Problem 51 - General Chemistry: Principles and Modern Applications

Question

What are the partial and total vapor pressures of a solution obtained by mixing 35.8 g benzene, \(\mathrm{C}_{6} \mathrm{H}_{6}\) and \(56.7 \mathrm{g}\) toluene, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{3},\) at \(25^{\circ} \mathrm{C} ? \mathrm{At} 25^{\circ} \mathrm{C}\) the vapor pressure of \(\mathrm{C}_{6} \mathrm{H}_{6}=95.1 \mathrm{mm} \mathrm{Hg} ;\) the vapor pressure of \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{3}=28.4 \mathrm{mmHg}\).

Step-by-Step Solution

Answer

To get a short answer, find the mole fractions for benzene and toluene, then use these mole fractions to find the partial vapor pressures via Raoult's Law. Finally, add these partial pressures to find the total vapor pressure.

Step 1: Calculate the Mole Fraction

First, the molar fractions of benzene and toluene in the solution need to be calculated. To find that, divide the mass of each substance by its molar mass to get the number of moles. The molar mass of benzene (C6H6) is approximately 78.11 g/mol while for toluene (C6H5CH3) it is approximately 92.14 g/mol. Now, compute the mole fractions, which are the ratio of the number of moles of a component to the total number of moles. The total number of moles is the sum of the moles of both benzene and toluene.

Step 2: Application of Raoult's Law

Now, apply Raoult's Law. The partial vapor pressure of a component in a solution equals to the product of mole fraction of the component and the vapor pressure of the pure component. Use the mole fractions calculated in the previous step. The vapor pressure of benzene is given as 95.1 mmHg and for toluene it's 28.4 mmHg. This will give you the partial pressures for both benzene and toluene.

Step 3: Calculate Total Vapor Pressure

The total vapor pressure is simply the sum of the partial vapor pressures. In the context of this problem, to find the total vapor pressure of the solution, sum up the partial pressure values calculated in the previous step, those of benzene and toluene respectively, to get the total vapor pressure of the mixture.

Key Concepts

Raoult's Law

Raoult's Law is a principle that helps us understand how the vapor pressure of a solution is related to its components. When you mix different substances, like benzene and toluene, each component still tries to evaporate. Raoult's Law gives us a way to predict how much each component will contribute to the overall vapor pressure of the solution. This is done by looking at its mole fraction in the mixture and its vapor pressure as a pure substance.
Raoult's Law states that the partial vapor pressure of any component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. Mathematically, this is expressed as:

• Partial vapor pressure of component A: \( P_A = \chi_A \times P_A^* \)
• Where \( \chi_A \) is the mole fraction of component A in the solution
• And \( P_A^* \) is the vapor pressure of the pure component A

The sum of these partial pressures gives the total vapor pressure of the solution. Using Raoult’s Law, we can calculate how each substance in a mixture contributes to the overall evaporation behavior derived from their original independent properties.

Mole Fraction

The mole fraction is a concept used to describe the composition of a solution. It's essentially a way to express the concentration of one component within a mixture without relying on units like grams or liters. Instead, it considers the number of moles.
To calculate the mole fraction of a component in a mixture, such as benzene or toluene, you first need to know how many moles of each are present. This is done by dividing the mass of each component by its respective molar mass.

• Moles of benzene = \( \frac{35.8 \, \text{g}}{78.11 \, \text{g/mol}} \)
• Moles of toluene = \( \frac{56.7 \, \text{g}}{92.14 \, \text{g/mol}} \)

Once you have the moles, you sum them to get the total moles in the solution. The mole fraction for a specific component, such as benzene, is then found by dividing the moles of benzene by the total moles.

• Mole fraction of benzene, \( \chi_{\text{benzene}} = \frac{\text{moles of benzene}}{\text{total moles}} \)

Mole fractions provide a handy way to easily convert between masses and the contribution to vapor pressure in Raoult's Law as shown previously.

Partial Vapor Pressure

Partial vapor pressure is a key concept when dealing with mixtures and their evaporative behaviors. Each component of a liquid mixture contributes to the total vapor pressure, which is the pressure exerted by vapors above the solution.
Using benzene and toluene as examples, when each is part of a liquid mixture, they both have particular partial vapor pressures. These values arise from their tendency to evaporate.
By using Raoult's Law, we determine these partial pressures through their respective mole fractions and pure vapor pressures. For benzene:

• \( P_{\text{benzene}} = \chi_{\text{benzene}} \times 95.1 \, \text{mmHg} \)

Similarly, the partial pressure for toluene is:

• \( P_{\text{toluene}} = \chi_{\text{toluene}} \times 28.4 \, \text{mmHg} \)

By adding both partial vapor pressures together, we obtain the total vapor pressure of the solution. This total pressure characterizes the mixture’s overall ability to evaporate compared to its individual substances.