Problem 68 - Chemistry: The Central Science

Question

At \(20^{\circ} \mathrm{C}\), the vapor pressure of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is 75 torr, and that of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right)\) is 22 torr. Assume that benzene and toluene form an ideal solution. (a) What is the composition in mole fraction of a solution that has a vapor pressure of 35 torr at \(20^{\circ} \mathrm{C}\) ? (b) What is the mole fraction of benzene in the vapor above the solution described in part (a)?

Step-by-Step Solution

Answer

The composition in mole fraction of a solution with a vapor pressure of 35 torr at \(20^{\circ} \mathrm{C}\) is approximately \(\frac{13}{53}\) for benzene and \(\frac{40}{53}\) for toluene. The mole fraction of benzene in the vapor above the solution is approximately 0.5375.

Step 1: Establish given data and Raoult's Law

We are given the vapor pressures of benzene and toluene at \(20^\circ \mathrm{C}\) as 75 torr and 22 torr, respectively. We are also given that the vapor pressure of the solution is 35 torr. Let benzene be component 1 and toluene be component 2. We can use Raoult's Law for each component as follows: \[P_1 = x_1 P_1^*\] \[P_2 = x_2 P_2^*\] Here, \(P_1\) and \(P_2\) are the partial pressures of the components, \(x_1\) and \(x_2\) are the mole fractions of the components in the solution, and \(P_1^*\) and \(P_2^*\) are the vapor pressures of the pure components.

Step 2: Calculate the mole fractions of the components in the solution

Raoult's law states that the total pressure of the solution is the sum of the partial pressures: \[P_\mathrm{total} = P_1 + P_2\] We know that \(P_\mathrm{total} = 35\,\mathrm{torr}\). We can substitute the expressions for \(P_1\) and \(P_2\) from Step 1: \(35\,\mathrm{torr} = x_1\cdot 75\,\mathrm{torr} + x_2\cdot 22\,\mathrm{torr}\) We also know that the sum of the mole fractions is equal to 1: \[x_1 + x_2 = 1\] We can solve these two equations for \(x_1\) and \(x_2\).

Step 3: Solve the equation system

Let us rearrange the second equation and substitute it into the first equation: \[x_2 = 1 - x_1\] \(35\,\mathrm{torr} = x_1\cdot 75\,\mathrm{torr} + (1-x_1)\cdot 22\,\mathrm{torr}\) Now, solve for \(x_1\): \(35 = 75x_1 + 22 - 22x_1\) \(13x_1 = 13\) This gives us: \[x_1 = 1\] Now we can compute \(x_2 = 1 - x_1 = 0\). So, the composition of the solution in mole fraction is \(x_1=1\) (pure benzene) and \(x_2=0\) (no toluene) to have a vapor pressure of 35 torr at \(20^\circ\mathrm{C}\).

Step 4: Calculate the mole fraction of benzene in the vapor

We are asked to find the mole fraction of benzene in the vapor above the solution described in part (a). We can use the formula: \[y_1 = \frac{x_1 P_1^*}{P_\mathrm{total}}\] Given that \(x_1=1\), \(P_1^*=75\,\mathrm{torr}\), and \(P_\mathrm{total}=35\,\mathrm{torr}\), we can now calculate the mole fraction of benzene in the vapor: \[y_1 = \frac{(1)\cdot(75\,\mathrm{torr})}{(35\,\mathrm{torr})}\] \[y_1 = 2.142857\] This indicates that the mole fraction of benzene in the vapor is approximately 2.14, which is unexpected as mole fractions can only be between 0 and 1. This means there must have been an error in our calculations from earlier steps. We will reexamine our analysis and calculations to find a solution consistent with the laws of chemistry. Considering the error in previous calculations, the correct answer should be:

Step 5: Reexamine the equations and resolve

It seems we have made a mistake in the calculations in Step 3. Let's revisit the first equation: \[35\,\mathrm{torr} = x_1\cdot 75\,\mathrm{torr} + (1-x_1)\cdot 22\,\mathrm{torr}\] \[35 = 75x_1 + 22 - 22x_1\] The correct calculation should be: \[53x_1 = 13\] Now, we get the correct value for \(x_1\): \[x_1 = \frac{13}{53}\] And for \(x_2\): \[x_2 = 1 - \frac{13}{53} = \frac{40}{53}\] Thus, the correct mole fractions for benzene and toluene in the solution are \(\frac{13}{53}\) and \(\frac{40}{53}\), respectively. Now let's recalculate the mole fraction of benzene in the vapor using the correct mole fraction value: \[y_1 = \frac{(\frac{13}{53})\cdot(75\,\mathrm{torr})}{(35\,\mathrm{torr})}\] \[y_1 = \frac{13}{53} \times \frac{75}{35}\] \[y_1 \approx 0.5375\] This gives us a mole fraction of benzene in the vapor as approximately 0.5375, which is a physically meaningful value. In conclusion, the correct composition in mole fraction for benzene and toluene in the solution is \(\frac{13}{53}\) and \(\frac{40}{53}\), respectively, and the mole fraction of benzene in the vapor above the solution is approximately 0.5375.

Key Concepts

Vapor Pressure

Vapor pressure is a crucial concept in understanding how liquids behave when exposed to air. It refers to the pressure exerted by the molecules of a liquid when they escape into the gaseous phase above the liquid in a closed container. Here, it's important to remember that this process continues until a dynamic equilibrium is achieved. In this scenario, the rate of evaporation equals the rate of condensation.
Factors influencing vapor pressure include temperature and the nature of the liquid. Higher temperatures can increase the kinetic energy of molecules, allowing more to escape into the vapor phase, thus increasing the vapor pressure. Each liquid has a unique vapor pressure at a given temperature, as illustrated by benzene and toluene having vapor pressures of 75 torr and 22 torr respectively, at 20°C.
For those diving into Raoult's Law, the concept of vapor pressure becomes even more significant. Raoult's Law helps us calculate the vapor pressure of an ideal solution by understanding the contributions of each component's partial pressure, based on their mole fractions and individual vapor pressures when pure.

Mole Fraction

The concept of mole fraction is central to understanding the composition of solutions. It represents the ratio of moles of one component in a mixture to the total moles of all components. The mole fraction, which is unitless, provides insight into the concentration and is essential for calculations involving colligative properties and Raoult's Law.
In the context of the original exercise, determining the mole fractions of benzene and toluene in the solution was crucial for calculating the total vapor pressure of the solution. In an ideal solution, the sum of the mole fractions equals one, allowing the calculations:

• Mole fraction of component 1, benzene, is denoted as \(x_1\).
• Mole fraction of component 2, toluene, is denoted as \(x_2\).

These relationships form the backbone for calculating each component's contribution to the total vapor pressure. Knowing one mole fraction allows us to determine the other, reinforcing the inherent balance in mixture compositions. This concept truly unravels the balance and dependency of various components in any given solution.

Ideal Solution

An ideal solution is a theoretical concept where the intermolecular forces between the mixed molecules are the same as those between molecules of the same kind. In other words, the interactions between different particles are identical to the interactions between similar particles. This assumption simplifies the study of mixtures, allowing us to use laws like Raoult's Law reliably.
In an ideal solution, Raoult's Law states that the partial vapor pressure of each component is directly proportional to its mole fraction in the solution. Thus, the total vapor pressure of the solution is simply the sum of these partial pressures. This was seen in the original exercise where the ideal solution assumption enabled the straight application of Raoult's Law, helping to solve for mole fractions.
While many mixtures approximate ideal behavior, especially in cases like closely related chemicals such as benzene and toluene, not all real-world solutions do. This is where deviations and more complex models might be necessary, but ideal solutions offer a fundamental starting point for understanding solution behavior and properties. When problems involve ideal solutions, it makes calculations more straightforward, as seen with the benzene-toluene mixture.