Question

How many grams of urea \(\left[\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}\right]\) must be added to \(658 \mathrm{~g}\) of water to give a solution with a vapor pressure \(2.50 \mathrm{mmHg}\) lower than that of pure water at \(30^{\circ} \mathrm{C} ?\) (The vapor pressure of water at \(30^{\circ} \mathrm{C}\) is \(31.8 \mathrm{mmHg} .)\)

Short answer

172.4 grams of urea.

Step-by-step solution

Understand Raoult's Law

Raoult's Law states that the vapor pressure lowering, \(\Delta P\), of a solvent in a solution is directly proportional to the mole fraction of the solute. It is given by \(\Delta P = P_0 - P = X_{solute} \cdot P_{0}\), where \(P_0\) is the vapor pressure of the pure solvent, and \(X_{solute}\) is the mole fraction of the solute.

Determine Vapor Pressure Lowering

Given the vapor pressure of pure water, \(P_0 = 31.8\, \mathrm{mmHg}\), and the vapor pressure of the solution, \(P = 31.8\, \mathrm{mmHg} - 2.50\, \mathrm{mmHg} = 29.3\, \mathrm{mmHg}\). Thus, \(\Delta P = 2.50\, \mathrm{mmHg}\).

Calculate Mole Fraction of Solute

Using \(\Delta P = X_{solute} \cdot P_{0}\), we get \(X_{solute} = \frac{\Delta P}{P_{0}} = \frac{2.50}{31.8}\). Solve to find \(X_{solute} \approx 0.0786\).

Express Mole Fraction Formula

The mole fraction \(X_{solute}\) is \(\frac{n_{solute}}{n_{solvent} + n_{solute}}\). Since \(X_{solute} \) is small, \(X_{solute} \approx \frac{n_{solute}}{n_{solvent}}\) because \(n_{solute} \ll n_{solvent}\).

Calculate Moles of Solvent

The molar mass of water is \(18.015\, \mathrm{g/mol}\). Therefore, \(n_{solvent} = \frac{658}{18.015} \approx 36.53\, \text{mol}\).

Find Moles of Urea Needed

Use \(X_{solute}\approx \frac{n_{solute}}{n_{solvent}}\), i.e., \(0.0786 = \frac{n_{solute}}{36.53}\). Solving for \(n_{solute}\), we find \(n_{solute} \approx 2.872\, \text{mol}\).

Convert Moles of Urea to Grams

The molar mass of urea \((\mathrm{NH}_2)_2\mathrm{CO}\) is \(60.06\, \mathrm{g/mol}\). Therefore, mass in grams = \(2.872 \times 60.06 \approx 172.4\, \mathrm{g}\).

Key concepts

Vapor Pressure Lowering

Vapor pressure lowering is a key phenomenon observed in solutions. It's based on Raoult's Law, which states that adding a non-volatile solute to a solvent decreases the solvent's vapor pressure.
This happens because solute particles occupy some of the surface area, reducing the number of solvent molecules escaping into the vapor phase.
The formula for vapor pressure lowering is:

• \( \Delta P = X_{\text{solute}} \cdot P_0 \)
• Where \( \Delta P \) is the reduction in vapor pressure of the solvent.

Understanding this concept helps in calculating how much a solution's vapor pressure differs compared to the pure solvent, which is crucial for tasks like determining solute concentrations.

Mole Fraction

Mole fraction measures how much of a certain substance exists in a mixture. It is a dimensionless number, providing the ratio of moles of solute to the total moles of all substances present.
For a solution, it's denoted as:

• \( X_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}} + n_{\text{solvent}}} \)

However, in cases where the solute is much less than the solvent, we can simplify the fraction:

• \( X_{\text{solute}} \approx \frac{n_{\text{solute}}}{n_{\text{solvent}}} \)

This simplification is often used to make calculations easier, especially when the solute is present in small quantities.

Molar Mass

Molar mass is the mass of one mole of a given substance and is expressed in grams per mole (g/mol). It connects the number of particles to their mass, essential for converting between moles and grams in chemical calculations.
For water and urea:

• Water's molar mass = \(18.015 \text{ g/mol}\)
• Urea's molar mass = \(60.06 \text{ g/mol}\)

Knowing the molar mass enables the conversion of calculated moles to grams, an essential step when determining how much solute to add to achieve a desired solution concentration.

Solution Concentration

Solution concentration indicates how much solute is present in a given quantity of solvent. In this context, it is determined by the ratio of moles of solute to moles of solvent, helping to predict the extent of vapor pressure lowering.
High concentration means more solute, affecting properties like boiling point, freezing point, and vapor pressure.
Accurate concentration calculations are central in fields like chemistry and pharmacology, as they ensure desired reactions and effects are consistently achieved. By understanding and manipulating concentration, chemists can control the properties and behaviors of solutions effectively.