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 fractions 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)?

Short answer

The composition in mole fractions of the solution is \(X_{benzene} = \frac{13}{53}\) and \(X_{toluene} = \frac{40}{53}\). The mole fraction of benzene in the vapor above the solution is approximately \(\frac{975}{1855}\).

Step-by-step solution

Recall Raoult's law

Raoult's law is given by the equation: \(P = X_{A} P_{A}^{*} + X_{B} P_{B}^{*}\) where \(P\) is the total vapor pressure, \(X_{A}\) and \(X_{B}\) are the mole fractions of components A and B, and \(P_{A}^{*}\) and \(P_{B}^{*}\) are the vapor pressures of the pure components A and B, respectively.

Set up the equation for the given problem

Given the vapor pressures of benzene (\(P_{benzene}^{*} = 75\) torr) and toluene (\(P_{toluene}^{*} = 22\) torr) and the total vapor pressure of the solution (\(P = 35\) torr), we can set up the equation for the system as: \(35 = X_{benzene} \cdot 75 + X_{toluene} \cdot 22\)

Introduce the mole fraction relationship

Since the mole fractions of the two components must add up to 1, we can say: \(X_{benzene} + X_{toluene} = 1\) Now, we can solve this equation for one of the mole fractions and substitute it into our Raoult's law equation. \(X_{benzene} = 1 - X_{toluene}\)

Solve for the mole fractions

Substitute the mole fraction relationship into the Raoult's law equation: \(35 = (1 - X_{toluene}) \cdot 75 + X_{toluene} \cdot 22\) Now, solve for the mole fraction of toluene (\(X_{toluene}\)): \(35 = 75 - 75 X_{toluene} + 22 X_{toluene}\) \(35 = 75 - 53 X_{toluene}\) \(X_{toluene} = \frac{75 - 35}{53} = \frac{40}{53}\) Now, find the mole fraction of benzene (\(X_{benzene}\)): \(X_{benzene} = 1 - X_{toluene} = 1 - \frac{40}{53} = \frac{13}{53}\) Thus, the composition in mole fractions of the solution is \(X_{benzene} = \frac{13}{53}\) and \(X_{toluene} = \frac{40}{53}\).

Calculate the mole fraction of benzene in the vapor

To find the mole fraction of benzene in the vapor, we will use the equation: \(y_{benzene} = \frac{X_{benzene} \cdot P_{benzene}^{*}}{P}\) Substitute the values into the equation: \(y_{benzene} = \frac{\frac{13}{53} \cdot 75}{35} = \frac{13 \cdot 75}{53 \cdot 35}\) \(y_{benzene} = \frac{975}{53 \cdot 35} = \frac{975}{1855}\) So, the mole fraction of benzene in the vapor above the solution is approximately \(\frac{975}{1855}\).

Key concepts

Vapor Pressure

Vapor pressure is an essential concept in understanding the behavior of liquids and solutions. It refers to the pressure exerted by the vapor in equilibrium with its liquid at a given temperature. In other words, it's the pressure created by the molecules that have escaped the liquid phase and entered the gas phase above the liquid. The vapor pressure of a pure substance is constant at a given temperature and increases as the temperature rises because more molecules have enough energy to escape into the gas phase.

When we mix two liquids to create a solution, the total vapor pressure of the solution is the sum of the pressure contributions from each component, which is predicted by Raoult's Law. This relationship assumes that the solution is ideal, meaning the interactions between the molecules of different components are similar to the interactions between molecules of the same component. Raoult's Law helps us calculate the vapor pressure of the solution if we know the pure substances' vapor pressures and the mole fractions of the components.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of that component to the total number of moles of all components in the mixture. The sum of all mole fractions in a mixture equals one. For instance, in a two-component system like the benzene-toluene mixture, if we know one component's mole fraction, we can find the other's by simply subtracting the known fraction from one.

To find a component's mole fraction in the vapor phase, which can differ from the liquid phase, we use an equation derived from Raoult's Law and Dalton's Law of partial pressures. This reflects the composition of the vapor based on the volatile nature of each component and their mole fractions in the liquid mixture. Understanding mole fractions is crucial as it can affect the solution's properties, such as boiling point, vapor pressure, and osmotic pressure.

Ideal Solution

An ideal solution is a mixture that follows Raoult's Law precisely across all concentrations for each component. In an ideal solution, the interactions between different molecules are similar to the interactions present in the pure substances, meaning there is no net change in energy when the pure substances are mixed. This similarity causes the vapor pressure of each component to be directly proportional to its mole fraction in the mixture.

An ideal solution also assumes that the components are non-reactive with each other and that the solution is homogeneous. However, it's important to note that not all mixtures behave ideally. Real solutions can exhibit positive or negative deviations from Raoult's Law when the intermolecular forces between molecules of different components are stronger or weaker than those in the pure components. These deviations reflect in the vapor pressure being higher or lower than what would be predicted by Raoult's Law for an ideal solution.