Question

A solution is made by dissolving \(25.8 \mathrm{g}\) urea \(\left(\mathrm{CH}_{4} \mathrm{N}_{2} \mathrm{O}\right),\) a nonelectrolyte, in \(275 \mathrm{g}\) water. Calculate the vapor pressures of this solution at \(25^{\circ} \mathrm{C}\) and \(45^{\circ} \mathrm{C}\). (The vapor pressure of pure water is \(\left.23.8 \text { torr at } 25^{\circ} \mathrm{C} \text { and } 71.9 \text { torr at } 45^{\circ} \mathrm{C} .\right)\)

Short answer

The vapor pressures of the urea-water solution at the given temperatures are: At 25°C, the vapor pressure is 23.1 torr, and at 45°C, the vapor pressure is 69.9 torr.

Step-by-step solution

(Step 1: Find the moles of urea and water)

First, we need to calculate the moles of urea and water in the solution. Calculate the moles of urea by dividing the mass of urea by its molar mass, and the moles of water by dividing the mass of water by the molar mass of water. Molar mass of urea: 12 (for C) + (4 x 1) (for H) + (2 x 14) (for N) + 16 (for O) = 60 g/mol Moles of urea: \(\frac{25.8\, g}{60\, g/mol} = 0.430\, mol\) Molar mass of water: (2 x 1) (for H) + 16 (for O) = 18 g/mol Moles of water: \(\frac{275\, g}{18\, g/mol} = 15.3\, mol\)

(Step 2: Calculate the mole fraction of urea and water)

Next, we need to determine the mole fraction of urea and water in the solution. The mole fraction is the ratio of the number of moles of a particular component to the total number of moles. Mole fraction of urea (\(X_{urea}\)) = \(\frac{moles\, urea}{moles\, urea + moles\, water} = \frac{0.430}{0.430 + 15.3} = 0.0273\) Mole fraction of water (\(X_{H_2O}\)) = \(\frac{moles\, water}{moles\, urea + moles\, water} = \frac{15.3}{0.430 + 15.3} = 0.9727\)

(Step 3: Apply Raoult's law)

Now, we can use Raoult's law to calculate the vapor pressures of the solution at 25°C and 45°C. According to Raoult's law, the partial pressure of a component in a non-ideal solution is equal to the product of its mole fraction and the vapor pressure of the pure component. At 25°C: Vapor pressure of the solution = \(X_{H_2O} * P^*_H{_2}O\) where \(P^*_{H_2O}\) is the vapor pressure of pure water at 25°C. Vapor pressure of the solution = \(0.9727 * 23.8\, torr = 23.1\, torr\) At 45°C: Vapor pressure of the solution = \(X_{H_2O} * P^*_H{_2}O\) where \(P^*_{H_2O}\) is the vapor pressure of pure water at 45°C. Vapor pressure of the solution = \(0.9727 * 71.9\, torr = 69.9\, torr\)

(Step 4: Report the vapor pressures)

Finally, we can report the vapor pressures of the urea-water solution at the given temperature: At 25°C: The vapor pressure of the solution is 23.1 torr. At 45°C: The vapor pressure of the solution is 69.9 torr.

Key concepts

Mole Fraction

The concept of mole fraction plays a vital role in solution chemistry. It is a way to express the concentration of a component in a mixture. To calculate the mole fraction, you divide the number of moles of the component by the total number of moles present in the solution. This ratio shows the proportion of each component within the mixture.

In the context of the given exercise involving urea and water, the mole fraction helps determine how the presence of urea affects the overall properties of the solution. For instance, you need to find the moles of urea and water separately using their respective molar masses:

• For urea, with a molar mass of 60 g/mol, divide the mass of urea (25.8 g) by 60 g/mol to get 0.430 moles.
• For water, with a molar mass of 18 g/mol, divide the mass of water (275 g) by 18 g/mol to get 15.3 moles.

Once you have the moles, you can easily calculate the mole fractions:

• The mole fraction of urea \(X_{urea}\) is \( \frac{0.430 \, \text{mol}}{0.430 \, \text{mol} + 15.3 \, \text{mol}} = 0.0273 \).
• The mole fraction of water \(X_{H_2O}\) is \( \frac{15.3 \, \text{mol}}{0.430 \, \text{mol} + 15.3 \, \text{mol}} = 0.9727 \).

Understanding the mole fraction helps in calculating other important properties like vapor pressure, as it reflects how much each component influences the behavior of the solution.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor in equilibrium with its liquid phase at a specific temperature. It is a crucial factor in understanding how substances behave in solutions. Raoult's Law is used to predict the vapor pressure of a solution, especially when dealing with nonelectrolyte solutions.

Raoult's Law states that the partial vapor pressure of a solvent in a solution is equal to the product of the mole fraction of the solvent and the vapor pressure of the pure solvent. This relationship helps us to find out how the addition of a solute like urea affects the vapor pressure of the solvent (water in this case).

• At 25°C, using the mole fraction of water \(X_{H_2O} = 0.9727\) and the pure water vapor pressure of 23.8 torr, the solution's vapor pressure is \(0.9727 \times 23.8 \, \text{torr} = 23.1 \, \text{torr}\).
• Similarly, at 45°C, with the pure water vapor pressure of 71.9 torr, the vapor pressure becomes \(0.9727 \times 71.9 \, \text{torr} = 69.9 \, \text{torr}\).

The vapor pressure of the solution is lower than the pure solvent because of the addition of a non-volatile solute like urea. This lowering of vapor pressure also results in an elevation of boiling point and depression of freezing point, properties which are often explored in solution chemistry.

Solution Chemistry

Solution chemistry examines how different substances interact when forming a homogenous mixture. It looks into how solutes dissolve in solvents, forming solutions, and how these solutions behave in various conditions.

In our exercise, urea acts as the solute and water as the solvent. Since urea is nonelectrolyte, it dissolves in water without breaking into ions. This results in a solution that still retains the inherent characteristics of the mixture, but with modified properties like vapor pressure.

The solution's properties, such as boiling point elevation or vapor pressure reduction, are colligative properties. This means they depend on the ratio of solute to solvent rather than the nature of the solute. By understanding the mole fraction, we use it to predict such changes accurately via Raoult's Law.

• Non-volatile solutes reduce the vapor pressure of the solution compared to the pure solvent.
• The extent of the change in properties hinges on the mole fraction of the solute.

Through the lens of solution chemistry, the effect of a solute on a solvent's physical properties can be predicted and calculated, helping in numerous practical applications ranging from industrial processes to everyday solutions.