Question

The vapor pressure of benzene is \(100.0 \mathrm{mmHg}\) at \(26.1^{\circ} \mathrm{C}\). Calculate the vapor pressure of a solution containing \(24.6 \mathrm{~g}\) of camphor \(\left(\mathrm{C}_{10} \mathrm{H}_{16} \mathrm{O}\right)\) dissolved in \(98.5 \mathrm{~g}\) of benzene. (Camphor is a low-volatility solid.)

Short answer

The vapor pressure of the solution is approximately 88.6 mmHg.

Step-by-step solution

Identify the components and their masses

The solution contains two components: camphor and benzene. We know that the mass of camphor is given as \(24.6 \, \text{g}\) and the mass of benzene is \(98.5 \, \text{g}\). We will use these values in our calculations.

Calculate molar masses

To find how many moles of each component are present, first calculate their molar masses. The molar mass of camphor (\(\text{C}_{10}\text{H}_{16}\text{O}\)) is calculated as: \(10(12.01) + 16(1.01) + 16.00 = 152.26 \, \text{g/mol}\). The molar mass of benzene \(\text{C}_6\text{H}_6\) is: \(6(12.01) + 6(1.01) = 78.12 \, \text{g/mol}\).

Calculate moles of camphor and benzene

Now calculate the moles of each component: For camphor, moles \(= \frac{24.6}{152.26} \approx 0.162 \, \text{mol}\). For benzene, moles \(= \frac{98.5}{78.12} \approx 1.261 \, \text{mol}\).

Determine the mole fraction of benzene

The mole fraction of benzene is calculated using the formula \(\chi_{\text{benzene}} = \frac{\text{moles of benzene}}{\text{moles of benzene} + \text{moles of camphor}}\). Substituting the values, \(\chi_{\text{benzene}} = \frac{1.261}{1.261 + 0.162} \approx 0.886\).

Raoult's Law application for vapor pressure

Using Raoult's Law, \(P = P_0 \chi\), where \(P_0\) is the pure solvent vapor pressure and \(\chi\) is the mole fraction. Here, \(P_0 = 100.0 \, \text{mmHg}\) and \(\chi_{\text{benzene}} = 0.886\). Then, vapor pressure of the solution \(P = 100.0 \times 0.886 = 88.6 \, \text{mmHg}\).

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle in solution chemistry, primarily concerning the vapor pressure of solutions. This law states that the vapor pressure of a solvent above a solution is directly proportional to the mole fraction of the solvent in the solution. In simpler terms, by adding a non-volatile solute to a volatile solvent, the overall vapor pressure decreases.

Mathematically, Raoult's Law is expressed as follows:

• The vapor pressure of the solution, \( P \), is equal to \( P_0 \chi \), where \( P_0 \) is the vapor pressure of the pure solvent, and \( \chi \) is the mole fraction of the solvent in the solution.

It's crucial to note that Raoult's Law applies mainly to ideal solutions, where interactions between different molecules are similar to those between identical molecules. This makes the calculation of the new vapor pressure straightforward by simply knowing the mole fraction of the solvent.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture or a solution. It is defined as the number of moles of one component divided by the total number of moles in the mixture.

In the case of a two-component solution, such as benzene and camphor, the mole fraction \( \chi \) can be calculated using:

• \( \chi_{\text{component}} = \frac{\text{moles of component}}{\text{moles of all components}} \)

This dimensionless number helps us understand how much of each substance is present in a mixture without referring to mass or volume. For example, the mole fraction of benzene tells us the proportion of benzene molecules relative to the total molecules in the solution.

Molar Mass

Molar mass is a property that signifies the mass of one mole of a given substance, expressed in grams per mole (g/mol). It is critically important in chemistry for converting between the mass of a substance and the amount in moles.

To calculate the molar mass of a compound, sum the atomic masses of all atoms in its formula. For instance, the molar mass of camphor, \( \text{C}_{10}\text{H}_{16}\text{O} \), is calculated by adding the total masses of carbon, hydrogen, and oxygen:

• 10 atoms of carbon \((10 \times 12.01 = 120.1 \text{ g/mol})\)
• 16 atoms of hydrogen \((16 \times 1.01 = 16.16 \text{ g/mol})\)
• 1 atom of oxygen \((16.00 \text{ g/mol})\)

Adding these together gives the molar mass of camphor as approximately 152.26 g/mol. Understanding molar mass helps in identifying how many moles are present in a given mass of material.

Solution Chemistry

Solution chemistry involves studying the properties and behaviors of solutes and solvents when mixed together. It is a crucial aspect of chemistry because it helps us understand how different substances interact when combined. Here are some important concepts in solution chemistry:

• **Solvent:** The component of a solution present in the greatest amount, usually a liquid.
• **Solute:** The substance dissolved into the solvent, typically present in a smaller amount.
• **Concentration:** A measure of how much solute is present in a given quantity of solvent or solution.
• **Vapor Pressure:** The pressure exerted by the vapor in equilibrium with its liquid or solid form in a closed system.

Understanding these concepts enables the prediction and explanation of changes in physical properties like boiling point and vapor pressure upon the mixing of different chemical substances.