucial for applying Raoult's Law to determine vapor pressures. It is calculated as the ratio of the number of moles of a component to the total number of moles in the solution.

The formula for mole fraction is:

• \[ \chi_{\text{component}} = \frac{\text{moles of component}}{\text{total moles of all components}} \]

In our exercise, we calculated the mole fraction of water in the solution. Firstly, we found the number of moles of ethylene glycol and water separately. Then, we summed them to find the total number of moles in the solution. Using these values, we obtained the mole fraction of water. The closer this fraction is to 1, the more similar the solution's vapor pressure will be to that of pure water.

Understanding mole fraction is key in predicting how a solution will behave in terms of its physical properties, like vapor pressure and boiling point. It offers a clear picture of how solutes dilute a solvent.

Nonvolatile Solutes

Nonvolatile solutes are important in the study of solutions because they do not vaporize easily. This property significantly affects the vapor pressure of a solution. When a nonvolatile solute is added to a solvent, it reduces the number of solvent molecules that can escape into the gas phase. As a result, the solution has a lower vapor pressure than the pure solvent.

Ethylene glycol, used in the exercise, is an example of a nonvolatile solute. When it is dissolved in water, the vapor pressure of the water is reduced. This is because ethylene glycol molecules interrupt the surface of the water, making it harder for water molecules to escape. In the case of the provided exercise, Raoult's Law helps us understand just how much the vapor pressure is affected.

Nonvolatile solutes have practical applications in everyday life. They are often used in antifreeze solutions for vehicles and in various other industries where altering the boiling and freezing points of water is beneficial. The presence of nonvolatile solutes can also prevent evaporation, making them useful in preserving substances over time.