Question

Hexane, \(\mathrm{C}_{6} \mathrm{H}_{14}\), and cyclohexane, \(\mathrm{C}_{6} \mathrm{H}_{12}\), form an ideal solution. The vapor pressure of hexane is 151 Torr and that of cyclohexane is 98 Torr at \(25.0^{\circ} \mathrm{C}\). Calculate the vapor pressure of each of the following solutions and the mole fraction of each substance in the vapor phase above those solutions at \(25^{\circ} \mathrm{C}\) : (a) \(0.25 \mathrm{~mol} \mathrm{} \mathrm{C}_{6} \mathrm{H}_{14}\) mixed with \(0.65 \mathrm{~mol} \mathrm{} \mathrm{C}_{6} \mathrm{H}_{12}\); (b) \(10.0 \mathrm{~g}\) of hexane mixed with \(10.0 \mathrm{~g}\) of cyclohexane.

Short answer

For Solution (a), the vapor pressure is calculated to be a sum of partial pressures obtained from Raoult's Law and the mole fractions in the vapor phase are determined by using Dalton's Law. For Solution (b), the mole fractions are first calculated by converting mass to moles and then the vapor pressure and vapor phase mole fractions are found similar to Solution (a).

Step-by-step solution

Calculate the mole fractions in Solution (a)

Find the total number of moles of both substances and then divide the number of moles of each component by the total number of moles to get their respective mole fractions. For hexane \(x_{\mathrm{C}_{6}\mathrm{H}_{14}}\) and for cyclohexane \(x_{\mathrm{C}_{6}\mathrm{H}_{12}}\).

Calculate the vapor pressure of Solution (a)

Use Raoult's Law to determine the partial vapor pressures of hexane and cyclohexane in the solution. The total vapor pressure is the sum of these partial pressures.

Calculate the mole fractions in the vapor phase for Solution (a)

Use Dalton's Law to calculate the mole fraction of each component in the vapor phase. This is the ratio of the component's partial pressure to the total vapor pressure.

Calculate the mass in moles for hexane and cyclohexane in Solution (b)

Convert the given mass of hexane and cyclohexane to moles using their respective molar masses.

Calculate the mole fractions in Solution (b)

Find the total number of moles of both substances as done in step 1 and calculate the mole fractions of hexane and cyclohexane.

Calculate the vapor pressure of Solution (b)

Apply Raoult's Law as in step 2, using the mole fractions obtained from step 5, to find the vapor pressure of the solution.

Calculate the mole fractions in the vapor phase for Solution (b)

Using Dalton's Law as in step 3, calculate the mole fraction of each component in the vapor phase for solution (b).

Key concepts

Understanding Raoult's Law

Raoult's Law is a fundamental principle in chemistry that relates the vapor pressure of a solution to the concentration of the solution's components. It states that the partial vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent in the solution.

The equation for Raoult's Law can be written as: \[ P_i = P_i^{*} \times X_i \]where \( P_i \) is the partial vapor pressure of component \( i \), \( P_i^{*} \) is the vapor pressure of the pure component \( i \) at the same temperature, and \( X_i \) is the mole fraction of component \( i \) in the solution.

This law is particularly useful when dealing with ideal solutions, where the interactions between different molecules are the same as the interactions between like molecules. In the provided exercise, Raoult's Law helps to determine the vapor pressure of hexane and cyclohexane when mixed in a solution by considering their respective mole fractions. This law is especially powerful as it lays the groundwork for predicting how a solution will behave solely based on the properties of its pure components and their quantities in the mixture.

Mole Fraction Basics

The mole fraction is a way of expressing the concentration of a component in a mixture. It's defined as the number of moles of one component divided by the total number of moles of all components in the mixture.

To calculate it, use the formula: \[ X_i = \frac{n_i}{n_{total}} \]where \( X_i \) is the mole fraction of component \( i \), \( n_i \) is the number of moles of component \( i \), and \( n_{total} \) is the total number of moles of all components in the mixture.

In the context of our exercise, the calculation of mole fractions is the first step in using Raoult's Law to determine the vapor pressure of a solution. For hexane and cyclohexane, knowing the mole fraction is crucial as it directly affects their contribution to the total vapor pressure of the solution.

Applying Dalton's Law

Dalton's Law, also known as Dalton's Law of Partial Pressures, is a principle in chemistry that deals with the pressures contributed by each gas in a mixture of non-reacting gases. The law states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases in the mixture.

It can be represented by the equation: \[ P_{total} = P_1 + P_2 + ... + P_n \]where \( P_{total} \) is the total pressure of the mixture, and \( P_1, P_2, ..., P_n \) are the partial pressures of the component gases.

In relation to the exercise, Dalton's Law is used to find the mole fraction of each substance in the vapor phase above the solution. The mole fraction in the vapor phase is critical because it differs from the mole fraction in the liquid phase due to the different volatilities of the substances involved. The student should note that vapor and liquid phase mole fractions will only be equal for individual components that adhere to Raoult's Law. In practice, the partial pressure of a component in the vapor phase can be found by multiplying the total vapor pressure by the mole fraction of that component in the vapor phase. Understanding Dalton's Law is key to solving problems involving the vapor pressure and composition of vapor over a solution.